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abstract: |
We study the pathwise regularity of the map $$\varphi \mapsto I(\varphi) = \int_0^T \langle \varphi(X_t), dX_t \rangle$$ where $\varphi$ is a vector function on ${\mathbb{R}}^d$ belonging to some Banach space $V$, $X$ is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A *stochastic current* is a continuous version of this map, seen as a random element of the topological dual of $V$. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient conditions when the process $X$ is a $d$-dimensional fractional Brownian motion (fBm); we identify regularity in Sobolev spaces for fBm with Hurst index $H \in (1/4,1)$. Next we provide some results about general Sobolev regularity of Brownian currents. Finally we discuss applications to a model of random vortex filaments in turbulent fluids.
**Key words:** Pathwise stochastic integrals, currents, forward and symmetric integrals, fractional Brownian motion, vortex filaments.
**MSC (2000): 76M35; 60H05; 60H30; 60G18; 60G15; 60G60; 76F55**
author:
- |
\
[ Franco Flandoli]{}\
[*Dipartimento di Matematica Applicata, Università di Pisa*]{}\
[*Via Bonanno 25B, I-56126 Pisa, Italia*]{}\
[ ]{}
- |
\
[ Massimiliano Gubinelli ]{}\
[*Laboratoire de Mathématiques, Université de Paris-Sud* ]{}\
[*Bâtiment 425, F-91405 Orsay Cedex, France*]{}\
[ ]{}
- |
\
[ Francesco Russo ]{}\
[*Institut Galilée, Mathématiques, Université Paris 13* ]{}\
[*99, av. J.-B. Clément - F-93430 Villetaneuse, France*]{}\
date: March 2007
title: |
On the regularity of stochastic currents,\
fractional Brownian motion\
and applications to a turbulence model
---
Introduction
============
We consider stochastic integrals, loosely speaking of the form $$\label{ECurr}
I\left( \varphi\right) =\int_{0}^{T}\left\langle \varphi\left(
X_{t}\right) ,dX_{t}\right\rangle,$$ where $\left( X_{t}\right) $ is a Wiener process or a fractional Brownian motion with Hurst parameter $H$ in a certain range. We are interested in the pathwise continuity properties with respect to $\varphi$: we would like to establish that the random generalized field $\varphi\mapsto I\left( \varphi\right) $ has a version that is a.s. continuous in $\varphi$ in certain topologies. In the language of geometric measure theory, such a property means that the stochastic integral defines pathwise a *current*, with the regularity specified by the topologies that we have found.
This problem is motivated by the study of fluidodynamical models. In [@Ffil], in the study of the energy of a vortex filament naturally appear some stochastic double integral related to Wiener process $$\label{EDint}
\int_{[0,T]^2} f(X_s - X_t) dX_s dX_t,$$ where $f(x) = K_{\alpha}\left( x\right)$ where $K_{\alpha}\left( x \right) $ is the kernel of the pseudo-differential operator $(1-\Delta)^{-\alpha}$ (precise definitions will be given in section \[preliminaries\]). $f$ is therefore a continuous singular function at zero.
The difficulty there comes from the appearance of anticipating integrands and from the singularity of $f$ at zero. [@Ffil] gives sense to this integral in some Stratonovich sense. Moreover that paper explores the connection with self-intersection local time considered for instance by J.-F. Le Gall in [@gall].
The work [@Nua] considers a similar double integral in the case of fractional Brownian motion with Hurst index $H > \frac{1}{2}$ using Malliavin-Skorohod anticipating calculus.
A natural approach is to interpret previous double integral as a symmetric (or eventually) forward integral in the framework of stochastic calculus via regularization, see [@RVSem] for a survey. We recall that when $X$ is a semimartingale, forward (respectively symmetric) integral $\int_0^t \varphi(X) d^- X$ (resp. $\int_0^t \varphi(X) d^\circ X$) coincides with the corresponding Itô (resp. Stratonovich) integral. The double stochastic integral considered by [@Ffil] coincides in fact with the symmetric integral introduced here. So (\[EDint\]) can be interpreted as $$\label{EDintsym}
\int_{[0,T]^2} f(X_s - X_t) d^\circ X_s d^\circ X_t,$$
In this paper, $X$ will be a fractional Brownian motion with Hurst index $ H > 1/4$ but a complete study of the existence of integrals (\[EDintsym\]) will be not yet performed here because of heavy technicalities. We will only essentially consider their regularized versions. Now, those double integrals are naturally in correspondence with currents related to $I$ defined in (\[ECurr\]). The investigation of those currents is strictly related to “pathwise stochastic calculus” in the spirit of *rough paths* theory by T. Lyons and coauthors, [@Lyons; @qianlyons]. Here we aim at exploring the “pathwise character” of stochastic integrals via regularization. A first step in this direction was done in [@GradNo], where the authors showed that forward integrals of the type $\int_0^T \varphi(X) d^- X$, when $X$ is a one-dimensional semimartingale or a fractional Brownian motion with Hurst index $H > \frac{1}{2}$, can be regarded as a.s. uniform approximations of their regularization $I^-_{\varepsilon}(\varphi)$ (see section 3), instead of the usual convergence in probability.
This analysis of currents related to stochastic integrals was started in [@FGGT] using an approach based on spectral analysis. The approach presented here is not based on Fourier transform and contains new general ideas with respect to [@FGGT]. Informally speaking, it is based on the formula $$\int_{0}^{T}\left\langle \varphi\left( X_{t}\right) ,dX_{t}\right\rangle
=\int_{\mathbb{R}^{d}}\left\langle (1-\Delta)^{\alpha}\varphi\left( x\right)
,\int_{0}^{T}K_{\alpha}\left( x-X_{t}\right) dX_{t}\right\rangle
dx .\label{decoupling}$$ This formula decouples $\varphi$ and $X$ and replaces the problem of the pathwise dependence of $I\left( \varphi\right) $ on the infinite dimensional parameter $\varphi$ with the problem of the pathwise dependence of $\int
_{0}^{T}K_{\alpha}\left( x-X_{t}\right) dX_{t}$ on the finite dimensional parameter $x\in\mathbb{R}^{d}$. Another form of decoupling is also one of the ingredients of the Fourier approach of [@FGGT] but the novelty here is that we can take better advantages from the properties of the underlying process (like, for example, the existence of a density). Moreover formula (\[decoupling\]) produces at least two new results.
First, we can treat in an essentially optimal way the case of fractional Brownian motion, making use of its Gaussian properties. For $H>1/2$ results in this direction can be extracted from the estimates proved in [@Nua] again by spectral analysis. However, with the present approach we may treat the case $H\in(1/4,1/2)$ as well.
Second, in the case of the Brownian motion, we may work with functions $\varphi$ in the Sobolev spaces of Banach type $H_{p}^{\alpha}$, with $p>1$, instead of only the Hilbert topologies $H_{2}^{\alpha}$ considered in [@FGGT], with the great advantage that it is sufficient to ask less differentiability on $\varphi$ (any $\alpha>1$ suffices), at the price of a larger $p$ (depending on $\alpha$ and the space dimension). In this way we may cover, for instance, the class $\varphi\in C^{1,\varepsilon}$ treated in [@Lyons], see also [@Gubi]; the approach here is entirely different and does not rely on rough paths, see Remark \[Rrough\].
Finally, we apply these ideas to random vortex filaments. In the case of the fractional Brownian motion we prove new results about the finiteness of the kinetic energy of the filaments (such a property is expected to be linked to the regularity of the pathwise current). In the case of the Brownian motion optimal conditions for a finite energy were already proved in [@Ffil; @FGub], while a sufficient condition when $H>1/2$ has been found in [@Nua]. The results of the present work provide new regularity properties of the random filaments, especially for the parameter range $H \in(1/4,1/2) $.
Generalities {#sec:hilbert}
============
Stochastic currents
-------------------
Let $\left( X_{t}\right) $ be a stochastic process such that $X_0 = 0$ a.s. on a probability space $\left( \Omega,\mathcal{F},P\right) $ with values in $\mathbb{R}^{d}$. Let $T > 0$. Let $V $ be a Banach space of vector fields $\varphi:\mathbb{R}^{d}\rightarrow
\mathbb{R}^{d}$ and $\mathcal{D}\subset V$ be a dense subset. Assume that a stochastic integral $$I\left( \varphi\right) =\int_{0}^{T}\left\langle \varphi\left(
X_{t}\right) ,dX_{t}\right\rangle$$ is well defined, in a suitable sense (Itô, etc.), for every $\varphi
\in\mathcal{D}$. Our first aim is to define it for every $\varphi\in V$. In addition, we would like to prove that it has a pathwise redefinition according to the following:
\[def:pathwise\] The family of r.v. $\{I(\varphi)\}_{\varphi\in V}$ has a [**pathwise redefinition**]{} on $V$ if there exists a measurable mapping $\xi
:\Omega\rightarrow V^{\prime}$ such that for every $\varphi\in\mathcal{D}$ $$I\left( \varphi\right) \left( \omega\right) =\left( \xi\left(
\omega\right) \right) \left( \varphi\right) \text{ for }P\text{-a.e.
}\omega\in\Omega.\label{pathwise}$$
Then, if we succeed in our objective,
- for every $\varphi\in V$ we consider the r.v. $\omega\mapsto\left(
\xi\left( \omega\right) \right) \left( \varphi\right) $ as the definition of the stochastic integral $I\left( \varphi\right) $ (now extended to the class $\varphi\in V$ )
- for $P$-a.e. $\omega\in\Omega$, we consider the linear continuous mapping $\varphi\mapsto\left( \xi\left( \omega\right) \right) \left(
\varphi\right) $ as a pathwise redefinition of stochastic integral on $V$.
Formally, the candidate for $\xi$ is the expression $$\xi\left( x\right) =\int_{0}^{T}\delta\left( x-X_{t}\right) dX_{t}.$$ where $\delta$ is here the $d$-dimensional Dirac measure. Indeed, always formally, $$\begin{aligned}
\xi\left( \varphi\right) & =\int_{\mathbb{R}^{d}}\langle\xi\left(
x\right) , \varphi\left( x\right) \rangle dx=\int_{0}^{T}\left\langle \left(
\int_{\mathbb{R}^{d}}\delta\left( x-X_{t}\right) \varphi\left( x\right)
dx\right) , dX_{t}\right\rangle \\
& =\int_{0}^{T}\langle\varphi( X_{t}), dX_{t}\rangle=I(\varphi).\end{aligned}$$ We remark that this viewpoint is inspired by the theory of currents; with other methods (spectral ones) it was developed in [@FGGT].
Decoupling by duality
---------------------
As we said in the introduction, our approach is based on a proper rigorous version of formula (\[decoupling\]). One way to interpret it by the following duality argument, that we describe only at a formal level.
Let $W$ be another Banach space and $\Lambda:V\rightarrow W$ be an isomorphism. Proceeding formally as above we have $$\begin{aligned}
\int_{0}^{T}\left\langle \varphi\left( X_{t}\right) ,dX_{t}\right\rangle
_{\mathbb{R}^{d}} & =\left\langle \varphi,\xi\right\rangle _{V,V^{\prime}}=\left\langle \Lambda^{-1}\Lambda\varphi,\xi\right\rangle _{V,V^{\prime}}\\
& =\left\langle \Lambda\varphi,\left( \Lambda^{-1}\right) ^{\ast}\xi\right\rangle _{W,W^{\prime}}$$ (notice that $\Lambda^{-1}:W\rightarrow V$, $\left( \Lambda^{-1}\right)
^{\ast}:V^{\prime}\rightarrow W^{\prime}$). Our aim essentially amounts to prove that $
\left( \Lambda^{-1}\right) ^{\ast}\xi:\Omega\rightarrow W^{\prime}$ is a well defined random variable.
This reformulation becomes useful if the spaces $W$, $W^{\prime}$ are easier to handle than $V$, $V^{\prime}$, and the operator $\left( \Lambda
^{-1}\right) ^{\ast}$ has a kernel $K(x,y)$ as an operator in function spaces: $$\left( \left( \Lambda^{-1}\right) ^{\ast}f\right) \left( x\right) =\int
K\left( x,y\right) f(y)dy.$$ In such a case, formally $$\begin{split}
\left( \left( \Lambda^{-1}\right) ^{\ast}\xi\right) \left( x\right) &
=\int K\left( x,y\right) \left( \int_{0}^{T}\delta\left( y-X_{t}\right)
dX_{t}\right) dy\\
& =\int_{0}^{T}\left( \int K\left( x,y\right) \delta\left( y-X_{t}\right) dy\right) dX_{t}\\
& =\int_{0}^{T}K\left( x,X_{t}\right) dX_{t}.
\end{split}$$ Below we make a rigorous version of this representation by choosing $V=H_{p}^{\alpha}\left( \mathbb{R}^{d}\right) $, $W=L^{p}\left( \mathbb{R}^{d}\right) $, $\Lambda=(1-\Delta)^{\frac{\alpha}{2}}$, $K\left( x,y\right)
=$ $K_{\alpha/2}\left( x-y\right) $ (notations are given in the next section).
Rigorous setting {#preliminaries}
----------------
Denote by $S\left( \mathbb{R}^{d}\right) $ the space of rapidly decreasing infinitely differentiable *vector fields* $\varphi:\mathbb{R}^{d}\rightarrow
\mathbb{R}^{d}$, by $S^{\prime}\left( \mathbb{R}^{d}\right) $ its dual (the space of tempered distributional fields) and by $\mathcal{F}$ the Fourier transform $$\left( \mathcal{F}\varphi\right) \left( \ell\right) =\left( 2\pi\right)
^{-d/2}\int_{\mathbb{R}^{d}}e^{-i\left\langle x,\ell\right\rangle }\varphi(x)dx,\quad\ell\in\mathbb{R}^{d}$$ which is an isomorphism in both $S\left( \mathbb{R}^{d}\right) $ and $S^{\prime}\left( \mathbb{R}^{d}\right) $. Let $\mathcal{F}^{-1}$ denote the inverse Fourier transform. For every $s\in\mathbb{R}$, let $\Lambda
_{s}:S^{\prime}\left( \mathbb{R}^{d}\right) \rightarrow S^{\prime}\left(
\mathbb{R}^{d}\right) $ be the pseudo-differential operator defined as $$\Lambda_{s}\varphi=\mathcal{F}^{-1}\left( 1+\left| \ell\right| ^{2}\right)
^{\frac{s}{2}}\mathcal{F}\varphi.$$ We shall also denote it by $(1-\Delta)^{\frac{s}{2}}$.
Let $H_{p}^{s}\left( \mathbb{R}^{d}\right) $, with $p>1$ and $s\in\mathbb{R}
$, be the Sobolev space of vector fields $\varphi\in S^{\prime}\left(
\mathbb{R}^{d}\right) $ such that $$\left\| \varphi\right\| _{H_{p}^{s}}^{p}:=\int_{\mathbb{R}^{d}}|(1-\Delta)^{\frac{s}{2}}\varphi(x)|^{p}dx<\infty,$$ see [@Triebel], sec. 2.3.3, where the definition chosen here for brevity is given as a characterization. From the very definitions of $\Lambda_{s}$ and $H_{p}^{s}\left( \mathbb{R}^{d}\right) $, the operator $\Lambda_{s}$ is an isomorphism from $H_{p}^{s}\left( \mathbb{R}^{d}\right) $ onto $L^{p}\left( \mathbb{R}^{d}\right)
$ (the Lebesgue space of $p$-integrable *vector fields*).
Another fact often used in the paper is that the dual space $\left( H_{p}^{s}\left( \mathbb{R}^{d}\right) \right) ^{\prime}$ is $H_{p^{\prime}}^{-s}\left( \mathbb{R}^{d}\right) $: $$\left( H_{p}^{s}\left( \mathbb{R}^{d}\right) \right) ^{\prime
}=H_{p^{\prime}}^{-s}\left( \mathbb{R}^{d}\right) ,\quad\frac{1}{p}+\frac
{1}{p^{\prime}}=1,$$ see [@Triebel], section 2.6.1. Moreover, being $\Lambda_{s}$ an isomorphism from $H_{p}^{s}\left( \mathbb{R}^{d}\right) $ onto $L^{p}\left(
\mathbb{R}^{d}\right) $, its dual operator $\Lambda_{s}^{\star}$ is an isomorphism from $L^{p^{\prime}}\left( \mathbb{R}^{d}\right) $ onto $H_{p^{\prime}}^{-s}\left( \mathbb{R}^{d}\right) $.
It is known that negative fractional powers of a positive selfadjoint operator $A$ in a Hilbert space $H$, such that $-A$ generates the semigroup $T(t)$, have the representation $$A^{-\alpha}=\frac{1}{\Gamma\left( \alpha\right) }\int_{0}^{\infty}t^{\alpha-1}T\left( t\right) dt,$$ where $\Gamma$ is the standard Gamma function, see [@Pa], formula (6.9). Taking $A=$ $(1-\Delta)$ in the Hilbert space $H=L^{2}\left( \mathbb{R}^{d}\right) $, we have $$\left( T\left( t\right) \varphi\right) \left( x\right) =\left( 4\pi
t\right) ^{-d/2}\int_{\mathbb{R}^{d}}e^{-\frac{\left| x-y\right| ^{2}}{4t}-t}\varphi(y)dy$$ and thus, for $\alpha>0$, $$\begin{aligned}
\left( (1-\Delta)^{-\alpha}\varphi\right) (x) & =\frac{\left( 4\pi
t\right) ^{-d/2}}{\Gamma\left( \alpha\right) }\int_{0}^{\infty}t^{\alpha
-1}\int_{\mathbb{R}^{d}}e^{-\frac{\left| x-y\right| ^{2}}{4t}-t}\varphi(y)dydt\\
& =\int_{\mathbb{R}^{d}}\left[ \frac{1}{\Gamma\left( \alpha\right) \left(
4\pi\right) ^{d/2}}\int_{0}^{\infty}t^{\alpha-1-\frac{d}{2}}e^{-\frac{\left|
x-y\right| ^{2}}{4t}-t}dt\right] \varphi(y)dy.\end{aligned}$$ In fact this formula can be proved more elementarily from the definition of $(1-\Delta)^{-\alpha}\varphi$ and the formula $$\lambda^{-\alpha}=\frac{1}{\Gamma\left( \alpha\right) }\int_{0}^{\infty
}t^{\alpha-1}e^{-\lambda t}dt,$$ then taking $\lambda=1+\left| \ell\right| ^{2}$ and the Fourier transform of the Gaussian density. This fact implies that the operator $(1-\Delta
)^{-\alpha}$, which originally is an isomorphism between $H_{2}^{-2\alpha
}\left( \mathbb{R}^{d}\right) $ and $L^{2}\left( \mathbb{R}^{d}\right) $, considered by restriction as a bounded linear operator in $L^{2}\left(
\mathbb{R}^{d}\right) $, has a kernel $K_{\alpha}\left( .\right) $, $$\left( (1-\Delta)^{-\alpha}\varphi\right) (x)=\int_{\mathbb{R}^{d}}K_{\alpha}\left( x-y\right) \varphi(y)dy\label{eq kernel}$$ given by $$K_{\alpha}\left( x\right) =\frac{1}{\Gamma\left( \alpha\right) \left(
4\pi\right) ^{d/2}}\int_{0}^{\infty}t^{\alpha-\frac{d}{2}}e^{-\frac{\left|
x\right| ^{2}}{4t}-t}\frac{dt}{t}.$$ The following estimates are not optimized as far as the exponential decay is concerned; we just state a version sufficient for our purposes. The proof of the two lemmas before are in Appendix A.
\[lemma su K\]There exists positive constants $c_{\alpha,d}$, $C_{\alpha,d}$ such that:
- For $0<\alpha<\frac{d}{2}$, we have $$\label{eq:asympK}
K_{\alpha}\left( x\right) =\left| x\right| ^{2\alpha-d}\rho\left(
x\right)$$ where $
c_{\alpha,d}e^{-2\left| x\right| ^{2}}\leq\rho\left( x\right) \leq
C_{\alpha,d}e^{-\frac{\left| x\right| }{8}}$;
- For $\alpha
>\frac{d}{2}$, we have $$\label{eq:asympK2new}
c_{\alpha,d} e^{-\frac{\left| x\right| ^{2}}{4}}\leq K_{\alpha
}\left( x\right) \leq C_{\alpha,d} e^{-\frac{\left| x\right| }{8}}$$ for two positive constants $c_{\alpha,d}^{\prime}$, $C_{\alpha,d}^{\prime}$. Moreover $$\label{eq:asympK2}
K_{\alpha}(x) = K_\alpha(0) - \rho'(x) |x|^{-d+2\alpha} \ge 0$$ with $0 < K_\alpha(0) < \infty$ and where $
c_{\alpha,d}e^{-2\left| x\right| ^{2}}\leq\rho'\left( x\right) \leq
C_{\alpha,d}e^{-\frac{\left| x\right| }{8}}$;
- Finally, when $\alpha=d/2$ we have $$\label{eq:log-asymp-K}
K_\alpha(x) \le C_{\alpha,d} \ \log |x| e^{- a_\alpha \ |x|}$$ where $a_\alpha$ is another positive constant.
\[RsuK\] In particular $\rho$ and $\rho'$ are bounded.
In the applications we will need also some control on $\Delta K_\alpha(x)$ which is provided by the next lemma.
\[lemma:deltaK\] It holds that $-\Delta K_{\alpha}(x) = K_\alpha(x) - K_{\alpha-1}(x)$. Then, when $\alpha<d/2+1$, $$|-\Delta K_{\alpha}(x)| \le \rho^{\prime\prime}(x)|x|^{2\alpha-d-2}\label{eq:asympDeltaK}$$ where $\rho^{\prime\prime}$ is positive, bounded above, locally bounded away from zero below and depends on $\alpha$.
Regularity of stochastic currents
---------------------------------
With these notations and preliminaries in mind, we may state a first rigorous variant of formula (\[decoupling\]). Given a continuous stochastic process $\left( X_{t}\right) _{t\geq0}$ on $\left( \Omega,\mathcal{F},P\right) $ with values in $\mathbb{R}^{d}$, given $\varepsilon>0 $, let $\left(
D_{\varepsilon}X_{t}\right) _{t\geq0}$ be any one of the following discrete derivatives: $$\frac{X_{t+\varepsilon}-X_{t}}{\varepsilon},\quad\frac{X_{t}-X_{t-\varepsilon
}}{\varepsilon},\quad\frac{X_{t+\varepsilon}-X_{t-\varepsilon}}{\varepsilon}$$ where we understand that $X_{t-\varepsilon}=0$ for $t<\varepsilon$. The following integral $$I_{\varepsilon}\left( \varphi\right) =\int_{0}^{T}\left\langle
\varphi\left( X_{t}\right) ,D_{\varepsilon}X_{t}\right\rangle dt$$ is well defined $P$-a.s. as a classical Riemann integral, at least for every continuous vector field $\varphi$.
\[L5\] Given $\alpha,\varepsilon>0$, with probability one the function $t\mapsto K_{\alpha/2}\left( x-X_{t}\right) D_{\varepsilon}X_{t}$ is integrable for a.e. $x\in\mathbb{R}^{d}$, the function $$\eta_{\varepsilon}\left( x\right) :=\int_{0}^{T}K_{\alpha/2}\left(
x-X_{t}\right) D_{\varepsilon}X_{t}dt$$ is in $L^{1}\left( \mathbb{R}^{d}\right) $ and for any $\varphi\in S\left( \mathbb{R}^{d}\right) $ we have $$\int_{0}^{T}\left\langle \varphi\left( X_{t}\right) ,D_{\varepsilon}X_{t}\right\rangle dt=\int_{\mathbb{R}^{d}}\left\langle (1-\Delta)^{\alpha
/2}\varphi\left( x\right) ,\int_{0}^{T}K_{\alpha/2}\left( x-X_{t}\right)
D_{\varepsilon}X_{t}dt\right\rangle dx.\label{decoupling epsilon}$$
Notice that $(1-\Delta)^{\alpha/2}\varphi\in S\left( \mathbb{R}^{d}\right) $ and, by (\[eq kernel\]), $$\begin{aligned}
\varphi=(1-\Delta)^{-\alpha/2}(1-\Delta)^{\alpha/2}\varphi
=\int_{\mathbb{R}^{d}}K_{\alpha/2}\left( \cdot-x\right) \left[
(1-\Delta)^{\alpha/2}\varphi\right] (x)dx.\end{aligned}$$ Thus $$\int_{0}^{T}\left\langle \varphi\left( X_{t}\right) ,D_{\varepsilon}X_{t}\right\rangle dt=\int_{0}^{T}\left\langle \int_{\mathbb{R}^{d}}K_{\alpha/2}\left( X_{t}-x\right) \left[ (1-\Delta)^{\alpha/2}\varphi\right] (x)dx,D_{\varepsilon}X_{t}\right\rangle dt.$$ Denote by $\widehat{K}_{\alpha/2}\left( x\right) $ the function equal to $K_{\alpha/2}\left( x\right) $ for $x\neq0$, infinite for $x=0$. Suppose for a moment that $$P\left( \int_{0}^{T}\int_{\mathbb{R}^{d}}\widehat{K}_{\alpha/2}\left(
X_{t}-x\right) dxdt<\infty\right) =1. \label{eq integrability}$$ Then the integrability properties stated in the lemma will hold. Since $$\sup_{ \left( t,x\right) \in\left[ 0,T\right]
\times\mathbb{R}^{d}} \left \vert
\left\langle \left[ (1-\Delta)^{\alpha/2}\varphi\right]
(x), D_{\varepsilon}X_{t}\right\rangle \right \vert <\infty \quad {a.s.},$$ using Fubini theorem we will get (\[decoupling epsilon\]) (notice that $K_{\alpha/2}\left(
X_{t}-x\right) =K_{\alpha/2}\left( x-X_{t}\right) $).
Thus we have only to prove (\[eq integrability\]). Since $K_{\alpha/2}$ is positive, we may apply again Fubini theorem and analyze $ \int_{0}^{T}( \int_{\mathbb{R}^{d}}\widehat{K}_{\alpha/2}(
X_{t}-x) dx) dt
$. But we have, for every $y\in\mathbb{R}^{d}$$$\int_{\mathbb{R}^{d}}\widehat{K}_{\alpha/2}\left( y-x\right) dx = \int_{\mathbb{R}^{d}}\widehat{K}_{\alpha/2}\left( x\right) dx.$$ This quantity is finite for every $\alpha>0$, from the estimates of lemma \[lemma su K\]. The proof is now complete.
Below we need a criterion to decide when $\eta_{\varepsilon}$ (defined in the previous lemma) belongs to $L^{2}\left( \mathbb{R}^{d}\right) $. It is thus useful to introduce the following condition which ensures the existence of the representation given in Lemma \[ltech5\] below.
\[cond:B\] $
\int_{0}^{T}\int_{0}^{T} \widehat
K_{\alpha}\left( X_{t}-X_{s}\right) dtds<\infty
$ ${P}$-a.s.
$\int_{0}^{T}\int_{0}^{T} \widehat K_{\alpha}\left( X_{t}-X_{s}\right) dtds<\infty$, implies that the function $\left( t,s\right) \mapsto X_{t}-X_{s}$ is different from zero except possibly on a zero measure set of $\left[ 0,T\right] ^{2}$, and that the well-defined function $\left( t,s\right) \mapsto \widehat K_{\alpha}\left(
X_{t}-X_{s}\right) $ is Lebesgue integrable on $\left[ 0,T\right] ^{2}$. From now on the notation $ \widehat
K_{\alpha} $ will simply be replaced by $K_{\alpha}$.
\[ltech5\] Under Condition \[cond:B\] we have the following double integral representation for the norm of $\eta_{\varepsilon}$ defined in Lemma \[L5\]: $$\left\| \eta_{\varepsilon}\right\| _{L^{2}(\mathbb{R}^{d})}^{2}=\int_{0}^{T}\int_{0}^{T}K_{\alpha}\left( X_{t}-X_{s}\right) \left\langle
D_{\varepsilon}X_{t},D_{\varepsilon}X_{s}\right\rangle dtds.$$
We have $$\begin{aligned}
\left\| \eta_{\varepsilon}\right\| _{L^{2}(\mathbb{R}^{d})}^{2} &
=\int_{\mathbb{R}^{d}}\left\langle \eta_{\varepsilon}\left( x\right)
,\eta_{\varepsilon}\left( x\right) \right\rangle dx\\
& =\int_{\mathbb{R}^{d}}\left\langle \int_{0}^{T}K_{\alpha/2}\left(
x-X_{t}\right) D_{\varepsilon}X_{t}dt,\int_{0}^{T}K_{\alpha/2}\left(
x-X_{s}\right) D_{\varepsilon}X_{s}ds\right\rangle dx\\
& \int_{0}^{T}\int_{0}^{T}\int_{\mathbb{R}^{d}}K_{\alpha/2}\left(
x-X_{t}\right) K_{\alpha/2}\left( x-X_{s}\right) \left\langle
D_{\varepsilon}X_{t},D_{\varepsilon}X_{s}\right\rangle dxdtds\end{aligned}$$ if we can apply Fubini Theorem. Then it is sufficient to use the property $$\int_{\mathbb{R}^{d}}K_{\alpha/2}\left( x-y\right) K_{\alpha/2}\left(
x-z\right) dx=K_{\alpha}\left( y-z\right) .$$ Since the process $X$ is continuous, both $(t,s) \mapsto D_{\varepsilon}X_s D_{\varepsilon}X_t$ is a continuous two-parameter process which on $[0,T]^2$ is a.s. bounded. Then a sufficient condition to apply Fubini Theorem is $$\int_{0}^{T}\int_{0}^{T}\int_{\mathbb{R}^{d}}K_{\alpha/2}\left(
x-X_{t}\right) K_{\alpha/2}\left( x-X_{s}\right) dxdtds<\infty.$$ Since the integrand is positive, it is equal to $$\begin{aligned}
& \int_{0}^{T}\int_{0}^{T}\left( \int_{\mathbb{R}^{d}}K_{\alpha/2}\left(
x-X_{t}\right) K_{\alpha/2}\left( x-X_{s}\right) dx\right) dtds
=\int_{0}^{T}\int_{0}^{T}K_{\alpha}\left( X_{t}-X_{s}\right) dtds.\end{aligned}$$
Invoking condition \[cond:B\] we can conclude the proof.
The double integral representation of the norm of $\eta_{\varepsilon}$ will play a major rôle in the following, so we introduce the notation $$Z_{\alpha,{\varepsilon}} := \int_{0}^{T}\int_{0}^{T}K_{\alpha}\left( X_{t}-X_{s}\right) \left\langle
D_{\varepsilon}X_{t},D_{\varepsilon}X_{s}\right\rangle dtds.$$
\[lemma:ordering\] Assume Condition \[cond:B\] holds for any $\alpha \ge \overline{\alpha}$. Then the function $\alpha \mapsto Z_{\alpha,{\varepsilon}} \ge 0$ is decreasing for $\alpha \ge \overline{\alpha}$.
Denote $\eta_{\alpha,{\varepsilon}}(x) = \int_0^T K_{\alpha/2}(x-X_t) D_{\varepsilon}X_t dt $ making explicit the dependence on $\alpha$. It is not difficult to prove that, if $\overline{\alpha} \le \alpha \le \beta $ we have $
\eta_{\beta,{\varepsilon}} = (1-\Delta)^{(\alpha-\beta)/2} \eta_{\alpha,{\varepsilon}}
$. Then $Z_{\beta,{\varepsilon}} = \|\eta_{\beta,{\varepsilon}}\|^2_{L^2({\mathbb{R}}^d)} = \|(1-\Delta)^{(\alpha-\beta)/2}\eta_{\alpha,{\varepsilon}}\|^2_{L^2({\mathbb{R}}^d)} \le \|\eta_{\alpha,{\varepsilon}}\|^2_{L^2({\mathbb{R}}^d)}= Z_{\alpha,{\varepsilon}}$, since being $\alpha-\beta\le 0$ , the operator $(1-\Delta)^{(\alpha-\beta)/2}$ has a norm bounded by one.
We will assume below the following condition on the convergence of the regularized integrals.
\[cond:A\] For every $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{d}\right) $, $I_{\varepsilon}\left( \varphi\right) $ converges in probability to some r.v., denoted by $I\left( \varphi\right) $.
Under Condition \[cond:A\], the mapping $\varphi\mapsto I\left(
\varphi\right) $ is a priori defined only on $C_{0}^{\infty}\left(
\mathbb{R}^{d}\right) $ with values in the set $L^{0}\left( \Omega\right) $ of random variables. Its extension to $\varphi\in H_{2}^{\alpha}\left(
\mathbb{R}^{d}\right) $ is a result of the next theorem.
\[th:l2result\] Assume Conditions \[cond:B\] and \[cond:A\], and the a priori bound $$\sup_{\varepsilon\in\left( 0,1\right) }E\left[ \int_{0}^{T}\int_{0}^{T}K_{\alpha}\left( X_{t}-X_{s}\right) \left\langle D_{\varepsilon}X_{t},D_{\varepsilon}X_{s}\right\rangle dtds\right] <\infty.$$ Then:
i\) the mappings $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{d}\right)
\mapsto I_{\varepsilon}\left( \varphi\right) ,I\left( \varphi\right) \in
L^{0}\left( \Omega\right) $ take values in $L^{2}\left( \Omega\right) $ and extend (uniquely) to linear continuous mappings from $H_{2}^{\alpha
}\left( \mathbb{R}^{d}\right) $ to $L^{2}\left( \Omega\right) $. Moreover, for every $\varphi\in H_{2}^{\alpha}\left( \mathbb{R}^{d}\right) $, $I_{\varepsilon}\left( \varphi\right) \rightarrow I\left( \varphi\right) $ in probability and in $L^{2-\delta}\left( \Omega\right) $ for every $\delta>0$.
ii\) In addition, there exist random elements $\xi_{\varepsilon},\xi
:\Omega\rightarrow H_{2}^{-\alpha}\left( \mathbb{R}^{d}\right) $ (in fact belonging to $L^{2}( \Omega;H_{2}^{-\alpha}( \mathbb{R}^{d})) $) that constitute pathwise redefinitions of $I_{\varepsilon}$ and $I$ over the functions $\varphi\in H_{2}^{\alpha}\left( \mathbb{R}^{d}\right) $, in the sense of Definition \[pathwise\].
**Step 1** (mean square results). By the assumptions and the previous lemma we have $
\sup_{\varepsilon\in\left( 0,1\right) }E\left[ \left\| \eta_{\varepsilon
}\right\| _{L^{2}(\mathbb{R}^{d})}^{2}\right] <\infty
$. From (\[decoupling epsilon\]), for $\varphi\in C_{0}^{\infty}\left(
\mathbb{R}^{d}\right) $ we have $$I_{\varepsilon}\left( \varphi\right) =\left\langle \left( 1-\Delta\right)
^{\alpha/2}\varphi,\eta_{\varepsilon}\right\rangle _{L^{2}(\mathbb{R}^{d})}.$$ Therefore, always for $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{d}\right)
$, $$\left| I_{\varepsilon}\left( \varphi\right) \right| \leq\left\|
\eta_{\varepsilon}\right\| _{L^{2}(\mathbb{R}^{d})}\left\| \left(
1-\Delta\right) ^{\alpha/2}\varphi\right\| _{L^{2}\left( \mathbb{R}^{d}\right) }\leq C_{\alpha}\left\| \eta_{\varepsilon}\right\|
_{L^{2}(\mathbb{R}^{d})}\left\| \varphi\right\| _{H_{2}^{\alpha}\left(
\mathbb{R}^{d}\right) }$$$$E\left[ \left| I_{\varepsilon}\left( \varphi\right) \right| ^{2}\right]
\leq C_{\alpha}\left\| \varphi\right\| _{H_{2}^{\alpha}\left(
\mathbb{R}^{d}\right) }^{2}E\left[ \left\| \eta_{\varepsilon}\right\|
_{L^{2}\left( \mathbb{R}^{d}\right) }^{2}\right] \leq C_{\alpha}^{\prime
}\left\| \varphi\right\| _{H_{2}^{\alpha}\left( \mathbb{R}^{d}\right)
}^{2}.$$
Immediately we have $I_{\varepsilon}\left( \varphi\right) \in L^{2}\left(
\Omega\right) $ for every $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{d}\right) $, and the mapping $\varphi\mapsto I_{\varepsilon}\left(
\varphi\right) $ extends (uniquely by density) to a linear continuous mapping from $H_{2}^{\alpha}\left( \mathbb{R}^{d}\right) $ to $L^{2}\left(
\Omega\right) $.
Given $\varphi\in C_{0}^{\infty}( \mathbb{R}^{d}) $, since $I_{\varepsilon}( \varphi) \rightarrow I( \varphi) $ in probability, uniform integrability arguments and $E[ |
I_{\varepsilon}( \varphi) | ^{2}] \leq C_{\alpha
}^{\prime}\| \varphi\| _{H_{2}^{\alpha}( \mathbb{R}^{d}) }^{2}$, yield $I_{\varepsilon}( \varphi) \rightarrow
I( \varphi) $ in $L^{2-\delta}( \Omega) $ for every $\delta>0$; moreover, it is not difficult to deduce $I( \varphi)
\in L^{2}( \Omega) $ and $
E[ | I( \varphi) | ^{2}] \leq
C_{\alpha}^{\prime}\| \varphi\| _{H_{2}^{\alpha}(
\mathbb{R}^{d}) }^{2}
$. As before, this implies that the mapping $\varphi\mapsto I\left(
\varphi\right) $ extends uniquely to a linear continuous mapping from $H_{2}^{\alpha}\left( \mathbb{R}^{d}\right) $ to $L^{2}\left(
\Omega\right) $. Now, with these extensions, it is not difficult to show that $I_{\varepsilon}\left( \varphi\right) \rightarrow I\left( \varphi\right) $ in $L^{2-\delta}\left( \Omega\right) $ for every $\delta>0$ also for every for $\varphi\in H_{2}^{\alpha}\left( \mathbb{R}^{d}\right) $.
**Step 2** (pathwise results). We still have to construct $\xi
_{\varepsilon}$ and $\xi$. Recalling that $\eta_{\varepsilon}\in L^{2}\left(
\Omega;L^{2}(\mathbb{R}^{d})\right) $, $\xi_{\varepsilon}$ is simply defined as $[ ( 1-\Delta) ^{\alpha/2}] ^{\star}\eta_{\varepsilon}$, element of $L^{2}\left( \Omega;H_{2}^{-\alpha}\left(
\mathbb{R}^{d}\right) \right) $, where $A^\star$ denotes the dual of an operator $A$.
To this end, recall that $\left(
1-\Delta\right) ^{\alpha/2}$ is an isomorphism between $H_{2}^{\alpha}\left(
\mathbb{R}^{d}\right) $ and $L^{2}(\mathbb{R}^{d})$, and thus the dual operator $[ ( 1-\Delta) ^{\alpha/2}] ^{\star}$ is an isomorphism between the dual spaces $L^{2}(\mathbb{R}^{d})$ and $H_{2}^{-\alpha}\left( \mathbb{R}^{d}\right) $ (we identify $L^{2}(\mathbb{R}^{d})$ with its dual).
The family $\left\{ \eta_{\varepsilon}\right\} $ is bounded in $L^{2}\left(
\Omega;L^{2}(\mathbb{R}^{d})\right) $, hence there exist a sequence $\eta_{\varepsilon_{n}}$ weakly convergent to some $\eta$ in $L^{2}\left(
\Omega;L^{2}(\mathbb{R}^{d})\right) $: $
E\left\langle \eta_{\varepsilon_{n}},Y\right\rangle _{L^{2}\left(
\mathbb{R}^{d}\right) }\rightarrow E\left\langle \eta,Y\right\rangle
_{L^{2}(\mathbb{R}^{d})}$ for every $Y\in L^{2}( \Omega;L^{2}(\mathbb{R}^{d})) $. We set $\xi:=[ ( 1-\Delta) ^{\alpha/2}] ^{\star}\eta$, random element of $H_{2}^{-\alpha}\left( \mathbb{R}^{d}\right) $. We shall see that this definition does not depend on the sequence $\varepsilon_{n}$. We have to prove that for every $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{d}\right) $, $
I\left( \varphi\right) \left( \omega\right) =\left( \xi\left(
\omega\right) \right) \left( \varphi\right) \text{ for }P\text{-a.s.
}\omega\in\Omega.
$ Equivalently we have to prove that for every $\varphi\in C_{0}^{\infty}\left(
\mathbb{R}^{d}\right) $
$$I\left( \varphi\right) \left( \omega\right) =
\eta
\left (\omega \right ) \left ( \left(
1-\Delta \right) ^{\alpha/2} \varphi\right) \text{ for
}P\text{-a.s. }\omega\in\Omega.\label{tobeproved}$$
We already know that $I_{\varepsilon}(\varphi) =\langle
( 1-\Delta) ^{\alpha/2}\varphi,\eta_{\varepsilon}\rangle
_{L^{2}(\mathbb{R}^{d})}$ for every $\varepsilon>0$. Choose $Y$ above of the form $Y=F\left( 1-\Delta\right) ^{\alpha/2}\varphi$ with generic $F\in
L^{2}\left( \Omega\right) $. Given $\varphi\in C_{0}^{\infty}\left(
\mathbb{R}^{d}\right) $, we know that $$E\left[ F\left\langle \eta_{\varepsilon_{n}},\left( 1-\Delta\right)
^{\alpha/2}\varphi\right\rangle _{L^{2}(\mathbb{R}^{d})}\right] \rightarrow
E\left[ F\left\langle \eta,\left( 1-\Delta\right) ^{\alpha/2}\varphi\right\rangle _{L^{2}(\mathbb{R}^{d})}\right]$$ for every $F\in L^{2}\left( \Omega\right) $. Hence $$E\left[ F I_{\varepsilon}\left( \varphi\right) \right] \rightarrow E\left[
F\left\langle \eta,\left( 1-\Delta\right) ^{\alpha/2}\varphi\right\rangle
_{L^{2}(\mathbb{R}^{d})}\right]$$ but we also know that $I_{\varepsilon}\left( \varphi\right) \rightarrow
I\left( \varphi\right) $ in $L^{2-\delta}\left( \Omega\right) $ for every $\delta>0$. We get $$E\left[ F I\left( \varphi\right) \right] =E\left[ F \left\langle
\eta,\left( 1-\Delta\right) ^{\alpha/2}\varphi\right\rangle _{L^{2}(\mathbb{R}^{d})}\right]$$ at least for every bounded random variable $F$, hence (\[tobeproved\]) holds true. This also implies that the definition of $\xi$ does not depend on the sequence $\varepsilon_{n}$. The proof is complete.
To state a possible converse of Th. \[th:l2result\] it is useful to introduce a weaker version of Def. \[def:pathwise\].
\[def:non-pathwise\] Let $A\in\mathcal{F}$ be such that $P(A)>0$. We say that $\{I(\varphi)\}_{\varphi\in V}$ has a [**pathwise redefinition on**]{} $A$ if there exists a measurable mapping $\xi:A\rightarrow
V^{\prime}$ such that for every $\varphi\in C_{0}^{\infty}\left(
\mathbb{R}^{d}\right) $ equation (\[pathwise\]) holds true for $P$-a.e. $\omega\in A$.
\[th:neg-res-l2\] Assume Conditions \[cond:B\] and \[cond:A\], and the a priori bound$$E\left[ \int_{0}^{T}\int_{0}^{T}K_{\alpha}\left( X_{t}-X_{s}\right)
\left\langle D_{\varepsilon}X_{t},D_{\varepsilon}X_{s}\right\rangle
dtds\right] <\infty$$ for every $\varepsilon>0$. If there exists $A\in\mathcal{F}$ with $P(A)>0$ such that $\{I(\varphi)\}_{\varphi\in V}$ has a pathwise redefinition on $A$, then $$\underset{\varepsilon\rightarrow0}{\lim\sup}\int_{0}^{T}\int_{0}^{T}K_{\alpha
}\left( X_{t}-X_{s}\right) \left\langle D_{\varepsilon}X_{t},D_{\varepsilon
}X_{s}\right\rangle dtds<\infty\quad\text{$P$-a.s. $\omega\in A$}.$$
The first part of step 1 of the previous proof is still valid (except for the uniformity in $\varepsilon$ of the constants). Thus in particular $E[
\| \eta_{\varepsilon}\| _{L^{2}(\mathbb{R}^{d})}^{2}]
<\infty$, $I_{\varepsilon}( \varphi) =\langle (
1-\Delta) ^{\alpha/2}\varphi,\eta_{\varepsilon}\rangle $, $| I_{\varepsilon}( \varphi) | \leq C_{\alpha
,\varepsilon}\| \varphi\| _{H_{2}^{\alpha}( \mathbb{R}^{d}) }^{2}$, $I_{\varepsilon}( \varphi) \in L^{2}(
\Omega) $ for every $\varphi\in C_{0}^{\infty}( \mathbb{R}^{d}) $ and the mapping $\varphi\mapsto I_{\varepsilon}\left(
\varphi\right) $ extends to a linear continuous mapping from $H_{2}^{\alpha
}\left( \mathbb{R}^{d}\right) $ to $L^{2}\left( \Omega\right) $.
We know, by Condition \[cond:A\] and the existence of $\xi$, that for every $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{d}\right) $$$\left\langle \left( 1-\Delta\right) ^{\alpha/2}\varphi,\eta_{\varepsilon
}\left( \omega\right) \right\rangle _{L^{2}\left( \Omega\right) }\underset{\varepsilon\rightarrow0}{\rightarrow}\left( \xi\left(
\omega\right) \right) \left( \varphi\right) \text{ for }P\text{-a.e.
}\omega\in A.$$ One can find a countable set $ {\cal D} \subset C_{0}^{\infty}\left( {\mathbb R}^{d}\right) $ with the following two properties: i) $\left( 1-\Delta\right)
^{\alpha/2}{\cal D}$ is dense in $L^{2}(\mathbb{R}^{d})$ and ii) for $P$-a.e. $\omega\in A$ $$\left\langle \left( 1-\Delta\right) ^{\alpha/2}\varphi,\eta_{\varepsilon
}\left( \omega\right) \right\rangle _{L^{2}\left( \Omega\right) }\underset{\varepsilon\rightarrow0}{\rightarrow}\left( \xi\left(
\omega\right) \right) \left( \varphi\right) \text{ for every }\varphi\in
\cal{D}.\label{limite aus}$$
Let us prove the claim by contradiction. Assume there is $A^{\prime}\in\mathcal{F}$, $A^{\prime}\subset A$, $P(A')>0$, such that $$\underset{\varepsilon\rightarrow0}{\lim\sup}\int_{0}^{T}\int_{0}^{T}K_{\alpha
}\left( X_{t}-X_{s}\right) \left\langle D_{\varepsilon}X_{t},D_{\varepsilon
}X_{s}\right\rangle dtds=\infty\quad\text{for every }\omega\in A^{\prime
}\text{.}$$ By lemma \[ltech5\], $
\underset{\varepsilon\rightarrow0}{\lim\sup}\left\| \eta_{\varepsilon}\left(
\omega\right) \right\| _{L^{2}(\mathbb{R}^{d})}^{2}=\infty$ for every $\omega\in A^{\prime}$.
Consequently, there is a subset $A^{\prime\prime}$ of $A^{\prime}$ with $P(A^{\prime\prime} ) > 0$ such that (\[limite aus\]) holds true for every $\omega\in
A^{\prime\prime}$. Thus, given $\omega\in A^{\prime\prime}$, there is an infinitesimal sequence $\left\{ \varepsilon_{n}\left( \omega\right)
\right\} $ such that $
\lim_{n\rightarrow\infty}\left\| \eta_{\varepsilon_{n}\left( \omega\right)
}\left( \omega\right) \right\| _{L^{2}(\mathbb{R}^{d})}^{2}=\infty
$ but at the same time $$\lim_{n\rightarrow\infty}\left\langle \left( 1-\Delta\right) ^{\alpha
/2}\varphi,\eta_{\varepsilon_{n}\left( \omega\right) }\left( \omega\right)
\right\rangle _{L^{2}\left( \Omega\right) }=\left( \xi\left(
\omega\right) \right) \left( \varphi\right)$$ for every $\varphi\in \cal{D}$. This is impossible because of the density of $\left( 1-\Delta\right) ^{\alpha/2} \cal{D} $ in $L^{2}(\mathbb{R}^{d})$. The proof is complete.
We say that a.s. a stochastic current $I$ does not belong to $V'$ if there is no $A \in \mathcal{F}$, ${P}(A) > 0$ such that $\{ I(\varphi) \}_{\varphi \in V}$ has a pathwise redefinition on $A$.
\[cor:neg-res\] Under the conditions of Theorem \[th:neg-res-l2\], if $$\underset{\varepsilon\rightarrow0}{\lim\sup}\int_{0}^{T}\int_{0}^{T}K_{\alpha
}\left( X_{t}-X_{s}\right) \left\langle D_{\varepsilon}X_{t},D_{\varepsilon
}X_{s}\right\rangle dtds= +\infty\quad\text{$P$-a.s.}$$ the current $I$ does not belong to $H^\alpha_2({\mathbb{R}}^d)$.
Application to the fractional Brownian motion
=============================================
We recall here that a fractional Brownian motion (fBm) $B = (B_t)$ with Hurst index $H\in\left( 0,1\right )$, is a Gaussian mean-zero real process whose covariance function is given by $$Cov(B_s,B_t) = \frac{1}{2}(|s|^{2H}+|t|^{2H}-|s-t|^{2H}),\,(s,t)\in \mathbb{R}_+^{2}.$$ This process has been widely studied: for some recent developments, we point to [@em] as a relevant monograph. For instance we recall that when $H = \frac{1}{2}$, $B$ is a classical Wiener process. Its trajectories are Hölder continuous with respect to any parameter $\gamma < H$. Recall that $$E( \vert B_t - B_s \vert^2) = \vert t - s \vert ^{2H}.$$ For our stochastic integral redefinition, we have chosen the framework of stochastic calculus via regularization; for survey about the topic and recent developments, see [@RVSem].
Let $X=\left( X^{1},...,X^{d}\right) $ be a $d$-dimensional fractional Brownian motion with Hurst index $H\in\left( 0,1\right)$, i.e. an $\mathbb{R}^d$-valued process whose components $X^{1},...,X^{d}$ are real independent fractional Brownian motions. In this section we study the regularity of the current generated by $X$ using the results of the previous section and in particular Theorem \[th:l2result\] which gives sufficient conditions for regularity in the Hilbert spaces $H^{\alpha}_{2}({\mathbb{R}}^d)$.
We will consider the *symmetric* and *forward* integrals, respectively defined as the limit in probability, as ${\varepsilon}\to 0$, of $$I^\circ_{\varepsilon}(\varphi) := \int_0^T < \varphi(X_t), \frac{X_{t+{\varepsilon}}-X_{t-{\varepsilon}}}{2{\varepsilon}}> dt$$ and $$I^-_{\varepsilon}(\varphi) := \int_0^T <\varphi(X_t), \frac{X_{t+{\varepsilon}}-X_{t}}{{\varepsilon}}> dt.$$ Whenever they exist we will write $$I^\circ (\varphi) = \int_0^T <\varphi(X_t), d^0 X_t>, \quad \text{and} \quad
I^-(\varphi) = \int_0^T <\varphi(X_t), d^- X_t>.$$
In [@Alos] the authors show that the symmetric integral exists for fBm with any $H > 1/4$ (the proof about the $d=1$ case but it extends without problem to higher dimensions). For $H > 1/6$ necessary and sufficient conditions for the existence of the symmetric integral are given in [@Grad]. As far as the forward integral is concerned, it is known that, in dimension $1$, it does not exist at least for some (very simple) functions $\varphi$. Existence of the forward integral in dimension one is guaranteed if and only if $H \ge 1/2$ [@RV; @RVSem]. Observe that, when $H < \frac{1}{2}$ the forward integral $\int_0^T B d^-B$ does not exist since $B$ is not of finite quadratic variation process.
When $H \ge 1/2$ and for $d > 1$ the forward integral (equal to the Young integral in the case $H > 1/2$), is equal to the symmetric integral minus the covariation $[\varphi(X),X]/2$. This exists if $X$ has all its *mutual covariations* and the $i$-th component of that bracket gives $$[f(X),X^i]_t
= \int_0^t \sum_{j=1}^d \partial_{j} f (X_s) d[X^i, X^j]_s,$$ see again [@RV].
Before entering into details concerning stochastic integration, we state an important preliminary result.
\[KAPPA-ALPHA\] If $\alpha > \max(0, d/2-1/(2H))$ then $ {E}\left (\int_0^T \int_0^T
K_\alpha(X_t - X_s) ds dt \right) < +\infty $.
Therefore Condition \[cond:B\] holds.
Since the corresponding random variable is non-negative, previous expectation equals $$\label{ETech}
\int_0^T \int_0^T {E}( K_\alpha(X_t - X_s) )ds dt =
2 \int_0^T \int_0^t {E}( K_\alpha(X_t - X_s) )ds dt.$$ Next we use the bound on $K_\alpha$ given in Lemma \[lemma su K\]:
- Suppose first that $ d/2-1/(2H) < \alpha < \frac{d}{2}$. Then the previous expression is bounded by $
{\textrm{const}}\int_0^T \int_0^t {E}( | X_t - X_s |^{2\alpha -d} )ds dt
$. For any $\gamma > -d$, the scaling property of fractional Brownian motion gives ${E}[| X_t - X_s |^\gamma] = {E}[| N |^{\gamma }]
(t-s)^{\gamma H}$, where $N$ is a standard $d$-dimensional Gaussian random variable. Then the right member of (\[ETech\]) equals $$\label{eq:tech}
{\textrm{const}}\ {E}[\vert N \vert^{2 \alpha -d} \int_0^T \int_0^t
\vert t-s \vert^{2 \alpha H - dH } ds dt$$ which is finite if $2 \alpha H - dH > -1$ and $\alpha > 0$. Therefore when $\alpha > \max(d/2-1/(2H),0)$.
- Suppose now that $\frac{d}{2} < \alpha$. In this case $K_\alpha \le {\textrm{const}}$ $(\ref{ETech})$ is trivially bounded.
- Finally, in the case $\alpha = d/2$ we have $K_\alpha(x) \le c
\log |x|$ and using again the scaling, it is easy to prove the boundedness of (\[ETech\]).
The Proposition above will allow us to verify the condition required by Lemma \[ltech5\] of previous section.
Symmetric integral
------------------
Let $D^0_{\varepsilon}X_t = (X_{t+{\varepsilon}}-X_{t-{\varepsilon}})/2{\varepsilon}$ and let $f:\mathbb{R}^{d} - \{0\}\rightarrow
\mathbb{R}_{+}$ such that $$\label{eq:f-class}
\int_0^T \int_0^T f(X_t-X_s) dt ds < \infty\qquad {P}-a.s.$$ and denote $$Z_{{\varepsilon}}(f):=\int_{0}^{T}\int_{0}^{T}f\left(
X_{t}-X_{s}\right)
\langle D^0_{\varepsilon}X_t, D^0_{\varepsilon}X_s \rangle
dsdt.$$
In this section we will study the r.v. $Z_{\alpha,{\varepsilon}} =
Z_{\varepsilon}(K_\alpha)$ in order to obtain necessary and sufficient conditions for the regularity of the symmetric current $I(\varphi)$ based on fBm.
The following lemma is proved in Appendix A.
\[lemma:bounds\]
1. We have the following estimates: $$\label{eq:bound1}
\vert{\textrm{Cov}}(D^0_{\varepsilon}X^i_t, D^0_{\varepsilon}X^i_s)|
\le {\textrm{const}}|t-s|^{2H-2};$$ and $$\label{eq:bound2}
|{\textrm{Cov}}(D^0_{\varepsilon}X^i_t, X^i_{t}-X^i_{s})| = |{\textrm{Cov}}(D^0_{\varepsilon}X^i_s, X^i_{t}-X^i_{s})|
\le {\textrm{const}}|t-s|^{2H-1}$$ uniformly in ${\varepsilon}>0$.
2. If $s \neq t$, one has $$\lim_{{\varepsilon}\to 0} |{\textrm{Cov}}(D^0_{\varepsilon}X^1_t,X^1_t- X^1_s)| = 2 H |t-s|^{2H-1}.$$
The main results of this section are contained in the next two theorems. Let $\alpha_H = d/2-1+1/(2H)$.
\[th:reg-symm\] For any $H>1/4$ and any $\alpha >\alpha_H$ we have $\sup_{{\varepsilon}} {E}Z_{\alpha,{\varepsilon}} < \infty$.
\[re:reg-symm\] We observe that $d/2-1/2H < \alpha_H$ so that for the range value for $\alpha$ in Theorem \[th:reg-symm\], Condition \[cond:B\] is verified.
\[th:opt-symm\] For any $H > 1/4$ and any $\alpha <\alpha_H$ we have $\liminf_{{\varepsilon}} E(Z_{\alpha,{\varepsilon}}) = +\infty$.
\[Ropt\] The statement of Theorem \[th:opt-symm\] and Lemma \[linter\] below allow us to formulate the following conjecture that we have not been able to prove: for $H>1/4$ and $\alpha < \alpha_H$ we should have $$\liminf_{{\varepsilon}\rightarrow 0} Z_{\alpha,{\varepsilon}} = + \infty, \quad \rm{a.s.}$$
For $\{I(\varphi)\}_{\varphi \in V}$ to have a pathwise redefinition on some $A \in \mathcal{F}$,${P}(A) > 0$ a necessary condition is that $\limsup_{{\varepsilon}\to 0} Z_{\alpha,{\varepsilon}} < \infty$ on $A$ (Th. \[th:neg-res-l2\]).
If this conjecture were true we could establish that a.s. $I(\varphi)$ does not belong to $H^{-\alpha}_{2}({\mathbb{R}}^d)$ when $\alpha < \alpha_H$.
Before proving the Theorems we deduce Sobolev regularity of fBm with any Hurst parameter between $(1/4,1)$.
\[cor:current-l2-reg\] The symmetric integral of a fractional Brownian motion with Hurst parameter $H > 1/4$ admits a pathwise redefinition on the space $H^{-\alpha}_{2}({\mathbb{R}}^d)$ for any $\alpha > \alpha_H$.
By Theorem \[th:reg-symm\] we know that ${E}[ Z_{\alpha,{\varepsilon}}]$ is uniformly bounded in ${\varepsilon}$ when $\alpha > \alpha_H$. Since the regularized integrals $I_{\varepsilon}(\varphi)$ converges in probability as ${\varepsilon}\to 0$ for any $H > 1/4$, Condition \[cond:A\] holds. So we can apply Theorem \[th:l2result\] to obtain a pathwise current with values in $H^{-\alpha}_{2}({\mathbb{R}}^d)$ for any $\alpha > \alpha_H$.
In the following proof we will use a basic result about Gaussian random variables recalled here:
\[th:wick\] Let $Z = (Z_{\ell})_{1 \le \ell \le N}$ be a mean-zero Gaussian random vector and $f \in C^{1}({\mathbb{R}}^{N} ;{\mathbb{R}})$, then we have $$\label{Ewick}
{E}[Z_{\ell} f(Z)] = \sum_{j=1}^{N} {\textrm{Cov}}(Z_{\ell},Z_{j}){E}[\nabla_{j} f(Z)].$$
The conclusion follows easily taking $f(z_1,...,z_N) = \exp(i \sum_{j=1}^N t_j z_j)$ for any $t=(t_1,...,t_N) \in {\mathbb{R}}^N $. In fact, in that case ${E}(f(Z)) $ is provided by the characteristic function. Therefore one has $${E}(\exp(it \cdot Z)) = \exp \left(- \frac{t \Gamma t'}{2} \right),$$ where $t'$ stands for the transposition and $\Gamma$ is the covariance matrix of $Z$. Differentiating the previous expression with respect to $t_\ell$ provides the result (\[Ewick\]) for the particular case of $f$. The general result follows by usual density arguments.
Using Lemma \[th:wick\], independence and equal distribution of different coordinates we have $$\label{eq:wick1}
\begin{split}
{E}Z_{\alpha,{\varepsilon}} & =
{E}\int_{0}^{T}\int_{0}^{T}K_\alpha\left(
X_{t}-X_{s}\right) \sum_{i}
{\textrm{Cov}}(D^0_{\varepsilon}X^i_t, D^0_{\varepsilon}X^i_s)
\,dt ds
\\ & \qquad +
{E}\int_{0}^{T}\int_{0}^{T} \Delta K_\alpha\left(
X_{t}-X_{s}\right)
{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s})
{\textrm{Cov}}(D^0_{\varepsilon}X^1_s, X^1_{t}-X^1_{s})
\,dt ds
\\ & =
{E}\int_{0}^{T}\int_{0}^{T}K_\alpha\left(
X_{t}-X_{s}\right) \sum_{i}
{\textrm{Cov}}(D^0_{\varepsilon}X^i_t, D^0_{\varepsilon}X^i_s)
\,dt ds
\\ & \qquad +
{E}\int_{0}^{T}\int_{0}^{T} (-\Delta) K_\alpha\left(
X_{t}-X_{s}\right)
|{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s})|^2
\,dt ds
\end{split}$$ where we used the fact that $${\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s}) = -
{\textrm{Cov}}(D^0_{\varepsilon}X^1_s, X^1_{t}-X^1_{s}) = \frac{1}{2{\varepsilon}}\left(|t-s+{\varepsilon}|^{2H}-|t-s-{\varepsilon}|^{2H}\right)$$ which can be verified by a straightforward computation.
Consider first the case $H > 1/2$. Assume $\alpha < d/2$ and note that when $H>1/2$ we have $\alpha_H < d/2$. By lemma \[lemma:bounds\] we have $$\begin{split}
{E}Z_{\alpha,{\varepsilon}} & \le {\textrm{const}}\,
{E}\int_{0}^{T}\int_{0}^{T}K_\alpha\left(
X_{t}-X_{s}\right) |t-s|^{2H-2}
\,dt ds
\\ & \qquad
+ {\textrm{const}}\,
{E}\int_{0}^{T}\int_{0}^{T} |(-\Delta) K_\alpha\left(
X_{t}-X_{s}\right)| |t-s|^{4H-2}
\,dt ds
\end{split}$$ and then, using lemma \[lemma su K\] 1) and 4), we get $$\label{eq:ubound}
\begin{split}
{E}Z_{\alpha,{\varepsilon}} & \le {\textrm{const}}\,
{E}\int_{0}^{T}\int_{0}^{T} |X_t-X_s|^{2\alpha-d} |t-s|^{2H-2}
\,dt ds
\\ & \qquad
+ {\textrm{const}}\,
{E}\int_{0}^{T}\int_{0}^{T} |X_t-X_s|^{2\alpha-d-2} |t-s|^{4H-2}
\,dt ds
\\ & = {\textrm{const}}\,
\int_{0}^{T}\int_{0}^{T} |t-s|^{(2\alpha+2-d)H-2}
\,dt ds
\\ & \qquad
+ {\textrm{const}}\,
\int_{0}^{T}\int_{0}^{T} |t-s|^{(2\alpha-d+2)H-2}
\,dt ds
\end{split}$$ which are uniformly bounded in ${\varepsilon}$ if $(2\alpha+2-d)H > 1$, i.e. when $\alpha > \alpha_H$ as required. We have established the uniform bound when $\alpha \in (\alpha_H, d/2)$ but now recall that $E Z_{\alpha,{\varepsilon}}$ is a decreasing function of $\alpha$ so that this bound extends to all $\alpha > \alpha_H$.
Let us now consider the case $H \le 1/2$ (so that now $\alpha_H >d/2$) and assume $\alpha > d/2$. Rewrite $Z_{\alpha,{\varepsilon}}$ as $$\label{eq:Zreg}
Z_{\alpha,{\varepsilon}} = Z_{{\varepsilon}}(h_\alpha) +
Z_{{\varepsilon}}(K_\alpha(0))$$ where $h_\alpha(x) = K_\alpha(x) - K_\alpha(0)$. Note that $0 < K_\alpha(0) < \infty$ when $\alpha > d/2$. Moreover $$Z_{{\varepsilon}}(K_\alpha(0)) = \int_0^T \int_0^T
K_\alpha(0) \langle D^0_{\varepsilon}X_s,D^0_{\varepsilon}X_t \rangle =
K_\alpha(0) \left|\int_0^T D^0_{\varepsilon}X_s ds\right|^2$$ and $$\int_0^T D^0_{\varepsilon}X_t dt = \int_0^T \frac{ X_{t+{\varepsilon}}- X_{t-{\varepsilon}}}{{\varepsilon}}
dt = {\varepsilon}^{-1} \int^{T+{\varepsilon}}_{T} X_t dt$$ so that by the continuity of the process $X$ we have the limit $$\lim_{{\varepsilon}\to 0} Z_{{\varepsilon}}(K_\alpha(0)) = K_\alpha(0) |X_T|^2$$ exists almost surely and $Z_{{\varepsilon}}(K_\alpha(0))$ is uniformly in $L^1$ (actually in all $L^p$).
So it remains to consider $Z_{{\varepsilon}}(h_\alpha)$. For $h_\alpha$ we have the estimate (\[eq:asympK2\]), provided $0 < 2\alpha-d \le 2$, so that we obtain an upper bound similar to eq. (\[eq:ubound\]) and the same condition on $\alpha$ follows. Note that $\alpha$ must satisfy $$d/2 < \alpha_H < \alpha < \frac{d}{2}+1$$ so that we must require $H > 1/4$. Again by monotonicity of $\alpha \mapsto Z_{\alpha,{\varepsilon}}$ we have uniform boundedness for any $\alpha > \alpha_H$.
We will perform a decomposition of $Z_{\alpha,{\varepsilon}}$ as follows. Write $$Z_{\alpha,{\varepsilon}} = A_{\alpha,{\varepsilon}} + B_{\alpha,{\varepsilon}} + Q_{\alpha,{\varepsilon}}$$ where $$A_{\alpha,{\varepsilon}} =
\int_{0}^{T}\int_{0}^{T}K_\alpha\left(
X_{t}-X_{s}\right) \sum_{i}
{\textrm{Cov}}(D^0_{\varepsilon}X^i_t, D^0_{\varepsilon}X^i_s)
\,dt ds$$ $$B_{\alpha,{\varepsilon}} =
\int_{0}^{T}\int_{0}^{T} (-\Delta) K_\alpha\left(
X_{t}-X_{s}\right)
|{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s})|^2
\,dt ds$$ and $Q_{\alpha,{\varepsilon}}$ is the remainder. Note that, by comparing this decomposition with eq. (\[eq:wick1\]), we have $${E}Z_{\alpha,{\varepsilon}} = {E}A_{\alpha,{\varepsilon}} + {E}B_{\alpha,{\varepsilon}}$$ so that ${E}Q_{\alpha,\beta} = 0$. This is a kind of Wick product decomposition, but not quite, since the terms $A,B$ are not constants, but still random variables.
A useful remark is that $A_{\alpha,{\varepsilon}} \ge 0$ since we can write $$A_{\alpha,{\varepsilon}} = \hat {E}\int_{0}^{T}\int_{0}^{T}K_\alpha\left(
X_{t}-X_{s}\right) \langle
D^0_{\varepsilon}\hat X_t, D^0_{\varepsilon}\hat X_s\rangle
\,dt ds$$ where we introduced an auxiliary independent $d$-dimensional fBm $\hat X$ with the same distribution of $X$ and where $\hat {E}$ denotes expectation with respect to this auxiliary fBm. So we have the formula $
A_{\alpha,{\varepsilon}} = \hat {E}\|\hat \eta_{{\varepsilon}}\|^2
$ where $
\hat \eta_{{\varepsilon}}(x) = \int_0^T K_{\alpha/2}(x-X_t) D_{\varepsilon}^0 \hat X_t dt
$ which shows that $A_{\alpha,{\varepsilon}}> 0$.
Next using the equality $-\Delta K_\alpha(x) = K_{\alpha-1}(x)-K_\alpha(x)$ we rewrite $B_{\alpha,{\varepsilon}}$ as $B^{(1)}_{\alpha,{\varepsilon}} - B^{(2)}_{\alpha,{\varepsilon}}$ where $$B^{(1)}_{\alpha,{\varepsilon}} =
\int_{0}^{T}\int_{0}^{T} K_{\alpha-1}\left(
X_{t}-X_{s}\right)
|{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s})|^2
\,dt ds$$ and $$B^{(2)}_{\alpha,{\varepsilon}} =
\int_{0}^{T}\int_{0}^{T} K_\alpha(X_t-X_s)
|{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s})|^2
\,dt ds.$$
Let us show first that $E| B^{(2)}_{\varepsilon}|$ is uniformly bounded in ${\varepsilon}$ when $\alpha > \alpha_H-1$ and $H> 1/4$. Indeed when $\alpha \le d/2$ by computations similar to those of Th. \[th:reg-symm\] we have that $E |B^{(2)}_{\varepsilon}|$ is uniformly bounded if $\alpha > \alpha_H-1$ and $\alpha > 0$. On the other hand, when $\alpha > d/2$, the kernel $K_\alpha$ is bounded, so Lemma \[lemma:bounds\] 1) allows to write $$E |B^{(2)}_{\alpha,{\varepsilon}}| \le {\textrm{const}}\int_{0}^{T}\int_{0}^{T}
|{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s})|^2
\,dt ds \le {\textrm{const}}\int_{0}^{T}\int_{0}^{T}
|t-s|^{4H-2}
\,dt ds \le C$$ uniformly in ${\varepsilon}$ provided $H > 1/4$.
Moreover below we will show the following.
\[linter\] If $H > 1/4$ and $\alpha <\alpha_H$ then $\liminf_{{\varepsilon}\rightarrow 0} B^{(1)}_{\alpha,{\varepsilon}} = + \infty $ a.s.
Then if we admit the result of previous lemma we can conclude with the use of Fatou lemma. In fact, for any $\alpha \in (\alpha_H-1,\alpha_H)$ and for some positive constant $c$ we have $$\begin{split}
\liminf_{{\varepsilon}\rightarrow 0} {E}(Z_{\alpha,{\varepsilon}}) & \ge
\liminf_{{\varepsilon}\rightarrow 0} {E}(B_{\alpha,{\varepsilon}}) \ge {E}(\liminf_{{\varepsilon}\rightarrow 0} B^{(1)}_{\alpha,{\varepsilon}}) - \sup_{\varepsilon}{E}( B^{(2)}_{\alpha,{\varepsilon}})
\\ & \ge
{E}(\liminf_{{\varepsilon}\rightarrow 0} B^{(1)}_{\alpha,{\varepsilon}}) - c = + \infty.
\end{split}$$ Moreover observe that this is enough since, from Lemma \[lemma:ordering\], we have that $Z_{\alpha,{\varepsilon}}$ is a decreasing function of $\alpha$ so the result will hold for any $\alpha < \alpha_H$.
By Fatou lemma we have $$\begin{split}
\liminf_{{\varepsilon}\to 0} B^{(1)}_{\alpha,{\varepsilon}} & \ge 2
\int_{0}^{T}\int_{0}^{t} K_{\alpha-1}\left(
X_{t}-X_{s}\right)
\liminf_{{\varepsilon}\to 0} |{\textrm{Cov}}(D^0_{\varepsilon}X^1_t,X^1_t- X^1_s)|^2
\,dt ds
\end{split}$$ But, when $t \neq s$, $$\liminf_{{\varepsilon}\to 0} |{\textrm{Cov}}(D^0_{\varepsilon}X^1_t,X^1_t- X^1_s)|^2 = 4 H^2 |t-s|^{4H-2}$$. Now assume $$\label{eq:cond1-alpha}
\alpha < d/2+1.$$ then by Lemma \[lemma su K\] there exist a small constant $r>0$ such that $K_{\alpha-1}(x) \ge C |x|^{2\alpha-d-2}1_{B(0,r)}(x)$. This allows us to bound from below as follows $$\begin{split}
\liminf_{{\varepsilon}\to 0} B^{(1)}_{\alpha,{\varepsilon}} & \ge {\textrm{const}}\,
\int_{0}^{T}\int_{0}^{t} K_{\alpha-1}\left(
X_{t}-X_{s}\right) |t-s|^{4H-2}
\,dt ds
\\ & \ge {\textrm{const}}\,
\int_{0}^{T}\int_{0}^{t} |X_t-X_s|^{2\alpha-d-2}
1_{B(0,r)}(X_t-X_s)
|t-s|^{4H-2}
\,dt ds.
\end{split}$$ Since the paths of fBm are Hölder continuous with parameter strictly smaller than $H$, for any $\gamma < H$ there exists a random constant $C_{X,\gamma}$ such that $$|X_t-X_s| \le C_{X,\gamma} |t-s|^\gamma, \qquad t,s \in [0,T]$$ By choosing a random time $S>0$ small enough such that $\sup_{t,s \in [0,S]} |X_t-X_s| < r$ we have $$\label{eq:div-B}
\begin{split}
\liminf_{{\varepsilon}\to 0}B^{(1)}_{\alpha,{\varepsilon}} & \ge {\textrm{const}}\,
\int_{0}^{S}\int_{0}^{t} |X_t-X_s|^{2\alpha-d-2}
|t-s|^{4H-2}
\,dt ds
\\ & \ge {\textrm{const}}\, C_{X,\gamma}^{d-2\alpha+2}
\int_{0}^{S}\int_{0}^{t}
|t-s|^{2H(\alpha-\alpha_H)-1+\delta}
\,dt ds
\end{split}$$ where $\delta = (d+2-2\alpha)(H-\gamma)$ is a arbitrarily small positive constant since $\gamma < H$ can be chosen arbitrarily near to $H$ and $$d+2-2\alpha > 4-\frac{1}{H} > 0$$ when $H>1/4$. Then when $\alpha < \alpha_H$ we can choose $\delta$ small enough to make the double integral in eq. (\[eq:div-B\]) diverge. Summing up we must have $
\alpha < \min (\alpha_H, d/2+1) \quad \text{and} \quad H > 1/4.
$ and $H> 1/4$. But when $H>1/4$ we have $\alpha_H < d/2-1$ so that sufficient conditions are $\alpha <\alpha_H$ and $H>1/4$. This observation concludes the proof.
The Forward integral
--------------------
Let $D^-_{\varepsilon}X_t = (X_{t+{\varepsilon}}-X_{t})/{\varepsilon}$, take $f:\mathbb{R}^{d}-\{0\}\rightarrow
\mathbb{R}_{+}$ satisfying (\[eq:f-class\]) and denote $${ Z^-}_{{\varepsilon}}(f)=\int_{0}^{T}\int_{0}^{T}f\left(
X_{t}-X_{s}\right)
\langle D^-_{\varepsilon}X_t, D^-_{\varepsilon}X_s \rangle
dsdt.$$
We can state similar theorems as in previous subsection.
\[th:reg-forw\] For any $H \ge 1/2$ and any $\alpha >\alpha_H$ we have $\sup_{{\varepsilon}} {E}Z_{\alpha,{\varepsilon}} < \infty$.
\[th:opt-forw\] For $H \ge 1/2$ and $\alpha < \alpha_H$ we have $\liminf_{{\varepsilon}} E(Z_{\alpha,{\varepsilon}}) = +\infty$.
The proofs follow the same line as the corresponding theorems about symmetric integrals. For this the following lemma will be crucial.
\[lemma:bounds-forw\]
1. We have the following estimates: $$\label{eq:bound1f}
\vert{\textrm{Cov}}(D^-_{\varepsilon}X^i_t, D^-_{\varepsilon}X^i_s)|
\le {\textrm{const}}|t-s|^{2H-2}$$ and $$\label{eq:bound2f}
|{\textrm{Cov}}(D^-_{\varepsilon}X^i_t, X^i_{t}-X^i_{s})| = |{\textrm{Cov}}(D^-_{\varepsilon}X^i_s, X^i_{t}-X^i_{s})|
\le {\textrm{const}}|t-s|^{2H-1}$$
2. If $s \neq t$, one has $$\lim_{{\varepsilon}\to 0} |{\textrm{Cov}}(D^-_{\varepsilon}X^1_t,X^1_t- X^1_s)| = 2 H |t-s|^{2H-1}.$$
Again the proof is similar to the one of Th. \[th:opt-symm\] where $D^-$ is replaced by $D^0$, Lemma \[lemma:bounds-forw\] is used instead of lemma \[lemma:bounds\]. In particular, according to Lemma \[lemma:bounds-forw\] if $s \neq t$ one has $$\liminf_{{\varepsilon}\to 0} |{\textrm{Cov}}(D^-_{\varepsilon}X^1_t,X^1_t- X^1_s)|^2 = 4 H^2 |t-s|^{4H-2}$$ when $H \ge \frac{1}{2}$.
In particular we have the following Corollary.
\[cor:current-l2-regforw\] The forward integral of a fractional Brownian motion with Hurst parameter $H \ge 1/2$ admits a pathwise redefinition on the space $H^{-\alpha}_{2}({\mathbb{R}}^d)$ for any $\alpha > \alpha_H$.
By Theorem \[th:reg-forw\] we know that ${E}Z_{\alpha,{\varepsilon}}$ is uniformly bounded in ${\varepsilon}$ when $\alpha > \alpha_H$. Since the regularized integrals $I^-_{\varepsilon}(\varphi)$ converges in probability as ${\varepsilon}\to 0$ for any $H \ge 1/2$, Condition \[cond:A\] holds and we can apply Theorem \[th:l2result\] to obtain a pathwise current with values in $H^{-\alpha}_{2}({\mathbb{R}}^d)$ for any $\alpha > \alpha_H$.
Brownian regularity in $H_{p}^{\alpha}$, $p\neq2$
=================================================
In this section we restrict ourselves to the case when $X$ is a $d$-dimensional classical Brownian motion, that we denote by $W$. The key ingredient is the following lemma.
\[lemma stima base su W\]If the dimension $d\geq2$ and the real numbers $\alpha>1$ and $p^{\prime}>1$ satisfy $$\left( d-\alpha+1\right) p^{\prime}<d$$ then $$\int_{\mathbb{R}^{d}}E\left[ \left( \int_{0}^{T}\frac{\exp\left(
-\varepsilon\left| x-W_{t}\right| \right) }{\left| x-W_{t}\right|
^{2d-2\alpha}}dt\right) ^{p^{\prime}/2}\right] dx<\infty$$ for every $\varepsilon>0$.
We shall prove below this lemma. Let us first describe its consequences.
From the bounds on $K_{\alpha/2}$, see \[lemma su K\], we have $$\int_{\mathbb{R}^{d}}E\left[ \left( \int_{0}^{T}K_{\alpha/2}^{2}\left(
x-W_{t}\right) dt\right) ^{p^{\prime}/2}\right] dx<\infty$$ and thus for a.e. $x\in\mathbb{R}^{d}$ we have $
P( \int_{0}^{T}K_{\alpha/2}^{2}( x-W_{t}) dt<\infty)
=1
$ which implies that the Itô integral $$\eta\left( x\right) :=\int_{0}^{T}K_{\alpha/2}\left( x-W_{t}\right) dW_{t}$$ is well defined, for a.e. $x\in\mathbb{R}^{d}$, as a limit in probability of $$\int_{0}^{T}K_{\alpha/2}\left( x-W_{t}\right) \frac{W_{t+\varepsilon}-W_{t}}{\varepsilon} dt,$$ see for instance [@RVSem]. These approximation integrals are measurable in the pair $\left(
x,\omega\right) $, hence they are measurable in $x$ as a mapping with values in the space of random variables with the metric of convergence in probability, and this way one can see that the limit object $\eta\left(
x\right) $ is measurable in the pair $\left( x,\omega\right) $. From Burkhoder-Davies-Gundy (BDG) inequality we have $
\int_{\mathbb{R}^{d}}E\left[ \left| \eta\left( x\right) \right|
^{p^{\prime}}\right] dx<\infty
$ and thus $\eta\in L^{p^{\prime}}\left( \Omega\times\mathbb{R}^{d}\right) $ and $$P\left( \omega\in\Omega:x\mapsto\eta\left( x,\omega\right) \in
L^{p^{\prime}}\left( \mathbb{R}^{d}\right) \right) =1.$$
To minimize the subtleties related to a direct use of $\eta\left( x\right)
$, we introduce a regularization. Let $T\left( t\right) $ be the semigroup on $L^{2}\left( \mathbb{R}^{d}\right) $ generated by $(\Delta-1)$ and we set $$K_{\alpha/2}^{\left( \delta\right) }\left( x\right) :=\left( T\left(
\delta\right) K_{\alpha/2}\right) \left( x\right) =\left( 4\pi
\delta\right) ^{-d/2}\int_{\mathbb{R}^{d}}e^{-\frac{\left| x-y\right| ^{2}}{4\delta}-\delta}K_{\alpha/2}\left( y\right) dy.$$ We have $K_{\alpha/2}^{\left( \delta\right) }\in S\left( \mathbb{R}^{d}\right) $. Set $$\eta^{\left( \delta\right) }\left( x\right) :=\int_{0}^{T}K_{\alpha
/2}^{\left( \delta\right) }\left( x-W_{t}\right) dW_{t}$$ which is obviously well defined for every $x$ and has a measurable version in the pair $\left( x,\omega\right) $. It is not difficult to justify that $\eta^{\left( \delta\right) }$ is square integrable in $\left(
x,\omega\right) $ and that $\eta^{\left( \delta\right) }=T\left(
\delta/2\right) \eta^{\left( \delta/2\right) }$ hence $\eta^{\left(
\delta\right) }\in S\left( \mathbb{R}^{d}\right) $ with probability one.
We have the following regularized version of (\[decoupling\]).
For $\delta>0$, $$\int_{0}^{T}\left\langle \left( T\left( \delta\right) \varphi\right)
\left( W_{t}\right) ,dW_{t}\right\rangle =\int_{\mathbb{R}^{d}}\left\langle
(1-\Delta)^{\alpha/2}\varphi\left( x\right) ,\eta^{\left( \delta\right)
}\left( x\right) \right\rangle dx.\label{decoupling W delta}$$
Given a vector field $\phi\in S\left( \mathbb{R}^{d}\right) $ and a continuous exponentially decreasing function $\psi$ on $\mathbb{R}^{d}$, we have the Fubini type identity $$\int_{0}^{T}\left\langle \int_{\mathbb{R}^{d}}\psi\left( W_{t}-x\right)
\phi(x)dx,dW_{t}\right\rangle =\int_{\mathbb{R}^{d}}\left\langle
\phi(x)dx,\int_{0}^{T}\psi\left( W_{t}-x\right) dW_{t}\right\rangle$$ with probability one. We omit the details of the proof. We have $$T\left( \delta\right) \varphi=T\left( \delta\right) (1-\Delta)^{-\alpha
/2}(1-\Delta)^{\alpha/2}\varphi=\int_{\mathbb{R}^{d}}K_{\alpha/2}^{\left(
\delta\right) }\left( \cdot-x\right) \left[ (1-\Delta)^{\alpha/2}\varphi\right] (x)dx$$ and thus $$\int_{0}^{T}\left\langle \left( T\left( \delta\right) \varphi\right)
\left( W_{t}\right) ,dW_{t}\right\rangle =\int_{0}^{T}\left\langle
\int_{\mathbb{R}^{d}}K_{\alpha/2}^{\left( \delta\right) }\left(
W_{t}-x\right) \left[ (1-\Delta)^{\alpha/2}\varphi\right] (x)dx,dW_{t}\right\rangle .$$ Here we can apply the Fubini rule because $T\left( \delta\right) \varphi\in
S\left( \mathbb{R}^{d}\right) $ for $\delta>0$ and $(1-\Delta)^{\alpha
/2}\varphi\in S\left( \mathbb{R}^{d}\right) $. This implies (\[decoupling W delta\]) and completes the proof.
For $d\geq2$, $\alpha>1$, $p^{\prime}>1$, such that $
\left( d-\alpha+1\right) p^{\prime}<d
$ we have $$\sup_{\delta>0}\int_{\mathbb{R}^{d}}E\left[ \left| \eta^{\left(
\delta\right) }\left( x\right) \right| ^{p^{\prime}}\right]
dx<\infty.\label{stima base su eta delta}$$
Let us restrict the argument to the most difficult case $0<\alpha<d$ where $K_{\alpha/2}$ has a singularity at zero. From BDG inequality we have $$\int_{\mathbb{R}^{d}}E\left[ \left| \eta^{\left( \delta\right) }\left(
x\right) \right| ^{p^{\prime}}\right] dx\leq C_{p^{\prime}}\int
_{\mathbb{R}^{d}}E\left[ \left( \int_{0}^{T}\left| K_{\alpha/2}^{\left(
\delta\right) }\left( x-W_{t}\right) \right| ^{2}dt\right) ^{p^{\prime
}/2}\right] dx.$$ From the definition of $K_{\alpha/2}^{\left( \delta\right) }$ in terms of $K_{\alpha/2}$ and estimate (\[eq:asympK\]) we get the inequality $$K_{\alpha/2}^{\left( \delta\right) }\left( x\right) \leq C_{\alpha
,d}\left( 4\pi\delta\right) ^{-d/2}\int_{\mathbb{R}^{d}}e^{-\frac{\left|
x-y\right| ^{2}}{4\delta}}\left| y\right| ^{\alpha-d}e^{-\frac{\left|
y\right| }{8}}dy.$$ Let us show that this implies$$K_{\alpha/2}^{\left( \delta\right) }\left( x\right) \leq C_{\alpha
,d}\left| x\right| ^{\alpha-d}e^{-\frac{\left| x\right| }{8}}\label{stima su K delta}$$ for a new constant $C_{\alpha,d}$, uniformly in $\delta\in\left( 0,1\right)
$. The proof of this result for $\left| x\right| >1$ is rather easy, so let us only deal with $\left| x\right| \leq1$. Write $x=re$ with $\left|
e\right| =1$ and change variable $y=rz$ in the integral, to get $$\begin{aligned}
K_{\alpha/2}^{\left( \delta\right) }\left( x\right) \leq r^{\alpha
-d}C_{\alpha,d}\left( 4\pi\delta\right) ^{-d/2}r^{d}\int_{\mathbb{R}^{d}}e^{-\frac{r^{2}\left| e-z\right| ^{2}}{4\delta}}\left| z\right|
^{\alpha-d}dz\\
=r^{\alpha-d}C_{\alpha,d}\left[ T\left( \frac{\delta}{r^{2}}\right) \left|
.\right| ^{\alpha-d}\right] \left( e\right)\end{aligned}$$ (see the definition of the semigroup $T(t)$). It is now easy to see that $\left[ T\left( t\right) \left| .\right| ^{\alpha-d}\right] \left(
e\right) $ is bounded above by a constant, uniformly in $t\geq0$. This proves (\[stima su K delta\]).
Having this estimate, it is sufficient to apply lemma \[lemma stima base su W\]. The proof is complete.
We can now prove the main result of this section.
The Itô integral $\int_{0}^{T}\left\langle \varphi\left( W_{t}\right)
,dW_{t}\right\rangle $ has a pathwise redefinition on the space $V=H_{p}^{\alpha}\left( \mathbb{R}^{d}\right) $ for every dimension $d$ and real numbers $\alpha>1$ and $p>1$ satisfying $$p>\frac{d}{\alpha-1}.$$ In particular, in any dimension $d$, given $\varepsilon>0$, for every $p>\frac{d}{\varepsilon}$ the integral $\int_{0}^{T}\left\langle
\varphi\left( W_{t}\right) ,dW_{t}\right\rangle $ has a pathwise redefinition on the space $H_{p}^{1+\varepsilon}\left( \mathbb{R}^{d}\right)
$.
**Step 1**. In the case $d=1$ we have $H_{p}^{\alpha}\left(
\mathbb{R}\right) \subset C^{1}\left( \mathbb{R}\right) $ by Sobolev embedding theorem (see [@Triebel], section 2.8.1, remark 2). Thus $$\int_{0}^{T}\varphi\left( W_{t}\right) dW_{t}=\Phi\left( W_{T}\right)
-\Phi\left( 0\right) -\frac{1}{2}\int_{0}^{T}\varphi^{\prime}\left(
W_{t}\right) dt$$ where $\Phi^{\prime}=\varphi$. This implies the result. We restrict now to the case $d\geq2$.
**Step 2**. We pass to the limit in (\[decoupling W delta\]). Let us treat the left-hand-side. With easy manipulations we see that $$\left( T\left( \delta\right) \varphi\right) \left( x\right) =\left(
2\pi\right) ^{-d/2}e^{-\delta}\int_{\mathbb{R}^{d}}e^{-\frac{\left|
z\right| ^{2}}{2}}\varphi\left( x-z\sqrt{2\delta}\right) dz$$ Hence, splitting the integral in a sufficiently large ball and the complementary, since $\varphi\in S\left( \mathbb{R}^{d}\right) $, we see that $T\left( \delta\right) \varphi\rightarrow\varphi$ uniformly over all $\mathbb{R}^{d}$ as $\delta\rightarrow0$. Thus $\int_{0}^{T}\left\langle
\left( T\left( \delta\right) \varphi\right) \left( W_{t}\right)
,dW_{t}\right\rangle $ easily converges to $\int_{0}^{T}\left\langle
\varphi\left( W_{t}\right) ,dW_{t}\right\rangle $, in mean square.
Given the value of $p$ in the statement of the theorem, under the assumption $\alpha>1$ the inequality $p>\frac{d}{\alpha-1}$ is equivalent to $\left(
d-\alpha+1\right) p^{\prime}<d$, where $1/p + 1/p^\prime = 1$, so the previous lemma applies. From (\[stima base su eta delta\]) there is a sequence $\delta_{n}\rightarrow0$ and an element $\eta^{\left( 0\right) }\in L^{p^{\prime}}\left(
\Omega\times\mathbb{R}^{d}\right) $ such that $\eta^{\left( \delta_n\right)
}\rightharpoonup\eta^{\left( 0\right) }$ weakly in $L^{p^{\prime}}\left(
\Omega\times\mathbb{R}^{d}\right) $, when $n \rightarrow +\infty$. From (\[decoupling W delta\]), for a given $\varphi\in S\left( \mathbb{R}^{d}\right) $, we thus have, in the limit as $n \rightarrow \infty$, $$E\left[ X\int_{0}^{T}\left\langle \varphi\left( W_{t}\right) ,dW_{t}\right\rangle \right] =E\left[ X\int_{\mathbb{R}^{d}}\left\langle
(1-\Delta)^{\alpha/2}\varphi\left( x\right) ,\eta^{\left( 0\right)
}\left( x\right) \right\rangle dx\right]$$ for every bounded r.v. $X$ and thus $$\int_{0}^{T}\left\langle \varphi\left( W_{t}\right) ,dW_{t}\right\rangle
=\int_{\mathbb{R}^{d}}\left\langle (1-\Delta)^{\alpha/2}\varphi\left(
x\right) ,\eta^{\left( 0\right) }\left( x\right) \right\rangle dx$$ with probability one.
**Step 3**. Therefore, given $\varphi\in S\left( \mathbb{R}^{d}\right)
$, with probability one we have $$\begin{aligned}
\left| \int_{0}^{T}\left\langle \varphi\left( W_{t}\right) ,dW_{t}\right\rangle \right| \leq\left\| (1-\Delta)^{\alpha/2}\varphi\right\|
_{L^{p}\left( \mathbb{R}^{d}\right) }\left\| \eta^{\left( 0\right)
}\right\| _{L^{p^{\prime}}\left( \mathbb{R}^{d}\right) }\\
\leq C\left\| \varphi\right\| _{H_{p}^{\alpha}\left( \mathbb{R}^{d}\right)
}\left\| \eta^{\left( 0\right) }\right\| _{L^{p^{\prime}}\left(
\mathbb{R}^{d}\right) }.\end{aligned}$$ The proof is complete.
\[remark stopping\] The same result is true for the stopped Brownian motion $$W_{t}^{R}=W_{t\wedge\tau_{R}},\quad\tau_{R}=\inf\left\{ t>0:\left|
W_{t}\right| \geq R\right\} .$$ with given $R>0$. The statement is that the Itô integral $\int_{0}^{T}\left\langle \varphi\left( W_{t}^{R}\right) ,dW_{t}^{R}\right\rangle $ has a pathwise redefinition on the space $H_{p}^{\alpha}\left( \mathbb{R}^{d}\right) $ under the same conditions on $d,\alpha,p$ as in the theorem. The proof is the same (even easier, since in the proof of lemma \[lemma stima base su W\] we do not have to care of the exponential term).
\[remark Walphap\] The same result is true in the Sobolev-Slobodeckij spaces $W_{p}^{\alpha}\left( \mathbb{R}^{d}\right) $ defined in [@Triebel], section 2.3. The statement is that the Itô integral $\int_{0}^{T}\left\langle \varphi\left( W_{t}\right) ,dW_{t}\right\rangle $ has a pathwise redefinition on the space $W_{p}^{\alpha}\left( \mathbb{R}^{d}\right) $ under the same conditions on $d,\alpha,p$ as in the theorem. Indeed, given a triple $d,\alpha,p$ as in the theorem, let $\alpha^{\prime
}<\alpha$ be such that also the triple $d,\alpha^{\prime},p$ satisfies the assumption of the theorem. Then $\int_{0}^{T}\left\langle \varphi\left(
W_{t}\right) ,dW_{t}\right\rangle $ has a pathwise redefinition on $H_{p}^{\alpha^{\prime}}\left( \mathbb{R}^{d}\right) $; by definition of pathwise redefinition, we see that this implies that $\int_{0}^{T}\left\langle
\varphi\left( W_{t}\right) ,dW_{t}\right\rangle $ has a pathwise redefinition on the space $W_{p}^{\alpha}\left( \mathbb{R}^{d}\right) $, because we have the continuous embedding $$W_{p}^{\alpha}\left( \mathbb{R}^{d}\right) \subset H_{p}^{\alpha^{\prime}}\left( \mathbb{R}^{d}\right),$$ see [@Triebel], remark 4 of section 2.3.3. The same result is of course true for the Itô integral $\int_{0}^{T}\left\langle \varphi\left(
W_{t}^{R}\right) ,dW_{t}^{R}\right\rangle $.
We can now elaborate the previous results in the direction of the Hölder topology. Given $\varepsilon\in\left( 0,1\right) $, denote by $C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $ the space of all continuously differentiable functions $f$ on $\mathbb{R}^{d}$ such that $$\left\| f\right\| _{C^{1+\varepsilon}}=\sup_{x\in\mathbb{R}^{d}}\left(
\left| f(x)\right| +\left| Df(x)\right| \right) +\sup_{x\neq y}\frac{\left| Df(x)-Df(y)\right| }{\left| x-y\right| ^{\varepsilon}}<\infty,$$ see [@Triebel], section 2.7. Endowed with the norm $\left\| .\right\|
_{C^{1+\varepsilon}}$, the space $C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $ is a Banach space.
In any dimension $d$, for every $\varepsilon\in\left( 0,1\right) $ the Itô integral $\int_{0}^{T}\left\langle \varphi\left( W_{t}\right)
,dW_{t}\right\rangle $ has a pathwise redefinition on the space $C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $.
**Step 1**. This preliminary step is devoted to a few details used below. Recall that the classical Sobolev space $W_{p}^{1}\left( \mathbb{R}^{d}\right) $ is defined as the space of all $f\in L^{p}\left(
\mathbb{R}^{d}\right) $ having distributional derivative $Df\in L^{p}\left(
\mathbb{R}^{d \times d } \right) $. Recall also (see remark 4 of section 2.5.1 of [@Triebel]) that, for every $\varepsilon\in\left( 0,1\right) $, the space $W_{p}^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $ of remark \[remark Walphap\] is characterized as the space of all $f\in W_{p}^{1}\left( \mathbb{R}^{d}\right) $ such that $$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{\left| Df(x)-Df(y)\right|
^{p}}{\left| x-y\right| ^{d+\varepsilon p}}dxdy<\infty$$ and as a norm on $W_{p}^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $ one can take the following one:$$\left\| f\right\| _{W_{p}^{1+\varepsilon}}^{p}=\left\| f\right\| _{L_{p}}^{p}+\left\| Df\right\| _{L_{p}}^{p}+\int_{\mathbb{R}^{d}\times
\mathbb{R}^{d}}\frac{\left| Df(x)-Df(y)\right| ^{p}}{\left| x-y\right|
^{d+\varepsilon p}}dxdy.$$ Then it is easy to verify that for every $\varepsilon,\varepsilon^{\prime}\in\left( 0,1\right) $ with $\varepsilon>\varepsilon^{\prime}$ the following assertion is true, where $B\left( 0,R\right) $ denotes the ball of center $0$ and radius $R>0$: $$f\in C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) \text{, }f\text{ with
support in }B\left( 0,R\right) \Rightarrow f\in W_{p}^{1+\varepsilon
^{\prime}}\left( \mathbb{R}^{d}\right)$$ and$$\label{Esob}
\left\| f\right\| _{W_{p}^{1+\varepsilon^{\prime}}}^{p}\leq C\left(
R,\varepsilon,\varepsilon^{\prime},p,d\right) \left\| f\right\|
_{C^{1+\varepsilon}}^{p}$$ where $C\left( R,\varepsilon,\varepsilon^{\prime},p,d\right) $ is a constant depending only on $R,\varepsilon,\varepsilon^{\prime},p,d$.
Indeed, we have $$\begin{aligned}
\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{\left| Df(x)-Df(y)\right|
^{p}}{\left| x-y\right| ^{d+\varepsilon^{\prime}p}}dxdy & =\int_{B\left(
0,R\right) \times B\left( 0,R\right) }\frac{\left| Df(x)-Df(y)\right|
^{p}}{\left| x-y\right| ^{d+\varepsilon^{\prime}p}}dxdy\\
& \leq\int_{\left| x-y\right| \leq1,\left| x\right| \leq R,\left|
y\right| \leq R}\frac{\left| Df(x)-Df(y)\right| ^{p}}{\left| x-y\right|
^{d+\varepsilon^{\prime}p}}dxdy \\
& + \int_{\left| x-y\right| >1,\left| x\right|
\leq R,\left| y\right| \leq R}\frac{\left| Df(x)-Df(y)\right| ^{p}}{\left| x-y\right| ^{d+\varepsilon^{\prime}p}}dxdy\\
& \leq\int_{\left| x-y\right| \leq1,\left| x\right| \leq R,\left|
y\right| \leq R}\frac{\left\| f\right\| _{C^{1+\varepsilon}}^{p}}{\left|
x-y\right| ^{d+\left( \varepsilon^{\prime}-\varepsilon\right) p}}dxdy+C\left( p,d\right) \left\| f\right\| _{C^{1+\varepsilon}}^{p}R^{d}$$ where $C\left( p,d\right) $ is a constant depending only on $p,d$. The claim (\[Esob\]) easily follows from this inequality.
**Step 2**. Let $d$ and $\varepsilon$ be given, as in the claim of the theorem. Choose $\varepsilon^{\prime}\in\left( 0,\varepsilon\right) $ and $p>\frac{d}{\varepsilon^{\prime}}$. Let $W$ be defined on a complete probability space $\left( \Omega,\mathcal{F},P\right) $. Remark \[remark Walphap\] states that there exists a random variable $C>0$ such that, for every $\varphi\in W_{p}^{1+\varepsilon^{\prime}}\left( \mathbb{R}^{d}\right) $, $$\left| \int_{0}^{T}\left\langle \varphi\left( W_{t}\right) ,dW_{t}\right\rangle \right| \leq C\left\| \varphi\right\| _{W_{p}^{1+\varepsilon
^{\prime}}}$$ on a full probability set $\Omega_{\varphi}$.
For every $R>0$, let $\theta_{R}:\mathbb{R}^{d}\rightarrow\lbrack0,\infty)$ be a $C^{\infty}$ function such that $\theta_{R}\left( x\right) =1$ for $\left| x\right| \leq R+1$, $\theta_{R}\left( x\right) =0$ for $\left|
x\right| \geq R+2$. Given $\varphi\in C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $, we have $\varphi\cdot\theta_{R}\in C^{1+\varepsilon}\left(
\mathbb{R}^{d}\right) $ and thus $\varphi\cdot\theta_{R}\in W_{p}^{1+\varepsilon^{\prime}}\left( \mathbb{R}^{d}\right) $. Therefore$$\left| \int_{0}^{T}\left\langle \left( \varphi\cdot\theta_{R}\right)
\left( W_{t}\right) ,dW_{t}\right\rangle \right| \leq C\left\|
\varphi\cdot\theta_{R}\right\| _{W_{p}^{1+\varepsilon^{\prime}}}$$ on a full probability set $\Omega_{\varphi\cdot\theta_{R}}$.
From step 1, there exists a random variable $C_{R}>0$, independent of $\varphi$, such that $$\left| \int_{0}^{T}\left\langle \left( \varphi\cdot\theta_{R}\right)
\left( W_{t}\right) ,dW_{t}\right\rangle \right| \leq C_{R}\left\|
\varphi\right\| _{C^{1+\varepsilon}}\text{ on }\Omega_{\varphi\cdot\theta
_{R}},$$ where we have also used the fact that $\left\| \varphi\cdot\theta_{R}\right\|
_{C^{1+\varepsilon}}\leq C_{\theta}\left\| \varphi\right\|
_{C^{1+\varepsilon}}$ for some constant $C_{\theta}>0$ depending on the function $\theta$ (and thus on $R$ again). Redefine, if necessary, $C_{R}$ in such a way that $R\mapsto C_{R}$ is non decreasing, with probability one.
Let $A_{R}$ be the set$$A_{R}=\left\{ \tau_{R}>T\right\},$$ where $\tau_{R}$ is defined in remark \[remark stopping\]. The sets $A_{R}$ increase with $R$. Given the family of events $A_{R}$ and random variables $C_{R}$, we can define a new random variable $C^{\prime}>0$ such that $C_{R}\leq C^{\prime}$ on $A_{R}$ (it is sufficient to put $C^{\prime}=C_{N+1}$ on $A_{N+1}\diagdown A_{N}$). Thus, given $\varphi\in
C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $, we have$$\left| \int_{0}^{T}\left\langle \left( \varphi\cdot\theta_{R}\right)
\left( W_{t}\right) ,dW_{t}\right\rangle \right| \leq C^{\prime}\left\|
\varphi\right\| _{C^{1+\varepsilon}}\text{ on }\Omega_{\varphi\cdot\theta
_{R}}\cap A_{R}.$$
For every $R>0$ and $\varphi\in C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $, there is a $P$-null set $N_{R,\varphi}$ such that $$\int_{0}^{T}\left\langle \left( \varphi\cdot\theta_{R}\right) \left(
W_{t}\right) ,dW_{t}\right\rangle =\int_{0}^{T}\left\langle \varphi\left(
W_{t}\right) ,dW_{t}\right\rangle \text{ on }A_{R}\diagdown N_{R,\varphi}.$$ Therefore, given $R>0$ and $\varphi\in C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $, we have$$\left| \int_{0}^{T}\left\langle \varphi\left( W_{t}\right) ,dW_{t}\right\rangle \right| \leq C^{\prime}\left\| \varphi\right\|
_{C^{1+\varepsilon}}\text{ on }\Omega_{\varphi\cdot\theta_{R}}\cap
A_{R}\diagdown N_{R,\varphi}.$$ It follows that$$\left| \int_{0}^{T}\left\langle \varphi\left( W_{t}\right) ,dW_{t}\right\rangle \right| \leq C^{\prime}\left\| \varphi\right\|
_{C^{1+\varepsilon}}\text{ on }\bigcup_{R>0}\left( \Omega_{\varphi\cdot
\theta_{R}}\cap A_{R}\diagdown N_{R,\varphi}\right) .$$ Since $P\left( \bigcup_{R>0}A_{R}\right) =1$ we have $P\left( \bigcup
_{R>0}\left( \Omega_{\varphi\cdot\theta_{R}}\cap A_{R}\diagdown N_{R,\varphi
}\right) \right) =1$. This means $\int_{0}^{T}\left\langle \varphi\left(
W_{t}\right) ,dW_{t}\right\rangle $ has a pathwise redefinition on the space $C^{1+\varepsilon}\left( \mathbb{R}^{d}\right) $. The proof is complete.
The strategy of step 2 in the previous proof can be used to deal with function spaces of Fréchet type that are not Banach spaces: by localization of the stochastic process, one can restrict the attention to compact support test functions and then prove the existence of a pathwise redefinition in topologies without decay at infinity. For this reason, even the uniformity in $x\in\mathbb{R}^{d}$ in the definition of $C^{1+\varepsilon}\left(
\mathbb{R}^{d}\right) $ is not necessary.
\[Rrough\] In rough path theory (see [@Lyons]), for every rough path $\gamma$ of a certain class which includes a.e. path of Brownian motion, a notion of integral $\int_{0}^{T}\left\langle \varphi\left( \gamma_{t}\right)
,d\gamma_{t}\right\rangle $ is defined for every function $\varphi$ with $\varepsilon$-Hölder first derivative (for arbitrary $\varepsilon>0$). The previous theorem is conceptually similar; a closer comparison, however, requires further investigation.
Finally, we have to prove lemma \[lemma stima base su W\].
Proof of lemma \[lemma stima base su W\]
----------------------------------------
If $\max_{\left[ 0,T\right] }\left| W_{t}\right| \leq\left| x\right| /2$ then, for every $t\in\left[ 0,T\right] $, $$\begin{aligned}
\frac{1}{2}\left| x\right| & \leq\left| x-W_{t}\right| \leq\frac{3}{2}\left| x\right| \\
\exp\left( -\varepsilon\left| x-W_{t}\right| \right) & \leq\exp\left(
-\varepsilon\left| x\right| /2\right) \\
\frac{1}{\left| x-W_{t}\right| ^{2d-2\alpha}} & \leq\frac{\left(
2/3\right) ^{2d-2\alpha}}{\left| x\right| ^{2d-2\alpha}}\text{ \quad if
}2d-2\alpha\leq0\\
\frac{1}{\left| x-W_{t}\right| ^{2d-2\alpha}} & \leq\frac{2^{2d-2\alpha}}{\left| x\right| ^{2d-2\alpha}}\text{ \quad if }2d-2\alpha>0\end{aligned}$$ and thus$$\frac{\exp\left( -\varepsilon\left| x-W_{t}\right| \right) }{\left|
x-W_{t}\right| ^{2d-2\alpha}}\leq\exp\left( -\varepsilon\left| x\right|
/2\right) \frac{C_{\alpha,d}}{\left| x\right| ^{2d-2\alpha}}$$ for a suitable constant $C_{\alpha,d}>0$. Therefore$$\int_{\mathbb{R}^{d}}E\left[ \left( \int_{0}^{T}\frac{\exp\left(
-\varepsilon\left| x-W_{t}\right| \right) }{\left| x-W_{t}\right|
^{2d-2\alpha}}dt\right) ^{p^{\prime}/2}\right] dx\leq I_{1}+I_{2}$$$$I_{1}:=\int_{\mathbb{R}^{d}}\exp\left( -\varepsilon p^{\prime}\left|
x\right| /4\right) E\left[ \left( \int_{0}^{T}\frac{C_{\alpha,d}}{\left|
x\right| ^{2d-2\alpha}}dt\right) ^{p^{\prime}/2}\right] dx$$$$I_{2} :=\int_{\mathbb{R}^{d}}E\left[ 1_{\max_{\left[ 0,T\right]
}\left| W_{t}\right| >\left| x\right| /2}\left( \int_{0}^{T}\frac
{1}{\left| x-W_{t}\right| ^{2d-2\alpha}}dt\right) ^{p^{\prime}/2}\right]
dx.$$
Obviously $I_{1}<\infty$, being $\left( d-\alpha+1\right) p^{\prime}<d$. Moreover$$I_{2}\leq\int_{\mathbb{R}^{d}}P\left( \max_{\left[ 0,T\right] }\left|
W_{t}\right| >\left| x\right| /2\right) ^{\frac{\delta}{1+\delta}}E\left[
\left( \int_{0}^{T}\frac{1}{\left| x-W_{t}\right| ^{2d-2\alpha}}dt\right)
^{\left( 1+\delta\right) p^{\prime}/2}\right] ^{\frac{1}{1+\delta}}dx$$ for every $\delta>0$. Recall the exponential inequality (see [@ry] Proposition 1.8) $$P\left( \max_{t\in\left[ 0,T\right] }W_{t}\geq\beta\right) \leq
e^{-\frac{\beta}{2T}}.$$ It easily implies, by symmetry, that $$P\left( \max_{t\in\left[ 0,T\right] }\left| W_{t}\right| \geq\left|
x\right| /2\right) \leq2e^{-\frac{\left| x\right| }{4T}}$$ and thus there exist $C_{\delta},\lambda_{\delta}>0$ (depending also on $T$) such that $$P\left( \max_{t\in\left[ 0,T\right] }\left| W_{t}\right| \geq\left|
x\right| /2\right) ^{\frac{\delta}{1+\delta}}\leq C_{\delta}e^{-\lambda
_{\delta}\left| x\right| }.$$ Moreover, by Young inequality, $$a^{\frac{1}{1+\delta}}=a^{\frac{1}{1+\delta}}\cdot1\leq a+C_{\delta}$$ for some constant $C_{\delta}>0$. Thus, for every $\delta>0$, $$I_{2}\leq C_{\delta}^{\prime}+C_{\delta}^{\prime}\int_{\mathbb{R}^{d}}e^{-\lambda_{\delta}\left| x\right| }E\left[ \left( \int_{0}^{T}\frac
{1}{\left| x-W_{t}\right| ^{2d-2\alpha}}dt\right) ^{\left( 1+\delta
\right) p^{\prime}/2}\right] dx$$ for some constant $C_{\delta}^{\prime}>0$.
The following lemma is inspired from the proof of Corollary 2.4 of Elworthy, Li, Yor [@ELY] and in fact it was suggested to us by K. D. Elworthy.
For every $d\geq2$, $q>1$, $\theta\in\mathbb{R}$, $x\in\mathbb{R}^{d}$, we have $$\begin{aligned}
E\left[ \left( \int_{0}^{T}\frac{dt}{\left| x+W_{t}\right| ^{2\left(
1-\theta\right) }}\right) ^{\frac q2}\right] \leq c_{q,\theta,T}\left\{ E\left[ \left| x+W_{T}\right| ^{\theta
q}\right] +\left| x\right| ^{\theta q}+\int_{0}^{T}E\left[ \frac
{dt}{\left| x+W_{t}\right| ^{\left( 2-\theta\right) q}}\right] \right\}
.\end{aligned}$$
Consider the process $Z_{t}=\left| x+W_{t}\right| ^{2}$ (squared Bessel process of dimension $d$). From Itô formula we have $$dZ_{t}=2\left\langle x+W_{t},dW_{t}\right\rangle +dt,\quad Z_{0}=\left|
x\right| ^{2}.$$ Introducing an auxiliary one-dimensional Brownian motion $\left( \beta
_{t}\right) $ we may also write $$dZ_{t}=2\sqrt{Z_{t}}d\beta_{t}+dt.$$ Since $d\geq2$, the one-point sets are polar sets for a $d$-dimensional Brownian motion, see Proposition 2.7, p. 191 of [@ry]. Therefore $P\left\{ Z_{t}>0,t\in\left[ 0,T\right] \right\} =1$ and we can develop $Z_{t}^{\theta/2}$ using Itô formula for any $\theta\in\mathbb{R}$. We obtain $$\begin{aligned}
d\left( Z_{t}^{\theta/2}\right) =\frac{\theta}{2}Z_{t}^{\left(
\theta-2\right) /2}\left( 2\sqrt{Z_{t}}d\beta_{t}+dt\right) +\frac{1}{2}\frac{\theta}{2}\frac{\theta-2}{2}Z_{t}^{\left( \theta-4\right) /2}4Z_{t}dt\\
=\theta Z_{t}^{\left( \theta-1\right) /2}d\beta_{t}+c_{\theta}Z_{t}^{\left(
\theta-2\right) /2}dt\end{aligned}$$ where $c_{\theta}=\frac{\theta\left( \theta-1\right) }{2}$. Therefore $$\int_{0}^{T}\theta Z_{t}^{\left( \theta-1\right) /2}d\beta_{t}=Z_{T}^{\theta/2}-Z_{0}^{\theta/2}-\int_{0}^{T}c_{\theta}Z_{t}^{\left(
\theta-2\right) /2}dt$$ and thus, from BDG inequality, for every $q>1$$$\begin{aligned}
E\left[ \left( \int_{0}^{T}\theta^{2}Z_{t}^{\theta-1}dt\right)
^{q/2}\right] & \leq c_{q}E\left[ \left( \int_{0}^{T}\theta Z_{t}^{\left( \theta-1\right)
/2}d\beta_{t}\right) ^{q}\right] \\
& \leq c_{q,\theta}\left\{ E\left[ Z_{T}^{\theta q/2}\right] +E\left[
Z_{0}^{\theta q/2}\right] +E\left[ \left( \int_{0}^{T}Z_{t}^{\left(
\theta-2\right) /2}dt\right) ^{q}\right] \right\} .\end{aligned}$$ This implies, by Hölder inequality, $$\begin{aligned}
E\left[ \left( \int_{0}^{T}\frac{1}{Z_{t}^{1-\theta}}dt\right)
^{q/2}\right] \leq c_{q,\theta,T}\left\{ E\left[ Z_{T}^{\theta q/2}\right] +E\left[
Z_{0}^{\theta q/2}\right] +E\left[ \int_{0}^{T}\frac{1}{Z_{t}^{\left(
2-\theta\right) q/2}}dt\right] \right\}\end{aligned}$$ and the proof is complete.
We go on with the proof of lemma \[lemma stima base su W\]. Simply by taking $1-\theta=d-\alpha$ and $q=\left( 1+\delta\right)
p^{\prime}$ we have:$$\begin{aligned}
& E\left[ \left( \int_{0}^{T}\frac{1}{\left| x-W_{t}\right| ^{2d-2\alpha
}}dt\right) ^{\left( 1+\delta\right) p^{\prime}/2}\right]
\\ &\qquad \qquad \leq C\left\{ E\left[ \frac{1}{\left| x+W_{T}\right| ^{\left(
d-\alpha-1\right) \left( 1+\delta\right) p^{\prime}}}\right] +\frac
{1}{\left| x\right| ^{\left( d-\alpha-1\right) \left( 1+\delta\right)
p^{\prime}}}\right\} \\
& \qquad\qquad +C\int_{0}^{T}E\left[ \frac{1}{\left| x+W_{t}\right| ^{\left(
d-\alpha+1\right) \left( 1+\delta\right) p^{\prime}}}\right] dt.\end{aligned}$$ With the notation $p_{t}\left( y\right) =\frac{1}{\sqrt{\left( 2\pi\right)
^{d}t^{d}}}\exp\left( -\frac{\left| y\right| ^{2}}{2t}\right) $, and the bound $\int_{0}^{T}p_{t}\left( y\right) dt\leq\frac{C_{d}\exp\left(
-\left| y\right| \right) }{\left| y\right| ^{d-2}}$ we have$$E\left[ \frac{1}{\left| x+W_{T}\right| ^{\left( d-\alpha-1\right) \left(
1+\delta\right) p^{\prime}}}\right] =\int_{\mathbb{R}^{d}}\frac{1}{\left|
x+y\right| ^{\left( d-\alpha-1\right) \left( 1+\delta\right) p^{\prime}}}p_{T}\left( y\right) dy$$ and $$\begin{aligned}
& \int_{0}^{T}E\left[ \frac{1}{\left| x+W_{t}\right| ^{\left(
d-\alpha+1\right) \left( 1+\delta\right) p^{\prime}}}\right] dt\\
&\qquad\qquad =\int_{\mathbb{R}^{d}}\frac{1}{\left| x+y\right| ^{\left( d-\alpha
+1\right) \left( 1+\delta\right) p^{\prime}}}\left( \int_{0}^{T}p_{t}\left( y\right) dt\right) dy\\
&\qquad\qquad \leq\int_{\mathbb{R}^{d}}\frac{1}{\left| x+y\right| ^{\left(
d-\alpha+1\right) \left( 1+\delta\right) p^{\prime}}}\frac{C_{d}\exp\left(
-\left| y\right| \right) }{\left| y\right| ^{d-2}}dy.\end{aligned}$$
Thus, with a new constant $C>0$ depending on $\delta$ and the other parameters,$$I_{2}\leq C\left( 1+I_{2}^{\left( 1\right) }+I_{2}^{\left( 2\right)
}+I_{2}^{\left( 3\right) }\right)$$ where$$I_{2}^{\left( 1\right) }:=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-\lambda_{\delta}\left| x\right| }\frac{1}{\left| x+y\right| ^{\left(
d-\alpha-1\right) \left( 1+\delta\right) p^{\prime}}}p_{T}\left( y\right)
dxdy$$$$I_{2}^{\left( 2\right) }:=\int_{\mathbb{R}^{d}}e^{-\lambda_{\delta}\left|
x\right| }\frac{1}{\left| x\right| ^{\left( d-\alpha-1\right) \left(
1+\delta\right) p^{\prime}}}dx$$$$I_{2}^{\left( 3\right) }:=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-\lambda_{\delta}\left| x\right| }\frac{1}{\left| x+y\right| ^{\left(
d-\alpha+1\right) \left( 1+\delta\right) p^{\prime}}}\frac{C_{d}\exp\left(
-\left| y\right| \right) }{\left| y\right| ^{d-2}}dxdy.$$Choose $\delta>0$ such that $\left( d-\alpha+1\right) \left( 1+\delta
\right) p^{\prime}<d$. Since $$\left( d-\alpha-1\right) \left(
1+\delta\right) p^{\prime}< \left( d-\alpha+1\right) \left(
1+\delta\right) p^{\prime}<d$$ the term $I_{2}^{\left( 2\right) }$ is finite. For $I_{2}^{\left( 1\right) }$ and $I_{2}^{\left( 3\right) }$ it is sufficient to integrate first in $x$, bound the result uniformly in $y$, then integrate in $y$; one proves that $I_{2}^{\left( 1\right) }$ and $I_{2}^{\left( 3\right) }$ are finite. The proof is complete.
The energy of a random vortex filament
======================================
In [@Ffil; @FGub; @Ascona] with the purpose of modeling turbulence in 3d fluids, the authors introduce and study a model of random vortex filaments based on Brownian motion. This model has been extended to the fBm with $H>1/2$ by [@Nua] and [@FlandoliMinnelli]. Here we recall briefly the model, emphasize the relationship of the vortex energy with the pathwise regularity of the current associated with the vortex *core* and obtain new conditions for the integrability of the vortex energy for the case $H \in (1/4,1/2)$.
For simplicity we consider only a single vortex since extension to a linear superposition of different vortexes is straightforward (and even to a random field of Poissonian vortexes, see for example [@FGubStat]). Let $(X_t)_{t\in[0,T]}$, $T>0$ be a 3d fBm with Hurst parameter $H \in (1/4, 1)$ and consider the associated vector current, formally written as $$\xi_{0}(x) = \int_0^T \delta(x-X_t) dX_t$$ where the integral is a symmetric (Stratonovich) integral. This object should be understood according to theorem \[th:l2result\] that is as a random distribution in the Sobolev space $H^{-\alpha}_{p}({\mathbb{R}}^d)$ of sufficiently large negative order. The vorticity field is then built by superposing translates of this core weighted according to a compactly supported signed measure $\rho$ with finite mass which determines the intensity of vorticity. For more details about those considerations, the reader can consult [@Ffil]. Then we end up with $$\xi(x) = \int_{{\mathbb{R}}^3} \xi_0(x-y) \rho(dy)$$ which is again a random distribution.
The velocity field $u$ is generated from $\xi$ according to the Biot-Savart relation $$u(x)= \int_{{\mathbb{R}}^3} \mathcal{K}(x-y) \wedge \xi(y) dy
= \int_{{\mathbb{R}}^3} \mathcal{K}*\rho (x-y) \wedge \xi_0(y) dy$$ where $\wedge$ is the vector product in ${\mathbb{R}}^3$ and the vector kernel $\mathcal{K}(x)$ is defined as $\mathcal{K}(x):=(4\pi)^{-1} x/|x|^3
$ and $\mathcal{K}*\rho$ denote the convolution $
(\mathcal{K}*\rho)(x) = \int_{{\mathbb{R}}^3} \mathcal{K} (x-z) \rho(dz).
$ The kinetic energy of the fluid is then defined as the $L^2({\mathbb{R}}^3)$ norm of $u$: $$\label{eq:energy}
\mathcal{E} = \int_{{\mathbb{R}}^3} |u(x)|^2 dx = \|u\|^2.$$ It is then interesting to find conditions on $\rho$ such that the kinetic energy of the fluid is finite. Abstractly we have $u = \Phi \xi_0$ where we introduced an operator $\Phi$ whose kernel is $\mathcal{K}*\rho$ having Fourier transform $$\mathcal{F}(\mathcal{K}*\rho)(q) = \frac{iq}{|q|^2}
\widehat \rho(q)$$ where we denoted $\widehat \rho$ the Fourier transform of the measure $\rho$.
From now on $L^2$ will stay for $L^2({\mathbb{R}}^3)$. Since, by Corollary \[cor:current-l2-reg\], $\xi_0$ belongs a.s. to the space $H^{-\alpha}_{2}({\mathbb{R}}^d)$ for any $\alpha > \alpha_H = 1/(2H) +
1/2$ (since $d=3$), the condition $u \in L^2$ a.s. can be satisfied if $\Phi :
H^{-\alpha}_{2}({\mathbb{R}}^3) \to L^2$ which in Fourier variables is sufficient to require that $$\|\Phi\|_{H^{-\alpha}_{2}\to L^2} = \|\Phi (1-\Delta)^{\alpha/2}\|_{L^2
\to L^2} = \text{ess}\sup_{q \in {\mathbb{R}}^3}
\frac{|\widehat \rho(q)|}{|q|} (1+|q|^2)^{\alpha/2} < \infty.$$ for some $\alpha > \alpha_H.$
We can now formulate the following result.
The kinetic energy of the vortex filament $\xi$ built upon a 3d fractional Brownian motion of Hurst index $H>1/4$ is a.s. finite and in $L^1$ if the measure $\rho$ satisfies $$\label{eq:our-finiteness}
\mathrm{ess}\sup_{q} |\widehat \rho(q)| |q|^{-1}(1+|q|^2)^{\alpha/2} < \infty$$ for some $\alpha > \alpha_H $.
1. Known conditions on $\rho$ which guarantee the integrability of the energy are given in [@Ffil; @FGub] for the case of Brownian motion and Itô Brownian processes, and in [@Nua] for the case of fractional Brownian motions with Hurst parameter $H> 1/2$. From [@Nua] it can be deduced that a sufficient condition for the integrability of the energy is $$\label{eq:nua-finiteness}
\int_{{\mathbb{R}}^3} dq \frac{|\widehat \rho(q)|^2}{|q|^{4-1/H}} <\infty$$ or, written in a different but equivalent form, $$\int_{{\mathbb{R}}^3} \int_{{\mathbb{R}}^3} \frac{\rho(dx) \rho(dy)}{|x-y|^{1/H-1}} < \infty.$$
2. Condition (\[eq:our-finiteness\]) implies Condition (\[eq:nua-finiteness\]) when $H > 1/2$.
In fact the left-hand side of Condition (\[eq:our-finiteness\]) gives $$\begin{aligned}
\int_{{\mathbb{R}}^3} dq |\widehat \rho(q)|^2 |q|^{-2}(1+|q|^2)^{\alpha} & &
|q|^{2}(1+|q|^2)^{-\alpha}
|q|^{1/H -4} =
\int_{{\mathbb{R}}^3} dq |\widehat \rho(q)|^2 |q|^{1/H - 2}
(1+|q|^2)^{-\alpha} \\
&\le&
A \int_0^{+\infty} r^{1/H} (1+r^2)^{-\alpha} dr ,\end{aligned}$$ where $A$ is the finite quantity of (\[eq:our-finiteness\]). Clearly previous expression is bounded for $\alpha > \alpha_H$.
Of course the converse is not true.
3. A way of finding similar conditions to (\[eq:nua-finiteness\]) is to follow the steps of Sec. \[sec:hilbert\] for the Hilbert space regularity of the stochastic currents and rewrite (formally) the kinetic energy as $$\label{eq:energy2}
\mathcal{E}= \int_0^T \int_0^T \langle dX_t, g(X_t-X_s) dX_s\rangle$$ where $g$ is a vector kernel with the following Fourier transform $$\begin{split}
\widehat g(q) &= \frac{|\widehat \rho(q)|^2}{|q|^2} \Pi_q
\end{split}$$ and $\Pi_q$ is the following matrix $$(\Pi_q)_{\alpha \beta} = \delta_{\alpha \beta} - \frac{q_\alpha
q_\beta}{|q|^2}, \qquad \alpha,\beta = 1,\dots,3,$$ which projects in directions orthogonal to $q$. Formula (\[eq:energy2\]) can be understood, formally according to Theorem \[th:l2result\], as being the limit of the expectations of ${\varepsilon}$-approximations $$\mathcal{E}_{\varepsilon}=
\int_{0}^{T}\int_{0}^{T} g( X_{t}-X_{s})
\left\langle D_{\varepsilon}X_{t},D_{\varepsilon}X_{s}\right\rangle
dtds$$ To obtain conditions for its finiteness in the spirit of eq. (\[eq:nua-finiteness\]), we need to follow again the computations involved in the proof of Theorem \[th:reg-symm\] and use a different strategy in bounding some terms. Then we can prove the following:
\[th:symmetric-vortex\] Let $H>1/4$ and let $\rho: {\mathbb{R}}^3 \to {\mathbb{R}}$ be a function with Fourier transform $\widehat \rho$ satisfying $$\label{eq:vortex-cond}
\int_{{\mathbb{R}}^3} dq \frac{|\widehat \rho(q)|^2}{ |q|^{4-1/H}} <\infty$$ then the family of random fields $\{ u_{\varepsilon}\}_{{\varepsilon}\in(0,1)}$ defined as $$u_{\varepsilon}(x) = \int_0^1 (\mathcal{K} *\rho)(x-X_t) \wedge D^0_{\varepsilon}X_t dt, \qquad x
\in {\mathbb{R}}^3$$ converges a.s. in $L^{2-\theta}(\Omega; L^2({\mathbb{R}}^3;{\mathbb{R}}^3))$ for any $\theta > 0$ to a random field $u \in L^2({\mathbb{R}}^3;{\mathbb{R}}^3)$.
We will prove that $\sup_{{\varepsilon}\in (0,1)} {E}\|u_{\varepsilon}\|^2 <
\infty$ following the lines of the proof of Theorem \[th:reg-symm\], then the conclusion follows applying Theorem \[th:l2result\]. Let $\mathcal{E}_{\varepsilon}= \|u_{\varepsilon}\|^2$.
We start by treating the case $H > 1/2$.
Using Theorem \[th:wick\] (Wick theorem) and independence of different coordinates we have $$\label{eq:wick1ter}
\begin{split}
{E}\mathcal{E}_{\varepsilon}& =
{E}\int_{0}^{T}\int_{0}^{T} \sum_i g_{ii}\left(
X_{t}-X_{s}\right)
{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, D^0_{\varepsilon}X^1_s)
\,dt ds
\\ & \qquad -
{E}\int_{0}^{T}\int_{0}^{T} \sum_{ij} \nabla_i \nabla_j g_{ij}\left(
X_{t}-X_{s}\right)
|{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, X^1_{t}-X^1_{s})|^2
\,dt ds
\\ & =
{E}\int_{0}^{T}\int_{0}^{T}\text{Tr} g\left(
X_{t}-X_{s}\right)
{\textrm{Cov}}(D^0_{\varepsilon}X^1_t, D^0_{\varepsilon}X^1_s)
\,dt ds
\end{split}$$ since a direct computation shows that $
\sum_i \nabla_i g_{ik}(x) = 0
$. Using the first bound in Lemma \[lemma:bounds\] we get
$$\label{eq:wick1bis}
\begin{split}
{E}\mathcal{E}_{\varepsilon}& \le {\textrm{const}}{E}\int_{0}^{T}\int_{0}^{t} \text{Tr} g\left(
X_{t}-X_{s}\right) |t-s|^{2H-2}
\,ds dt
\\ & = {\textrm{const}}\int_{{\mathbb{R}}^3} dq \text{Tr} \widehat g(q)
\int_{0}^{T}\int_{0}^{t} |t-s|^{2H-2} {E}e^{-i \langle q,X_t-X_s\rangle}
\,ds dt
\\ & = {\textrm{const}}\int_{{\mathbb{R}}^3} dq \text{Tr} \widehat g(q)
\int_{0}^{T}\int_{0}^{t} |t-s|^{2H-2} e^{-|q|^2 (t-s)^{2H}/2}
\,ds dt
\\ &\le \text{const} \int_{{\mathbb{R}}^3} dq |\widehat g(q)| \int_{0}^T dt
\int_{0}^{\infty}
d\tau \tau^{2H-2} e^{-|q|^2 \tau^{2H}/2}
\\ & = \text{const} \int_{{\mathbb{R}}^3} dq |\widehat g(q)| |q|^{1/H-2} \int_{0}^T dt \int_{0}^{\infty}
dy y^{1-1/H} e^{-y^2/2}
\\ & \le \text{const} T \int_{{\mathbb{R}}^3} dq |\widehat g(q)| |q|^{1/H-2}
\end{split}$$
since $$\int_{0}^{\infty}
dy y^{1-1/H} e^{-y^2/2} < \infty$$ for $1-1/H > -1$, that is $H > 1/2$. Sufficient condition for uniform boundedness of ${E}\mathcal{E}_{\varepsilon}$ is that $$\int_{{\mathbb{R}}^3} dq |\widehat g(q)| |q|^{1/H-2} < \infty.$$
Let us now consider the case $H \le 1/2$ and rewrite the approximated energy as $$\mathcal{E}_{\varepsilon}= \|u_{\varepsilon}\|^2 = - \int_0^1 \int_0^1 \langle h(X_t-X_s)
D_{\varepsilon}X_s,D_{\varepsilon}X_t \rangle + \int_0^1 \int_0^1 \langle
g(0) D_{\varepsilon}X_s,D_{\varepsilon}X_t \rangle$$ where $h(x) = g(0) - g(x)\ge 0$. Note that $g(0)$ is well defined using the hypothesis of the theorem about the integrability of its Fourier transform, moreover, as in Thm. \[th:reg-symm\] (in the $H<1/2$ part) we have the limit $$\int_0^1 \int_0^1 \langle
g(0) D_{\varepsilon}X_s,D_{\varepsilon}X_t \rangle \to \langle (X_1-X_0)
g(0) ,(X_1-X_0) \rangle.$$ So let us focus on the double integral with the kernel $h$. Proceeding as in the $H > 1/2$ case we have $$\begin{split}
J & = - \int_0^T \int_0^T \langle h(X_t-X_s)
D_{\varepsilon}X_s,D_{\varepsilon}X_t \rangle
\\ & \le - {\textrm{const}}\int_0^T \int_0^T \int_{{\mathbb{R}}^3} dq \text{Tr} \widehat h(q) {E}[ e^{i
\langle q,X_t-X_s \rangle} ] |t-s|^{2H-2 }\, dtds
\\ & = - {\textrm{const}}\int_0^T \int_0^T \int_{{\mathbb{R}}^3} dq \text{Tr} \widehat h(q) e^{-|q|^2/2(t-s)^{2H}}
|t-s|^{2H-2 }\, dtds
\end{split}$$ but since $\widehat h(q) = g(0) \delta(q) - \widehat g(q) $ we have $$\begin{split}
& \int_{{\mathbb{R}}^3} dq \text{Tr} h(q) e^{-|q|^2/2(t-s)^{2H}}
= \int_{{\mathbb{R}}^3} dq \text{Tr} [g(0) \delta(q) - \widehat g(q)] e^{-|q|^2/2(t-s)^{2H}}
\\ & \qquad = \int_{{\mathbb{R}}^3} dq \text{Tr} [g(0) \delta(q) - \widehat g(q)] [e^{-|q|^2/2(t-s)^{2H}}-1]
= - \int_{{\mathbb{R}}^3} dq \text{Tr} \widehat g(q) [e^{-|q|^2/2(t-s)^{2H}}-1]
\end{split}$$ Then $$\begin{split}
|J| & \le {\textrm{const}}\int_0^T \int_0^T \int_{{\mathbb{R}}^3} dq |\widehat g(q)| (1- e^{-|q|^2/2(t-s)^{2H}})
|t-s|^{2H-2 }\, dtds
\\&\le \text{const} \int_{{\mathbb{R}}^3} dq |\widehat g(q)| \int_{0}^T dt \int_{0}^{t}
ds (t-s)^{2H-2} (1-e^{-2 |q|^2 (t-s)^{2H}})
\\ &\le \text{const} \int_{{\mathbb{R}}^3} dq |\widehat g(q)| \int_{0}^T dt \int_{0}^{\infty}
d\tau \tau^{2H-2} (1-e^{-|q|^2 \tau^{2H}/4})
\\ & = \text{const} \int_{{\mathbb{R}}^3} dq |\widehat g(q)| |q|^{1/H-2} \int_{0}^1 dt \int_{0}^{\infty}
dy y^{1-1/H} (1-e^{-y^2/4}-1)
\\ & \le \text{const} \int_{{\mathbb{R}}^3} dq |\widehat g(q)| |q|^{1/H-2}
\end{split}$$ where we made a change of variables $y = |q|\tau^H$ and we used the fact that $$\int_{0}^{\infty}
dy y^{1-1/H} (1-e^{-y^2/4}) \le \int_{0}^{\infty}
dy y^{1-1/H} \min(y^2,1) < \infty$$ since $-3 < 1 - 1/H < -1$ when $1/4 < H < 1/2$. So we obtain the uniform boundedness of ${E}\mathcal{E}_{\varepsilon}$ when eq. (\[eq:vortex-cond\]) is satisfied.
Analogously, the case $H=1/2$ does not pose any additional problem.
Note that for $H \ge 1/2$ we recover condition (\[eq:nua-finiteness\]). However while in [@Ffil; @Nua] only the existence and the integrability properties of the energy are studied, here we have also informations about convergence of ${\varepsilon}$-approximations of the velocity field generated by the random vortexes.
Some proofs and auxiliary results {#sec:appA}
=================================
Denote $\frac{1}{\Gamma\left( \alpha\right) \left( 4\pi\right) ^{d/2}}$ by $\gamma$, for shortness. Notice that for $x=0$ we have $$K_{\alpha}\left( 0\right) =\gamma\int_{0}^{\infty}t^{\alpha-\frac{d}{2}}e^{-t}\frac{dt}{t}<\infty\text{ if and only if }\alpha>\frac{d}{2}.$$ For $x\neq0$ we may use the change of variables $t=\left| x\right| ^{2}s$ and get $$\begin{aligned}
K_{\alpha}\left( x\right) =\left| x\right| ^{2\alpha-d}\rho\left(
x\right) \\
\rho\left( x\right) :=\gamma\int_{0}^{\infty}s^{\alpha-\frac{d}{2}}e^{-\frac{1}{4s}-\left| x\right| ^{2}s}\frac{ds}{s}$$ where the integral converges for every value of the parameters, thanks to the exponentials. For $0<\alpha<\frac{d}{2}$, we have $$c_{\alpha,d}e^{-2\left| x\right| ^{2}}\leq\gamma\int_{1}^{2}s^{\alpha
-\frac{d}{2}}e^{-\frac{1}{4s}-\left| x\right| ^{2}s}\frac{ds}{s}\leq
\rho\left( x\right)$$ for a positive constant $c_{\alpha,d}$. Moreover, $$\begin{aligned}
\rho\left( x\right) & \leq\gamma\int_{0}^{\frac{1}{\left| x\right| }}s^{\alpha-\frac{d}{2}}e^{-\frac{1}{4s}}\frac{ds}{s}+\gamma\int_{\frac
{1}{\left| x\right| }}^{\infty}s^{\alpha-\frac{d}{2}}e^{-\frac{\left|
x\right| }{4}-\left| x\right| ^{2}s}\frac{ds}{s}\\
& \leq C_{\alpha,d}\int_{0}^{\frac{1}{\left| x\right| }}e^{-\frac{1}{8s}}\frac{ds}{s^{2}}+C_{\alpha,d}e^{-\frac{\left| x\right| }{4}}\int_{\frac
{1}{\left| x\right| }}^{\infty}s^{\alpha-\frac{d}{2}}\frac{ds}{s}\\
& \leq C_{\alpha,d}e^{-\frac{\left| x\right| }{8}}+C_{\alpha,d}e^{-\frac{\left| x\right| }{4}}\left| x\right| ^{\frac{d}{2}-\alpha}\leq
C_{\alpha,d}e^{-\frac{\left| x\right| }{8}}$$ for a positive constant $C_{\alpha,d}$ that we do not rename at every step.
If $\alpha>\frac{d}{2}$, we directly have from the original formula $$\begin{aligned}
c_{\alpha,d}^{\prime}e^{-\frac{\left| x\right| ^{2}}{8}}\leq\gamma\int
_{1}^{2}t^{\alpha-\frac{d}{2}}e^{-\frac{\left| x\right| ^{2}}{4t}-t}\frac{dt}{t}\leq K_{\alpha}\left( x\right) \\
\leq\gamma\int_{0}^{\left| x\right| }t^{\alpha-\frac{d}{2}}e^{-\frac{\left|
x\right| ^{2}}{4t}}\frac{dt}{t}+\gamma e^{-\frac{\left| x\right| }{4}}\int_{\left| x\right| }^{\infty}t^{\alpha-\frac{d}{2}}e^{-t}\frac{dt}{t}\\
\leq C_{\alpha,d}^{\prime}e^{-\frac{\left| x\right| }{8}}$$ for a positive constants $c_{\alpha,d}^{\prime}$, $C_{\alpha,d}^{\prime}$. To estimate $K_{\alpha}(0)-K_{\alpha}(x)$ we write $$\begin{split}
K_{\alpha}(0)-K_{\alpha}(x) & = \gamma \int_{0}^{\infty}t^{\alpha-\frac{d}{2}}e^{-t} (1-e^{-\frac{\left|
x\right| ^{2}}{4t}})\frac{dt}{t}
\\ & = \gamma |x|^{\alpha-\frac d2} \int_{0}^{\infty}s^{\alpha-\frac{d}{2}}e^{-s|x|^{2}} (1-e^{-\frac{1}{4s}})\frac{ds}{s}
\end{split}$$ and use the same arguments as above. In the case $\alpha = d/2$ we simply split the integral as above and by straightforward estimation we can prove that $K_\alpha(x) \le {\textrm{const}}\ \log|x|$ for small $|x|$ and that $K_\alpha(x)$ decay exponentially for large $|x|$. The proof is complete.
We observe that, since $K_{\alpha}$ is the kernel of the operator $(1-\Delta)^{-\alpha}$ we have the identity $
K_{\alpha-1}(x) = (1-\Delta) K_{\alpha}(x)
$ so that $$-\Delta K_{\alpha}(x) = K_{\alpha-1}(x) - K_{\alpha}(x) \qquad x \neq 0.$$ Then $
|-\Delta K_{\alpha}(x)| \le| K_{\alpha-1}(x)| +| K_{\alpha}(x)|$ which gives the required bound using Lemma \[lemma su K\].
We start with the first estimate in 1. A direct computation shows $${\textrm{Cov}}(D^0_{\varepsilon}X^i_t, D^0_{\varepsilon}X^i_s) = \frac{|t-s|^{2H-2}}{2} \Phi\left(\frac{2 {\varepsilon}}{t-s}\right)$$ where $$\Phi(x) = \frac{|1+x|^{2H}+|1-x|^{2H}-2}{x^2}.$$
The function $\Phi$ is continuous in $(0,\infty)$, $\lim_{x\to
0}\Phi(x) = 2H-1$ so, when $|t-s| \le 2{\varepsilon}$ we have $$\frac{|t-s|^{2H-2}}{2} \Phi\left(\frac{2 {\varepsilon}}{t-s}\right) \le {\textrm{const}}\, |t-s|^{2H-2}.$$ Moreover $\lim_{x\to \pm\infty} |x|^{2-2H} \Phi(x) = 2$, so when $|t-s|> 2{\varepsilon}$ there exists a constant not depending on ${\varepsilon}$ such that $$\frac{|t-s|^{2H-2}}{2} \Phi\left(\frac{2 {\varepsilon}}{t-s}\right) \le {\textrm{const}}\ {\varepsilon}^{2H-2} \le {\textrm{const}}\ |t-s|^{2H-2}$$ which proves the first claim.
We discuss now the second estimate in 1. and point 2. A direct computation gives $$\begin{aligned}
{\textrm{Cov}}(D^0_{\varepsilon}X^i_t, X^i_{t}-X^i_{s}) &=& -
{\textrm{Cov}}(D^0_{\varepsilon}X^i_s, X^i_{t}-X^i_{s}) =
\frac{1}{2{\varepsilon}}\left(|t-s+{\varepsilon}|^{2H}-|t-s-{\varepsilon}|^{2H}\right) \\
&=& |t-s|^{2H - 1} \psi \left(\frac{{\varepsilon}}{t-s}\right)\end{aligned}$$ where $ \psi(x) = \frac{|1+x|^{2H} - |1-x|^{2H}}{2x}$. It is easy to show that $\psi$ is continuous and $\psi(0+) = 2H$, moreover $|x|^{2-2H} \psi(x) \to 2H$ when $x \to \pm\infty$. This allows to conclude the proof.
The first estimate in 1. is very similar to the previous lemma. A direct computation shows $$\label{EPhi-forw}
{\textrm{Cov}}(D^-_{\varepsilon}X^i_t, D^-_{\varepsilon}X^i_s) = \frac{|t-s|^{2H-2}}{2} \Phi\left(\frac{ {\varepsilon}}{t-s}\right)$$ where $\Phi$ is the same as previously. As for the other points a direct computation gives $$\begin{aligned}
{\textrm{Cov}}(D^-_{\varepsilon}X^i_t, X^i_{t}-X^i_{s}) &=& -
{\textrm{Cov}}(D^-_{\varepsilon}X^i_s, X^i_{t}-X^i_{s}) =
\frac{1}{2{\varepsilon}}\left(|t-s+{\varepsilon}|^{2H}-|t-s-{\varepsilon}|^{2H} - {\varepsilon}^{2H}\right) \\
&=& (t-s)^{2H - 1} \tilde \psi \left(\frac{{\varepsilon}}{t-s} \right)\end{aligned}$$ where $ \tilde \psi(x) = \frac{x^{2H} +1 - (1-x)^{2H}}{2x}$. Then, it is not difficult, arguing as in lemma \[lemma:bounds\] to conclude. In particular one can evaluate the limit in (\[EPhi-forw\]).
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{
"pile_set_name": "ArXiv"
}
|
---
title: 'On the Distribution of the Sum of $n$ Non-Identically Distributed Uniform Random Variables'
---
[David M. Bradley and Ramesh C. Gupta]{}
[*Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, U.S.A.*]{}
.1in
[*e-mail: [dbradley]{}@[e-math.ams.org]{}, [rcgupta]{}@[maine.maine.edu]{}* ]{}
.5in
[[*Key words and phrases:*]{} Uniform distribution, probability density, convolution, Fourier transform, sine integrals.]{}
.1in
[Abstract.]{} The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formul[æ]{} for the distribution of the sum of $n$ non-identically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formul[æ]{} some special cases which have appeared in the literature.
Introduction {#sect:Intro}
============
The classical uniform distribution is perhaps the most versatile statistical model: applications abound in nonparametric statistics and Bayes procedures. Chu (1957) and Leone (1961) utilized uniform distributions in connection with sample quasi-ranges. Naus (1966) applied uniform distribution in a power comparison of tests of random clustering. For additional applications and examples, see Johnson et al (1995).
Here, we are concerned with the distribution of the sum of $n$ independent non-identically distributed uniform random variables. It is well-known that the probability density function of such a sum, in which the summands are uniformly distributed in a common interval $[-a,a]$, can be obtained via standard convolution formul[æ]{}: see Feller (1966, p. 27) or Renyi (1970, pp.196–197), for example. However, it is sometimes necessary to analyze data which have been drawn from non-identical uniform distributions. For example, measurements accurate to the nearest foot may be combined with measurements accurate to the nearest inch. In such cases, the distribution of the sum is more complicated. Tach (1958) gives tables to five decimal places of the cumulative distribution of the sum for $n= 2, 3$ and 4 for some special cases.
The first general result in this direction seems to have been made by Olds (1952), who derived the distribution of the sum $\sum_{j=1}^n X_j$, in which each $X_j$ is uniformly distributed in an interval of the form $[0,a_j)$ with $a_j>0$. The proof is by induction, and in that respect is somewhat unsatisfactory, since in general inductive proofs require knowing beforehand the formula to be proved. Subsequently, Roach (1963) deduced what is essentially Olds’ formula using $n$-dimensional geometry. Later Mitra (1971), apparently unaware of these previous results, derived the distribution of the sum in which each random variable is uniformly distributed in an interval of the form $[-\omega_j,\omega_j]$ using Nörlund’s (1924) difference calculus.
Here, we derive an explicit formula for the slightly more general situation of the distribution of the sum $\sum_{j=1}^n X_j$, in which each $X_j$ is uniformly distributed in an interval of the form $[c_j-a_j,c_j+a_j]$ with $a_j>0$ and $c_j$ an arbitrary real number. Of course, each of the aforementioned results can be obtained from ours by specializing the parameters $c_j$ and $a_j$ accordingly.
Our approach is via Fourier theory and is quite straightforward; specifically we invert the characteristic function. As a result, our formula differs somewhat in form from the special cases alluded to previously. However, the inversion technique is quite flexible, and readily lends itself to the study of other types of distributions, such as the discrete case, which seems not to have been discussed in the literature. Thus, in a similar fashion, we derive the distribution of the sum of $n$ random variables with point mass at the integers in intervals of the form $[-m_j,m_j]$, in which each $m_j$ is a positive integer. The formula in the discrete case is somewhat more complicated than the corresponding formula in the continuous case; nevertheless, they are clearly closely related, and there is a certain charm and elegance to both. Of course, the same results may be obtained using the standard transformation methods.
The Continuous Case {#sect:Cont}
===================
Fix a positive integer $n$, and let $\vec a=(a_1,a_2,\dots,a_n)$ and $\vec c=(c_1,c_2,\dots,c_n)$ be vectors of real numbers with each $a_j>0$. For each $j=1,2,\dots,n$, we consider a random variable $X_j$ uniformly distributed on the closed interval $[c_j-a_j,c_j+a_j]$. The step function $\chi_j:{\mathbf{R}}\to{\mathbf{R}}$ defined for real $x$ by $$\qquad 2a_j\,\chi_j(x) = \left\{\begin{array}{lll}1 &\mbox{if
$|x-c_j|<a_j$,}\\ \tfrac12 &\mbox{if $|x-c_j|=a_j$,}\\ 0 &\mbox{if
$|x-c_j|>a_j$}\end{array}\right.
\label{ChiDef}$$ represents the density of the random variable $X_j$ for each $j=1,2,\dots,n$. (When employing techniques from Fourier theory, it is convenient to define the densities at jump discontinuities so that the equation $\chi_j(x+)+\chi_j(x-)=2\chi_j(x)$ is satisfied for all real $x$.) The corresponding characteristic function (Fourier transform) is given by $$\qquad \widehat{\chi}_j(t):=\frac{1}{2a_j}\int_{c_j-a_j}^{c_j+a_j}
e^{itx}\,dx
= e^{i c_j t}\,{\mathrm{sinc}}(a_j t),\qquad t\in{\mathbf{R}},$$ where ${\mathrm{sinc}}(x):=x^{-1}\sin x$ if $x\ne 0$, and ${\mathrm{sinc}}(0):=1$. For real $x$, the density of the sum $\sum_{j=1}^n X_j$ is given by the $n$-fold convolution $f_n(x) :=
(\chi_1*\chi_2*\dots*\chi_n)(x)$. Thus, $$\qquad f_n(x) =
\int_{-\infty}^\infty\chi_1(x-y_2)\int_{-\infty}^\infty
\chi_2(y_2-y_3)
\cdots\int_{-\infty}^\infty\chi_{n-1}(y_{n-1}-y_n)
\chi_n(y_n)\,dy_2\dots dy_n.$$ In particular, if each of the $n$ intervals is centered at $0$, and we write $x_1=x-y_2$, $x_n=y_n$ and $x_j=y_j-y_{j+1}$ for $1<j<n$, then the conditions on the variables $x_1,x_2,\dots,x_n$ are that each $|x_j|<a_j$ and $\sum_{j=1}^n x_j = x.$ Thus, $f_n(x)$ is simply the volume (in the sense of Lebesgue measure) of the $(n-1)$-dimensional region $$\qquad\{(x_1,x_2,\dots,x_{n-1})\in{\mathbf{R}}^{n-1} :
\big|x-\textstyle\sum_{j=1}^{n-1}x_j\big|<a_n
\;{\mathrm{and}}\; |x_j|<a_j\;
{\mathrm{for}}\; 1\le j<n\}$$ divided by the volume $\prod_{j=1}^n 2a_j$ of the $n$-dimensional hyperbox $$\qquad
\{(x_1,x_2,\dots,x_n)\in{\mathbf{R}}^n : |x_j|<a_j \quad {\mathrm{for}}
\quad j=1,2,\dots,n\}.
\label{hypercube}$$ Despite the utility of these representations, it is desirable to have an explicit formula for $f_n$. In this vein, we have the following
\[Thm:Cts\] The density of the sum of $n$ independent random variables, uniformly distributed in the intervals $[c_j-a_j,c_j+a_j]$ for $j=1,2,\dots,n$, is given by $$\label{CtsFormula}
\begin{split}
\qquad f_n(x) &= \bigg[
\sum_{\vec\varepsilon\in\{-1,1\}^n}\bigg(x+\sum_{j=1}^n (\varepsilon_j
a_j-c_j)\bigg)^{n-1}\\
&\qquad\times{\mathrm{sign}}\bigg(x+\sum_{j=1}^n (\varepsilon_j
a_j-c_j)\bigg)\prod_{j=1}^n\varepsilon_j\bigg]
\bigg/\bigg[(n-1)! 2^{n+1}\prod_{j=1}^n a_j\bigg],
\end{split}$$ in which the sum is over all $2^n$ vectors of signs $$\qquad\vec\varepsilon=(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n)\in
\{-1,1\}^n \qquad \mbox{i.e. each $\varepsilon_j=\pm 1$}$$ and $$\qquad
{\mathrm{sign}}(y) := \left\{\begin{array}{rrr} 1 &\mbox{if $y>0$,}\\
0 &\mbox{if $y=0$,}\\
-1 &\mbox{if $y<0$.}\end{array}\right.$$
Since the random variables are assumed to be independent, the characteristic function of the distribution of the sum is the product of the characteristic functions of their distributions: $$\qquad \widehat{f}_n(t) = \prod_{j=1}^n \widehat{\chi}_j(t)
=\prod_{j=1}^n e^{i tc_j}\,{\mathrm{sinc}}(a_j t).
\label{sincprod}$$ Mitra (1971, p. 195) remarks that “It is difficult to obtain an inverse Fourier transform of this product in a neat form.” Indeed, his approach is to expand the product of sincs into a power series using generalized Bernoulli polynomials. However, we shall see that the inverse Fourier transform of (\[sincprod\]) has an elegant representation involving a sum over the vertices of the hyperbox (\[hypercube\]).
Since for all real $x$, $f_n(x) = \tfrac12 f_n(x+) + \tfrac12
f_n(x-)$ by continuity of $f_n$ for $n>1$, and by definition of $\chi_1$ when $n=1$, Fourier inversion gives $$\qquad f_n(x) = \frac1{2\pi}\int_{-\infty}^\infty e^{-itx}
\widehat{f}_n(t)\,dt =
\frac1{2\pi}\int_{-\infty}^\infty e^{-ity}\prod_{j=1}^n
{\mathrm{sinc}}(a_j t)\,dt,$$ where $y:=x-\sum_{j=1}^n c_j$. Since ${\mathrm{sinc}}$ is an even function, making the change of variable $t\mapsto-t$ yields $$\qquad f_n(x) =
\frac1{2\pi}\int_{-\infty}^\infty e^{ity}\prod_{j=1}^n
{\mathrm{sinc}}(a_j t)\,dt.
\label{SincIntegral}$$ It remains only to evaluate the integral (\[SincIntegral\]). Related integrals are studied in Borwein and Borwein (2001) using a version of the Parseval/Plancherel formula, and by expanding the product of sincs into a sum of cosines. Our approach is somewhat more direct. We first express the ${\mathrm{sinc}}$ functions using complex exponentials, so that $$\qquad f_n(x) = \frac1{2\pi}\left(\frac1{2i}\right)^n
\bigg(\prod_{j=1}^n a_j^{-1}\bigg)\int_{-\infty}^\infty t^{-n}
e^{ity}\prod_{j=1}^n\left(e^{ita_j}-e^{-ita_j}\right)\,dt.
\label{RequiredIntegral}$$ For each of the $2^n$ vectors of signs $\vec\varepsilon=(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n)\in\{-1,1\}^n$, let $$\qquad\rho_{\varepsilon}:=\prod_{j=1}^n \varepsilon_j,
\qquad\mbox{and}\qquad \vec\varepsilon\cdot\vec a=\sum_{j=1}^n
\varepsilon_j a_j.$$ By carefully expanding the product of exponentials in (\[RequiredIntegral\]), we find that $$\qquad\prod_{j=1}^n \left(e^{ita_j}-e^{-ita_j}\right)
=\sum_{\vec\varepsilon\in\{-1,1\}^n} \rho_{\varepsilon}
\exp(it\vec\varepsilon\cdot\vec a).
\label{ProdtoSum}$$ It follows that $$\label{PVIntegral}
\qquad f_n(x) = \frac1{2\pi}\left(\frac1{2i}\right)^n
\bigg(\prod_{j=1}^n a_j^{-1}\bigg)^n
\sum_{\vec\varepsilon\in\{-1,1\}^n}
\rho_{\varepsilon}\, {\mathrm{P.V.}}\int_{-\infty}^\infty
t^{-n}\exp(it(y+\vec\varepsilon\cdot\vec a))\,dt.$$ Although each of the individual integrals in (\[PVIntegral\]) is divergent, the singularities must cancel because (\[RequiredIntegral\]) converges. Therefore, the required finite integral (\[RequiredIntegral\]) is equal to its principal value, and hence by linearity is given by (\[PVIntegral\]).
In view of the fact that (\[ProdtoSum\]) is entire with a zero of order $n$ at $t=0$, we may integrate (\[PVIntegral\]) by parts $n-1$ times and thereby obtain $$\label{PVParts}
\begin{split}
\qquad f_n(x) &=
\frac1{2\pi}\left(\frac1{2i}\right)^n\bigg(\prod_{j=1}^n
a_j^{-1}\bigg)\frac{i^{n-1}}{(n-1)!}
\sum_{\vec\varepsilon\in\{-1,1\}^n}\rho_{\varepsilon}
(y+\vec\varepsilon\cdot\vec a)^{n-1}\\
& \qquad\times
{\mathrm{P.V.}}\int_{-\infty}^\infty t^{-1}
\exp(it(y+\vec\varepsilon\cdot\vec a))\,dt.
\end{split}$$ But for any real number $b$, we have $$\begin{aligned}
\qquad&{\mathrm{P.V.}}\int_{-\infty}^\infty
t^{-1}\exp(itb)\,dt\\
&= \lim_{\varepsilon\to0+}\left\{\int_{\varepsilon}^\infty
t^{-1}\exp(itb)\,dt
+\int_{-\infty}^{-\varepsilon}t^{-1}\exp(itb)\,dt\right\}\\
&= \lim_{\varepsilon\to0+}\left\{\int_{\varepsilon}^\infty
t^{-1}\exp(itb)\,dt - \int_{\varepsilon}^\infty
t^{-1}\exp(-itb)\,dt\right\}\\
&= 2i\int_0^\infty t^{-1}\sin(tb)\,dt\\
&= i\pi\,{\mathrm{sign}}(b).\end{aligned}$$ Applying this latter result to (\[PVParts\]) yields (\[CtsFormula\]) and completes the proof of Theorem \[Thm:Cts\].
In some applications, it may be easier to work with powers of expressions involving the maximum function $y_+:=\max(y,0)$ as opposed to the ${\mathrm{sign}}$ functions in (\[CtsFormula\]). To this end, we make the following
\[TauDef\] Let $\tau:{\mathbf{R}}\to{\mathbf{R}}$ be given by $$\qquad\tau(x) = \left\{\begin{array}{lll} 1,& \mbox{if $x>0$,}\\
\tfrac12, & \mbox{if $x=0$,}\\ 0, &\mbox{if
$x<0$,}\end{array}\right.
\label{taudef}$$ and for $y$ real and $n$ a positive integer, let $y^{n-1}_+:=
y^{n-1}\tau(y)$. Note that $y^0_+ = \tau(y)$ and $y^n_+=(\max(y,0))^n$ for $n>0$.
Then we have the following corollary to Theorem \[Thm:Cts\].
\[Cor:tau\] Let $f_n$ be as in Theorem \[Thm:Cts\]. Then $$\label{CtsFormula2}
\begin{split}
\qquad f_n(x) &=
\bigg[\sum_{\vec\varepsilon\in\{-1,1\}^n}\bigg(x+\sum_{j=1}^n (\varepsilon_j
a_j-c_j)\bigg)^{n-1}_+\;\prod_{j=1}^n\varepsilon_j\bigg]\bigg/
\bigg[(n-1)!\,2^n\prod_{j=1}^n a_j\bigg].
\end{split}$$
Note that ${\mathrm{sign}}(x)=2\tau(x)-1$ holds for all real $x$. Hence, substituting the $\tau$ function for the ${\mathrm{sign}}$ function in (\[CtsFormula\]), we see that it suffices to prove the identity $$\label{CoolIdentity}
\qquad\sum_{\vec\varepsilon\in\{-1,1\}^n} \bigg(x+
\sum_{j=1}^n(\varepsilon_ja_j-c_j)\bigg)^{n-1}\;\prod_{j=1}^n
\varepsilon_j = 0.$$ Since $e^{a_j t}-e^{-a_j t}=2a_j t+O(t^2)$ as $t\to0$, (\[CoolIdentity\]) follows easily on comparing coefficients of $t^{n-1}$ in $$\qquad\sum_{\vec\varepsilon\in\{-1,1\}^n}
\exp\bigg\{\bigg(x+\sum_{j=1}^n \varepsilon_j a_j-c_j\bigg)t\bigg\}
\;\prod_{j=1}^n \varepsilon_j
= e^{xt}\prod_{j=1}^n e^{-c_j
t}\left(e^{a_jt}-e^{-a_jt}\right).$$ Alternatively, note that if $x\ge \sum_{j=1}^n (c_j+a_j)$, then $f_n(x)=0$ by definition: since each $X_j$ is $u[c_j-a_j,c_j+a_j]$, the sum $\sum_{j=1}^n X_j$ must fall within the interval $[\sum_{j=1}^n (c_j-a_j),\sum_{j=1}^n (c_j+a_j)]$. But, if $x\ge \sum_{j=1}^n (c_j+a_j)$, then we can drop the subscripted “+” from (\[CtsFormula2\]) since $x+\sum_{j=1}^n
(\varepsilon_ja_j-c_j)\ge 0$ for each $\vec\varepsilon\in\{-1,1\}^n$. It follows that (\[CoolIdentity\]) holds for all $x\ge \sum_{j=1}^n
(c_j+a_j)$. Since the left hand side of (\[CoolIdentity\]) is a polynomial in $x$ which vanishes for all sufficiently large values of $x$, it must in fact vanish for all real $x$ by the identity theorem.
When $n=1$, formul[æ]{} (\[CtsFormula\]) and (\[CtsFormula2\]) give the central difference representations $$\qquad\chi_1(x) = \frac{{\mathrm{sign}}(x-c_1+a_1)-{\mathrm{sign}}(x-c_1-a_1)}{4a_1}
= \frac{\tau(x-c_1+a_1)-\tau(x-c_1-a_1)}{2a_1}$$ respectively. Both are equivalent to the definition (\[ChiDef\]) with $j=1$.
When $n=2$, we know that $$\qquad f_2(x) =
(\chi_1*\chi_2)(x)=\int_{-\infty}^\infty
\chi_1(x-y)\chi_2(y)\,dy.$$ If $c_1=c_2=0$, the convolution reduces to $$\qquad f_2(x)
=\frac{1}{4a_1a_2}\int_{\max(x-a_1,\,-a_2)}^{\min(x+a_1,\,a_2)}\,dy
=\frac{\min(x+a_1,a_2)-\max(x-a_1,-a_2)}{4a_1a_2},$$ so that, in particular, $$\qquad f_2(0)
= \frac{1}{2\pi}\int_{-\infty}^\infty
\frac{\sin(a_1t)}{a_1t}\cdot\frac{\sin(a_2t)}{a_2t}\,dt =
\frac{\min(a_1,a_2)}{2a_1a_2}. \label{Checkn=2}$$ On the other hand, in light of the fact that $y\,{\mathrm{sign}}(y)=|y|$ for $y$ real, formula (\[CtsFormula\]) gives $$\begin{aligned}
\qquad f_2(0) &=
\frac{|a_1+a_2|-|a_1-a_2|-|-a_1+a_2|+|-a_1-a_2|}{8a_1a_2}\\
&= \frac{2(a_1+a_2)-2|a_1-a_2|}{8a_1a_2}\\
&= \frac{\min(a_1,a_2)}{2a_1a_2},\end{aligned}$$ in agreement with (\[Checkn=2\]). Alternatively, since $y\tau(y)=y_+=\max(y,0)$, formula (\[CtsFormula\]) gives $$\begin{aligned}
\qquad f_2(0) &=
\frac{\max(a_1+a_2,0) -\max(a_1-a_2,0)-\max(-a_1+a_2,0)+\max(-a_1-a_2,0)}
{4a_1a_2}\\
&= \frac{ a_1+a_2-\max(a_1-a_2,0)-\max(a_2-a_1,0)}{4a_1a_2}\\
&= \frac{\min(a_1,a_2)}{2a_1a_2}.\end{aligned}$$
If the random variables comprising the sum are uniformly distributed in a common interval centered at 0—say $[-a,a]$ with $a>0$—then formula (\[CtsFormula2\]) gives $$\qquad f_n(x) =
\frac1{(n-1)!\,(2a)^n}
\sum_{\vec\varepsilon\in\{-1,1\}^n}
\bigg(x+a\sum_{j=1}^n\varepsilon_j\bigg)^{n-1}_+\;
\prod_{j=1}^n\varepsilon_j.$$ If $k$ of $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n$ are negative and the remaining $n-k$ are positive, then summing over $k$ yields $$\qquad f_n(x) = \frac1{(n-1)!\,(2a)^n}\sum_{k=0}^n (-1)^k \binom{n}{k}
\left(x+(n-2k)a\right)^{n-1}_+,$$ in agreement with Feller (1966).
If for each $j=1,2,\dots,n$ we set $c_j=a_j$ and then replace $a_j$ by $a_j/2$, then $X_j$ will be uniformly distributed in $[0,a_j]$ and will have density $\chi_j$ now given by $$\label{NewChiDef}
\qquad a_j\,\chi_j(x) :=\left\{\begin{array}{lll}
1, &\mbox{if $0<x<a_j$,}\\
\tfrac12, &\mbox{if $x=a_j$ or if $x=0$,}\\
0, &\mbox{if $x>a_j$ or if $x<0$.}\end{array}\right.$$ Formula (\[CtsFormula2\]) of Corollary \[Cor:tau\] now gives $$\begin{split}
\qquad f_n(x) &=\bigg[
\sum_{\vec\varepsilon\in\{-1,1\}^n}\bigg(x-\sum_{j=1}^n \bigg(
\frac{1-\varepsilon_j}{2}\bigg)a_j\bigg)^{n-1}_+\;
\prod_{j=1}^n\varepsilon_j\bigg]
\bigg/\bigg[(n-1)!\,\prod_{j=1}^n a_j\bigg]\\
&=\bigg[\sum_{\vec s\in\{0,1\}^n} (-1)^{\Sigma \vec s}
\left(x-\vec s\cdot\vec a\right)^{n-1}_+\bigg]
\bigg/\bigg[(n-1)!\,\prod_{j=1}^n a_j\bigg],
\end{split}$$ where the sum is now over all $2^n$ vectors $\vec
s=(s_1,s_2,\dots,s_n)$ in which each component takes the value $0$ or $1$, $\Sigma \vec s$ denotes the sum of the components $s_1+s_2+\cdots +s_n$ and $\vec s\cdot \vec a$ denotes the dot product $s_1 a_1+s_2a_2+\cdots+s_na_n$. If we now break up the sum according to the number of non-zero components in the vector $\vec s$, we find that $$\begin{split}
\label{Olds}
\qquad f_n(x) &=
\bigg[ x^{n-1}_+ - \sum_{1\le j_1\le n}
\left(x-a_{j_1}\right)^{n-1}_+ +\sum_{1\le j_1 < j_2\le n}
\left(x-a_{j_1}-a_{j_2}\right)^{n-1}_+ -+\cdots\\
&\qquad\qquad+(-1)^n
\sum_{1\le j_1 < j_2 <\cdots < j_n\le n}
\bigg(x-\sum_{k=1}^n a_{j_k}\bigg)^{n-1}_+\,\bigg]
\bigg/\bigg[(n-1)!\,\prod_{j=1}^n a_j\bigg],
\end{split}$$ which is equivalent to the formula of Olds (1952, p. 282 (1)) when $n>1$. When $n=1$, the two formul[æ]{} differ at $x=0$ and at $x=a_1$ because Olds uses the density which is $1/a_1$ for $0\le x<a_1$ and $0$ otherwise, in contrast to our more symmetrical density (\[NewChiDef\]).
The Discrete Case {#sect:Discrete}
=================
Here, we fix $n$ positive integers $m_1,m_2,\dots,m_n$. For each $j=1,2,\dots,n$, we now consider a random variable $X_j$ uniformly distributed on the set of $(2m_j+1)$ integers contained in the closed interval $[-m_j,m_j]$, i.e. the set $\{-m_j,1-m_j,\dots,m_j-1,m_j\}$. The mass function of $X_j$ is the rational-valued function of an *integer* variable given by $$\qquad(2m_j+1)\chi_j(p) := \left\{\begin{array}{ll}1 &\mbox{if $|p|\le
m_j$,}\\ 0 &\mbox{if $|p|>m_j$.}\end{array}\right.
\label{newChiDef}$$ We seek a formula analogous to (\[CtsFormula\]) for the probability mass function of the sum $\sum_{j=1}^n X_j$, namely the $n$-fold convolution $$\qquad g_n(p) := (\chi_1*\chi_2*\cdots*\chi_n)(p)
= \sum_{k_1+k_2+\cdots+k_n=p}\; \prod_{j=1}^n \chi_j(k_j),
\label{DiscreteConvolution}$$ where the sum is over all integers $k_1,k_2,\dots,k_n$ such that $k_1+k_2+\cdots+k_n=p$. Although (\[DiscreteConvolution\]) is a formula of sorts, the various conditions on the summation indices make it inconvenient to apply. For example, when $n=3$, let $u_k=u_k(p):=\min(m_2,p-k+m_3)$ and $v_k=v_k(p):=\max(-m_2,p-k-m_3)$. Then $$\qquad g_3(p) =\bigg(\prod_{j=1}^3 (2m_j+1)^{-1}\bigg)
\sum_{\substack{|k|\le m_1\\u_k<v_k}}
(u_k-v_k+1),$$ where the sum is over all integers $k$ for which $u_k<v_k$ and $|k|\le m_1$. In general, one needs to consider cases which depend on the size of $p$ in relation to various signed sums of subsets of the parameters $m_1,m_2,\dots,m_n$. The number of cases to be delineated increases exponentially with $n$. Thus, the situation becomes rapidly unwieldy. Fortunately, there is alternative approach provided by Fourier theory. With the convolution $g_n:{\mathbf{Z}}\to{\mathbf{Z}}$ defined as above, we have
\[Thm:Discrete\] For all integers $p$, $$\begin{split}
\label{DiscreteFormula}
\qquad g_n(p) &= \frac{M}{2^n}\sum_{\vec\varepsilon\in\{-1,1\}^n}
{\mathrm{sign}}\bigg(2p+\sum_{j=1}^n (2m_j+1)\varepsilon_j\bigg)
\bigg(\prod_{j=1}^n\varepsilon_j\bigg)\\
&\qquad\qquad\times\sum_{k=0}^{(n-1)/2}
(-1)^k
b_{2k}^{(n)}
\frac{\left(2p+\sum_{j=1}^n(2m_j+1)\varepsilon_j\right)^{n-2k-1}}
{(n-2k-1)!},
\end{split}$$ where $$\qquad M:=\prod_{j=1}^n (2m_j+1)^{-1},$$ and the rational numbers $b_{2k}^{(n)}$ are the coefficients in the Laurent series expansion $$\qquad \bigg(\frac1{\sin x}\bigg)^n = \sum_{k=0}^\infty x^{2k-n}
b_{2k}^{(n)}.
\label{CscLaurent}$$ Explicitly, $$\qquad b_{2k}^{(n)} = (-1)^k \binom{n+2k}{n}\sum_{m=0}^{2k}
\frac{n}{n+m}\binom{2k}{m}\frac{1}{2^m(2k+m)!}
\sum_{r=0}^m (-1)^r\binom{m}{r}(2r-m)^{2k+m}.
\label{LaurentCoefficientFormula}$$
The corresponding characteristic functions for $j=1,2,\dots,n$ are now given by $$\qquad(2m_j+1)\widehat{\chi}_j(t) = \sum_{|k|\le m_j} e^{itk}
= \left\{\begin{array}{ll}
\displaystyle{\frac{\sin((m_j+1/2)t)}{\sin(t/2)}},
&\mbox{if $t\notin 2\pi{\mathbf{Z}}$,}\\ 2m_j+1, &\mbox{if $t\in
2\pi{\mathbf{Z}}$,}\end{array}\right.$$ which we recognize as the familiar Dirichlet kernel—see eg. Korner (1988, p. 68). Since for $t\notin 2\pi{\mathbf{Z}}$, we have $$\qquad\left(\chi_1*\chi_2*\cdots*\chi_n\right)\sphat (t)
=\prod_{j=1}^n \widehat{\chi}_j(t)
=M\prod_{j=1}^n \frac{\sin((m_j+1/2)t)}{\sin(t/2)},$$ it follows by orthogonality of the exponential that $$\qquad g_n(p) = \frac{M}{2\pi}\int_0^{2\pi} e^{ipt}\prod_{j=1}^n
\frac{\sin((m_j+1/2)t)}{\sin(t/2)}\,dt
= \frac{M}{\pi}\int_0^\pi e^{2ipx}\prod_{j=1}^n
\frac{\sin((2m_j+1)x)}{\sin x}\,dx.$$ We remark in passing that the factors $\sin((2m_j+1)x)/\sin x$ in this latter representation are simply the Chebyshev polynomials $U_{2m_j}(\cos x)$ of the second kind—see eg. Abramowitz and Stegun (1972, p. 766). The partial-fraction expansion $$\qquad\bigg(\frac{1}{\sin x}\bigg)^n
= \sum_{k=0}^{(n-1)/2} b_{2k}^{(n)}
\sum_{r=-\infty}^\infty\bigg(\frac{(-1)^r}{x+r\pi}\bigg)^{n-2k},
\label{CscParfrac}$$ is a modified version of the formula in Schwatt (1924, pp.209–210), and yields $$\label{GnParfrac}
\qquad g_n(p) = \sum_{k=0}^{(n-1)/2} b_{2k}^{(n)}
\sum_{r=-\infty}^\infty\frac{M}{\pi}
\int_0^\pi e^{2ipx}\bigg(\frac{(-1)^r}{x+r\pi}\bigg)^{n-2k}
\prod_{j=1}^n \sin((2m_j+1)x)\,dx.$$ When $n-2k>1$, the bilateral series $$\qquad \sum_{r=-\infty}^\infty \bigg(\frac{(-1)^r}{x+r\pi}\bigg)^{n-2k}$$ converges absolutely, and the interchange of summation and integration is easily justified using either Lebesgue’s dominated convergence theorem or Fubini’s theorem with both Lebesgue and counting measure. If $n-2k=1$, absolute convergence can be recovered by recasting the bilateral series in the form $$\qquad \sum_{r=-\infty}^\infty \frac{(-1)^r}{x+r\pi}
= \frac1x+\sum_{r=1}^\infty (-1)^r\bigg(\frac{ 2 x}{x^2-r^2\pi}\bigg),$$ and the justification proceeds as in the previous case.
We shall see that the inner sum of integrals in (\[GnParfrac\]) can be expressed as a single integral over the whole real line. To this end, we compute $$\begin{aligned}
\qquad &\sum_{r=-\infty}^\infty\frac1{\pi}
\int_0^\pi e^{2ipx}\bigg(\frac{(-1)^r}{x+r\pi}\bigg)^{n-2k}
\prod_{j=1}^n \sin((2m_j+1)x)\,dx\\
=&\sum_{r=-\infty}^\infty\frac{(-1)^{(n-2k)r}}{\pi}
\int_{r\pi}^{(r+1)\pi}
t^{2k-n} e^{2ip(t-r\pi)}\prod_{j=1}^n
\sin\left((2m_j+1)(t-r\pi)\right)\,dt\\
=&\sum_{r=-\infty}^\infty\frac{(-1)^{(n-2k)r}}{\pi}
\int_{r\pi}^{(r+1)\pi} t^{2k-n}e^{2ipt}\prod_{j=1}^n (-1)^{(2m_j+1)r}
\sin((2m_j+1)t)\,dt\\
=&\;\;\frac1{\pi}\int_{-\infty}^\infty t^{2k-n} e^{2ipt}\prod_{j=1}^n
\sin((2m_j+1)t)\,dt.\end{aligned}$$ This latter integral can be evaluated just as we evaluated the integral (\[SincIntegral\]). After expanding the product of sines as a sum over the constituent exponentials and integrating by parts $n-2k-1$ times, one finds that $$\begin{gathered}
\frac1{\pi}\int_{-\infty}^\infty t^{2k-n} e^{2ipt}\prod_{j=1}^n
\sin((2m_j+1)t)\,dt\\
=
\frac{(-1)^k}{2^n(n-2k-1)!}
\sum_{\vec\varepsilon\in\{-1,1\}^n}
\bigg(2p+\sum_{j=1}^n(2m_j+1)\varepsilon_j
\bigg)^{n-2k-1}\\
\times{\mathrm{sign}}\bigg(2p+\sum_{j=1}^n(2m_j+1)\varepsilon_j\bigg)
\prod_{j=1}^n\varepsilon_j.
\label{latterformula}\end{gathered}$$ Substituting (\[latterformula\]) into (\[GnParfrac\]) and interchanging the order of summation completes the proof of Theorem \[Thm:Discrete\].
[**Remark.**]{} The equations (\[CscLaurent\]), (\[LaurentCoefficientFormula\]), (\[CscParfrac\]) can also be obtained using Jordan’s (1979, §74, p. 216) general formula for the higher derivatives of a power of a reciprocal function and then applying Mittag-Leffler’s theorem.
As in the continuous case, we can replace the ${\mathrm{sign}}$ function in Theorem \[Thm:Discrete\] with the $\tau$ function of Definition \[TauDef\]. Thus we obtain
\[Cor:DiscreteFormula2\] Let $g_n$, $M$, and $b_{2k}^{(n)}$ be as in Theorem \[Thm:Discrete\]. Then, for all positive integers $n$ and integer $p$, $$\label{DiscreteFormula2}
\qquad g_n(p) = \frac{M}{2^{n-1}}
\sum_{k=0}^{(n-1)/2} \frac{(-1)^k b_{2k}^{(n)}}{(n-2k-1)!}
\sum_{\vec\varepsilon\in\{-1,1\}^n}
\bigg(2p+\sum_{j=1}^n(2m_j+1)\varepsilon_j\bigg)^{n-2k-1}_+\;
\prod_{j=1}^n\varepsilon_j,$$ where $y^{n-2k-1}_+=y^{n-2k-1}\tau(y)$ as in Definition \[TauDef\].
Making the substitution ${\mathrm{sign}}(y)=2\tau(y)-1$ in Theorem \[Thm:Discrete\] and noting that the coefficient of $t^{n-2k-1}$ in $$\qquad\sum_{\vec\varepsilon\in\{-1,1\}^n}\exp\bigg\{\bigg(
2p+\sum_{j=1}^n(2m_j+1)\varepsilon_j\bigg)t\bigg\}=
e^{2pt}\prod_{j=1}^n\left(e^{(2m_j+1)t}-e^{-(2m_j+1)t}\right)$$ vanishes for $0\le k\le (n-1)/2$, we obtain the desired result.
When $n=1$, formul[æ]{} (\[DiscreteFormula\]) and (\[DiscreteFormula2\]) give the representations $$\begin{split}
\qquad(2m_1+1)\chi_1(p) &= \tfrac12{\mathrm{sign}}(p+m_1+\tfrac12)
-\tfrac12{\mathrm{sign}}(p-m_1-\tfrac12)\\
&= \tau(p+m_1+\tfrac12)-\tau(p-m_1-\tfrac12),
\end{split}$$ respectively. Both are equivalent to the definition (\[newChiDef\]) with $j=1$.
When $n=2$, we have $M:=(2m_1+1)^{-1}(2m_2+1)^{-1}$, and $$\qquad g_2(p) =
\chi_1*\chi_2(p)= \sum_{k+j=p}\chi_1(k)\chi_2(j).$$ Since this is simply the coefficient of $t^p$ in the product $$\qquad M \sum_{|k|\le m_1} t^k \sum_{|j|\le m_2} t^j,$$ letting $u(p):=\min(m_1,p+m_2)$ and $v(p):=\max(-m_1,p-m_2)$, we have $$\qquad g_2(p) = M\times\left\{\begin{array}{ll} u(p)-v(p)+1 &\mbox{if
$u(p)\ge v(p)$,}\\ 0 &\mbox{if $u(p)<v(p)$.}\end{array}\right.$$ On the other hand formula (\[DiscreteFormula\]) gives the elegant representation $$\begin{aligned}
\qquad g_2(p) &=
\tfrac12 M\left(|p+m_1+m_2+1|
-|p+m_1-m_2|
-|p-m_1+m_2|\right.\\
&\left.\qquad\qquad+|p-m_1-m_2-1|\right),\end{aligned}$$ in which we have used the relation $y\,{\mathrm{sign}}(y)=|y|$ for $y$ real.
[References]{}
Abramowitz, M. and Stegun, I. A. (1972). [*Handbook of Mathematical Functions*]{}, Dover Publications, New York. .2in
Borwein, D. and Borwein, J. M. (2001). Some remarkable properties of sinc and related integrals. [*The Ramanujan Journal*]{}, **5**, (1), 73–89. .2in
Chu, J. T. (1957). Some uses of quasi-ranges, [*Annals of Mathematical Statistics*]{}, **28**, 173–180. .2in
Feller, W. (1966). [*An Introduction to Probability Theory and its Applications*]{}, Vol. II, John Wiley & Sons, New York..2in
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). [*Continuous univariate distributions*]{}, (2nd ed.) John Wiley, New York..2in
Jordan, C. (1979). [*Calculus of Finite Differences*]{}, Chelsea Publishing, (3rd ed. reprinted). .2in
Körner, T. W. (1988). [*Fourier Analysis*]{}, Cambridge University Press..2in
Leon, F. C. (1961). The use of sample quasi-ranges in setting confidence intervals for the population standard deviation, *Journal of the American Statistical Association*, **56**, 260–272..2in
Mitra, S. K. (1971). On the probability distribution of the sum of uniformly distributed random variables, [*SIAM J. Appl. Math.*]{}, **20**, (2), 195–198..2in
Naus, I. (1966). A power comparison of two tests of non-random clustering, [*Technometrics*]{}, **8**, 493–517. .2in
Nörlund, N. E. (1924). [*Vorlesungen über Differenzenrechnung*]{}, Springer, Berlin..2in
Olds, E. G. (1952). A note on the convolution of uniform distributions, [*Annals of Mathematical Statistics*]{}, **23**, 282–285..2in
Rényi, A. (1970). [*Probability Theory*]{}, North-Holland Publishing, Amsterdam..2in
Roach, S. A. (1963). The frequency distribution of the sample mean where each member of the sample is drawn from a different rectangular distribution, [*Biometrika*]{}, **50**, 508–513..2in
Schwatt, I. J. (1924). [*An Introduction to the Operations with Series*]{}, University of Pennsylvania Press, Philadelphia..2in
Tach, L. T. (1958). Tables for cumulative distribution function of a sum of independent random variables, [*Convair Aeronautics Report M*]{}, ZU-7-119-TN, San Diego.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is known that two Reissner-Nordstrom black holes or two overextreme Reissner-Nordstrom sources cannot be in physical equilibrium. In the static case such equilibrium is possible only if one of the sources is a black hole and another one is a naked singularity. We define the notion of physical equilibrium in general (stationary) case when both components of a binary system are rotating and show that such system containing a Kerr-Newman black hole and a Kerr-Newman naked singularity also can stay in physical equilibrium. The similar question about the system of two charged rotating black holes or two rotating overextreme charged sources still remaines open.'
author:
- |
G.A. Alekseev$^{\text{*)}}$ and V. A. Belinski$^{\text{**)}}$\
$^{\text{*)}}$Steklov Mathematical Institute, Gubkina 8, Moscow 119991,\
Moscow, Russia, *[email protected]*\
$^{\text{**)}}$ICRANet, Piazzale della Repubblica, 10, 65122 Pescara, Italy;\
Rome University “La Sapienza”, 00185 Rome, Italy;\
IHES, F-91440 Bures-sur-Yvette, France, *[email protected]*
title: |
Superposition of fields of two rotating charged\
masses in General Relativity and existence\
of equilibrium configurations
---
[*Keywords*]{}: [Einstein - Maxwell equations; solitons; exact solutions; black holes; naked singularities]{}
Introduction
============
In the non-relativistic physics two particles can be in equilibrium if the product of their masses is equal to the product of their charges (we use the units $G=c=1$). However, the question on the existence of an analogue of such equilibrium state in General Relativity is far from being trivial. Besides the natural mathematical complications, in General Relativity there arise two different types of the “point” centers, namely Kerr - Newman black holes and Kerr - Newman naked singularities. Therefore, seeking equilibrium configurations, one has to consider all three possible types of binary systems $$\begin{array}{l}
\bullet\hskip1ex\text{black hole - black hole}\\
\bullet\hskip1ex\text{black hole - naked singularity}\\
\bullet\hskip1ex\text{naked singularity - naked singularity}
\end{array}$$ and analyse in each case all physical and geometrical conditions which are necessary for equilibrium. In particular, in each case a physically defined distance between these objects should exist in the corresponding solution.
When the Inverse Scattering Method (ISM) has been adopted for integration of the Einstein and Einstein-Maxwell equations, it was shown that Kerr-Newman black holes and Kerr-Newman naked singularities represent nothing else but stationary axially symmetric solitons. Then by the ISM machinery one can obtain the families of exact stationary axially symmetric solutions of these equations containing any number of such solitons centralized at different points of the symmetry axis. The mathematical construction of such solutions do not represents any principal difficulties apart from the routine calculations in the framework of the well developed soliton generating procedure which allows to insert any number of solitons into a given background space-time. However, it is quite intricate task to single out from these families the *physically* reasonable cases which correspond to a real equilibrium states of charged rotating black holes and naked singularities interacting with each other because in general, formally constructed stationary axisymmetric soliton solutions possess some features unacceptable from the physical point of view. These unwanted traits are due to the presence in the solutions of exotic peculiarities such as:
- [non-vanishing global NUT parameter,]{}
- [closed time-like curves around those parts of the symmetry axis which are outside the sources,]{}
- [angle deficit (or excess) at the points of the symmetry axis,]{}
- [non-zero physical magnetic charges of the sources,]{}
- [existence of some additional singularities on or off the axis]{}
The global NUT parameter is incompatible with asymptotic flatness of the space-time at spatial infinity. Keeping in mind the physical applications, we should exclude also the appearance outside the sources of closed timelike curves which may arise in formally constructed solutions. Besides that, the angle deficit or excess at the points of the symmetry axis give rise to the well known conical singularities violating the local Euclidness of space at these points (it can be treated as some singular external strut or string preventing the sources to fall onto or to run away from each other). Also the magnetic charges of each of the sources should be excluded since their existence contradicts to the present physical experience.[^1] All five aforementioned phenomena have nothing to do with a real equilibrium of the physical bodies and the corresponding equilibrium solution should be free of such pathologies. To single out from the families of soliton solutions the solutions without all listed above physically unacceptable properties, one needs to impose on the parameters of these families of solutions some constraints which lead to a system of algebraic equations. The problem is that these equations, even for the simplest case of two objects, are extremely complicated and it is difficult to resolve them in an exact analytical form in order to show directly whether they have physically appropriate solutions compatible with the condition of existence of a positive distance between the sources.
The aforementioned nuisances constitute the real troubles only in the general case of rotating sources. The static case is more simple and it would be not an overstatement to say that for the case of two non-rotating charged objects the problem have been solved completely. The first indications that two static charged masses can stay in real physical equilibrium without any struts between them and without any other pathologies came from the results of Bonnor [@Bon] and Perry and Cooperstock [@Per]. In [@Bon] it was analyzed the static equilibrium condition for a charged test particle in the Reissner-Nordstrom field and it was shown that such test body can be at rest in the field of the Reissner-Nordstrom source only if they both are either extreme (the charge equal to mass) or one of them is of a black hole type (the charge is less than the mass) and the other is of naked singularity type (the charge is grater than the mass). There is no way for equilibrium in cases when both masses are either of naked singularity type or both are of black hole type. The more solid arguments in favour of existence of a static equilibrium configuration for the black hole - naked singularity system was presented in [@Per], where both sources have been treated exactly, that is no one of the components was considered as test particle. These results have been obtained there by numerical calculations and three examples of numerical solutions of the equilibrium equation have been demonstrated. These solutions can correspond to the equilibrium configurations free of struts, though the authors have not been able to show the existence of a positive definite distance between the sources. The authors of [@Per] also reported that a number of numerical experiments for two black holes and for two naked singularities showed the negative outcomes, i.e. all tested sets of the parameters was not in power to satisfy the equilibrium equation. These findings were in full agreement with Bonnor’s test particle analysis.
The explicit solution of the problem in static case have been presented in paper [@Alekseev-Belinski:2007] where it was constructed the exact analytic static solution of the Einstein-Maxwell equations for two charged massive sources separated by the well defined positive distance and free of struts or of any other unphysical properties [^2]. We showed also that such solution indeed exists only for the black hole – naked singularity system and it is impossible to have the similar static equilibrium state for the pairs black hole – black hole or naked singularity – naked singularity. After these results the natural question arises whether the analogous physical equilibrium exists for two *rotating* sources. It turns out that the answer is affirmative and in the present paper we demonstrate the *six-parametric* exact equilibrium solution of the Einstein-Maxwell equations for two rotating charged objects one of which is a black hole and another one is a naked singularity.
It is worth mentioning that in the literature one can find many formal mathematical constructions of particular cases of doble-soliton stationary solutions. However, all of them have one and the same weakness: they cannot be accepted as physical equilibrium states because each of them contains at least one of the five pathological traits listed above. The present article proves the existence of the physical equilibrium states of two rotating sources of the black hole – naked singularity type which is free of all these pathologies and depends on *six* independent parameters. We restrict ourselves namely by the system of rotating bodies of the black hole - naked singularity type, because it is known that for such system the physical equilibrium in statc case indeed is possible. As we said already this is only the case for the static physical equilibrium configuration and it is natural to expect that it can be extended also for the rotating sources of these types. Nevertheless, it is necessary to emphasize that in spite of the known non-existence of physical equilibrium for the systems black hole - black hole and naked singularity - naked singularity in static case, we cannot assert the non-existence of physical equilibrium for such systems also for rotating configurations. May be the rotations can create the additional forces which can overcome the “no go” result for the static case for the systems black hole - black hole and naked singularity - naked singularity and permit them to stay in the physical equilibrium (although it seems to be a little bit strange, because such equilibrium would have no limit to the case of vanishing rotations). Thus, for binary systems black hole - black hole and naked singularity - naked singularity, the question on the possibility of physical equilibrium at present remaines open.
Besides that, we had to clarify also an important question concerning the absence in our solution of any other singularities anywhere outside two Kerr-Newman sources. At the last section of the manuscript, we construct an asymptotic representation of our solution in the whole space-time region outside two Kerr - Newman sources – the black hole and naked singularity for the case of rather large coordinate distance separating these two sources. The regular character of this asymptotic representation indicates the non-existence in this solution of other singularities which could be considered as supplement sources of gravitational and electromagnetic fields in our solution.
General properties of soliton solutions
=======================================
In the context of the Inverse Scattering Method (ISM), the gravitational solitons as exact solutions of pure gravity Einstein equations have been introduced in the papers [@BZ1; @BZ2]. The generalization of this technique for the coupled gravitational and electromagnetic fields was constructed in [@A1]. Its more detailed description can be found in [@A2] and in the book [@BV]. In this generalized approach one starts from some given background solution of the Einstein-Maxwell equations and generates on this background any desired number of solitons. We have to do here with the linear spectral differential equations (Lax pair) for the $3\times3$ matrix function $\Psi(\rho,z,w)$, where $w$ is a complex spectral parameter independent of coordinates $\rho,z$. First of all, for chosen background solution of the Einstein-Maxwell equations we have to find from the Lax pair the corresponding background spectral matrix $\Psi_{0}(\rho,z,w)$. Using the ISM dressing procedure it is possible to find explicitly the spectral matrix $\Psi_{n}(\rho,z,w),$ corresponding to the new solution containing $n$ solitons generating on the background space-time and extract then the new metric and new electromagnetic potential from this $\Psi
_{n}.$ The solitonic field added to the background can be characterized by the matrix $\Psi_{n}\Psi_{0}^{-1}-I$ which is a meromorphic (with respect to the spectral parameter $w$) matrix function tending to zero in the limit $w\rightarrow\infty$ and having $n$ simple poles in the complex plane of the parameter $w$ (one pole for each soliton).
In pure gravity case some of these poles can be located at the real axis of the $w$-plane and the corresponding sources have horizons (that is they are of the black hole type) while complex poles generate objects with naked singularities. However, in the presence of electromagnetic field the formal machinery of the ISM developed in [@A1] in general does not allow poles to be located at real axis which means that by this method one can produce solutions containing sources without the horizons, i.e. only of the naked singularity type.[^3] Nevertheless, also in this case after one obtains the final form of solution it is possible to forget the way how it was derived and to continue the solution analytically in the space of its parameters in order to get the complete family containing solutions with real metric of the physical signature and with horizons as well. However, the technical procedure how to do this is simple only for the case of one-solitonic solution (that is for the Kerr-Newman case) and some simple enough generalization of such procedure was found also for two static solitonic objects [@AB3]. In the general case of two rotating sources (the corresponding 12-parametric solitonic solution have been constructed in [@A3]) this task is much more complicated. Fortunately, there is an effective way to get over this difficulty. Because we need to construct solution of the black hole - naked singularity type, we can consider the Kerr-Newman black hole as new background (instead of the flat space-time) and insert to it one soliton of naked singularity type. This is exactly what can be done easily with the generating technique proposed in [@A1] and what we are interested in. The exact expressions (in terms of the Ernst potentials) for the solution together with the proof that all conditions of the physical equilibrium can be satisfied are given below.
Superposition of fields of the Kerr - Newman black hole and naked singularity
=============================================================================
For stationary axisymmetric soliton solutions of Einstein - Maxwell equations, metric and electromagnetic potential components in Weyl coordinates $\left( t,\rho,z,\varphi\right)$ take the forms: $$\label{FieldComponents}
\begin{array}{c}
ds^{2}=-f\left( d\rho^{2}+dz^{2}\right) +g_{tt}dt^{2}+2g_{t\varphi
}dtd\varphi+g_{\varphi\varphi}d\varphi^{2},\\[1ex]
g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}=-\rho^{2}, \\[1ex]
A_{t}=A_{t}(\rho,z),\text{ \ }A_{\varphi}=A_{\varphi}(\rho,z),\text{
\ }A_{\rho}=0,\text{ \ }A_{z}=0\text{ },
\end{array}$$ where all metric coefficients depend only on the variables $\rho,z$. The Lorentz signature of this metric implies that the conformal factor $f>0$.
Our solution depends on eight real and two complex constant parameters $$\label{Parameters}
\{m_{\scriptscriptstyle{0}},\,a_{\scriptscriptstyle{0}},\, b_{\scriptscriptstyle{0}},\,e_{\scriptscriptstyle{0}}\},\quad
\{m_{s},\,a_{s},\,b_{s},\,e_{s}\},\quad l=z_{2}-z_{1}>0\quad\text{and}\quad
c_{\scriptscriptstyle{0}}\text{ },%$$ where the parameters $e_{\scriptscriptstyle{0}}$ and $e_{s}$ are complex and the others are real. The parameters with the suffix $0$ are related to the background solution (black hole) and the parameters with the suffix ${s}$ are the parameters of a soliton we add to the black hole background. The parameter $l$ (which was chosen positive for definiteness) characterizes a $z$-distance between the sources because $z_{1}$ and $z_{2}$ determine respectively the location of a black hole and a naked singularity on the axis. The constant $c_{0}$ is an arbitrary multiplier in front of the metric coefficient $f$ in (\[FieldComponents\]) which should be chosen in accordance, e.g., with the condition of regularity of the axis at spatial infinity. It is convenient to use two functions of these parameters – the real $\sigma_{0}$ and pure imaginary $\sigma_{s}$, such that $$\label{sigmas}
\sigma_{0}^{2}=m_{0}^{2}+b_{0}^{2}-a_{0}^{2}-e_{0} \overline{e}_0\geq
0,\qquad\sigma_{{s}}^{2}=m_{{s}}^{2}+b_{{s}}^{2}-a_{{s}}^{2}-e_{s} \overline{e}_s\leq0.$$
Though our stationary axisymmetric solution depends on two Weyl coordinates $\rho,z$ only, it is more convenient to express it in terms of the so called bipolar coordinates – two pairs of polar coordinate $(x_1, y_1)$ and $(x_2, y_2)$ centered respectively at the location of a black hole (the coordinates with the suffix $1$) and at the location of a naked singularity (the coordinates with the suffix $2$). Of course, these four coordinates should satisfy two additional constraints and each of these four coordinates can be expressed in terms of Weyl coordinates $\rho,z$. The corresponding defining relations take the forms $$\label{Weyl}
\left\{\begin{array}{l}
\rho=\sqrt{x_{1}^{2}-\sigma_{0}^{2}}\sqrt{1-y_{1}^{2}}\\[0.5ex]
z=z_{1}+x_{1} y_{1}
\end{array}\right.\quad\text{and}\quad
\left\{\begin{array}{l}
\rho=\sqrt{x_{2}^{2}-\sigma_{{s}}^{2}}\sqrt{1-y_{2}^{2}}\\[0.5ex]
z=z_{1}+l+x_{2}y_{2}
\end{array}\right.$$ It is worth to note that $z_{1}$ is not an essential parameter because it determines a shift of the whole configuration of the sources and their fields along the axis. The inverse relations for coordinates $(x_{1},y_{1})$ corresponding to real $\sigma_{0}$ are $$\label{x1y1}
\begin{array}{l}
x_{1}=\dfrac{1}{2}\left[ \sqrt{(z-z_{1}+\sigma_{0})^{2}+\rho^{2}}%
+\sqrt{(z-z_{1}-\sigma_{0})^{2}+\rho^{2}}\right] ,\\[1ex]%
y_{1}=\dfrac{2(z-z_1)}{\sqrt{(z-z_{1}+\sigma_{0})^{2}+\rho^{2}}
+\sqrt{(z-z_{1}-\sigma_{0})^{2}+\rho^{2}}} .
\end{array}$$ For the coordinates $(x_{2},y_{2})$ corresponding to imaginary $\sigma_{s}$ ($\sigma_{{s}}^{2}<0$) the similar relations are more complicated ($z_{2}=l+z_{1}$): $$\begin{array}{l}\label{x2y2}
x_{2}=\sqrt{\dfrac{1}{2}\left[(z-z_{2})^{2}+\rho^{2}+\sigma_{{s}}%
^{2}\right]+\dfrac{1}{2}\sqrt{\left[ (z-z_{2})^{2}+\rho^{2}+\sigma_{{s}%
}^{2}\right] ^{2}-4\sigma_{{s}}^{2}(z-z_{2})^{2}}},\\[2ex]
y_{2}=\dfrac{z-z_2}{\sqrt{\dfrac{1}{2}\left[(z-z_{2})^{2}+\rho^{2}+\sigma_{{s}}%
^{2}\right]+\dfrac{1}{2}\sqrt{\left[ (z-z_{2})^{2}+\rho^{2}+\sigma_{{s}%
}^{2}\right] ^{2}-4\sigma_{{s}}^{2}(z-z_{2})^{2}}}}.
\end{array}$$ Sometimes it is convenient also to use instead of pairs of coordinates $(x_{1},y_{1})$ and $(x_{2},y_{2})$ the pairs of quasi-spherical coordinates $(r_{1},\theta_{1})$ and $(r_{2},\theta_{2})$: $$\label{r1r2}
\left\{\begin{array}{l}
x_{1}=r_{1}-m_{0},\\
y_{1}=\cos\theta_{1},
\end{array}\right.
\qquad\qquad
\left\{\begin{array}{l}
x_{2}=r_{2}-m_{{s}},\\
y_{2}=\cos\theta_{2}.
\end{array}\right.$$ The Ernst potentials and the conformal factor $f$ for our solution are: $$\label{ErnstPotentials}
\mathcal{E}=1-\dfrac{2(m_{0}-ib_{0})}{\mathcal{R}_{1}}- \dfrac{2(m_{s}-ib_{s})}{\mathcal{R}_{2}%
},\qquad\Phi=\dfrac{e_{0}}{\mathcal{R}_{1}}+ \dfrac{e_{s}}{\mathcal{R}_{2}},$$ $$\begin{aligned}
\dfrac{1}{\mathcal{R}_{1}} & =\dfrac{x_{2}+ia_{s}y_{2}+K_{1}(x_{2}-\sigma_{s}
y_{2})+L_{1}(x_{1}+\sigma_{0}y_{1})+S_{0}\left( x_{2}+\sigma_{s}y_{2}\right)
}{D},\label{CalR1R2}\\
\dfrac{1}{\mathcal{R}_{2}} & =\dfrac{x_{1}+ia_{0}y_{1}+K_{2}(x_{1}-\sigma_{0}
y_{1})+L_{2}(x_{2}+\sigma_{s}y_{2})}{D},\nonumber\end{aligned}$$ $$\begin{aligned}
\label{Dvalue}
D & =(x_{1}+ia_{0}y_{1}+m_{0}-ib_{0})\left[ x_{2}+ia_{s}y_{2}+m_{s}%
-ib_{s}+S_{0}\left( x_{2}+\sigma_{s}y_{2}\right) \right]-\\
& -\left[ m_{0}-ib_{0}-K_{2}(x_{1}-\sigma_{0}y_{1})-L_{2}(x_{2}+\sigma
_{s}y_{2})\right] \times\nonumber\\
& \times\left[ m_{s}-ib_{s}-K_{1}(x_{2}-\sigma_{s}y_{2})-L_{1}(x_{1}%
+\sigma_{0}y_{1}\right] ,\nonumber\end{aligned}$$ $$\begin{aligned}
K_{1} & =\dfrac{ia_{s}-\sigma_{s}}{\sigma_{0}+\sigma_{s}+l},
\quad L_{1}=\dfrac{(m_{0}+ib_{0})(m_{s}-ib_{s})-\overline{e}_{0} e_{s}
}{(ia_{0}-\sigma_{0})(\sigma_{1}+\sigma_{2}%
+l)},\\
K_{2} & =\dfrac{ia_{0}-\sigma_{0}}{\sigma_{0}+\sigma_{s}-l},
\quad L_{2}=\dfrac{(m_{0}-ib_{0})(m_{s}+ib_{s})-e_{0}
\overline{e}_{s}}{(ia_{s}-\sigma_{s})(\sigma_{0}+\sigma_{s}%
-l)},\nonumber\end{aligned}$$ $$\label{S0}
S_{0}=\frac{\sigma_{0}Y_{s}\bar{Y}_{s}}{\sigma_{s}\left( \sigma_{0}^{2}+a_{0}^{2}\right) (ia_{s}-\sigma_{s})(l-\sigma_{0}-\sigma_{s})},$$ $$\label{Ys}
Y_{s}=\left( m_{0}-i b_{0}\right) e_{s}-\left(
m_{s}-i b_{s}\right) e_{0},$$ $$\label{factorf}
f=c_{0}\frac{D\bar{D}}{\left( x_{1}^{2}-\sigma_{0}^{2}y_{1}^{2}\right)
\left( x_{2}^{2}-\sigma_{s}^{2}y_{2}^{2}\right) }.$$ Here and in what follows the bar over a letter means complex conjugation. In (\[c0\]), $c_{0}$ is an arbitrary real constant which should be chosen as $$\label{c0}
c_{0}=\left\vert\vphantom{I^I_I} 1+S_{0}-(K_{1}+L_{1})(K_{2}+L_{2})\right\vert ^{-2}\text{ }$$ in order to provide a correct limit value of the conformal factor $f$ at spatial infinity where we should have $f\to 1$.
It seems useful to recall here that in Weyl coordinates the Kerr-Newman black hole horizon corresponds to the segment on the axis of symmetry $\{\rho=0,z_1-\sigma_0\le z\le z_1+\sigma_0\}$, while the naked singularity of the Kerr-Newman type in these coordinates is represented by a segment $\{\rho=\vert\sigma_s\vert\sin\theta,\,z=z_2,\,0\le\theta\le \pi \}$ which is orthogonal to the symmetry axis. In the Kerr-Newman naked singularity geometry this segment corresponds to a “critical” sphere $r_2=m_2$.
Searches for equilibrium configurations
=======================================
The solution described above was constructed as the nonlinear superposition of gravitational and electromagnetic fields of a Kerr - Newman black hole and a Kerr - Newman naked singularity. Using the previous experience of studies of static configurations of charged massive sources in equilibrium (see [@Alekseev-Belinski:2007]), one may expect that in the rotating case, the equilibrium configurations of a black hole and a naked singularity also can be expected existing. However, for arbitrary choice of (in total) twelve real parameters of our solution, this solution can not be considered as describing the physical equilibrium of these two sources in their common fields, because for such physical interpretation of the solution it is necessary that in this solution any physically pathological properties (such as closed timelike curves, conical points on the axis, the total magnetic charge and the magnetic charges of each of the sources and NUT parameters) as well as any other sources of fields (such as additional curvature singularities on the axis or outside it) are absent. Further in this section, we find the constraints on the parameters of our solution which provides the absence in this solution of the pathological properties mentioned above, i.e. the conditions of equilibrium of the Kerr - Newman black hole and naked singularity in their common gravitational and electromagnetic fields. The absence of any other sources of these fields in the solution (\[ErnstPotentials\]) – (\[c0\]) and the solution of the equilibrium conditions will be described in the next section where we use an additional assumption that the coordinate distance $l$ separating the sources is large enough.
#### *Asymptotic flatness of the solution and its physical parameters.*
At spatial infinity, i.e. for $r=\sqrt{\rho^{2}+z^{2}}
\rightarrow\infty,$ the Ernst potentials determined by the expressions (\[ErnstPotentials\]) – (\[c0\]) have the following asymptotical behaviour: $$\label{ErnstEF}
\begin{array}{l}
\mathcal{E}=1-\dfrac{2(M-iB)}{r}+\dfrac{(z_\ast+2 i J)y+const}{r^2}+O(\dfrac{1}{r^{3}}),\\[2ex]
\Phi=\dfrac{Q_{e}+iQ_{m}}{r}+\dfrac{(D_e+i D_m)y+const}{r^2}+O(\dfrac{1}{r^{3}}),
\end{array}$$ where $M$ and $B$ are the total gravitational mass and total NUT parameter of the configuration, $Q_{e}$ and $Q_{m}$ are its total electric and magnetic charges, $J$ is the total angular momentum and $D_e$ and $D_m$ are the electric and magnetic dipole moments. The constant $z_\ast$ depends on the location of the origin of the quasi-spherical coordinate system $(r,\theta)$ on the axis of symmetry and $y=\cos\theta$.
Direct calculations show that some of these physical parameters possess the following expressions in terms of the parameters (\[Parameters\]) and (\[sigmas\]): $$\label{Physparameters}
\begin{array}{l}
M=\text{Re}\left[ \dfrac{(m_{0}-ib_{0})(1+K_{1}+L_{1}+S_{0})+(m_{{s}}
-ib_{{s}})(1+K_{2}+L_{2})}{1+S_{0}-(K_{1}+L_{1})(K_{2}+L_{2})}\right],\\[2ex]
B=-\text{Im}\left[ \dfrac{(m_{0}-ib_{0})(1+K_{1}+L_{1}+S_{0})+(m_{{s}
}-ib_{{s}})(1+K_{2}+L_{2})}{1+S_{0}-(K_{1}+L_{1})(K_{2}+L_{2})}\right],\\[2ex]
Q_{e}=\operatorname{Re}\left[\dfrac{e_{0}
(1+K_{1}+L_{1}+S_{0})+e_{s} (1+K_{2}+L_{2})}
{1+S_{0}-(K_{1}+L_{1})(K_{2}+L_{2})}\right],\\[2ex]
Q_{m}=\operatorname{Im}\left[ \dfrac{e_{0}
(1+K_{1}+L_{1}+S_{0})+e_{s}(1+K_{2}+L_{2})}
{1+S_{0}-(K_{1}+L_{1})(K_{2}+L_{2})}\right].
\end{array}$$ The other physical parameters of our solution, such as $J$, $D_e$, $D_m$ possess more complicate expressions and we do not present them here. From the expressions (\[Physparameters\]) we conclude immediately that to have a configuration without a total NUT and magnetic charge, we should impose on the parameters the restrictions $$\label{BQm}
B=0\quad \text{and}\quad Q_{m}=0$$ where the expressions (\[Physparameters\]) should be taken into account.
#### *Closed time-like curves.*
If at some points of the axis (where $\rho=0$) we have $g_{t\varphi}
\neq 0$, this implies (in accordance with the relation between the metric coefficients in Weyl coordinates $g_{tt}g_{\varphi\varphi}-g_{t\varphi}
^{2}=-\rho^{2}$) that near these points $g_{\varphi\varphi}>0$. Such inequality means that near these points of the axis the coordinate lines of the periodic (azimuth angle) coordinate, being closed lines, are time-like. To avoid such trouble it is necessary to demand that on every part of the axis outside the sources and between them $g_{t\varphi}$ should vanish. As it follows directly from the Einstein - Maxwell equations, on the axis of symmetry $\rho=0$ the value $\Omega =g_{t\varphi}/g_{tt}$ is independent of $z$ and therefore, it is constant. However, this constant can be different on different disconnected parts of the axis separated by the sources. Therefore, to exclude the existence of closed time-like curves near the axis, first of all we should impose two conditions $$\Omega_{-}=\Omega_{i}=\Omega_{+}\text{ }, \label{20}%$$ where $\Omega_{-}$, $\Omega_{i}$ and $\Omega_{+}$ are the constants which are the values of $g_{t\varphi}/g_{tt}$ on the negative, intermediate and positive parts of the axis respectively. If the conditions (\[20\]) are satisfied, the corresponding common constant value of $\Omega$ on the axis can be reduced to zero by a coordinate transformation of the form $t^{\prime}=t+a\varphi$, and $\varphi^{\prime}=\varphi$ with an appropriate constant $a$. In order to satisfy the conditions (\[20\]) we calculate constants $\Omega_{+}-\Omega_{-}$ and $\Omega_{i}-\Omega_{-}$ and put both of them to zero. Calculations show that the first constant take simple form: $$\Omega_{+}-\Omega_{-}=-4B\text{ }, \label{20-0}$$ where $B$ is the total NUT parameter given by the formula (\[Physparameters\]), while the expression for the second parameter is much more complicated: $$\label{Omegai}
\Omega_{i}-\Omega_{-}\equiv-4B-\dfrac{\omega_{{\times}}\overline{\omega
}_{{\times}}}{(a_{{\times}}+i\sigma_{{s}})}+\dfrac{(1+2\delta)}{(1-2\delta
)}\dfrac{\mathcal{H}_{0}\overline{\mathcal{H}}_{0}}{(a_{{\times}}+i\sigma
_{{s}})\mathcal{W}_{o}}.$$ The explicit expression for $\delta$ takes the form: $$\label{delta}
\delta=\dfrac{\sigma_{0}(m_{0}m_{{s}}+b_{0}b_{{s}}-q_{0}q_{s}-\mu_{0}\mu_{s}%
)}{\sigma_{0}(l^{2}-\sigma_{0}^{2}-\sigma_{{s}}^{2}-2a_{0}a_{{s}}%
)+(l\sigma_{0}+\sigma_{0}^{2}-ia_{0}\sigma_{{s}})(l-\sigma_{0}-\sigma_{{s}%
})S_{0}},$$ where $S_{0}$ was defined in (\[S0\]) and $$\omega_{{\times}}=m_{{\times}}-ib_{{\times}}+i(a_{{\times}}+i\sigma_{{s}%
}),\qquad\mathcal{W}_{o}=(l^{2}-\sigma_{0}^{2}-\sigma_{{s}}^{2})^{2}%
-4\sigma_{0}^{2}\sigma_{{s}}^{2}\text{ }, \label{27-1}%$$$$\begin{array}{l}
\mathcal{H}_{0}=-2i(l-\sigma_{{s}}-ia_{0}+m_{0}+ib_{0})\overline{X}_{{\times}
}\\
\phantom{\mathcal{H}_{0}=}+(a_{{\times}}+i\sigma_{{s}}-im_{{\times}}+b_{{\times}})[(l-\sigma_{{s}}%
)^{2}-\sigma_{0}^{2}]+\\
\phantom{\mathcal{H}_{0}=}+2(a_{{\times}}+i\sigma_{{s}})\left[ (m_{0}+ib_{0})(l-\sigma_{{s}}
+ia_{0})+\sigma_{0}^{2}+a_{0}^{2}\right],
\end{array}$$ $$\label{Xtimes}
X_{{\times}}=(m_{0}+ib_{0})(m_{{\times}}-ib_{{\times}})-\overline{e}_{0} e_{\times}.
\qquad\qquad\qquad\qquad\quad$$ In these formulas we used the new parameters denoted by the same letters but with subscript “$\times$”. These new constants are defined by the relations:$$m_{\times}=M-m_{0},\quad b_{\times}=B-b_{0},\quad e_{\times}=Q_{e}+i Q_{m}-e_{0}
\label{20-3}%$$$$a_{\times}+i\sigma_{s}=\frac{\Gamma\bar{\Gamma}\sqrt{c_{0}}}{\left(
a_{s}+i\sigma_{{s}}\right) \sqrt{\mathcal{W}_{o}}}\text{ }, \label{20-4}%$$$$\Gamma=\left( m_{0}+ib_{0}\right) \left( m_{{s}}-ib_{{s}}\right) -\overline{e}_{0} e_{s} -\left( a_{{s}}%
+i\sigma_{{s}}\right) \left( a_{0}-i\sigma_{{s}}-il\right),$$ where parameters $c_{0,}M,B,Q_{e},Q_{m}$ have been defined by the relations (\[c0\]), (\[Physparameters\]).
#### *Conical singularities on the axis.*
[If the conditions (\[20\]) are satisfied, this does not mean yet that the geometry on each part of the axis is regular since at the points of different parts of the axis the local Euclidness of spatial geometry still may occur to be violated. This behaviour of geometry on the sections $z=const$ looks like on the surface of a cone near its vortex, where the ratio of the length of a circle (surrounding the vortex) to its radius is not equal to $2\pi$. In the solution, on any surface $z=const$ intersecting the axis, the length and radius of the circles $\rho=const$ are represented asymptotically for $\rho\rightarrow0$ by the expressions $L=2\pi\sqrt{-g_{\varphi\varphi}}$ and $R=\sqrt{f}\rho$ respectively. Besides that we note that the local Euclidness of the geometry on the parts of the axis of symmetry outside the sources, besides the condition $g_{t\varphi}\to 0$ implies for $\rho\to 0$ even more stronger condition $g_{t\varphi}=O(\rho^2)$. Using this last condition and the mentioned above relation for metric coefficients $g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}=-\rho^{2}$, we obtain that for $\rho\rightarrow0$, the condition $L/(2\pi R)\rightarrow1$ is equivalent to the constraints: $$\label{PmPiPp}
P_{-}=P_{i}=P_{+}=1,\qquad P\equiv fg_{tt}\text{ },$$ where $P_{-}$, $P_{i}$ and $P_{+}$ are the values of the product $fg_{tt}$ respectively on the negative, intermediate and positive parts of the axis. (In accordance with the Einstein - Maxwell equations, the product $fg_{tt}$ is constant on a part of the axis where ]{}$g_{t\varphi}=0$[, however, these constants again may occur to be different for different disconnected parts of the axis.) ]{} To obtain the constrains on the parameters of our solution implied by (\[PmPiPp\]), we have to analyze the behaviour of metric components on different parts of the axis of symmetry.
$\underline{\text{\textit{Negative semi-infinite part of the axis:}
} \{\rho=0,\,-\infty<z<z_{1}-\sigma_{0}\}}$. On this part of the axis for bipolar coordinates we have the expressions $$x_{1}=z_{1}-z,\quad x_{2}=l+z_{1}-z,\quad y_{1}=y_{2}=-1 \label{22}%$$ and the metric component $g_{tt}$ and the conformal factor $f$ take the values $$\label{gttmfm}
\begin{array}{l}
g_{tt} =\dfrac{[(z-z_{1})^{2}-\sigma_{0}^{2}][(z-z_{1}-l)^{2}-\sigma_{{s}%
}^{2}]}{c_{0}D_{-}\overline{D}_{-}},\\
\text{ }f =\dfrac{c_{0}D_{-}\overline{D}_{-}}{[(z-z_{1})^{2}-\sigma
_{0}^{2}][(z-z_{1}-l)^{2}-\sigma_{{s}}^{2}]},
\end{array}$$ where $c_{0}$ has been defined in (\[c0\]) and $D_{-}$ denotes the value of $D$ (defined by (\[Dvalue\])) on the negative semi-infinite part of the axis. These expressions show that on this part of the axis the corresponding condition from (\[PmPiPp\]), i.e. $P_-=1$ is satisfied automatically.
$\underline{\text{\textit{Positive semi-infinite part of the axis:}} \{\rho=0,\,z_{1}+l<z<\infty\}}$. On this part of axis the bipolar coordinates possess the expressions $$x_{1}=z-z_{1},\quad x_{2}=z-z_{1}-l,\quad y_{1}=y_{2}=1 \label{24}%$$ and here for the metric component $g_{tt}$ and for the conformal factor we have $$\label{gttpfp}
\begin{array}{l}
g_{tt} =\dfrac{[(z-z_{1})^{2}-\sigma_{0}^{2}][(z-z_{1}-l)^{2}-\sigma_{{s}
}^{2}]}{c_{0}D_{+}\overline{D}_{+}},\\
f =\dfrac{c_{0}D_{+}\overline{D}_{+}}{[(z-z_{1})^{2}-\sigma_{0}
^{2}][(z-z_{1}-l)^{2}-\sigma_{{s}}^{2}]}\text{ },
\end{array}$$ where $D_{+}$ denotes the value of $D$ on the positive semi-infinite part of the axis. From these expressions we see that the corresponding condition from (\[PmPiPp\]) is satisfied identically, i.e. $P_+=1$ on the positive semi-infinite part of the axis.
$\underline{\text{\textit{Intermediate part of the axis:}}
\{\rho=0,\,z_{1}+\sigma_{0}<z<z_{1}+l\}}$. At these points for bipolar coordinates we have the expressions $$x_{1}=z-z_{1},\quad x_{2}=l+z_{1}-z,\quad y_{1}=1,\quad y_{2}=-1, \label{25}%$$ and the corresponding expressions for $g_{tt}$ and the conformal factor $f$ are $$\begin{aligned}
g_{tt} & =\dfrac{[(z-z_{1})^{2}-\sigma_{0}^{2}][(z-z_{1}-l)^{2}-\sigma_{{s}%
}^{2}]}{c_{0}D_{i}\overline{D}_{i}}\left( \dfrac{1-2\delta}{1+2\delta
}\right) ^{2},\text{ \ }\label{26}\\
f & =\dfrac{c_{0}D_{i}\overline{D}_{i}}{[(z-z_{1})^{2}-\sigma_{0}%
^{2}][(z-z_{1}-l)^{2}-\sigma_{{s}}^{2}]}\text{ },\nonumber\end{aligned}$$ where $\delta$ is the same as defined already by the expression (\[delta\]). As it follows from these expressions, on the intermediate part of the axis the corresponding condition from (\[PmPiPp\]), i.e. the equation $P_i=1$ is equivalent to the constraint $$\label{Zerodelta}
\delta=0.$$
#### *Magnetic and electric charges of the sources.*
To obtain more realistic configurations, we have to exclude from the solution the total magnetic charge, i.e. to set $Q_m=0$ (see the expressions (\[Physparameters\])) as well as the magnetic charges of both sources. To calculate the physical values of magnetic charges we should consider the magnetic fluxes coming through closed space-like surfaces surrounding each charged center and apply the Gauss theorem. In this way we can find the physical magnetic charges $\mu_{0},\mu_{s}$ (as well as physical electric charges $q_{0},q_{s}$ ) of each source calculating the corresponding Komar-like integrals. The detailed procedure how to do this have been described in the section “Physical parameters of the sources” in paper [@AB2]. The results of these calculations are: $$\label{q0mu0}
q_{0}=\text{Re} (e_{0})+\operatorname{Re}F\text{ },\quad\mu_{0}=\operatorname{Im} (e_{0})+\operatorname{Im}F,$$ $$q_{s}=\operatorname{Re}(e_{\times})-\operatorname{Re}F,\quad\mu_{s}
=\operatorname{Im}(e_{\times})-\operatorname{Im}F\text{ },$$ where $$F=e_{\times}\dfrac{(a_{{\times}}+i\sigma_{{s}%
}-im_{{\times}}+b_{{\times}})}{2(a_{{\times}}+i\sigma_{{s}})}-\dfrac
{\mathcal{L}_{0}\mathcal{H}_{0}}{2\mathcal{W}_{o}(a_{{\times}}+i\sigma_{{s}}%
)}\dfrac{(1+2\delta)}{(1-2\delta)}\text{ }, \label{30}%$$ and we introduced here the new parameter polynomial: $$\label{L0}
\mathcal{L}_{0} =\left[ (l+\sigma_{{s}})^{2}-\sigma_{0}^{2}\right] e_{\times}
+2\left[ X_{{\times}}-i(a_{{\times}}+i\sigma_{{s}})(l+\sigma_{{s}}%
-ia_{0})\right] e_{0},$$ where $ X_{{\times}}$ was defined in (\[Xtimes\]). The physically acceptable solution should satisfy $$\label{mu0mus}
\mu_{0}=\mu_{s}=0.$$ These conditions mean that the physical magnetic charges of both sources vanish. It is useful to note here also that the total magnetic charge of the configuration $Q_m=\operatorname{Im} e_{0}+\operatorname{Im}e_{\times}=\mu_{0}+\mu_{s}$ and therefore, the conditions (\[mu0mus\]) is sufficient for the condition $Q_m=0$ would be satisfied.
#### Summary for the equilibrium conditions.
Here we present a list of the constraints on the parameters of our solution providing a physical equilibrium of two interacting Kerr-Newman sources – a Kerr-Newman black hole and a Kerr-Newman naked singulariry in their common gravitational and electromagnetic fields. Choosing the multiplier $c_0$ in $f$ as it was determined in (\[c0\]), we obtain an eleven-parameter solution which parameters for equilibrium configurations should satisfy the following five constraints (equilibrium conditions):
- [$\underline{\text{\textit{Absence of closed timelike curves on semi-infinite parts of the axis.}}}$ This\
equilibrium conditions is equivalent to the vanishing of a NUT parameter $B$, and it implies in accordance with (\[Physparameters\]) the constraint: $$\text{Im}\left[ \dfrac{(m_{0}-ib_{0})(1+K_{1}+L_{1}+S_{0})+(m_{{s}}-ib_{{s}%
})(1+K_{2}+L_{2})}{1+S_{0}-(K_{1}+L_{1})(K_{2}+L_{2})}\right] =0 \label{Eq1}%$$]{}
- [$\underline{\text{\textit{Absence of closed timelike curves on the axis between the sources.}}}$ From\
(\[Omegai\]), with the condition $B=0$ taken into account, we obtain $$\mathcal{W}_{o}\,\omega_{{\times}}\overline{\omega
}_{{\times}}=\mathcal{H}_{0}\overline{\mathcal{H}}_{0}\,\dfrac{(1+2\delta)}{(1-2\delta
)}$$]{} This simplifies obviously, if we use here the constraint $\delta=0$ (see below)
- [$\underline{\text{\textit{Absence of conical points on the axis of symmetry.}}}$ The choice (\[c0\]) of the constant $c_0$ provides the absence of conical points on both semi-infinite parts of the axis. The cosntraint $\delta=0$ provides the absence of conical points on the part of the axis between the sources. This constraint, due to (\[delta\]), can be presented in the form $$m_{0}m_{{s}}+b_{0}b_{{s}}=\operatorname{Re}(e_{0}\overline{e}_{s})$$]{}
- [$\underline{\text{\textit{Vanishing of magnetic charges of each of the sources.}}}$ To eliminate the physical magnetic charges $\mu_{0}, \mu_{s}$ of the sources, using (\[q0mu0\])-(\[L0\]) and bearing in mind $\delta=0$, we obtain the following two conditions: $$\label{magcharges}
\begin{array}{l}
\operatorname{Im}(e_{0})+\operatorname{Im}\left[e_{\times}
\dfrac{(a_{{\times}}+i\sigma_{{s}}-im_{{\times}}+b_{{\times}})}{2(a_{{\times}
}+i\sigma_{{s}})}-\dfrac{\mathcal{L}_{0}\mathcal{H}_{0}}{2\mathcal{W}_{o} (a_{{\times}}+i\sigma_{{s}})}\right] =0\\[2ex]
\operatorname{Im}(e_{\times})-\operatorname{Im}\left[e_{\times}
\dfrac{(a_{{\times}}+i\sigma_{{s}}-im_{{\times}}+b_{{\times}})}{2(a_{{\times}
}+i\sigma_{{s}})}-\dfrac{\mathcal{L}_{0}\mathcal{H}_{0}}{2\mathcal{W}_{o} (a_{{\times}}+i\sigma_{{s}})}\right] =0
\end{array}$$]{}
All these constraints possess rather complicated forms which do not allow to expect their simple and explicit solution in exact form. Because of that we consider the solution of these constraints using a supplementary assumption that the distance $l$ separating the sources is rather large.
Black hole–naked singularity system with\
large separating distance
=========================================
The existence and properties of equilibrium configurations of two interacting Kerr - Newman sources can be studied effectively if we use some rather natural assumption that the $z$-distance $l$ which separates a black hole and a naked singularity is large enough in comparison with the linear sizes which characterize the sources. Namely, we assume that the parameters (\[Parameters\]) satisfy the condition: $$\label{Assumption}
d\equiv\max\{\vert m_{\scriptscriptstyle{0}}\vert,\,\vert a_{\scriptscriptstyle{0}}\vert,\, \vert b_{\scriptscriptstyle{0}}\vert,\,\vert e_{\scriptscriptstyle{0}}\vert, \vert m_{s}\vert,\,\vert a_{s}\vert,\,\vert b_{s}\vert,\,\vert e_{s}\vert\} \ll l$$
#### *On the absence in the solution (\[ErnstPotentials\]) – (\[factorf\]) of other sources besides Kerr - Newman black hole and Kerr - Newman naked singularity.*
We construct our solution generating a one soliton of the Kerr-Newman black hole backgound. By a construction, this solution (at list in the case of rather large z-distance separating the black hole and the location of the soliton (Kerr-Newman naked singularity) should not include other singularities which could be additional sources of gravitational and electromagnetic fields in this solution. To exhibit the absence of other curvature singularities (on the axis of symmetry or outside it) more clearly, we construct asymptotic representations of our solution in three space-time domains which, being considered together, cover the whole space-time region outside the two Kerr-Newman sources. In all three series expansions the same small parameter is used (see (\[Assumption\]))) $$\varepsilon=\dfrac{d}{l}\ll 1$$ and we consider a few first terms of the corresponding series expansions. However, we do not prove here the convergency of these $\varepsilon$-series because it could be rather tedious and it is not in the scope of our paper. However, it seems “highly likely” that these series expansions converge, because for large separating distance, one of these expansions represents a post-Newtonian approximations of different orders of the fields of one of the sources perturbed by the field of another source, while the other expansions describe small perturbations of the nearest zone of each of the sources by the external field of the other source.[^4]
To construct the asymptotic representations of our solution, we introduce two functions which are the “Eucledean” distances on the plane of Weyl coordinates $(\rho,z)$ from two points $(\rho=0,z=z_1)$ and $(\rho=0,z=z_2=z_1+l)$: $$R_1=\sqrt{\rho^2+(z-z_1)^2},\qquad R_2=\sqrt{\rho^2+(z-z_1-l)^2}.$$ The asymptotic representations of our solution we consider in three domains $$(S0):\hskip2ex R_1\gg d\hskip1ex\text{and}\hskip1ex R_2\gg d,\qquad
(S1):\hskip2ex R_1\ll l,\qquad (S2):\hskip2ex R_2\ll l$$ It is clear that pairs $(S_0, S_1)$ and $(S_0, S_2)$ of these domains overlap in the regions $$\hbox{$(S0\cap S1):\hskip1ex d\ll R_1\ll l$}\quad\qquad\text{and}\quad\qquad
\hbox{$(S0\cap S2):\hskip1ex d\ll R_2\ll l$}$$ and therefore, the regularity of asymptotic representations of our solution in the domains $S_0$, $S_1$ and $S_2$ will indicate the absence of any curvature singularities in the entire space-time region outside two Kerr-Newman sources which superposition of gravitational and electromagnetic fields is described by our solution.
#### *Asymptotic structure of the solution in the domain $S_0$*.
In this domain, i.e. at large distances from each of the sources (in comparison with their typical “sizes” $\sim d$), we use an approximation in spirit of Bonnor [@Bonnor:2001], in which we introduce (unlike Bonnor’s paper), a small parameter $\varepsilon$ explicitly: $$\label{epsilon}
\left.\begin{array}{lclclcl}
m_0=\varepsilon \widehat{m}_0& a_0=\varepsilon \widehat{a}_0& b_0=\varepsilon \widehat{b}_0& e_0=\varepsilon \widehat{e}_0\\
m_s=\varepsilon \widehat{m}_s& a_s=\varepsilon \widehat{a}_s& b_s=\varepsilon \widehat{b}_s& e_s=\varepsilon \widehat{e}_s
\end{array}\quad\right\Vert\quad \varepsilon=\dfrac{d}{l} \ll 1,$$ where, in accordance with our basic assumption (\[Assumption\]), the parameter $d\ll l$. The parameter $l$, all new constant parameters with hats as well as the distances $R_1$ and $R_2$ also take some finite values. In this domain, the Weyl coordinates $(\rho,z)$ can be expressed in terms of $R_1$ and $R_2$, and for bipolar coordinates we obtain $$\begin{array}{lcl}
x_1=R_1+\dfrac{\rho^2\widehat{\sigma}_0^2}{2 R_1^3}\varepsilon^2+O(\varepsilon^4)&&
y_1=\dfrac{z-z_1}{R_1}-\dfrac{\rho^2(z-z_1)\widehat{\sigma}_0^2}{2 R_1^5}\varepsilon^2+O(\varepsilon^4)\\[2ex]
x_2=R_2+\dfrac{\rho^2\widehat{\sigma}_s^2}{2 R_2^3}\varepsilon^2+O(\varepsilon^4)&&
y_2=\dfrac{z-z_2}{R_2}-\dfrac{\rho^2(z-z_2)\widehat{\sigma}_s^2}{2 R_2^5}\varepsilon^2+O(\varepsilon^4)
\end{array}$$ Substituting these expansions together with (\[epsilon\]) into (\[ErnstPotentials\])–(\[Ys\]), we obtain the following asymptotic representation for the Ernst potentials of our solution: $$\begin{array}{l}\label{EFepsilon}
\mathcal{E}=1-2\left(\dfrac{\widehat{m}_0-i \widehat{b}_0}{R_1}+
\dfrac{\widehat{m}_s-i \widehat{b}_s}{R_2}\right)\varepsilon+\dfrac{\mathcal{A}(R_1,R_2)}{l (\widehat{a}_0^2+\widehat{\sigma}_0^2)(\widehat{a}_s+i\widehat{\sigma}_s) \widehat{\sigma}_s R_1^3 R_2^3}\, \varepsilon^2+O(\varepsilon^3),\\[3ex]
\Phi=\left(\dfrac{\widehat{e}_0}{R_1}+
\dfrac{\widehat{e}_s}{R_2}\right)\varepsilon+\dfrac{\mathcal{B}(R_1,R_2)}{l (\widehat{a}_0^2+\widehat{\sigma}_0^2)(\widehat{a}_s+i\widehat{\sigma}_s) \widehat{\sigma}_s R_1^3 R_2^3}\, \varepsilon^2+O(\varepsilon^3),
\end{array}$$ where $\mathcal{A}(R_1,R_2)$ and $\mathcal{B}(R_1,R_2)$ are polynomials of the parameters with hats and of the distances $R_1$ and $R_2$. It is easy to check that coefficients in higher terms of these asymptotics possess the similar structures. The regular structures of the coefficients of asymptotic representations (\[EFepsilon\]) indicates obviously the absence of any singularities of the Ernst potentials outside the $\varepsilon$-vicinities of the points $R_1=0$ and $R_2=0$, i.e. outside the small regions surrounding the sources – a black hole and a naked singularity of Kerr - Newman types.
#### *Asymptotic structure of the solution near the black hole (domain $S_1$)*
Near the black hole, we consider the domain with Weyl coordinates such that $$\label{asymp1}
R_1\ll l, \qquad\quad\Longrightarrow\quad\qquad
\rho\ll l\quad \text{and}\quad \vert z-z_1\vert \ll l$$ In this domain, the values of $\rho$ and $z-z_1$ are of the order $\sim d$ where $d\ll l$ is a typical “size” of each of the sources (see (\[Assumption\])). Therefore, we can introduce here, instead of (small enough) $\rho$ and $z-z_1$, the dimensionless coordinates with hats which take in this domain the finite velues: $$\label{rozhats}
\rho=\widehat{\rho}\, d\quad \text{and}\quad z-z_1= \widehat{z}\, d$$ Then, in accordance with (\[x2y2\]), for bipolar coordinates $(x_2,y_2)$, we have $$\label{sx2sy2}
\left.\begin{array}{l}
x_2=l \left[1-\widehat{z}\varepsilon+\dfrac{1}{2}\widehat{\rho}{}^2\varepsilon^2+ \dfrac{1}{2}\widehat{\rho}{}^2 \widehat{z}\varepsilon^3+O(\varepsilon^4)\right],\\
y_2=-1+\dfrac{1}{2}\widehat{\rho}{}^2\varepsilon^2+ \widehat{\rho}{}^2 \widehat{z}\,\varepsilon^3+O(\varepsilon^4),
\end{array}\quad\right\Vert\quad \varepsilon=\dfrac{d}{l}\ll 1.$$ It is convenient to return now to the Weyl coordinates using (\[rozhats\]) and express these coordinates in accordance with (\[Weyl\]) in terms of polar coordinates $(x_1,y_1)$ most adapted for the first source. Then we obtain from (\[sx2sy2\]) the series $$\label{x2y2-x1y1}
\begin{array}{l}
x_2=l\left[-\dfrac{x_1 y_1}{d}\,\varepsilon+\dfrac{(x_1^2-\sigma_0^2)(1-y_1^2)}{2 d^2}\varepsilon^2+\dfrac{x_1 y_1(x_1^2-\sigma_0^2)(1-y_1^2)}{2 d^3}\varepsilon^3+O(\varepsilon^4)\right] \\[2ex]
y_2=-1+\dfrac{(x_1^2-\sigma_0^2)(1-y_1^2)}{2 d^2}\,\varepsilon^2+\dfrac{x_1 y_1 (x_1^2-\sigma_0^2)(1-y_1^2)}{d^3}\,\varepsilon^3+O(\varepsilon^4)
\end{array}$$ Suibstituting these series into the Ernst potentials (\[ErnstPotentials\]) of our solution and taking into account the expressions (\[CalR1R2\])–(\[Ys\]), we obtain the following expansions $$\label{Source1}
\begin{array}{l}
\mathcal{E}=\dfrac{x_1-m_0+i b_0+i a_0 y_1}{x_1+m_0-i b_0+i a_0 y_1}+
\dfrac{\mathcal{E}_1(x_1,y_1)}{(a_s+i\sigma_s)(x_1+m_0-i b_0+i a_0 y_1)^2 d} \,\varepsilon+O(\varepsilon^2)\\[2ex]
\Phi=\dfrac{e_0}{x_1+m_0-i b_0+i a_0 y_1}+
\dfrac{\mathcal{F}_1(x_1,y_1)}{(a_s+i\sigma_s)(x_1+m_0-i b_0+i a_0 y_1)^2 d}\,\varepsilon +O(\varepsilon^2)
\end{array}$$ where $\mathcal{E}_1(x_1,y_1)$ and $\mathcal{F}_1(x_1,y_1)$ are quadratic polynomials of coordinates $x_1$ and $y_1$ with coefficients which are polynomial of parameters of the solution: $$\label{E1F1}
\begin{array}{l}
\mathcal{E}_1(x_1,y_1)=2 i \overline{e}_s[e_0(m_s-i b_s)-e_s(m_0-i b_0)](x_1+i a_0 y_1)\\
\phantom{\mathcal{E}_1(x_1,y_1)}+2(m_0-i b_0)(a_s+i\sigma_s)[(m_0-i b_0)(m_s+i b_s)-e_0 \overline{e}_s ]\\
\phantom{\mathcal{E}_1(x_1,y_1)}-2(a_s+i\sigma_s)(x_1+i a_0 y_1)[(m_s-i b_s)(x_1+i a_0 y_1)+2 i a_s(m_0-i b_0)]\\[1ex]
\mathcal{F}_1(x_1,y_1)=e_s(a_s+i\sigma_s)(x_1+m_0-i b_0+i a_0 y_1)^2\\
\phantom{\mathcal{F}_1(x_1,y_1)}+i e_s[(m_0-i b_0)(m_s+i b_s)-e_0 \overline{e}_s](x_1+m_0-i b_0+i a_0 y_1)\\
\phantom{\mathcal{F}_1(x_1,y_1)}-2(a_s+i\sigma_s)(m_0-i b_0)[e_s(x_1+m_0-i b_0+i a_0 y_1)+i e_0(a_s+b_s)]\\
\phantom{\mathcal{F}_1(x_1,y_1)}+[e_0(m_s-i b_s-i a_s+\sigma_s)-e_s(m_0-i b_0)]\\
\phantom{\mathcal{F}_1(x_1,y_1)}\times [i e_0 \overline{e}_s-(a_s+i \sigma_s)(x_1+m_0-i b_0+i a_0 y_1)]
\end{array}$$
In each of the series (\[Source1\]) the first terms ($\sim \varepsilon^0$) correspond exactly to the Ernst potentials of the Kerr-Newman black hole ($\sigma_0^2>0$), while the next terms ($\sim \varepsilon,\varepsilon^2,\ldots$) describe small perturbations of this black hole by a distant Kerr-Newman naked singularity ($\sigma_s^2<0$). The equations (\[Source1\]), (\[E1F1\]) show that this perturbation of the Ernst potentials near the black hole is completely regular.
#### *Asymptotic structure of the solution near the naked singularity (domain $S_2$).*
Similarly to the consideration given just above of the region near the black hole, we consider now the region near the naked singularity (the source 2) which can be described in the Weyl coordinates by the conditions $$\label{asymp2}
R_2\ll l, \qquad\quad\Longrightarrow\quad\qquad
\rho\ll l\quad \text{and}\quad \vert z-z_2\vert \ll l$$ In this domain, in accordance with (\[x1y1\]), for bipolar coordinates, $(x_1,y_1)$ we have $$\label{sx1sy1}
\begin{array}{l}
x_1= l\left[1+\dfrac{z-z_2}{d}\varepsilon+\dfrac{\rho^2}{2 d^2}\varepsilon^2-\dfrac{(z-z_2)\rho^2}{2 d^3}\varepsilon^3+O(\varepsilon^4)\right] \\[1ex]
y_1=1-\dfrac{\rho^2}{2 d^2}\,\varepsilon^2+\dfrac{(z-z_2)\rho^2}{d^3}\,\varepsilon^3+O(\varepsilon^4)
\end{array}$$ Using in (\[sx1sy1\]) the expressions (\[Weyl\]), we obtain $$\label{x1y1-x2y2}
\begin{array}{l}
x_1=l\left[1+\dfrac{x_2 y_2}{d}\varepsilon+\dfrac{(x_2^2-\sigma_s^2)(1-y_2^2)}{2 d^2}\,\varepsilon^2-\dfrac{x_2 y_2(x_2^2-\sigma_s^2)(1-y_2^2)}{2 d^3}\,\varepsilon^3+O(\varepsilon^4)\right] \\[2ex]
y_1=1-\dfrac{(x_2^2-\sigma_s^2)(1-y_2^2)}{2 d^2}\,\varepsilon^2+\dfrac{x_2 y_2 (x_2^2-\sigma_s^2)(1-y_2^2)}{d^3}\,\varepsilon^3+O(\varepsilon^4)
\end{array}$$ Substituting these series into (\[CalR1R2\])–(\[Ys\]), we obtain the following asymptotics for the Ernst potentials of our solution in the neighbourhood of the source 2: $$\label{Source2}
\begin{array}{l}
\mathcal{E}=\dfrac{x_2-m_s+i b_s+i a_s y_2}{x_2+m_s-i b_s+i a_s y_2}+
\dfrac{\mathcal{E}_2(x_2,y_2)}{k_0 (x_2+m_s-i b_s+i a_s y_2)^2 d}\,\varepsilon+O(\varepsilon^2)\\[2ex]
\Phi=\dfrac{e_s}{x_2+m_s-i b_s+i a_s y_2}+
\dfrac{\mathcal{F}_2(x_2,y_2)}{k_0 (x_2+m_s-i b_s+i a_s y_2)^2 d}\,\varepsilon+O(\varepsilon^2)
\end{array}$$ where $k_0=\sigma_s (a_s+i\sigma_s)(a_0^2+\sigma_0^2)$ and the coefficients $\mathcal{E}_2(x_2,y_2)$ and $\mathcal{F}_2(x_2,y_2)$ are quadratic polynomials of coordinates $x_2$ and $y_2$ with coefficients which are polynomials of parameters of the solution $m_0$,$m_s$,$a_0$,$a_s$,$b_0$,$b_s$,$e_0$,$e_s$ and $\sigma_0$,$\sigma_s$.
In the series (\[Source2\]) the first terms ($\sim \varepsilon^0$) correspond to a Kerr-Newman naked singularity ($\sigma_s^2<0$), while the subsequent terms ($\sim \varepsilon, \varepsilon^2,\ldots$) describe small perturbations of the gravitational and electromagnetic fields of this naked singularity by a distant Kerr-Newman black hole. The explicit expressions for $\mathcal{E}_2(x_2,y_2)$ and $\mathcal{F}_2(x_2,y_2)$ are very similar to the expressions for the coefficients (\[E1F1\]), but these are more longer and we do not present them here. However, it is important that like the expressions (\[E1F1\]), these expressions show that the rather distant Kerr-Newman black hole (Source 1) produces a regular perturbation in the vicinity of the “critical surface” (i.e. in the region $x_2 \ge 0$) surrounding the Kerr-Newman naked singularity (Source 2).
Thus, in the above we had shown that, at least for rather large values of the coordinate $z$-distance $l$ separating the black hole and the naked singularity, the asymptotical behaviour of the Ernst potentials is regular everywhere outside the black hole horizon ($x_1\ge \sigma_0$) and “critical surface” of the naked singularity ($x_2\ge 0$) and therefore, there are no any additional singularity can be expected in this solution besides two Kerr-Newman sources mentioned above.
#### *Solution of equilibrium conditions (\[Eq1\])-(\[magcharges\])*
In this section, we describe an asymptotic construction of solution of equilibrium conditions for our black hole - naked singularity system described by the solution (\[ErnstPotentials\]) - (\[factorf\]) of Einstein - Maxwell equations. Our construction is based on the main assumption (\[Assumption\]) but we use now more detail asymptotic representation of the parameters of the solution as power series in terms of a small parameter $\varepsilon \ll 1$: $$\label{epsseries}
\begin{array}{ll}
m_0=m_{0\cdot}\varepsilon+m_{0\cdot\cdot}\varepsilon^2+ m_{0\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4),&\hskip-1ex
m_s=m_{s\cdot}\varepsilon+m_{s\cdot\cdot}\varepsilon^2+ m_{s\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4)\\[1ex]
b_0=b_{0\cdot}\varepsilon+b_{0\cdot\cdot}\varepsilon^2+ b_{0\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4),&\hskip-1ex
b_s=m_{s\cdot}\varepsilon+b_{s\cdot\cdot}\varepsilon^2+ b_{s\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4)\\[1ex]
a_0=a_{0\cdot}\varepsilon+a_{0\cdot\cdot}\varepsilon^2+ a_{0\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4),&\hskip-1ex
a_s=a_{s\cdot}\varepsilon+a_{s\cdot\cdot}\varepsilon^2+ a_{s\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4)\\[1ex]
e_0=e_{0\cdot}\varepsilon+e_{0\cdot\cdot}\varepsilon^2+ e_{0\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4),&\hskip-1ex
e_s=e_{s\cdot}\varepsilon+e_{s\cdot\cdot}\varepsilon^2+ e_{s\cdot\cdot\cdot}\varepsilon^3+O(\varepsilon^4)
\end{array}$$ where the number of dots in the suffices of coefficients characterizes the order of approximation. The z-distance $l$ is considered as given parameter of some finite value. Substitution of these expansions into the equilibrium conditions (\[Eq1\])-(\[magcharges\]) leads to the following leading order equations: $$\begin{array}{lcl}
\text{(I)}\quad b_{0\cdot}+b_{s\cdot}=0,&&\text{(II)}\quad b_{s\cdot}=0\\[1ex]
\text{(III)}\quad
m_{0\cdot}m_{s\cdot}+b_{0\cdot}b_{s\cdot}-\text{Re}(e_{0\cdot} \overline{e}_{s\cdot})=0&&
\text{(IV)}\quad \text{Im}(e_{0\cdot})= \text{Im}(e_{s\cdot})=0
\end{array}$$ It is easy to see from these leading order equations that in this approximation we obtain the equilibrium configurations which depend on seven real parameters $m_{0\cdot}$, $a_{0\cdot}$, $e_{0\cdot}$, $m_{s\cdot}$, $a_{s\cdot}$, $e_{s\cdot}$ and $l$, which should satisfy the only relation $$m_{0\cdot}m_{s\cdot}-e_{0\cdot} e_{s\cdot}=0.$$ To conclude our present consideration, we note that in the leading approximation, the parameters mentioned just above represent respectively the physical parameters of the system – masses, angular momentums per unit masses and electric charges of the sources respectively. In this case, the total mass, electric charge and angular momentum of equilibrium configuration are $$M_{tot\cdot}= m_{0\cdot}+m_{s\cdot},\quad
Q_{tot\cdot}= e_{0\cdot}+e_{s\cdot},\quad
J_{tot\cdot}=m_{0\cdot}a_{0\cdot}+m_{s\cdot}a_{s\cdot}$$ However, in the subsequent approximations of the field of interacting sources these parameters loose their “individual” physical interpretations. (Just this phenomenon was observed (in the exact form) in our paper [@AB3] where the static limit of the field of two Kerr - Newman sources was considered.)
The equations of the subsequent orders which arise from the equilibrium conditions (\[Eq1\])-(\[magcharges\]), allow to calculate subsequently the other coefficients of the expansions (\[epsseries\]). This leads to more precise description of equilibrium configurations of two Kerr - Newman sources under consideration and of influence of their interaction on the physical paremeters of each of these sources. However, the detail analysis of higher orders of the used approximation is not in the scope of the present paper, in which our purpose was to present the corresponding exact solution and to show the existence of six-parametric family of equilibrium configurations. More detail description of this family of equilibrium configurations as well as consideration of superposition of fields of other types of Kerr-Newman sources we are going to postpone for a future work.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank the Referee and the Editor respectively for useful questions and comments, which urged us to clarify some points and to include into the manuscript more detail description of the structure of our solution.
The work of GAA was supported in parts by the Russian Foundation for Basic Research (grant 18-01-00273 a).
VAB would like to express his gratitude to the Max-Planck-Institute for Gravitational Physics(Albert Einstein Institute) at Golm (Germany) for the fruitful collaboration, hospitality and financial support.
[99]{} W. B. Bonnor “The equilibrium of a charged test particle in the field of a spherical charged mass in general relativity”, Class. Quant. Grav., **10**, 2077 (1993).
G. P. Perry and F. I. Cooperstock “Electrostatic equilibrium of two spherical charged masses in general relativity”, Class. Quant. Grav., **14**, 1329 (1997); arXiv:gr-qc/9611066.
G. A. Alekseev and V. A. Belinski “Equilibrium configurations of two charged masses in General Relativity”, Phys.Rev*.* **D76,** 021501(R) (2007); arXiv:gr-qc/0706.1981.
G. A. Alekseev and V. A. Belinski “Superposition of Fields of Two Reissner-Nordstrom Sources”, in Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity (Berlin, Germany 2007), Part A: Plenary and Review talks, eds. H. Kleinert, R.T. Jantzen and R. Ruffini (World Scientific, Singapore, 2008), p. 543; arXiv:gr-qc/0710.2515.
V. Belinski and V. Zakharov “Integration of the Einstein Equations by means of the inverse scattering problem technique and construction of exact soliton solutions”, Sov. Phys. JETP, **48,** 985 (1978).
V. Belinski and V. Zakharov “Stationary gravitational solitons with axial symmetry”, Sov. Phys. JETP **50,** 1 (1979).
G. A. Alekseev “N-soliton solutions of Einstein-Maxwell equations”, Pis’ma Zh. Eksp. Teor. Fiz. **32**, 301 (1980); English transl. JETP Lett. **32,** 277 (1981).
G. A. Alekseev “Exact Solutions in the General Theory of Relativity” *Trudy Metem. Inst. Steklova*, **176,** 211 (1987); English transl. *Proceedings of the Steklov Institute of Mathematics* (American Mathematical Society, Providence, Rhode Island), issue 3 of 4 p. 215 (1988).
V.Belinski and E.Verdaguer “Gravitational Solitons”, Cambridge University Press (2001).
G. Alekseev and V. Belinski Soliton Nature of Equilibrium State of Two Charged Masses in General Relativity, International Journal of Modern Physics, Conference Series, vol.12, p.10 (2012); arXiv:1103.0582 \[gr-qc\].
G. A. Alekseev “Twelve-parametric electrovacuum two-soliton solution - the external field of two interacting Kerr-Newman sources”, in *Abstracts of contributed papers 11th International Conference on General Relativity and Gravitation*, Vol. **1** (Stockholm, Sweeden, 1986), p. 227.
W.B. Bonnor, “The interactions of charged, spinning, magnetized masses”, Class. Quantum Grav. [**18**]{}, 2853-2863 (2001)
[^1]: In our pesent consideration we use the condition that the total magnetic charge as well as the magnetic charges of each of the sources are absent. However, if one is interested in consideration of the equilibrium configurations with magnetic charges also, it is easy to exclude from our equilibrium conditions the corresponding conditions for their absence, because such magnetic charges in the solution (\[ErnstPotentials\]) – (\[c0\]) are present initially.
[^2]: More details of derivation of this solution an interested reader can find in the paper [@AB2].
[^3]: We said “in general” because it can be shown that the ISM considered in [@A1] can be adjust also to the real $w$-poles but only for that special restriction on the parameters of the solutions which correspond to the extreme black holes.
[^4]: It seems rather natural to expect that the leading terms of the asymptotic representations of our solution, constructed below, for small enough $\varepsilon$ describe correctly the asymptotic behaviour of the solution in the corresponding overlapping space-time domains. However, if one has a sharp desire to prove the convergency of these series, we can mention some (presumably helpful) idea of this proof. Namely, our $\varepsilon$-series possess the forms $\sum a_n \varepsilon^n$, where each coefficient $a_n$ is a sum of terms of finite values which number grows with $n$. Therefore, for convergency of the series it is enough, for example, that the sums representing the coefficients $a_n$ would consist of the terms which number grows with $n$ not so fast as fixed power $k$ of $n$, i.e. if for all $n$ we have $a_n\lesssim n^k$ with some $k>0$. Then we can construct a majorant series which sum is the known Polylogarithm $(-k,\varepsilon)$ function. This function takes finite values for $0<\varepsilon<1$ and this proves a regular convergency of the corresponding series $\sum a_n \varepsilon^n$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We give sharp limiting case Hardy inequalities on the sphere $\mathbb{S}^{2}$ and show that their optimal constants are unattainable by any $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$. The singularity of the problem is related to the geodesic distance from a point on the sphere.'
address: |
Mathematics Department, Faculty of Science, Assiut University, Assiut 71516 - Egypt\
Email: [email protected]
author:
- 'Ahmed A. Abdelhakim'
bibliography:
- '<your-bib-database>.bib'
title: Limiting case Hardy inequalities on the sphere
---
critical Hardy inequality ,sharp constant ,2-sphere,Sobolev spaces
Introduction
============
The classical Hardy inequality $$\label{ie1}
\int_{\mathbb{R}^n} |\nabla u|^{2} dx
\geq \frac{(n-2)^{2}}{4}\int_{\mathbb{R}^n} \frac{u^{2}}{|x|^2} dx$$ is valid in dimensions $n\geq 3$ for all functions $u\in H^{1}\left(\mathbb{R}^{n}\right)$ ([@Balinsky]). It obviously fails on $\mathbb{R}^{2}$ as the right hand side of (\[ie1\]) no longer makes sense. In order to obtain a version of (\[ie1\]) in the critical case $n=2$ on bounded domains, a logarithmic weight can be introduced to tame the singularity. In [@Ioku1; @Ioku2; @Machihara; @Ruzhansky; @Sano; @Takahashi1; @Takahashi2], for instance, inequalities of the type $$\int_{B} |\nabla u|^{n} dx
\geq C_{n}(\Omega)\int_{B} \frac{|u|^{n}}{|x|^n
\left(\log{\frac{1}{|x|}}\right)^{n}} dx$$ were analysed for $u \in W_{0}^{1,n}(B)$ where $B$ is the unit ball in $\mathbb{R}^{n}$.\
Let $n\geq 3$ and $\,\mathbb{S}^{n}$ be the unit sphere equipped with its Lebesgue surface measure $\sigma_{n}$ in $\mathbb{R}^{n+1}$. Denote by $d(.,p):\mathbb{S}^{n}\rightarrow [0,\pi]$ the geodesic distance from $p\in \mathbb{S}^{n}$, and by $\nabla_{\mathbb{S}^{2}}$ the gradient on $\mathbb{S}^{n}$. Recently, Xiao [@Xiao] proved that if $f\in C^{\infty}\left(\mathbb{S}^{2}\right)$ then $$\label{xiao}
{\bar{c}_{n}}\int_{\mathbb{S}^{n}} f^{2} d \sigma_{n}+
\int_{\mathbb{S}^{n}} |\nabla_{\mathbb{S}^{2}} f|^{2}
d\sigma_{n} \geq c_{n}^{2}
\int_{\mathbb{S}^{n}}
\left(\frac{f^{2}}{d\left(x,p\right)^{2}}+
\frac{f^{2}}{\left(\pi-d(x,p)\right)^{2}} \right) d \sigma_{n}$$ with $ \bar{c}_{n}
=\left(\frac{2}{3}+\frac{1}{\pi^2}\right)
c_{n}^{2}+c_{n},\,$ $c_{n}=\frac{n-2}{2}.$ It was also shown in [@Xiao] that the constant $c_{n}$ in (\[xiao\]) is sharp in the sense that $$c_{n}^{2}=
\inf_{f\in C^{\infty}\left(\mathbb{S}^{n}\right)\setminus\{0\}}
\frac{D_{n}(f)}
{\int_{\mathbb{S}^{n}}
\frac{f^{2}}{d\left(x,p\right)^{2}}d \sigma_{n}} \,=\,
\inf_{f\in C^{\infty}\left(\mathbb{S}^{n}\right)\setminus\{0\}}
\frac{D_{n}(f)}
{\int_{\mathbb{S}^{n}}
\frac{f^{2}}{\left(\pi-d(x,p)\right)^{2}} d \sigma_{n}}$$ where $$D_{n}(f):={c_{n}}\int_{\mathbb{S}^{n}} f^{2} d \sigma_{n}+
\int_{\mathbb{S}^{n}} |\nabla_{\mathbb{S}^{2}} f|^{2}
d\sigma_{n},\quad f\in C^{\infty}\left(\mathbb{S}^{n}\right).$$ We prove $L^{2}$ Hardy inequalities with optimal constants on the sphere $\mathbb{S}^{2}$ in $\mathbb{R}^{3}$. This is a critical exponent case as the integral $\int_{\mathbb{S}^{2}}{\theta^{-1+\lambda}}
{d\sigma_{2}}$, where $\theta$ is the polar angle, diverges for $\lambda \leq -1$. We also argue the lack of maximizers for our inequalities. Our approach denies the possibility of an equality in Xiao’s inequality (\[xiao\]) as well.
Preliminaries
=============
A point on the sphere $\mathbb{S}^{2}$ will have the standard spherical coordinate parametrization $\left(\sin{\theta}\cos{\varphi},\sin{\theta}\sin{\varphi},
\cos{\theta}\right)$ where $\theta \in [0,\pi]$ refers to the polar angle and $\varphi\in [0,2\pi[$ is the azimuthal angle. Then the surface measure induced by the Lebesgue measure on $\mathbb{R}^3$ is $d\sigma_{2}=\sin{\theta} d\theta d\varphi$, the gradient and the Laplace-Beltrami operator, respectively, are given by $$\nabla_{\mathbb{S}^{2}}=
\hat{\theta}\,\frac{\partial}{\partial \theta}+\hat{\varphi}\,\frac{1}{\sin{\theta}}\frac{\partial}{\partial \varphi},\;
\Delta_{\mathbb{S}^{2}}=\frac{1}{\sin{\theta}}
\frac{\partial}{\partial \theta}\left(\sin{\theta}\,\frac{\partial}{\partial \theta}\right)+\frac{1}{\sin^{2}{\theta}}
\frac{\partial^{2}}{\partial \varphi^{2}}.$$ The Sobolev space $H^{1}\left(\mathbb{S}^{2}\right)$ is the completion of $C^{\infty}\left(\mathbb{S}^{2}\right)$ in the norm $$\parallel f\parallel_{H^{1}\left(\mathbb{S}^{2}\right)}:=
\left(\parallel f\parallel^{2}_{L^{2}\left(\mathbb{S}^{2}\right)}+
\parallel \nabla f \parallel^{2}_{L^{2}\left(\mathbb{S}^{2}\right)}
\right)^{\frac{1}{2}}.$$ In order to find the geodesic distance $d(x,p)$ from a point $x\in \mathbb{S}^{2}$ to a given a point $p\in \mathbb{S}^{2}$, we rotate the axes, if necessary, to put $p$ on the zenith direction then place the great circle passing through $p$ and $x$ in the azimuth reference direction so that we have $d(x,p)=\theta$.\
For simplicity, we henceforth denote $d\sigma_{2}$, $\nabla_{\mathbb{S}^{2}}$ and $\Delta_{\mathbb{S}^{2}}$ by $d\sigma$, $\nabla$ and $\Delta$, respectively.
Main results
============
Let $\phi:[0,\pi]\rightarrow [1,\infty[$ be defined by $\phi(t):=\log\left(\pi e/t\right),\,$ $\psi:[0,\pi]\rightarrow [1+\log{\pi},\infty[$ be such that $\psi(t):=\phi\left(\sin{t}\right),$ and $\,\rho_{\phi}(t):=t\phi(t)$. Let $A>0$. Denote by $S,$ $\,T_{A},$ and $\,Q\left(.;\phi\right)$ the positive nonlinear functionals on $H^{1}\left(\mathbb{S}^{2}\right)$ given by $$\begin{aligned}
S(f):=&\int_{\mathbb{S}^{2}}
|\hat{\theta}.\nabla f|^{2} d\sigma+\frac{1}{2\pi^{2}}\int_{\mathbb{S}^{2}}
f^{2}\,d\sigma,\\
T_{A}(f):=&\int_{\mathbb{S}^{2}} |\nabla \, f|^{2}
d\sigma_{2}+\frac{A}{4}\int_{\mathbb{S}^{2}} f^{2} d \sigma_{2},\\
Q\left(f;\phi\right):=&
\frac{1}{4}
\int_{\mathbb{S}^{2}}
\bigg(
\frac{f^{2}}{{{\rho}_{\phi}}^{2}\left(d\left(x,p\right)\right)}+
\frac{f^{2}}{{{\rho}_{\phi}}^{2}\left(\pi-d\left(x,p\right)\right)}\bigg)
d \sigma_{2}.\end{aligned}$$
\[thm1\] Assume that $f\in H^{1}\left(\mathbb{S}^{2}\right).$ Then there exists constants $A,\,B>0$, independent of $f$, such that $$\begin{aligned}
\label{main11} Q\left(f;\phi\right)&\leq& T_{A}(f),\\
\label{main12} Q\left(f;\psi\right)&\leq& T_{B}(f).\end{aligned}$$
Both inequalities (\[main11\]) and (\[main12\]) are optimal, but an equality is impossible in either one:
\[thm2\] $$\begin{gathered}
\label{main21}
\sup_{f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}}
\frac{Q\left(f;\phi\right)}
{T_{A}(f)} = 1, \\
\label{main22}
\sup_{f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}}
\frac{Q\left(f;\psi\right)}
{T_{B}(f)} = 1.\end{gathered}$$
\[thm3\] There does not exist $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$ such that $\,Q\left(f;\phi\right)= T_{A}(f),$ or $\,Q\left(f;\psi\right)= T_{B}(f)$.
A variant of the abovementioned results follows via a different approach:
\[thm4\] Let $f\in H^{1}\left(\mathbb{S}^{2}\right).$ Then $$\begin{aligned}
\label{w01}
&\frac{1}{4}\int_{\mathbb{S}^{2}} \frac{f^{2}}{{{\rho}_{\phi}}^{2}\left(d\left(x,p\right)\right)}
\, d\sigma
\leq S(f)+\frac{1}{2\pi}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{\pi-d\left(x,p\right)}\, d\sigma,\\
\label{w02}
&\frac{1}{4}
\int_{\mathbb{S}^{2}} \frac{f^{2}}{{{\rho}_{\phi}}^{2}\left(\pi-d\left(x,p\right)\right)}
\, d\sigma \leq S(f)+\frac{1}{2\pi}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{d\left(x,p\right)}\, d\sigma.\end{aligned}$$ Moreover $$\label{w03}
\sup_{f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}}
\frac{\frac{1}{4}\int_{\mathbb{S}^{2}} \frac{f^{2}}{{{\rho}_{\phi}}^{2}\left(d\left(x,p\right)\right)}
\, d\sigma}
{S(f)+\frac{1}{2\pi}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{\pi-d\left(x,p\right)}\, d\sigma} =
\sup_{f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}}
\frac{\frac{1}{4}\int_{\mathbb{S}^{2}}\frac{f^{2}}{{{\rho}_{\phi}}^{2}\left(\pi-d\left(x,p\right)\right)}
\, d\sigma}
{S(f)+\frac{1}{2\pi}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{d\left(x,p\right)}\, d\sigma} =
1,$$ and the suprema in (\[w03\]) are not attained in $H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$.
Proof of Theorem 1
==================
Let $f\in C^{\infty}\left(\mathbb{S}^{2}\right)$. Notice that $\psi>1$ and write $f(\theta,\varphi)=\sqrt{\psi(\theta)} g(\theta,\varphi)$. We have $$\begin{aligned}
\nonumber
|\nabla \, f|^{2}&=& |\psi^{\frac{1}{2}} \nabla \, g+
g \nabla \, \psi^{\frac{1}{2}}|^{2}\\
\nonumber &=& \psi |\nabla \, g|^{2} +\langle\psi^{\frac{1}{2}} \nabla \, g, g\psi^{-\frac{1}{2}} \nabla \,\,\psi\rangle+ |\frac{1}{2}
{\psi}^{-\frac{1}{2}}\nabla \,\psi|^{2} g^{2}\\
\label{p1}&=& \psi |\nabla \, g|^{2}
+\frac{1}{2}\langle
\nabla \,\psi,\,
\nabla \, g^2\rangle+\frac{1}{4}
\frac{1}{\psi}
|\nabla \,\psi|^{2} g^{2}.\end{aligned}$$ Integrating both sides of (\[p1\]) over $\mathbb{S}^{2}$ we get $$\begin{aligned}
\nonumber
\int_{\mathbb{S}^{2}} |\nabla \, f|^{2} d\sigma
&=& \int_{\mathbb{S}^{2}} \left( \psi |\nabla \, g|^{2}
+\frac{1}{2}\langle
\nabla \,\psi,\,
\nabla \, g^2\rangle+\frac{1}{4}
\frac{1}{\psi}
|\nabla \,\psi|^{2} g^{2}
\right) d\sigma\\
\label{notattain} &\geq& \frac{1}{4}\int_{\mathbb{S}^{2}}
\frac{1}{\psi}
|\nabla \,\psi|^{2} g^{2}
d\sigma
+\frac{1}{2}\int_{\mathbb{S}^{2}} \langle
\nabla \,\psi,\,
\nabla \, g^2\rangle
d\sigma\\
\label{p2} &=& \frac{1}{4}\int_{\mathbb{S}^{2}}
\frac{1}{\psi}
|\psi^{\prime}|^{2} g^{2}
d\sigma
-\frac{1}{2}\int_{\mathbb{S}^{2}}
g^{2} \Delta \psi
d\sigma\end{aligned}$$ by partial integration over the closed manifold $\mathbb{S}^{2}$. Calculating, we find $$\begin{aligned}
\label{delt}
\Delta \psi=
\frac{1}{\sin{ \theta}}\frac{\partial}{\partial \theta}
\left( \sin{\theta}\frac{\partial }{\partial \theta}\psi \right)=
1.\end{aligned}$$ Returning $g$ to $f/\sqrt{\psi}$ and substituting for $\Delta \psi$ from (\[delt\]) into (\[p2\]), we obtain $$\begin{aligned}
\label{p3}
\int_{\mathbb{S}^{2}} |\nabla \, f|^{2} d\sigma &\geq& \frac{1}{4}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{\psi^{2}}
\frac{\cos^{2}{\theta}}{\sin^{2}{\theta}}
d\sigma- \frac{1}{2}
\int_{\mathbb{S}^{2}}
\frac{f^{2}}{\psi}
d\sigma.\end{aligned}$$ Adding the finite integral $\,\displaystyle
\frac{1}{4}\int_{\mathbb{S}^{2}}
\left(\frac{1}{\theta^{2}\phi^{2}\left(\theta\right)}+
\frac{1}{\left(\pi-\theta\right)^{2}\phi^{2}
\left(\pi-\theta\right)}\right)f^{2}
d\sigma\,$ to both sides of (\[p3\]) transforms it into the inequality $$\begin{aligned}
&\nonumber \frac{1}{4}\int_{\mathbb{S}^{2}}
\left(\frac{1}{\theta^{2}\phi^{2}\left(\theta\right)}+
\frac{1}{\left(\pi-\theta\right)^{2}\phi^{2}
\left(\pi-\theta\right)}\right)f^{2}
d\sigma\\
\label{p4}&\qquad\qquad\qquad\leq\int_{\mathbb{S}^{2}} |\nabla \, f|^{2} d\sigma+\frac{1}{4}\int_{\mathbb{S}^{2}} F(\theta)\, f^{2} d\sigma,\end{aligned}$$ where $$F(t):=
\frac{1}{t^{2}\phi^{2}\left(t\right)}+
\frac{1}{\left(\pi-t\right)^{2}\phi^{2}
\left(\pi-t\right)}-
\frac{\cos^{2}{t}}{\sin^{2}{t}}
\frac{1}{\phi^{2}\left(\sin{t}\right)}
+\frac{2}{\phi\left(\sin{t}\right)}.$$ Obviously, $F$ is continuous on $]0,\pi[$ and, as expected from the facts that $\phi(t)\rightarrow +\infty$ when $t\rightarrow 0^{+},\,$ $\,\sin{t} = t+o(t)$ as $t\rightarrow 0,\,$ it turns out $$\lim_{t\rightarrow 0^{+}} F(t)=
\lim_{t\rightarrow \pi^{-}} F(t)=\frac{1}{\pi^{2}}.$$ Hence, $F$ can be extended to a uniformly continuous, consequently a bounded, function on $[0,\pi]$. Noting this in (\[p4\]) implies (\[main11\]). Direct computation also shows $$A=\sup_{[0,\pi]}|F|=F(\frac{\pi}{2})=
\frac{2}{1+\log{\pi}}+\frac{8}{\left(1+\log{2}\right)^{2}}
\frac{1}{\pi^{2}}.$$ To prove (\[main12\]), we add to both sides of (\[p3\]) the well-defined integral\
\
$\,\displaystyle
\frac{1}{4}\int_{\mathbb{S}^{2}}
\left(\frac{1}{\theta^{2}}+
\frac{1}{\left(\pi-\theta\right)^{2}}\right)
\frac{f^{2}}{\psi^{2}\left(\theta\right)}
d\sigma.$ We then obtain the following analogue of (\[p4\]): $$\begin{aligned}
&\nonumber \frac{1}{4}\int_{\mathbb{S}^{2}}
\left(\frac{1}{\theta^{2}}+
\frac{1}{\left(\pi-\theta\right)^{2}}\right)
\frac{f^{2}}{\psi^{2}\left(\theta\right)}
d\sigma\\
\label{p5}&\qquad\qquad\qquad\leq\int_{\mathbb{S}^{2}} |\nabla \, f|^{2} d\sigma+\frac{1}{4}\int_{\mathbb{S}^{2}} G(\theta)\, f^{2} d\sigma,\end{aligned}$$ where $$\begin{aligned}
\nonumber
G(t)&:=&
\frac{M(t)}{\psi^{2}(t)}
+\frac{2}{\psi(t)},\\
\label{mm1}
M(t)&:=&\frac{1}{t^{2}}+
\frac{1}{(\pi-t)^{2}}
-\frac{\cos^{2}{t}}{\sin^{2}{t}}.\end{aligned}$$ Once the boundedness of $G$ is ensured, we see that (\[p5\]) yields the inequality (\[main12\]). Evidently, $G$ has the same features as $F$. Since $$\label{mm2}
\lim_{\theta\rightarrow 0} M(\theta)
\,=\,\lim_{\theta\rightarrow \pi} M(\theta)
\,=\,\frac{2}{3}+\frac{1}{\pi^2},\;
\lim_{\theta\rightarrow 0^{+}} \psi(t)
\,=\,\lim_{\theta\rightarrow \pi^{-}} \psi(t)
\,=\,+\infty$$ then $\,M\in C[0,\pi],$ and $\;\lim_{t\rightarrow 0^{+}} G(t)=
\lim_{t\rightarrow \pi^{-}} G(t)=0,\,$ which makes $G$ bounded on $[0,\pi]$. Moreover $$B=\sup_{[0,\pi]}|G|=G(\frac{\pi}{2})=
\frac{2}{1+\log{\pi}}+\frac{8}{\left(1+\log{\pi}\right)^{2}}
\frac{1}{\pi^{2}}.$$
Proof of Theorem \[thm2\] {#proofthm2}
=========================
First, we would like to define the weak laplace-Beltrami gradient of a function $f\in L^{1}\left(\mathbb{S}^{2}\right)$. Suppose $f\in C^{\infty}\left(\mathbb{S}^{2}\right)$ and $\,v(\theta,\varphi)=v_{\theta}(\theta,\varphi)
\hat{\theta}+v_{\varphi}(\theta,\varphi)\hat{\varphi}\,$ with $v_{\theta},v_{\varphi}\in C^{\infty}\left(\mathbb{S}^{2}\right)$. Then $$\begin{aligned}
&\int_{\mathbb{S}^{2}} \frac{\partial f}{\partial \theta}
\,v_{\theta} d\sigma =
\int_{\mathbb{S}^{2}}
\nabla f \cdot \hat{\theta}
\,v_{\theta}d\sigma=-\int_{\mathbb{S}^{2}}
f\, \nabla \cdot (\,v_{\theta}\hat{\theta})
d\sigma,
\\
&\int_{\mathbb{S}^{2}}\frac{1}{\sin{\theta}} \frac{\partial f}{\partial \varphi}
\,v_{\varphi} d\sigma =
\int_{\mathbb{S}^{2}}
\nabla f \cdot \hat{\varphi}
\,v_{\varphi}d\sigma=-\int_{\mathbb{S}^{2}}
f\, \nabla \cdot (\,v_{\varphi}\hat{\varphi})
d\sigma.\end{aligned}$$ Adding these identities we get $$\label{mot1}
\int_{\mathbb{S}^{2}}\nabla f \cdot V \, d\sigma
=-\int_{\mathbb{S}^{2}} f\, \nabla\cdot V \, d\sigma$$ for any vector field $V\in C^{\infty}\left(\mathbb{S}^{2}\rightarrow
T\left(\mathbb{S}^{2}\right)\right)$ where $T\left(\mathbb{S}^{2}\right)$ is the tangent bundle of the smooth manifold $\mathbb{S}^{2}$. Motivated by (\[mot1\]), $f$ is weakly differentiable if there is a vector field $\vartheta_{f}\in L^{1}\left(\mathbb{S}^{2}\rightarrow T\left(\mathbb{S}^{2}\right)\right)$ such that $$\label{mot2}
\int_{\mathbb{S}^{2}} \vartheta_{f} \cdot V \, d\sigma
=-\int_{\mathbb{S}^{2}} f\, \nabla\cdot V \, d\sigma,
\quad \forall\, V\in C^{\infty}\left(\mathbb{S}^{2}\rightarrow
T\left(\mathbb{S}^{2}\right)\right).$$ This, unique up to a set of zero measure, vector field $\vartheta_{f}$ is the weak surface gradient of $f$. According to ([@Eichhorn], Proposition 3.2., page 15) $$H^{1}\left(\mathbb{S}^{2}\right)=
W^{1,2}\left(\mathbb{S}^{2}\right):=
\left\{f\in L^{2}(\mathbb{S}^{2}):
|\vartheta_{f}|\in L^{2}\left(\mathbb{S}^{2}\right)\right\}.$$ We start with (\[main21\]). By Theorem \[thm1\], it suffices to prove the existence of a sequence $\left\{f_{n}\right\}_{n\geq 1}$ in $ H^{1}\left(\mathbb{S}^{2}\right)$ such that $$\label{limq}
\lim_{n\rightarrow \infty}\frac{Q\left(f_{n};\phi\right)}
{T_{A}(f_{n})}=1.$$ Consider the functions $$\label{fn}
f_{n}(\theta,\varphi):=
{\phi(\theta)}^{\frac{1}{2}-\frac{1}{n}}.$$ The functions $f_{n}$ are independent of $\varphi$, hence $$\label{plug}
\frac{Q\left(f_{n};\phi\right)}
{T_{A}(f_{n})}=
\frac{
\int_{0}^{\pi}
\frac{f_{n}^{2}\,\sin{\theta}}{\theta^{2}\,\phi^{2}(\theta) }\,d\theta+
\int_{0}^{\pi}
\frac{f_{n}^{2}\,\sin{\theta}}
{\left(\pi-\theta\right)^{2}
\,\phi^{2}\left(\pi-\theta\right)
}\,d\theta}
{4\int_{0}^{\pi} \left(\frac{\partial f_{n}}{\partial \theta} \right)^{2}\,\sin{\theta}
d\theta+A \int_{0}^{\pi} f_{n}^{2} \,\sin{\theta}
d\theta}$$ where the derivative $\partial f_{n}/\partial \theta$ is understood in the week sense discussed above. Since $\phi \in L^{1}_{\text{loc}}
\left(\mathbb{R}\right)$ and $\phi \geq 1$ on $[0,\pi]$, then $$\begin{aligned}
\label{calc1}
\int_{0}^{\pi} f^{2}_{n} \,\sin{\theta}
d \theta\,=\,
\int_{0}^{\pi} {\phi(\theta)}^{1-\frac{2}{n}} \,\sin{\theta}
d\theta
\leq \int_{0}^{\pi} {\phi(\theta)} \,d\theta
\,\approx\, 1.\end{aligned}$$ Thus $f_{n} \in L^{2}
\left(\mathbb{S}^{2}\right)$ for all $n\geq 1$. Notice also that $f_{n}$ is smooth on $[0,\pi]\setminus \{0\}$ and its weak derivative $$\label{fnprime}
\frac{\partial f_{n}}{\partial \theta} =\frac{\frac{1}{n}-\frac{1}{2}}
{\theta\,{\phi^{\frac{1}{2}+\frac{1}{n}}}}.$$ Therefore $$\begin{aligned}
\nonumber
\int_{0}^{\pi} \left(\frac{\partial f_{n}}{\partial \theta}\right)^{2} \,\sin{\theta}
d \theta\,=\,
\frac{a_{n}}{4}
\int_{0}^{\pi}
\frac{1}
{\theta\,{\phi^{1+\frac{2}{n}}}}
\,\frac{\sin{\theta}}{\theta}
d \theta,\quad a_{n}:=\left(1-\frac{2}{n}\right)^{2}.\end{aligned}$$ And since $\displaystyle \,\int_{0}^{\pi}
\frac{d \theta}{\theta\,{\phi^{1+\frac{2}{n}}}}
=\frac{n}{2},\,$ $\,{\sin{\theta}}\leq {\theta},\,$ then $\,{\partial f_{n}}/{\partial \theta} \in L^{2}\left(\mathbb{S}^{2}\right)$ for all $n\geq 1$. Substituting for $f_{n}$ from (\[fn\]) and for ${\partial f_{n}}/{\partial \theta}$ from (\[fnprime\]) into (\[plug\]) implies $$\label{lim}
\frac{Q\left(f_{n};\phi\right)}
{T_{A}(f_{n})}=
\frac{\alpha_{n}+\beta_{n}}{a_{n}\alpha_{n}+\gamma_{n}}=
\frac{1}{a_{n}}\left(
1+\frac{\beta_{n}-
\gamma_{n}/a_{n}}{\alpha_{n}+\gamma_{n}/a_{n}}\right)$$ where $$\begin{aligned}
\alpha_{n} &:=&\int_{0}^{\pi}
\frac{1}
{\theta\,{\phi^{1+\frac{2}{n}}}}
\,\frac{\sin{\theta}}{\theta}
d \theta,
\\
\beta_{n} &:=&
\int_{0}^{\pi}
\frac{\,\phi^{1-\frac{2}{n}}(\theta)\,\sin{\theta}}
{\left(\pi-\theta\right)^{2}
\,\phi^{2}\left(\pi-\theta\right)
}\,d \theta,
\\
\gamma_{n} &:=&
A\int_{0}^{\pi} \phi^{1-\frac{2}{n}}\,\sin{\theta}
d \theta.\end{aligned}$$ Observe that $\lim_{n\rightarrow +\infty}
a_{n}=1.\,$ We shall show that, while $\lim_{n\rightarrow +\infty}
\alpha_{n}=+\infty,\,$ the sequences $\left\{\beta_{n}\right\}_{n\geq 1}$ and $\left\{\gamma_{n}\right\}_{n\geq 1}$ are both convergent. Using this in (\[lim\]) proves (\[limq\]).\
Exploiting the continuity and positivity of $\,{\sin{\theta}}/
\left({\theta^2\,{\phi^{1+\frac{2}{n}}}}\right)\,$ on $[\pi/2,\pi]$, then applying the inequality $\,{\sin{\theta}}/{\theta}\geq {2}/{\pi}\,$ when $\,0\leq \theta \leq \pi/2,\,$ we obtain $$\begin{aligned}
\nonumber \alpha_{n}&=&
\int_{0}^{\pi/2}
\frac{1}
{\theta\,{\phi^{1+\frac{2}{n}}}}
\,\frac{\sin{\theta}}{\theta}
d \theta+
\int_{\pi/2}^{\pi}
\frac{\sin{\theta}}
{\theta^2\,{\phi^{1+\frac{2}{n}}}}
\,d \theta\\
\label{alpha11}&\geq&\frac{2}{\pi}
\int_{0}^{\pi/2}
\frac{1}
{\theta\,{\phi^{1+\frac{2}{n}}}}
\,d \theta\,=\,\frac{n}{\pi (1+\log(2))^{\frac{2}{n}}}.
\end{aligned}$$ This proves the divergence of $\{\alpha_{n}\}$. Next, by the dominated convergence theorem and (\[calc1\]) we readily find $$\begin{aligned}
\lim_{n\rightarrow +\infty} \gamma_{n} \,=\,
A\lim_{n\rightarrow +\infty} \int_{0}^{\pi} {\phi^{1-\frac{2}{n}}(\theta)} \,\sin{\theta}
d\theta\,=\,\int_{0}^{\pi} {\phi(\theta)} \,\sin{\theta} d\theta\,\lesssim 1.\end{aligned}$$ Finally, since $\displaystyle \,\theta \mapsto{\sin{\theta}}/{\left(\left(\pi-\theta\right)^{2}
\,\phi^{2}\left(\pi-\theta\right)\right)
}\in C\left([0,\pi/2]\right),\,$ then using the local integrability of $\phi$ and the dominated convergence theorem again implies $$\label{beta1}
\lim_{n\rightarrow \infty}
\int_{0}^{\pi/2}
\frac{\phi^{1-\frac{1}{n}}(\theta)\,\sin{\theta}}
{\left(\pi-\theta\right)^{2}
\,\phi^{2}\left(\pi-\theta\right)
}\,d \theta\,=\,\int_{0}^{\pi/2}
\frac{\phi(\theta)\,\sin{\theta}}
{\left(\pi-\theta\right)^{2}
\,\phi^{2}\left(\pi-\theta\right)
}\,d \theta\,\lesssim\,1.$$ Furthermore, since $\,\phi\in C\left([\pi/2,\pi]\right),\,$ and $\,\displaystyle \frac{\sin{\theta}}{\pi-\theta}=
\frac{\sin{\left(\pi-\theta\right)}}{\pi-\theta}\leq 1,\,$ on $[\pi/2,\pi],\,$ then $$\label{beta2}
\int_{\pi/2}^{\pi}
\frac{\phi^{1-\frac{1}{n}}(\theta)\,\sin{\theta}}
{\left(\pi-\theta\right)^{2}
\,\phi^{2}\left(\pi-\theta\right)
}\,d \theta \,\lesssim\,
\int_{\pi/2}^{\pi}
\frac{d \theta}
{\left(\pi-\theta\right)
\,\phi^{2}\left(\pi-\theta\right)}\,\approx\,1.$$ The convergence of $\{\beta_{n}\}$ follows from (\[beta1\]) together with (\[beta2\]).\
\
The proof of (\[main22\]) shares the main idea of (\[main21\]). The functions $\,g_{n}(\theta,\varphi):=
{\psi(\theta)}^{\frac{1}{2}-\frac{1}{n}}\in
L^{2}\left(\mathbb{S}^{2}\right),\,n\geq 1,\,$ and satisfy $\,\displaystyle \lim_{n\rightarrow \infty}\frac{Q\left(g_{n};\psi\right)}
{T_{B}(g_{n})}=1$. Indeed, we have $$\begin{aligned}
\frac{Q\left(g_{n};\psi\right)}
{T_{B}(g_{n})}&=&
\frac{
\int_{0}^{\pi}
\frac{g_{n}^{2}\,\sin{\theta}}{\theta^{2}\,\psi^{2}(\theta) }\,d\theta+
\int_{0}^{\pi}
\frac{g_{n}^{2}\,\sin{\theta}}
{\left(\pi-\theta\right)^{2}
\,\psi^{2}\left(\pi-\theta\right)
}\,d\theta}
{4\int_{0}^{\pi} \left(\frac{\partial g_{n}}{\partial \theta} \right)^{2}\,\sin{\theta}
d\theta+B \int_{0}^{\pi} g_{n}^{2} \,\sin{\theta}
d\theta}\\
&=&\frac{\tilde{\alpha}_{n}}
{a_{n}\tilde{\alpha}_{n}+\tilde{\beta}_{n}}=
\frac{1}{a_{n}}\left(
1-\frac{\tilde{\beta}_{n}/a_{n}}{\tilde{\alpha}_{n}+
\tilde{\beta}_{n}/a_{n}}\right)\end{aligned}$$ where $$\begin{aligned}
\tilde{\alpha}_{n} &:=&\int_{0}^{\pi}
\frac{\sin{\theta}\,d \theta}
{\theta^2\,{\psi^{1+\frac{2}{n}}}}
+\int_{0}^{\pi}
\frac{\sin{\theta}d \theta}
{(\pi-\theta)^2\,{\psi^{1+\frac{2}{n}}}}
=2\int_{0}^{\pi}
\frac{\sin{\theta}\,d \theta}
{\theta^2\,{\psi^{1+\frac{2}{n}}}},
\\
\tilde{\beta}_{n} &:=&
B\int_{0}^{\pi} \psi^{1-\frac{2}{n}}\,\sin{\theta}
d \theta-
a_{n}\int_{0}^{\pi}
M(\theta)
\frac{\sin{\theta}}{\psi^{1+\frac{2}{n}}}
\,d \theta.\end{aligned}$$ Similarly to (\[alpha11\]), we have $$\begin{aligned}
\tilde{\alpha}_{n}
&=&2\int_{0}^{1}
\frac{\sin{\theta}}{\theta^{2}}
\frac{1}{\,{\psi^{1+\frac{2}{n}}}}\,d \theta+
2\int_{1}^{\pi}
\frac{\sin{\theta}}{\theta^{2}}
\frac{1}{\,{\psi^{1+\frac{2}{n}}}}\,d \theta\\
&\geq&
2\int_{0}^{1}
\frac{\sin{\theta}}{\theta^{2}}
\frac{1}{\,{\psi^{1+\frac{2}{n}}}}\,d \theta
\,=\,2\int_{0}^{1}
\frac{\sin^{2}{\theta}}{\theta^{2}\cos{\theta}}
\frac{1}{\,{\psi^{1+\frac{2}{n}}}}
\frac{\cos{\theta}}{\sin{\theta}}
\,d \theta\\
&\geq&\frac{8}{\pi^{2}}
\int_{0}^{1}
\frac{1}{\,{\psi^{1+\frac{2}{n}}}}
\frac{\cos{\theta}}{\sin{\theta}}\,d \theta
\,=\,\frac{4n}{\pi^2}
\frac{1}{\left(1+\log{\pi}\right)^{\frac{2}{n}}}.\end{aligned}$$ Hence $\,\lim_{n\rightarrow \infty} \tilde{\alpha}_{n}=
\infty$. Recall from (\[mm1\]) and (\[mm2\]) that $M\in C([0,\pi])$. Also, since $\psi\in L^{1}_{\text{loc}} \left(\mathbb{R}\right),\,$ $\psi> 1$ uniformly, then $\,\lim_{n\rightarrow \infty} \tilde{\beta}_{n}$ exists by the dominated convergence theorem.
Proof of Theorem \[thm3\]
=========================
The transition to the inequalities (\[main11\]) and (\[main12\]) from their respective stronger versions, (\[p4\]) and (\[p5\]), comes from the bounds $$\int_{\mathbb{S}^{2}} F(\theta)\, f^{2} d\sigma\,\leq\, A\int_{\mathbb{S}^{2}} f^{2} d\sigma,\quad
\int_{\mathbb{S}^{2}} G(\theta)\, f^{2} d\sigma\,\leq\, B\int_{\mathbb{S}^{2}} f^{2} d\sigma$$ where the bounded functions $F$ and $G$ are both positive and independent of $f$. Interestingly, as seen in Section \[proofthm2\], the size of $\,0<A,B<\infty\,$ played no role in optimising (\[main11\]) and (\[main12\]).\
Up to the inequality (\[p4\]) or (\[p5\]) an equality relation persists except for the only inequality (\[notattain\]). So a sufficient and necessary condition for an equality in (\[p4\]) or (\[p5\]) (and a necessary condition for an equality in (\[main11\]) and (\[main12\])) is an equality in (\[notattain\]). But an equality in (\[notattain\]) occurs if and only if $$\label{iff1}
\int_{\mathbb{S}^{2}}
\psi |\nabla \, g|^{2}
d\sigma=0.$$ Recalling that $g=f/\sqrt{\psi},$ we compute $$\begin{aligned}
\nonumber
\psi|\nabla g|^{2}&=&
\psi\left|\frac{\nabla f}{\sqrt{\psi}}-
\frac{1}{2}\frac{f}{\psi^{\frac{3}{2}}}\frac{\partial \psi}{\partial \theta} \hat{\theta}
\right|^{2}
\\
\nonumber &=&|\nabla f|^{2}-
\frac{f}{\psi}\frac{\partial \psi}{\partial \theta} \, \nabla f\cdot\hat{\theta}+
\frac{1}{4}\frac{f^2}{\psi^{2}}\left(\frac{\partial \psi}{\partial \theta}\right)^{2}\\
\nonumber &=&|\nabla f|^{2}-\left(\frac{\partial f}{\partial \theta}\right)^{2}
+\left(\frac{\partial f}{\partial \theta}\right)^{2}
-\frac{f}{\psi}\frac{\partial \psi}{\partial \theta} \, \frac{\partial f}{\partial \theta}+
\frac{1}{4}\frac{f^2}{\psi^{2}}\left(\frac{\partial \psi}{\partial \theta}\right)^{2}\\
\label{iff2}&=&|\nabla f|^{2}-\left(\frac{\partial f}{\partial \theta}\right)^{2}
+\left(\frac{\partial f}{\partial \theta}
-\frac{1}{2}\frac{f}{\psi}\frac{\partial \psi}{\partial \theta}\right)^{2}.\end{aligned}$$ Since $\,\displaystyle |\nabla f|^{2}-\left(\frac{\partial f}{\partial \theta}\right)^{2}=\frac{1}{\sin^{2}{\theta}}
\left(\frac{\partial f}{\partial \varphi}\right)^{2}\,\geq\,0,\,$ then, by (\[iff2\]), the equality (\[iff1\]) is equivalent to $$\label{iff3}
\int_{\mathbb{S}^{2}}
|\nabla f|^{2}-\left(\frac{\partial f}{\partial \theta}\right)^{2}d\sigma\,=\,
\int_{\mathbb{S}^{2}}
\left(\frac{\partial f}{\partial \theta}
-\frac{1}{2}\frac{f}{\psi}\frac{\partial \psi}{\partial \theta}\right)^{2}d\sigma\,=\,0.$$ The equalities (\[iff3\]) are, in their turn, equivalent to $$\label{iff4}
\frac{1}{\sin{\theta}}
\left|\frac{\partial f}{\partial \varphi}\right|\,=\,
\left|\frac{\partial f}{\partial \theta}
-\frac{1}{2}\frac{f}{\psi}\frac{\partial \psi}{\partial \theta}\right|\,=\,0.$$ Suppose that $f$ is not the zero function. Then (\[iff4\]) are possible if and only if $$f\,=\,f(\theta),\,
\frac{d f}{f}
\,=\,\frac{1}{2}\frac{d\psi}{\psi}.$$ That is $f\,=\,c\sqrt{{\psi}},$ $c$ is a constant. But such $f\notin H^{1}\left(\mathbb{S}^{2}\right)\,$ because $$\begin{aligned}
\int_{\mathbb{S}^{2}}
|\nabla \, f|^{2}
d\sigma&=&2\pi\int_{0}^{\pi}
\left(\frac{\partial f}{\partial \theta}\right)^{2}
d\theta\,\gtrsim
\int_{0}^{1}
\frac{\cos^{2}{\theta}}{\sin{\theta}}
\frac{1}{\psi} d\theta
\\
&\gtrsim&\int_{0}^{1}
\frac{d\theta}{\sin{\theta}\,\phi(\sin{\theta})}
\,\approx\,
\int_{0}^{1}
\frac{d\theta}{{\theta}\,\phi({\theta})}
\,=\,+\infty.\end{aligned}$$
Proof of Theorem \[thm4\]
=========================
Write $$\begin{aligned}
\frac{1}{\theta}\frac{1}{\phi^{2}\left(\theta\right)} =
\nabla \left( \frac{1}{\phi\left(\theta\right)}\right)\cdot \hat{\theta}.\end{aligned}$$ Assume that $f$ is smooth. Then integrating by parts w.r.t. the surface measure $\sigma$ we get $$\begin{aligned}
\nonumber \int_{\mathbb{S}^{2}} \frac{f^{2}}{\theta^{2}\phi^{2}\left(\theta\right)}
\, d\sigma =& \int_{\mathbb{S}^{2}} \nabla \left( \frac{1}{\phi\left(\theta\right)}\right)\cdot \frac{f^{2}}{\theta} \hat{\theta} d\sigma \\
\nonumber =&- \int_{\mathbb{S}^{2}} \frac{1}{\phi\left(\theta\right)} \nabla \cdot \left(\frac{f^{2}}{\theta} \hat{\theta}\right)d\sigma\\
\nonumber =&- 2\int_{\mathbb{S}^{2}} \frac{f\,\nabla f.\hat{\theta}}{\theta\,\phi\left(\theta\right)}d\sigma+
\int_{\mathbb{S}^{2}} \frac{f^{2}}{\theta^{2}\,\phi\left(\theta\right)}d\sigma+\\
\label{w1} &-
\int_{\mathbb{S}^{2}} \frac{f^{2}}{\theta\,\phi\left(\theta\right)}
\frac{\cos{\theta}}{\sin{\theta}}d\sigma.\end{aligned}$$ Observe here that each of the last two integrals on the right hand side of (\[w1\]) can diverge. They suffer nonintegrable singularities at $\theta=0$. The reality is, put together, their sum $$\begin{aligned}
\label{wc1}
I:=\int_{\mathbb{S}^{2}} \frac{f^{2}}{\theta^{2}\,\phi\left(\theta\right)}d\sigma-
\int_{\mathbb{S}^{2}} \frac{f^{2}}{\theta\,\phi\left(\theta\right)}
\frac{\cos{\theta}}{\sin{\theta}}d\sigma=
\int_{\mathbb{S}^{2}} \frac{1}{\theta\,\phi\left(\theta\right)}
\left(\frac{1}{\theta}-\frac{\cos{\theta}}{\sin{\theta}} \right)f^{2}d\sigma\end{aligned}$$ is convergent. In fact $$\begin{aligned}
\lim_{\theta\rightarrow 0^{+}}\frac{1}{\theta\,\phi\left(\theta\right)}
\left(\frac{1}{\theta}-\frac{\cos{\theta}}{\sin{\theta}} \right)=0.\end{aligned}$$ Also,$\,\theta\mapsto1/\left({\theta^{2}\,\phi\left(\theta\right)}\right)\,$ is continuous on a neighborhood of $\theta=\pi$. Furthermore, if we fix $\delta>0$ and let $\,D:=\left\{x(\theta,\varphi)\in \mathbb{S}^{2}:
0\leq \theta < \delta \right\}$, then the integral $
\displaystyle \int_{\mathbb{S}^{2}\setminus D} \frac{f^{2}}{\theta\,\phi\left(\theta\right)}
\frac{\cos{\theta}}{\sin{\theta}} d\sigma$ does exist. Unfortunately, we can not control the integral $I$ by $\int_{\mathbb{S}^{2}} f^{2} d\sigma$, up to a constant factor. The reason is $$\begin{aligned}
\lim_{\theta\rightarrow \pi^{-}}\frac{1}{\theta\,\phi\left(\theta\right)}
\frac{\cos{\theta}}{\sin{\theta}} =\infty.\end{aligned}$$ But since $$\begin{aligned}
\lim_{\theta\rightarrow \pi^{-}}\left(\frac{1}{\theta\,\phi\left(\theta\right)}
\frac{\cos{\theta}}{\sin{\theta}}+
\frac{1}{\pi}\frac{1}{(\pi-\theta)}\right)
=0\end{aligned}$$ then, we may introduce the convergent integral $\displaystyle J:=\frac{1}{\pi}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{\pi-\theta}\, d\sigma$ to the integral $I$ to get $$\label{wc2}
I=I-J+J=\int_{\mathbb{S}^{2}}
K(\theta)\,f^{2}\, d\sigma+J$$ where $$\begin{aligned}
K(\theta):=\frac{1}{\theta\,\phi\left(\theta\right)}
\left(\frac{1}{\theta}-\frac{\cos{\theta}}{\sin{\theta}} \right)-\frac{1}{\pi}\frac{1}{(\pi-\theta)}.\end{aligned}$$ By the continuity of $K$ on $]0,\pi[$ and since $$\begin{aligned}
\lim_{\theta\rightarrow 0^{+}}
K(\theta)=-\lim_{\theta\rightarrow \pi^{-}}
K(\theta)=-\frac{1}{\pi^{2}}\end{aligned}$$ then $K$ is bounded on $[0,\pi]$. Actually, $K$ is monotonically increasing. Thus $$\begin{aligned}
\label{wc3}
\sup_{[0,\pi]}|K|=\frac{1}{\pi^{2}}.\end{aligned}$$ Using (\[wc3\]) in (\[wc2\]) we deduce that $$\begin{aligned}
\label{wc4}
I\leq \frac{1}{\pi^{2}}\int_{\mathbb{S}^{2}}
f^{2}\, d\sigma+J.\end{aligned}$$ Returning with (\[wc4\]) to the inequality (\[w1\]) in the light of (\[wc1\]) we obtain $$\label{w2} \int_{\mathbb{S}^{2}} \frac{f^{2}}{\theta^{2}\phi^{2}\left(\theta\right)}
\, d\sigma\,\leq\,- 2\int_{\mathbb{S}^{2}} \frac{f\,\nabla f.\hat{\theta}}{\theta\,\phi\left(\theta\right)}d\sigma+
\frac{1}{\pi^{2}}\int_{\mathbb{S}^{2}}
f^{2}\, d\sigma+\frac{1}{\pi}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{\pi-\theta}\, d\sigma.$$ Applying Cauchy’s inequality with an $\epsilon$ we find $$\label{w4}
- 2\int_{\mathbb{S}^{2}} \frac{f\,\nabla f.\hat{\theta}}{\theta\,\phi\left(\theta\right)}d\sigma
\leq 2\epsilon \int_{\mathbb{S}^{2}}
\frac{f^{2}}{\theta^{2}\phi^{2}\left(\theta\right)}
\, d\sigma+\frac{1}{2\epsilon}\int_{\mathbb{S}^{2}}
|\hat{\theta}.\nabla f|^{2} d\sigma.$$ Therefore, it follows from (\[w2\]) and (\[w4\]) that $$\begin{aligned}
\nonumber
&2\epsilon(1-2\epsilon)
\int_{\mathbb{S}^{2}} \frac{f^{2}}{\theta^{2}\phi^{2}\left(\theta\right)}
\, d\sigma \leq \int_{\mathbb{S}^{2}}
|\hat{\theta}.\nabla f|^{2} d\sigma+\\
\label{w5}&\qquad\qquad\qquad
+\frac{2\epsilon}{\pi^{2}}\int_{\mathbb{S}^{2}}
f^{2}\, d\sigma+\frac{2\epsilon}{\pi}\int_{\mathbb{S}^{2}}
\frac{f^{2}}{\pi-\theta}\, d\sigma,\quad 0<\epsilon<\frac{1}{2}.\end{aligned}$$ The choice $\epsilon={1}/{4}$ maximizes the factor $2\epsilon(1-2\epsilon)$ and, consequently, the left hand side of (\[w5\]). This proves (\[w01\]). The inequality (\[w02\]) can be obtained analogously.\
In the fashion of the proof of Theorem \[thm2\], the sequence $f_{n}=\phi^{\frac{1}{2}-\frac{1}{n}}$ clearly satisfies $$\lim_{n\rightarrow \infty}
\frac{\frac{1}{4}\int_{0}^{\pi} \frac{f_{n}^{2}}{{{\rho}_{\phi}}^{2}
\left(\theta\right)}
\, \sin{\theta}\,d \theta}
{U(f_{n}
)+\frac{1}{2\pi}\int_{0}^{\pi}
\frac{f_{n}^{2}}{\pi-\theta}\, \sin{\theta}\,d \theta} =
\lim_{n\rightarrow \infty}
\frac{\frac{1}{4}\int_{0}^{\pi}
\frac{f_{n}^{2}}{{{\rho}_{\phi}}^{2}
\left(\pi-\theta\right)}
\, \sin{\theta}\,d \theta}
{U(f_{n})+\frac{1}{2\pi}\int_{0}^{\pi}
\frac{f_{n}^{2}}{\theta}\, \sin{\theta}\,d \theta} =
1$$ where $$U(f)=\int_{0}^{\pi}
\left(\frac{\partial f}{\partial{\theta}}\right)^{2}\, \sin{\theta} \,d \theta+\frac{1}{2\pi^{2}}\int_{0}^{\pi}
f^{2}\, \sin{\theta}\,d \theta.$$ One only needs to inspect the convergence of $\,\int_{0}^{\pi}
\left(
\phi^{1-\frac{2}{n}}\sin{\theta}/{\theta}\,\right)d \theta,$\
$\;\int_{0}^{\pi}
\left(\phi^{1-\frac{2}{n}}\sin{\theta}/\left(\pi-{\theta}\right)
\right)d \theta
$ as $n\rightarrow \infty.\,$ This is obvious from the bound $\sin{\theta}\leq \min\{{\theta},{\pi-\theta}\}\,$ on $[0,\pi]$ and the fact $\phi \in L^{1}\left([0,\pi]\right)$.\
Finally, careful review of the proof of (\[w01\]) above reveals that a necessary condition for a function $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$ to achieve an equality in (\[w01\]) is that it yields an equality in (\[w4\]). This is equivalent to $$\label{ww}
\nabla f.\hat{\theta}\,=\,
- \frac{1}{2}\frac{f}{\theta\,\phi\left(\theta\right)}.$$ Suppose (\[ww\]) was true. Then by (\[w1\]) and (\[wc1\]) we must have $$\label{ww1}
\int_{\mathbb{S}^{2}} \frac{h(\theta)\,f^{2}}{\theta\,\phi\left(\theta\right)}
\,d\sigma\,=\,0$$ where $$h(\theta):=\frac{1}{\theta}-\frac{\cos{\theta}}{\sin{\theta}}.$$ On the other hand $$\lim_{\theta\rightarrow 0^{+}} h(\theta)=0,\;\;
h^{\prime}(\theta)\,=\,
\frac{{\theta}^{2}-
\sin^{2}{\theta}}{{\theta}^{2}\,\sin^{2}{\theta}}>0,\;
0<\theta<\pi.$$ This shows $h$ is strictly positive on $]0,\pi]$ and since $\,\theta \phi(\theta)\geq 0$ then (\[ww1\]) is a contradiction.
**References**
[10]{} A. Balinsky, W. D. Evans, R. T. Lewis, The analysis and geometry of hardy’s inequality. Springer, New York 2015. Jürgen Eichhorn, Global analysis on open manifolds. Nova Science Publishers, Inc., New York, 2007. N. Ioku, M. Ishiwata, A scale invariant form of a critical hardy inequality, International Mathematics Research Notices, 18 (2015), 8830 - 8846. N. Ioku, M. Ishiwata and T. Ozawa, Sharp remainder of a critical Hardy inequality, Archiv der Mathematik, 106 (2016), 65 - 71. S. Machihara, T. Ozawa and H. Wadade, Hardy type inequalities on balls, Tohoku Mathematical Journal, 65, (2013), No. 3, 321 - 330. M. Ruzhansky and D. Suragan, Critical Hardy inequalities. arXiv:1602.04809, 2016. M. Sano, F. Takahashi, Scale invariance structures of the critical and the subcritical Hardy inequalities and their improvements, Calculus of variations and partial differential equations, (2017) 56: 69. F. Takahashi, A simple proof of Hardy’s inequality in a limiting case, Archiv der Mathematik, 104 (2015), Issue 1, 77 - 82. Hardy’s inequality in a limiting case on general bounded domains. arXiv: 1707.04018, 2017. Y. Xiao, Some Hardy inequalities on the sphere, J. Math. Inequal. 10 (2016), 793 - 805.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We first prove some basic properties of Okounkov bodies, and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Küronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.'
address:
- 'Department of Mathematics, Yonsei University, Seoul, Korea'
- 'School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea'
- 'Center for Geometry and Physics, Institute for Basic Science, Pohang, Korea'
author:
- Sung Rak Choi
- Jinhyung Park
- Joonyeong Won
title: Okounkov bodies associated to pseudoeffective divisors II
---
[^1]
Introduction
============
This paper is a continuation of our investigation on Okounkov bodies associated to pseudoeffective divisors ([@CHPW1], [@CHPW2], [@CPW]). Let $X$ be a smooth projective variety of dimension $n$, and $D$ be a divisor on $X$. Fix an admissible flag $Y_\bullet$ on $X$, that is, a sequence of irreducible subvarieties $$Y_\bullet: X=Y_0\supseteq Y_1\supseteq \cdots\supseteq Y_{n-1}\supseteq Y_n=\{ x\}$$ where each $Y_i$ is of codimension $i$ in $X$ and is smooth at $x$. The Okounkov body ${\Delta}_{Y_\bullet}(D)$ of a big divisor $D$ with respect to $Y_\bullet$ is a convex body in the Euclidean space ${\mathbb{R}}^n$ which carries rich information of $D$. Okounkov first defined the Okounkov body associated to an ample divisor in [@O1], [@O2]. After this pioneering work, Lazarsfeld-Mustaţă [@lm-nobody] and Kaveh-Khovanskii [@KK] independently generalized Okounkov’s work to big divisors (see [@B2] for a survey). We then further extended the study of Okounkov bodies to pseudoeffective divisors in [@CHPW1]. More precisely, we have introduced and studied two convex bodies, called the *valuative Okounkov body* ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ and the *limiting Okounkov body* ${\Delta^{\lim}}_{Y_\bullet}(D)$ associated to a pseudoeffective divisor $D$. See Sections \[okbdsubsec\] and \[nakpvssec\] for definitions and basics on Okounkov bodies.
In this paper, we first prove supplementary results to [@CHPW1]. Main theorems of [@CHPW1] and the subsequent results in this paper depend on the following property of the Okounkov body. This theorem is a generalization of [@lm-nobody Theorem 4.26] and [@Jow Theorem 3.4].
\[newthrm\] Let $X$ be a smooth projective variety of dimension $n$, and $D$ be a big divisor on $X$. Fix an admissible flag $Y_\bullet$ such that $Y_{n-k} \not\subseteq {\mathbf B_+}(D)$. Then we have $${\Delta}_{Y_{n- k\bullet}}(D) = {\Delta}_{Y_\bullet}(D) \cap (\{ 0\}^{n-k} \times {\mathbb{R}}_{\geq 0}^{k}).$$
In [@CHPW1], we proved that the Okounkov bodies ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ and ${\Delta^{\lim}}_{Y_\bullet}(D)$ encode nice properties of the divisor $D$ if the given admissible flag $Y_\bullet$ contains a Nakayama subvariety of $D$ or a positive volume subvariety of $D$ (see Theorem \[chpwmain\]). We show the following characterization of those special subvarieties in terms of Okounkov bodies.
\[critintro\] Let $X$ be a smooth projective variety of dimension $n$, and $D$ be an ${\mathbb{R}}$-divisor on $X$. Fix an admissible flag $Y_\bullet$ such that $Y_n$ is a general point in $X$. Then we have the following:
1. If $D$ is effective, then $Y_\bullet$ contains a Nakayama subvariety of $D$ if and only if ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D) \subseteq \{0 \}^{n-\kappa(D)} \times {\mathbb{R}}^{\kappa(D)}$.
2. If $D$ is pseudoeffective, then $Y_\bullet$ contains a positive volume subvariety of $D$ if and only if ${\Delta^{\lim}}_{Y_\bullet}(D) \subseteq \{0 \}^{n-{\kappa_\nu}(D)} \times {\mathbb{R}}^{{\kappa_\nu}(D)}$ and $\dim {\Delta^{\lim}}_{Y_\bullet}(D)={\kappa_\nu}(D)$.
One of the most important properties one can probably expect a convex set in ${\mathbb{R}}^n$ to satisfy is rational polyhedrality. However, the geometric structure of Okounkov body is rather wild. It can be non-polyhedral even if the variety $X$ is a Mori dream space and a divisor $D$ is ample (see [@lm-nobody Subsection 6.3], [@KLM Section 3]). However, Anderson-Küronya-Lozovanu proved that if a big divisor $D$ has a finitely generated section ring $R(X, D):=\bigoplus_{m \geq 0} H^0(X, mD)$, then there exists an admissible flag $Y_\bullet$ such that the Okounkov body ${\Delta}_{Y_\bullet}(D)$ is a rational polytope ([@AKL Theorem 1]). We also refer to [@CPW Theorems 1.1 and 4.17] and [@S Corollary 4.5] for more related results.
Our next aim is to generalize [@AKL Theorem 1] to the valuative and limiting Okounkov bodies. We recall that when a divisor $D$ is big, it has a finitely generated section ring if and only if it admits the birational good Zariski decomposition (see [@nakayama III.1.17.Remark]). However, for a pseudoeffective divisor $D$, such equivalence no longer holds in general; $D$ admits the birational good Zariski decomposition if and only if $D$ has a finitely generated section ring and is abundant (see Proposition \[zdabfg\]). For the rational polyhedrality of the Okounkov bodies of pseudoeffective divisors, we assume the existence of good Zariski decomposition on some birational model instead of the finite generation condition. See Subsection \[zdsubsec\] for our definition of (good) Zariski decomposition.
\[main1\] Let $X$ be a smooth projective variety, and $D$ be a pseudoeffective ${\mathbb{Q}}$-divisor on $X$ which admits the good birational Zariski decomposition. Then each Okounkov bodies ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ and ${\Delta^{\lim}}_{Y_\bullet}(D)$ is rational polyhedral with respect to some admissible flag $Y_\bullet$.
We expect that the rational polyhedrality of Okounkov body holds in more general situations. There are examples of divisors which do not admit birational good Zariski decompositions, but whose associated Okounkov bodies are rational polyhedral (see Remark \[ratrem\]).
To prove Theorem \[main1\] for the case of valuative Okounkov bodies, we use the same idea as [@AKL Proposition 4]. Using only the finite generation of section ring, we show the rational polyhedrality of the valuative Okounkov body with respect to an admissible flag taken by the intersections of general members of the linear series (see Theorem \[ratsimval\]). For the case of limiting Okounkov bodies, under the given assumption, we prove the statement by reducing to the rationality problem of the limiting Okounkov body on some high model $f \colon Y\to X$ where the good Zariski decomposition of $f^*D$ exists (see Theorem \[ratsimlim\]).
The organization of the paper is as follows. In Section \[prelimsec\], we collect basic facts on various notions that are used in the proofs. Next, in Section \[okbdsubsec\], we recall basic properties of Okounkov bodies, and prove Theorem \[newthrm\]. Then we study some properties of Nakayama subvarieties and positive volume subvarieties to show Theorem \[critintro\] in Section \[nakpvssec\]. Section \[ratsec\] is devoted to showing Theorem \[main1\].
Acknowledgment {#acknowledgment .unnumbered}
--------------
We would like to thank the referee for providing numerous helpful suggestions and comments and for pointing out several gaps in earlier versions of this manuscript.
Preliminaries {#prelimsec}
=============
In this section, we collect relevant facts which will be used later. Throughout the paper, $X$ is a smooth projective variety of dimension $n$, and we always work over an algebraically closed field of characteristic zero.
Asymptotic invariants
---------------------
We review basic asymptotic invariants of divisors, namely, the asymptotic base loci and volume functions. The *stable base locus* of an ${\mathbb{R}}$-divisor $D$ is defined as $\operatorname{SB}(D):= \bigcap_{D \sim_{{\mathbb{R}}} D' \geq 0} \operatorname{Supp}(D')$. The *augmented base locus* of an ${\mathbb{R}}$-divisor $D$ is defined as ${\mathbf B_+}(D):=\bigcap_A\text{SB}(D-A)$ where the intersection is taken over all ample divisors $A$. The *restricted base locus* of an ${\mathbb{R}}$-divisor $D$ is defined as ${\mathbf B_-}(D):=\bigcup_{A}\operatorname{SB}(D+A)$ where the union is taken over all ample divisors $A$. Note that ${\mathbf B_+}(D)$ and ${\mathbf B_-}(D)$ depend only on the numerical class of $D$. For details, we refer to [@elmnp-asymptotic; @inv; @of; @base] and [@lehmann-red].
Now, let $V$ be an irreducible subvariety of $X$ of dimension $v$. The *restricted volume* of a ${\mathbb{Z}}$-divisor $D$ along $V$ is defined as $
\operatorname{vol}_{X|V}(D):=\limsup_{m \to \infty} \frac{h^0(X|V,mD)}{m^v/v!}
$ where $h^0(X|V,mD)$ is the dimension of the image of the natural restriction map $\varphi \colon H^0(S,{\mathcal}O_X(D))\to H^0(V,{\mathcal}O_V(D))$. The restricted volume $\operatorname{vol}_{X|V}(D)$ depends only on the numerical class of $D$, and one can uniquely extend it to a continuous function $$\operatorname{vol}_{X|V} \colon \text{Big}^V (X) \to {\mathbb{R}}$$ where $\text{Big}^V(X)$ is the set of all ${\mathbb{R}}$-divisor classes $\xi$ such that $V$ is not properly contained in any irreducible component of ${\mathbf B_+}(\xi)$. When $V=X$, we simply let $\operatorname{vol}_X(D):=\operatorname{vol}_{X|X}(D)$, and we call it the *volume* of an ${\mathbb{R}}$-divisor $D$. For more details on volumes and restricted volumes, see [@pos] and [@elmnp-restricted; @vol; @and; @base; @loci]. Now assume that $V\not\subseteq{\mathbf B_-}(D)$ for an ${\mathbb{R}}$-divisor $D$. The *augmented restricted volume* of $D$ along $V$ is defined as $\operatorname{vol}_{X|V}^+(D):=\lim_{{\varepsilon}\to 0+} \operatorname{vol}_{X|V}(D+{\varepsilon}A)$ where $A$ is an ample divisor on $X$. The definition is independent of the choice of $A$. Note that $\operatorname{vol}_{X|V}^+(D)=\operatorname{vol}_{X|V}(D)$ for $D \in \text{Big}^V (X)$. This also extends uniquely to a continuous function $$\operatorname{vol}_{X|V}^+ \colon \overline{\text{Eff}}^V(X) \to {\mathbb{R}}$$ where $\overline{\text{Eff}}^V(X) := \text{Big}^V(X) \cup \{ \xi \in \overline{\text{Eff}}(X) \setminus \text{Big}(X) \mid V \not\subseteq {\mathbf B_-}(\xi) \}$. For $D\in \overline{\text{Eff}}^V(X)$, we have $\operatorname{vol}_{X|V}(D) \leq \operatorname{vol}_{X|V}^+(D) \leq \operatorname{vol}_{V}(D|_V)$, and both inequalities can be strict in general. See [@CHPW1] for more details on augmented restricted volumes.
Iitaka dimension
----------------
Let $D$ be an ${\mathbb{R}}$-divisor on $X$. Let $\mathbb N(D)=\{m\in{\mathbb{Z}}_{>0}|\; |\lfloor mD\rfloor|\neq\emptyset\}$. For $m\in\mathbb N(D)$, we consider the rational map $\phi_{mD} \colon X \dashrightarrow Z_m \subseteq \mathbb P^{\dim|\lfloor mD\rfloor|}$ defined by the linear system $|\lfloor mD\rfloor |$. The *Iitaka dimension* of $D$ is defined as $$\kappa(D):=\left\{
\begin{array}{ll}
\max\{\dim\text{Im}(\phi_{mD}) \mid \;m\in\mathbb N(D)\}& \text{if }\mathbb N(D)\neq\emptyset\\
-\infty&\text{if }\mathbb N(D)=\emptyset.
\end{array}
\right.$$ We remark that the Iitaka dimension $\kappa(D)$ is not really an invariant of the ${\mathbb{R}}$-linear equivalence class of $D$. Nonetheless, it satisfies the property that $\kappa(D)=\kappa(D')$ for effective divisors $D,D'$ such that $D\sim_{{\mathbb{R}}}D'$.
For another important invariant, we fix a sufficiently ample ${\mathbb{Z}}$-divisor $A$ on $X$. The *numerical Iitaka dimension* of $D$ is defined as $${\kappa_\nu}(D):= \max\left\{k \in {\mathbb{Z}}_{\geq0} \left|\; \limsup_{m \to \infty} \frac{h^0(X, \lfloor mD \rfloor + A)}{m^k}>0 \right.\right\}$$ if $h^0(X, \lfloor mD \rfloor + A)\neq\emptyset$ for infinitely many $m>0$, and we let ${\kappa_\nu}(D):=-\infty$ otherwise. The numerical Iitaka dimension ${\kappa_\nu}(D)$ depends only on the numerical class $[D]\in\operatorname{N^1}(X)_{{\mathbb{R}}}$.
An ${\mathbb{R}}$-divisor $D$ is said to be *abundant* if $\kappa(D)={\kappa_\nu}(D)$.
By definition, $\kappa(D) \leq {\kappa_\nu}(D)$ holds and the inequality can be strict in general. However, ${\kappa_\nu}(D)=\dim X$ if and only if $\kappa(D)=\dim X$. We refer to [@E], [@lehmann-nu], [@nakayama] for more detailed properties of $\kappa$ and ${\kappa_\nu}$.
Recall that the *section ring of an ${\mathbb{R}}$-divisor $D$* is defined as $R(X, D):=\bigoplus_{m \geq 0} H^0(X, \lfloor mD \rfloor)$.
\[semiampleabundant\] A ${\mathbb{Q}}$-divisor $D$ on $X$ is semiample if and only if it is nef, abundant, and its section ring is finitely generated.
Zariski decomposition {#zdsubsec}
---------------------
We now briefly recall several notions related to Zariski decompositions in higher dimension. For more details, we refer to [@B1], [@nakayama], [@P].
To define the divisorial Zariski decomposition, we first consider a divisorial valuation $\sigma$ on $X$ with the center $V:=\operatorname{Cent}_X \sigma$ on $X$. If $D$ is a big ${\mathbb{R}}$-divisor on $X$, we define *the asymptotic valuation* of $\sigma$ at $D$ as $\operatorname{ord}_V(||D||):=\inf\{\sigma(D')\mid D\equiv D'\geq 0\}$. If $D$ is only a pseudoeffective ${\mathbb{R}}$-divisor on $X$, we define $\operatorname{ord}_V(||D||):=\lim_{{\varepsilon}\to 0+}\operatorname{ord}_V(||D+{\varepsilon}A||)$ for some ample divisor $A$ on $X$. This definition is independent of the choice of $A$. The *divisorial Zariski decomposition* of a pseudoeffective ${\mathbb{R}}$-divisor $D$ is the decomposition $$D=P_{\sigma}+N_{\sigma}$$ into the *negative part* $N_{\sigma}:=\sum_{\operatorname{codim}E=1} \operatorname{ord}_E(||D||)E$ where the summation is over the codimension one irreducible subvarieties $E$ of $X$ such that $ \operatorname{ord}_E(||D||)>0$ and the *positive part* $P_{\sigma}:=D-N_{\sigma}$. Let $D$ be an ${\mathbb{R}}$-divisor on $X$ which is effective up to $\sim_{{\mathbb{R}}}$. The *$s$-decomposition* of $D$ is the decomposition $$D=P_s+N_s$$ into the *negative part* $N_s:=\inf\{L \mid L \sim_{{\mathbb{R}}} D, L \geq 0\}$ and the *positive part* $P_s:=D-N_s$. The positive part $P_s$ is also characterized as the smallest divisor such that $P_s \leq D$ and $R(X, P_s) \simeq R(X, D)$ ([@P Proposition 4.8]). Note that $P_s \leq P_\sigma$ and $P_s, P_\sigma$ do not coincide in general.
\[abundantdiv=s\] Let $D$ be an abundant ${\mathbb{R}}$-divisor on $X$ with the divisorial Zariski decomposition $D=P_\sigma + N_\sigma$ and the $s$-decomposition $D=P_s+N_s$. Then $P_\sigma=P_s$.
Let $\sigma$ be a divisorial valuation on $X$ with $V=\operatorname{Cent}_X \sigma$. [@lehmann-red Proposition 6.4] implies that $\inf_{m \in {\mathbb{Z}}_{>0}, D' \in |\lfloor mD\rfloor|} \frac{1}{m}\sigma(D) = \operatorname{ord}_V(||D||)$ holds. Since $\inf_{m \in {\mathbb{Z}}_{>0}, D' \in |\lfloor mD\rfloor|} \frac{1}{m}\sigma(D)=\sigma(N_s)$, we see that $D=P_s + N_s$ is the divisorial Zariski decomposition.
The *Fujita-Zariski decomposition* of a pseudoeffective ${\mathbb{R}}$-divisor $D$ is the decompositon $$D=P_f+N_f$$ into the effective *negative part* $N_f$ and the nef *positive part* $P_f$ such that if $f \colon Y \to X$ is a birational morphism from a smooth projective variety and $f^*D=P'+N'$ with $P'$ nef and $N' \geq 0$, then $P' \leq f^*P$. By definition, the divisorial Zariski decomposition and $s$-decomposition uniquely exist, and the Fujita-Zariski decomposition is also unique if it exists. Recall that the Fujita-Zariski decomposition does not exist in general even if we take the pullback on a sufficiently high model $f \colon \widetilde{X} \to X$ (see [@nakayama Chapter IV]).
It is unclear in general whether the Fujita-Zariski decomposition is the divisorial Zariski decomposition (cf. [@nakayama III.1.17.Remark (2)]). However, this holds when the divisor is abundant and the positive part is semiample.
\[goodzd\] Let $D$ be an abundant ${\mathbb{Q}}$-divisor on $X$ having a decomposition $D=P+N$ into a nef divisor $P$ and an effective divisor $N$. Then the following are equivalent:
1. It is the divisorial Zariski decomposition with $P=P_\sigma$ semiample.
2. It is the Fujita-Zariski decomposition with $P=P_f$ semiample.
3. It is the $s$-decomposition with $P=P_s$ semiample.
$(1) \Rightarrow (2)$: It is easy to check that the divisorial Zariski decomposition with the nef positive part is the Fujita-Zariski decomposition (see [@nakayama III.1.17.Remark]).\
$(2) \Rightarrow (3)$: Let $D=P_s+N_s$ be the $s$-decomposition. Then $P_f \geq P_s$ by definition. Since $P_f$ is semiample, we also have $P_f \leq P_s$. Therefore $P_f=P_s$.\
$(3) \Rightarrow (1)$: It follows from Lemma \[abundantdiv=s\].
\[defbirgoodzd\] If one of the conditions in Proposition \[goodzd\] holds for an abundant ${\mathbb{Q}}$-divisor $D$, then we say that $D$ *admits the good Zariski decomposition*, and denote it by $D=P+N$. We say that $D$ *admits the birational good Zariski decomposition* if there exists a birational morphism $f \colon \widetilde{X} \to X$ from a smooth projective variety such that $f^*D$ admits the good Zariski decomposition.
\[qgoodzd\] Let $D$ be a pseudoeffective ${\mathbb{Q}}$-divisor with the good Zariski decomposition $D=P+N$. Then $P, N$ are also ${\mathbb{Q}}$-divisors.
Since $P$ is semiample, there exists a morphism $f \colon X \to Y$ such that $P \sim_{{\mathbb{R}}} f^* A$ where $A$ is an ample divisor on $Y$. The ample divisor $A$ can be written as a finite sum of ample Cartier divisors on $Y$ with positive real coefficients. Thus we can write $P \sim_{{\mathbb{R}}} \sum_{i=1}^k a_i P_i$ for some semiample Cartier divisors $P_i$ and some positive real numbers $a_i$. Now we write $N = \sum_{j=1}^m b_j N_j$ for prime divisors $N_1, \ldots, N_m$ and positive real numbers $b_j$. Then $N_1, \ldots, N_m$ are linearly independent in $\operatorname{N^1}(X)_{{\mathbb{R}}}$ by [@nakayama III.1.10.Proposition]. Let $V_P$ and $V_N$ be the subspaces of $\operatorname{N^1}(X)_{\mathbb{R}}$ spanned by $\{P_i\}_{i=1}^k$ and $\{N_j\}_{j=1}^m$, respectively. We now claim that $V_P\cap V_N=\{0\}$. Suppose that the claim does not hold. Then there exists a nonzero class $\eta\in V_P\cap V_N$ such that $\eta\equiv P'\equiv N'$ where $P'\in \bigoplus_{i=1}^k {\mathbb{R}}\cdot P_i$ and $N'\in \bigoplus_{j=1}^m {\mathbb{R}}\cdot N_j$. Note that there exists a positive number ${\varepsilon}>0$ such that for any real number $r$ satisfying $|r|<{\varepsilon}$, the divisor $P-r P'$ is nef and $N+r N'$ is effective. Thus [@nakayama Proposition III.1.14 (2)] implies that in the following decompositions $$\begin{array}{rl}
D&=P+N\\
&\equiv (P-r P')+(N+r N'),
\end{array}$$ we have $N\leq N+r N'$, hence $0\leq r N'$ for any $r$ such that $|r|<{\varepsilon}$. However, since $N'$ is a nonzero divisor, this is a contradiction. The claim implies that if $D$ is a ${\mathbb{Q}}$-divisor, then so is $N$ in the decomposition $D = P+N$. Therefore $P, N$ are both ${\mathbb{Q}}$-divisors.
Now, we characterize when a divisor admits the birational good Zariski decomposition.
\[zdabfg\] Let $D$ be a pseudoeffective ${\mathbb{Q}}$-divisor on $X$. Then $D$ admits the birational good Zariski decomposition if and only if $D$ is abundant and $R(X, D)$ is finitely generated.
Suppose that there exists a birational morphism $f \colon \widetilde{X} \to X$ from a smooth projective variety such that $f^*D = P+N$ is the good Zariski decomposition. By definition, $D$ is abundant. Note that $R(X, D) \simeq R(\widetilde{X}, f^*D) \simeq R(\widetilde{X}, P)$. Since $P$ is a semiample ${\mathbb{Q}}$-divisor by Proposition \[qgoodzd\], it follows from Proposition \[semiampleabundant\] that $R(X, D)$ is finitely generated. Conversely, suppose that $D$ is abundant and $R(X, D)$ is finitely generated. For a sufficiently large and divisible integer $m>0$, we take a resolution $f \colon \widetilde{X} \to X$ of the base locus of $|mD|$ and consider the decomposition $f^*(mD)=M+F$ into the base point free $M$ and the fixed part $F$ of $|f^*mD|$ By the finite generation of $R(X, D)$, we see that $f^*D=\frac{1}{m}M+\frac{1}{m}F$ is the $s$-decomposition with semiample positive part. By Proposition \[goodzd\], $f^*D$ admits the good Zariski decomposition.
Okounkov bodies {#okbdsubsec}
===============
In this section, we recall the construction of Okounkov bodies associated to pseudoeffective divisors in [@lm-nobody], [@KK], and [@CHPW1] and basic results. In the end, we prove Theorem \[newthrm\] (=Theorem \[newtheorem\]).
First, fix an admissible flag on $X$ $$Y_\bullet: X=Y_0\supseteq Y_1\supseteq\cdots \supseteq Y_{n-1}\supseteq Y_n=\{x\}$$ where each $Y_i$ is an irreducible subvariety of codimension $i$ in $X$ and is smooth at $x$. Let $D$ be an ${\mathbb{R}}$-divisor on $X$ with $|D|_{{\mathbb{R}}}:=\{ D' \mid D \sim_{{\mathbb{R}}} D' \geq 0 \}\neq\emptyset$. We define a valuation-like function $$\nu_{Y_\bullet}:|D|_{{\mathbb{R}}}\to {\mathbb{R}}_{\geq0}^n$$ as follows. For $D'\in |D|_{\mathbb{R}}$, let $$\nu_1=\nu_1(D'):=\operatorname{ord}_{Y_1}(D').$$ Since $D'-\nu_1(D')Y_1$ is effective, we can define $$\nu_2=\nu_2(D'):=\operatorname{ord}_{Y_2}((D'-\nu_1Y_1)|_{Y_1}).$$ If $\nu_i=\nu_i(D')$ is defined, then we define $\nu_{i+1}=\nu_{i+1}(D')$ inductively as $$\nu_{i+1}(D'):=\operatorname{ord}_{Y_{i+1}}((\cdots((D'-\nu_1Y_1)|_{Y_1}-\nu_2Y_2)|_{Y_2}-\cdots-\nu_iY_i)|_{Y_{i}}).$$ The values $\nu_i(D')$ for $1 \leq i$ obtained as above define $\nu_{Y_\bullet}(D')=(\nu_1(D'),\nu_2(D'),\cdots,\nu_n(D'))$.
The *Okounkov body* ${\Delta}_{Y_\bullet}(D)$ of a big ${\mathbb{R}}$-divisor $D$ with respect to an admissible flag $Y_\bullet$ is defined as the closure of the convex hull of $\nu_{Y_\bullet}(|D|_{{\mathbb{R}}})$ in ${\mathbb{R}}^n_{\geq 0}$.
More generally, a similar construction can be applied to a graded linear series $W_\bullet$ on $X$ to construct the Okounkov body ${\Delta}_{Y_\bullet}(W_\bullet)$ of $W_\bullet$. For more details, we refer to [@lm-nobody].
When $D$ is not big, we have the following extension introduced in [@CHPW1].
Let $D$ be an ${\mathbb{R}}$-divisor on $X$.
1. When $D$ is effective up to $\sim_{\mathbb{R}}$, i.e., $|D|_{{\mathbb{R}}}\neq \emptyset$, the *valuative Okounkov body* ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ of $D$ with respect to an admissible flag $Y_\bullet$ is defined as the closure of the convex hull of $\nu_{Y_\bullet}(|D|_{{\mathbb{R}}})$ in ${\mathbb{R}}^n_{\geq 0}$. If $|D|_{\mathbb{R}}=\emptyset$, then we set ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D):=\emptyset$.
2. When $D$ is pseudoeffective, the *limiting Okounkov body* ${\Delta^{\lim}}_{Y_\bullet}(D)$ of $D$ with respect to an admissible flag $Y_\bullet$ is defined as $${\Delta^{\lim}}_{Y_\bullet}(D):=\lim_{{\varepsilon}\to 0+}{\Delta}_{Y_\bullet}(D+{\varepsilon}A) = \bigcap_{{\varepsilon}>0} {\Delta}_{Y_\bullet}(D+{\varepsilon}A),$$ where $A$ is an ample divisor on $X$. (Note that ${\Delta^{\lim}}_{Y_\bullet}(D)$ is independent of the choice of $A$.) If $D$ is not pseudoeffective, we set ${\Delta^{\lim}}_{Y_\bullet}(D)
:=\emptyset$.
Boucksom’s numerical Okounkov body ${\Delta^{\text{num}}}_{Y_\bullet}(D)$ in [@B2] is the same as our limiting Okounkov body ${\Delta^{\lim}}_{Y_\bullet}(D)$.
Suppose that $D$ is effective. By definition, ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D) \subseteq {\Delta^{\lim}}_{Y_\bullet}(D)$, and the inclusion can be strict in general (see [@CHPW1 Examples 4.2 and 4.3]). Moreover, by [@B2 Proposition 3.3 and Lemma 4.8], we have $$\dim {\Delta^{\operatorname{val}}}_{Y_\bullet}(D) = \kappa (D) \leq \dim {\Delta^{\lim}}_{Y_\bullet}(D) \leq {\kappa_\nu}(D).$$
The following lemmas will be useful for computing Okounkov bodies.
\[okbdbir\] Let $D$ be an ${\mathbb{R}}$-divisor on $X$. Consider a birational morphism $f : \widetilde{X} \to X$ with $\widetilde{X}$ smooth and an admissible flag $$\widetilde{Y}_\bullet : \widetilde{X}=\widetilde{Y}_0 \supseteq \widetilde{Y}_1 \supseteq \cdots \supseteq \widetilde{Y}_{n-1} \supseteq \widetilde{Y}_n=\{ x' \}.$$ on $\widetilde{X}$. Suppose that $Y_n$ is a general point in $X$ and $$Y_\bullet:=f(\widetilde{Y}_\bullet) : X=Y_0 \supseteq Y_1=f(\widetilde{Y}_1) \supseteq \cdots \supseteq Y_{n-1}=f(\widetilde{Y}_{n-1}) \supseteq Y_n=f(\widetilde{Y}_n)=\{ f(x') \}.$$ is an admissible flag on $X$. Then we have ${\Delta^{\operatorname{val}}}_{\widetilde{Y}_\bullet}(f^*D)={\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ and ${\Delta^{\lim}}_{\widetilde{Y}_\bullet}(f^*D) = {\Delta^{\lim}}_{Y_\bullet}(D)$.
The limiting Okounkov body case is shown in [@CHPW2 Lemma 3.3]. The proof for the valuative Okounkov body case is almost identical and we leave the details to the readers as an exercise.
\[okbdzd\] Let $D$ be an ${\mathbb{R}}$-divisor on $X$ with the $s$-decomposition $D=P_s+N_s$ and the divisorial Zariski decomposition $D=P_\sigma+N_\sigma$. Fix an admissible flag $Y_\bullet$ on $X$ such that $Y_n$ is a general point in $X$. Then we have ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)={\Delta^{\operatorname{val}}}_{Y_\bullet}(P_s)$ and ${\Delta^{\lim}}_{Y_\bullet}(D)={\Delta^{\lim}}_{Y_\bullet}(P_\sigma)$, respectively.
The first assertion follows from the fact that $R(X, D) \simeq R(X, P_s)$ and the construction of the valuative Okounkov body. The second assertion is nothing but [@CHPW2 Lemma 3.5].
Finally, we give a proof of the main result of this section. The following key result is implicitly used in [@CHPW1] (especially in the proof of [@CHPW1 Theorem B]) and in this paper as well. We include the complete proof here.
\[newtheorem\] Let $X$ be a smooth projective variety of dimension $n$, and $D$ be a big divisor on $X$. Fix an admissible flag $Y_\bullet$ such that $Y_{n-k} \not\subseteq {\mathbf B_+}(D)$. Then we have $${\Delta}_{Y_{n- k\bullet}}(D) = {\Delta}_{Y_\bullet}(D) \cap (\{ 0\}^{n-k} \times {\mathbb{R}}_{\geq 0}^{k}).$$
We may assume that each $Y_i$ is a smooth variety. Let $\{ A_i \}$ be a sequence of ample divisors on $X$ such that each $D+A_i$ is a ${\mathbb{Q}}$-divisor and $\lim\limits_{i \to \infty} A_i = 0$. Then we have $${\Delta}_{Y_\bullet}(D) = \bigcap_{i=1}^{\infty} {\Delta}_{Y_\bullet}(D+A_i) \text{ and } {\Delta}_{Y_{n-k\bullet}}(D) = \bigcap_{i=1}^{\infty} {\Delta}_{Y_{n-k\bullet}}(D+A_i).$$ Furthermore, $Y_{n-k} \not\subseteq {\mathbf B_+}(D+A_i)$ for all $i$. Note that it is enough to prove the statement for the ${\mathbb{Q}}$-divisors $D+A_i$ for all sufficiently large $i$. Thus we assume below that $D$ is a ${\mathbb{Q}}$-divisor.
It is easy to check that ${\Delta}_{Y_{n-k\bullet}}(D) \subseteq {\Delta}_{Y_\bullet}(D)$. This implies that ${\Delta}_{Y_{n-k\bullet}}(D) \subseteq {\Delta}_{Y_\bullet}(D) \cap (\{ 0\}^{n-k} \times {\mathbb{R}}_{\geq 0}^{k})$ by definition. Suppose that the inclusion is strict: $${\Delta}_{Y_{n-k\bullet}}(D) \subsetneq {\Delta}_{Y_\bullet}(D) \cap (\{ 0\}^{n-k} \times {\mathbb{R}}_{\geq 0}^{k}).$$ Then there exists a point $(0^{n-k}, x_1, \ldots, x_k) \in {\Delta}_{Y_\bullet}(D) \cap (\{ 0\}^{n-k} \times {\mathbb{R}}_{\geq 0}^{k})$, but $(0^{n-k}, x_1, \ldots, x_k) \not\in {\Delta}_{Y_{n-k\bullet}}(D)$.
Let $A$ be an ample ${\mathbb{Q}}$-divisor on $X$. Note that ${\Delta}_{Y_{n-k\bullet}}(D) \subseteq {\Delta}_{Y_{n-k\bullet}}(D+{\varepsilon}A)$ for any ${\varepsilon}\geq 0$. Since $Y_{n-k} \not\subseteq {\mathbf B_+}(D+{\varepsilon}A)$, we have $\operatorname{vol}_{{\mathbb{R}}^k} {\Delta}_{Y_{n-k\bullet}}(D+{\varepsilon}A)= \frac{1}{(n-k)!}\operatorname{vol}_{X|Y_{n-k}}(D+{\varepsilon}A)$. Recall that by [@elmnp-restricted; @vol; @and; @base; @loci Theorem A], the function $\operatorname{vol}_{X|Y_{n-k}} \colon \text{Big}^{Y_{n-k}}(X) \to {\mathbb{R}}$ is continuous, where $\text{Big}^{Y_{n-k}}(X)$ denotes the cone in $\operatorname{N^1}(X)_{\mathbb{R}}$ consisting of the real divisor classes $\eta$ such that $Y_{n-k}$ is not properly contained in any of the irreducible components of ${\mathbf B_+}(\eta)$. Thus we can find a rational number ${\varepsilon}>0$ such that $(x_1, \ldots, x_k)\not\in{\Delta}_{Y_{n-k\bullet}}(D+{\varepsilon}A)$ and $$\operatorname{vol}_{{\mathbb{R}}^k} {\Delta}_{Y_{n-k\bullet}}(D+{\varepsilon}A) < \operatorname{vol}_{{\mathbb{R}}^k} \Delta$$ where $\Delta\subseteq {\mathbb{R}}^k$ is the convex hull of the set ${\Delta}_{Y_{n-k\bullet}}(D)$ and the point $(x_1, \ldots, x_k)$. Note that we can fix a small neighborhood $U$ of $(x_1, \ldots, x_k)$ in ${\mathbb{R}}^k$ which is disjoint from ${\Delta}_{Y_{n-k\bullet}}(D+{\varepsilon}A)$. There exists a sufficiently small $\delta >0$ such that the divisors $$\begin{array}{rcl}
A_1=A_1(\delta_1)& \sim_{{\mathbb{Q}}} &\frac{1}{2}{\varepsilon}A+\delta_1 Y_1,\\
A_2=A_2(\delta_1,\delta_2)& \sim_{{\mathbb{Q}}} &A_1|_{Y_1}+\delta_2 Y_2,\\
&\vdots&\\
A_{n-k}=A_{n-k}(\delta_1,\delta_2,\ldots,\delta_{n-k})& \sim_{{\mathbb{Q}}} &A_{n-k-1}|_{Y_{n-k-1}}+\delta_{n-k} Y_{n-k}
\end{array}$$ are successively ample for any $\delta_j$ satisfying $\delta \geq \delta_1, \delta_2, \ldots, \delta_{n-k} >0$. Since $(0^{n-k}, x_1, \ldots, x_k) \in {\Delta}_{Y_\bullet}(D)$, there exists a sequence of valuative points $$\mathbf x_i=(\delta_1^i, \ldots, \delta_{n-k}^i, x_1^i, \ldots, x_k^i)\in{\Delta}_{Y_\bullet}(D)$$ such that $$\lim_{i \to \infty} \delta_j^i = 0 \text{ for $1 \leq j \leq n-k$\;\; and\; }\lim_{i \to \infty} x_l^i = x_l \text{ for $1 \leq l \leq k$}.$$ Since it is known that the set of rational valuative points $\{\nu_{Y_\bullet}(D')| D\sim_{\mathbb{Q}}D'\geq 0\}$ is dense in ${\Delta}_{Y_\bullet}(D)$, we may assume that $\mathbf x_i\in \{\nu_{Y_\bullet}(D')| D\sim_{\mathbb{Q}}D'\geq 0\}$ so that $\mathbf x_i\in{\mathbb{Q}}^n$ for all $i$. We now fix a sufficiently large $i$ such that $0\leq \delta_j^i < \delta$ for all $1 \leq j \leq n-k$ and $(x_1^i, \ldots, x_k^i)$ lies in the small neighborhood $U$ in ${\mathbb{R}}^k$ of $(x_1, \ldots, x_k)$. Since $\mathbf x_i$ is a rational valuative point of ${\Delta}_{Y_\bullet}(D)$, there exist an effective divisor $D'\sim_{\mathbb{Q}}D$ such that $\nu_{Y_\bullet}(D')=\mathbf x_i$. Namely, we have $$\begin{array}{rcl}
D'&=&D_1 + \delta_1^i Y_1,\\
D_1|_{Y_1}&=&D_2 + \delta_2^i Y_2,\\
&\vdots& \\
D_{n-k-1}|_{Y_{n-k-1}}&=&D_{n-k}+\delta_{n-k}^i Y_{n-k}
\end{array}$$ where $D_j$ on $Y_{j-1}$ ($j=1,\ldots, n-k$) are effective divisors. Now note that we have $$D'+\frac{1}{2}{\varepsilon}A = D_1 + \left( \frac{1}{2}{\varepsilon}A + \delta_1^i Y_1 \right) \sim_{{\mathbb{Q}}} D_1 + A'_1$$ where we may assume that $A'_1$ is an effective ample divisor such that $\operatorname{mult}_{Y_1} A'_1=0$. We also have $$(D_1+A'_1)|_{Y_1} = D_2 + (A'_1|_{Y_1} + \delta_2^i Y_2) \sim_{{\mathbb{Q}}} D_2 + A'_2$$ where we may assume that $A'_2$ is an effective ample divisor such that $\operatorname{mult}_{Y_2} A'_2=0$. By continuing this process, we finally obtain $$(D_{n-k-1} + A'_{n-k-1})|_{Y_{n-k-1}} = D_{n-k} + (A'_{n-k-1}|_{Y_{n-k-1}} + \delta_{n-k}^i Y_{n-k}) \sim_{{\mathbb{Q}}} D_{n-k} + A'_{n-k}$$ where we may assume that $A'_{n-k}$ is an effective ample divisor such that $\operatorname{mult}_{Y_{n-k}} A'_{n-k}=0$.
We now claim that there exists an effective divisor $D'' \sim_{{\mathbb{Q}}} D+{\varepsilon}A$ such that $D''|_{Y_{n-k-1}} = D_{n-k}+E$ for some effective divisor $E$ with $\operatorname{mult}_{Y_{n-k}}E=0$ and $\nu_{Y_{n-k\bullet}}(E|_{Y_{n-k}}) = (x_1', \ldots, x_k')$ where we may assume that $x_j'\geq 0$ are arbitrarily small. Note that such $D''$ defines a rational valuative point $\nu_{Y_{\bullet}}(D'') = (0^{n-k}, x_1^i+x_1', \ldots, x_k^i + x_k') \in {\Delta}_{Y_\bullet}(D + {\varepsilon}A)$. Thus $(x_1^i+x_1', \ldots, x_k^i + x_k') \in {\Delta}_{Y_{n-k \bullet}}(D+{\varepsilon}A)$. If our claim holds, then we can conclude that $(x_1^i+x_1', \ldots, x_k^i + x_k')$ belongs to the small neighborhood $U$ of $(x_1, \ldots, x_k)$ in ${\mathbb{R}}^k$, which is a contradiction since $U$ is disjoint from ${\Delta}_{Y_{n-k\bullet}}(D+{\varepsilon}A)$. Therefore we finally obtain ${\Delta}_{Y_{n-k\bullet}}(D) = {\Delta}_{Y_\bullet}(D) \cap (\{ 0\}^{n-k} \times {\mathbb{R}}_{\geq 0}^{k})$.
It now remains to show the claim. For a sufficiently divisible and large integer $m>0$, we take a log resolution $f_m \colon \widetilde{X}_m \to X$ of the base ideal of $|m(D+\frac{1}{2}{\varepsilon}A)|$ so that we obtain a decomposition $f_m^*(m(D+\frac{1}{2}{\varepsilon}A)) = M_m' + F_m'$ into a base point free divisor $M_m'$ and the fixed part $F_m'$ of $|f_m^*(m(D+\frac{1}{2}{\varepsilon}A))|$. Let $M_m:=\frac{1}{m}M_m'$. We may assume that $f_m$ is isomorphic outside ${\mathbf B_+}(D+\frac{1}{2}{\varepsilon}A)$. We can take smooth strict transforms $\widetilde{Y}_i^m$ on $\widetilde{X}_m$ of $Y_i$ for $1 \leq i \leq n-k$. For a general point $y$ in $\widetilde{Y}_{n-k}^m$, we have the positive moving Seshadri constant ${\varepsilon}(||D+\frac{1}{2}{\varepsilon}A||; f_m(y)) > 0$. Thus we also have the positive Seshadri constant ${\varepsilon}(M_m; y) >0$ for $m \gg 0$ so that $\widetilde{Y}_{n-k}^m \not\subseteq {\mathbf B_+}(M_m)$. Let $g_m \colon \widetilde{X}_m \to Z_m$ be the birational morphism defined by $|M_m'|$. Possibly by taking a further blow-up of $\widetilde{X}_m$, we may assume that every irreducible component of the exceptional locus of $g_m$ is a divisor. We can still assume that $f_m$ is isomorphic over a general point in $Y_{n-k}$. The divisor $H_m:=M_m - E_m$ is ample for any sufficiently small effective divisor $E_m$ whose support is the $g_m$-exceptional locus. Note that $\operatorname{mult}_{\widetilde{Y}_{n-k}^m}(E_m)=0$. Let $f_m^*(D+\frac{1}{2}{\varepsilon}A)=P_m+N_m$ be the divisorial Zariski decomposition. As in [@lehmann-nu Proof of Proposition 3.7], by applying [@elmnp-asymptotic; @inv; @of; @base Proposition 2.5], we see that $P_m - M_m$ is arbitrarily small if we take a sufficiently large $m>0$. Since we may take an arbitrarily small $E_m$, so is $P_m - H_m$ for a sufficiently large $m>0$.
For simplicity, we fix a sufficiently large integer $m>0$ and we denote $f=f_m$, $\widetilde{X}=\widetilde{X}_m$ and $\widetilde{Y}_i=\widetilde{Y}_i^m$. Let $f^*(D+\frac{1}{2}{\varepsilon}A) = P+N$ be the divisorial Zariski decomposition. Then as we have seen above, we can assume that $P$ can be arbitrarily approximated by an ample divisor $H$ on $\widetilde{X}$ such that $F=f^*(D+\frac{1}{2}{\varepsilon}A)-H$ is an effective divisor satisfying $\operatorname{mult}_{\widetilde{Y}_{n-k}}(F)=0$. Note that $F-N$ is an arbitrarily small effective divisor such that $\operatorname{mult}_{\widetilde{Y}_{n-k}}(F-N)=0$. Thus we can find an effective divisor $A_0 \sim_{{\mathbb{Q}}} A$ such that $\operatorname{mult}_{Y_{n-k-1}}A_0=0$, $E_0:=\frac{1}{2}{\varepsilon}f^*A_0|_{\widetilde{Y}_{n-k-1}} - (F-N)|_{\widetilde{Y}_{n-k-1}}$ is effective, and $\operatorname{mult}_{\widetilde{Y}_{n-k}}E_0=0$. Let $f^*D=P'+N'$ be the divisorial Zariski decomposition. Since $P'+f^*(\frac{1}{2}{\varepsilon}A)$ is movable, we get $P \geq P'+f^*(\frac{1}{2}{\varepsilon}A)$ and so $N' \geq N$. Since $Y_{n-k} \not\subseteq {\mathbf B_+}(D)$, every irreducible component of $N'$ cannot contain $\widetilde{Y}_{n-k-1}$. Clearly, $f^*D_{n-k} - N'|_{\widetilde{Y}_{n-k-1}}$ is effective, and so is $f^*D_{n-k} - N|_{\widetilde{Y}_{n-k-1}}$. Thus $$E_1:=f^*(D_{n-k} + A_{n-k}') - N|_{\widetilde{Y}_{n-k-1}} + E_0 = f^*(D_{n-k}+A_{n-k}') - F|_{\widetilde{Y}_{n-k-1}} + \frac{1}{2}{\varepsilon}f^*A_0|_{\widetilde{Y}_{n-k-1}}$$ is an effective divisor on $\widetilde{Y}_{n-k-1}$. Note that $E_1 \sim_{{\mathbb{Q}}} (H+\frac{1}{2}{\varepsilon}f^*A)|_{\widetilde{Y}_{n-k-1}}$. Since $$H^0\left(\widetilde{X}, m\left(H+\frac{1}{2}{\varepsilon}f^*A\right)\right) \to H^0\left(\widetilde{Y}_{n-k-1}, m\left(H+\frac{1}{2}{\varepsilon}f^*A\right)\Bigm|_{\widetilde{Y}_{n-k-1}}\right)$$ is surjective for all sufficiently divisible integers $m>0$, it follows that there exists $H' \sim_{{\mathbb{Q}}} H+\frac{1}{2}{\varepsilon}f^*A$ such that $H'|_{\widetilde{Y}_{n-k-1}} = E_1$. Then we have $$(H' + F)|_{\widetilde{Y}_{n-k-1}} = E_1 + F|_{\widetilde{Y}_{n-k-1}} = f^*D_{n-k} +E'$$ where $$E':=f^*A_{n-k}' + (F-N)|_{\widetilde{Y}_{n-k-1}} + E_0 = f^*A_{n-k}' + \frac{1}{2}{\varepsilon}f^*A_0|_{\widetilde{Y}_{n-k-1}}$$ is an effective divisor. Note that $\operatorname{mult}_{\widetilde{Y}_{n-k}}E'=0$. We may also assume that each $x_j'\geq 0$ is arbitrarily small in $\nu_{\widetilde{Y}_{n-k\bullet}}(E'|_{\widetilde{Y}_{n-k}}) = (x_1', \ldots, x_k')$. By letting $D'':=f_*(H'+F) \sim_{{\mathbb{Q}}} D + {\varepsilon}A$ and $E:=f_*E'$, we obtain the divisors satisfying the required properties. This shows the claim, and hence, we complete the proof.
Nakayama subvarieties and positive volume subvarieties {#nakpvssec}
======================================================
In [@CHPW1], we introduced Nakayama subvarieties and positive volume subvarieties of divisors. We now further study those subvarieties, and prove Theorem \[critintro\](=Theorem \[geomcrit\]) in this section. We first recall the definitions of those subvarieties.
Let $D$ be an ${\mathbb{R}}$-divisor on $X$.
1. When $D$ is effective, a *Nakayama subvariety of $D$* is an irreducible subvariety $U \subseteq X$ such that $\dim U=\kappa(D)$ and for every integer $m \geq 0$ the natural map $$H^0(X, \lfloor mD \rfloor) \to H^0(U, \lfloor mD|_U \rfloor)$$ is injective (or equivalently, $H^0(X, {\mathcal}I_U \otimes {\mathcal}O_X(\lfloor mD \rfloor))=0$ where ${\mathcal}I_U$ is an ideal sheaf of $U$ in $X$).
2. When $D$ is pseudoeffective, a *positive volume subvariety of $D$* is an irreducible subvariety $V \subseteq X$ such that $\dim V = {\kappa_\nu}(D)$ and $\operatorname{vol}_{X|V}^+(D)>0$.
In [@CHPW1], we required an additional condition $V \not \subseteq {\mathbf B_-}(D)$ for the definition of positive volume subvariety. However, we can drop this condition by Lemma \[notinbm\]. Note that $V \not \subseteq {\mathbf B_-}(D)$ does not imply $\operatorname{vol}_{X|V}^+(D)>0$ (see [@CHPW1 Example 2.14]).
\[notinbm\] Let $D$ be a pseudoeffective ${\mathbb{R}}$-divisor on $X$. If $V$ is a positive volume subvariety of $D$, then $V \not\subseteq {\mathbf B_-}(D)$.
If $V \subseteq {\mathbf B_-}(D)$, then there is a sequence $\{ A_i \}$ of ample divisors on $X$ such that $\lim_{i \to \infty} A_i = 0$ and $V \subseteq \operatorname{SB}(D+A_i)$. Then $\operatorname{vol}_{X|V}(D+A_i)=0$, so $\operatorname{vol}_{X|V}^+(D)=0$. Thus $V$ is not a positive volume subvariety of $D$.
Even if $V$ is a positive volume subvariety of $D$, it is possible that $V \subseteq \operatorname{SB}(D)$. For instance, consider a ruled surface $S$ carrying a nef divisor $D$ such that $D\cdot C>0$ for every irreducible curve $C \subseteq S$, but $D$ is not ample (see e.g., [@pos Example 1.5.2]). Since $\kappa(D)=-\infty$, we have $\operatorname{SB}(D)=S$. Thus every positive volume subvariety of $D$ is contained in $\operatorname{SB}(D)$.
\[gensub\] When $\kappa(D)=0$ (resp. ${\kappa_\nu}(D)=0$), every point not in $\operatorname{Supp}(D)$ (resp. ${\mathbf B_-}(D)$) is a Nakayama (resp. positive volume) subvariety of $D$. When $\kappa(D)>0$, any $\kappa(D)$-dimensional general subvariety (e.g., intersection of general ample divisors) is a Nakayama subvariety of $D$ ([@CHPW1 Proposition 2.9]). Similarly, when ${\kappa_\nu}(D)>0$, any ${\kappa_\nu}(D)$-dimensional intersection of sufficiently ample divisors is a positive volume subvariety of $D$ ([@CHPW1 Proposition 2.17]). In particular, we can always construct an admissible flag $Y_\bullet$ on $X$ containing a Nakayama subvariety of $D$ or a positive volume subvariety of $D$ such that $Y_n$ is a general point in $X$.
The importance of such special subvarieties associated to divisors is that one can read off interesting asymptotic properties of divisors from Okounkov bodies with respect to admissible flags containing those subvarieties. The following theorem is the main result of [@CHPW1], which can be regarded as a generalization of [@lm-nobody Theorem A].
\[chpwmain\] We have the following:
1. Let $D$ be an effective ${\mathbb{R}}$-divisor on $X$. Fix an admissible flag $Y_\bullet$ containing a Nakayama subvariety $U$ of $D$ such that $Y_n$ is a general point in $X$. Then ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D) \subseteq \{0 \}^{n-\kappa(D)} \times {\mathbb{R}}^{\kappa(D)}$ so that one can regard ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D) \subseteq {\mathbb{R}}^{\kappa(D)}$. Furthermore, we have $$\dim {\Delta^{\operatorname{val}}}_{Y_\bullet}(D)=\kappa(D) \text{ and } \operatorname{vol}_{{\mathbb{R}}^{\kappa(D)}}({\Delta^{\operatorname{val}}}_{Y_\bullet}(D))=\frac{1}{\kappa(D)!} \operatorname{vol}_{X|U}(D).$$
2. Let $D$ be a pseudoeffective ${\mathbb{R}}$-divisor on $X$, and fix an admissible flag $Y_\bullet$ containing a positive volume subvariety $V$ of $D$. Then ${\Delta^{\lim}}_{Y_\bullet}(D) \subseteq \{0 \}^{n-{\kappa_\nu}(D)} \times {\mathbb{R}}^{{\kappa_\nu}(D)}$ so that one can regard ${\Delta^{\lim}}_{Y_\bullet}(D) \subseteq {\mathbb{R}}^{{\kappa_\nu}(D)}$. Furthermore, we have $$\dim {\Delta^{\lim}}_{Y_\bullet}(D)={\kappa_\nu}(D) \text{ and } \operatorname{vol}_{{\mathbb{R}}^{{\kappa_\nu}(D)}}({\Delta^{\lim}}_{Y_\bullet}(D))=\frac{1}{{\kappa_\nu}(D)!} \operatorname{vol}_{X|V}^+(D).$$
To extract asymptotic properties of divisors from ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ as in Theorem \[chpwmain\] (1), we need to assume that $Y_n$ is a general point in $X$. When considering ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ (resp. ${\Delta^{\lim}}_{Y_\bullet}(D)$, we say that $Y_n$ is *general* if $Y_n$ is not contained in $\operatorname{SB}(D)$ (resp. ${\mathbf B_-}(D)$) (see [@lm-nobody Lemma 2.6] and [@CHPW1 Subsection 3.2]).
As an application of Theorem \[chpwmain\], we now prove the following Theorem \[critintro\].
\[geomcrit\] Let $D$ be an ${\mathbb{R}}$-divisor on $X$. Fix an admissible flag $Y_\bullet$ such that $Y_n$ is a general point in $X$. We have the following:
1. If $D$ is effective, then $Y_\bullet$ contains a Nakayama subvariety of $D$ if and only if ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D) \subseteq \{0 \}^{n-\kappa(D)} \times {\mathbb{R}}^{\kappa(D)}$.
2. If $D$ is pseudoeffective, then $Y_\bullet$ contains a positive volume subvariety of $D$ if and only if ${\Delta^{\lim}}_{Y_\bullet}(D) \subseteq \{0 \}^{n-{\kappa_\nu}(D)} \times {\mathbb{R}}^{{\kappa_\nu}(D)}$ and $\dim {\Delta^{\lim}}_{Y_\bullet}(D)={\kappa_\nu}(D)$.
The $(\Rightarrow)$ direction of both $(1)$ and $(2)$ at once follows from Theorem \[chpwmain\]. For the $(\Leftarrow)$ direction of $(1)$, note that $\operatorname{ord}_{Y_{n-\kappa(D)}}(D')=0$ for every effective divisor $D' \sim_{{\mathbb{R}}} D$ under the assumption that ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D) \subseteq \{0 \}^{n-\kappa(D)} \times {\mathbb{R}}^{\kappa(D)}$. This means that $H^0(X, \mathcal{I}_{Y_{n-\kappa(D)}} \otimes \mathcal{O}_X(\lfloor mD \rfloor)) = 0$ for every integer $m \geq 0$. Thus $Y_{n-\kappa(D)}$ is a Nakayama subvariety of $D$.
For the $(\Leftarrow)$ direction of $(2)$, take an arbitrary ample divisor $A$ on $X$. Since ${\Delta}_{Y_\bullet}(D+A) \supseteq {\Delta^{\lim}}_{Y_\bullet}(D)$, it follows that $${\Delta}_{Y_\bullet}(D+A)\cap(\{0\}^{n-{\kappa_\nu}(D)}\times{\mathbb{R}}_{\geq0}^{{\kappa_\nu}(D)})\supseteq{\Delta^{\lim}}_{Y_{\bullet}}(D).
$$ Since $Y_n$ is general, we have $Y_{n-{\kappa_\nu}(D)} \not\subseteq {\mathbf B_-}(D)$. Thus $Y_{n-{\kappa_\nu}(D)} \not\subseteq {\mathbf B_+}(D+A)$ and using Theorem \[newtheorem\], we obtain ${\Delta}_{Y_{n-{\kappa_\nu}(D)\bullet}}(D+A)\supseteq{\Delta^{\lim}}_{Y_{\bullet}}(D)$. Therefore, by [@lm-nobody (2.7)] we have $$\begin{array}{rl}
\operatorname{vol}_{X|Y_{n-{\kappa_\nu}(D)}}(D+A)&= {\kappa_\nu}(D)!\cdot \operatorname{vol}_{{\mathbb{R}}^{{\kappa_\nu}(D)}}{\Delta}_{Y_{n-{\kappa_\nu}(D)\bullet}}(D+A)\\
&\geq {\kappa_\nu}(D)!\cdot \operatorname{vol}_{{\mathbb{R}}^{{\kappa_\nu}(D)}}{\Delta^{\lim}}_{Y_{\bullet}}(D).
\end{array}$$ The given condition implies that $\operatorname{vol}_{{\mathbb{R}}^{{\kappa_\nu}(D)}}{\Delta^{\lim}}_{Y_{\bullet}}(D)>0$. Hence, $\operatorname{vol}_{X|Y_{n-{\kappa_\nu}(D)}}^+(D)>0$, and by definition $Y_{n-{\kappa_\nu}(D)}$ is a positive volume subvariety of $D$.
Regarding Theorem \[geomcrit\] (1), we recall that $\dim {\Delta^{\operatorname{val}}}_{Y_\bullet}(D)=\kappa(D)$ always holds whenever $D$ is effective by [@B2 Proposition 3.3].
Rational polyhedrality of Okounkov bodies {#ratsec}
=========================================
This section is devoted to showing the rational polyhedrality of Okounkov bodies of pseudoeffective divisors. We then finally prove Theorem \[main1\] (=Corollary \[ratpolval\] and Theorem \[ratsimlim\]). First, we study the Okounkov bodies under surjective morphisms.
\[morokbd\] Let $f \colon X \to \overline{X}$ be a surjective morphism of projective varieties of the same dimension $n$, and fix an admissible flag $$Y_\bullet: X=Y_0\supseteq Y_1\supseteq\cdots \supseteq Y_{n-1}\supseteq Y_n=\{x\}$$ on $X$ such that $$\overline{Y}_\bullet : \overline{X}=f(Y_0)\supseteq f(Y_1)\supseteq \cdots \supseteq f(Y_{n-1}) \supseteq f(Y_n)=\{f(x) \}$$ is an admissible flag on $\overline{X}$. For a big ${\mathbb{Z}}$-divisor $D$ on $\overline{X}$, consider a graded linear series $W_\bullet$ associated to $f^*D$ on $X$ with $W_k:=H^0(\overline{X}, kD) \subseteq H^0(X, kf^*D)$ for any integer $k \geq 0$. Then ${\Delta}_{Y_\bullet}(W_\bullet)={\Delta}_{\overline{Y}_\bullet}(D)$.
It follows from the construction of Okounkov body associated to a graded linear series.
The following lemma plays a crucial role in proving Theorem \[main1\].
\[simplex\] Let $W_\bullet$ be a graded linear series on a smooth projective variety $X$ generated by a base point free linear series $W_1$. Suppose also that $W_1$ defines a surjective morphism $f \colon X \to \overline{X}$ of projective varieties of the same dimension $n$. Let $Y_\bullet$ be an admissible flag on $X$ defined by successive intersection of sufficiently general members $E_1, \ldots, E_n$ of $W_1$ ; $Y_i := E_1 \cap \cdots \cap E_i$ for $1 \leq i \leq n-1$ and $Y_n = \{x \}$ is a general point in $X$. Then ${\Delta}_{Y_\bullet}(W_\bullet)$ is a $n$-dimensional simplex in ${\mathbb{R}}_{\geq 0}^n$ whose verticies are $0, e_1, \ldots, e_{n-1}, \operatorname{vol}_X(W_\bullet)e_n$.
There exists a very ample ${\mathbb{Z}}$-divisor $D$ on $\overline{X}$ so that we may assume $W_k=H^0(\overline{X}, kD) \subseteq H^0(X, kf^*D)$ for any integer $k \geq 0$. By the genericity assumption on $E_j$ for defining $Y_i$, we may assume that $$\overline{Y}_\bullet : \overline{X}=f(Y_0)\supseteq f(Y_1)\supseteq \cdots \supseteq f(Y_{n-1}) \supseteq f(Y_n)$$ is an admissible flag on $\overline{X}$. By Lemma \[morokbd\], ${\Delta}_{Y_\bullet}(W_\bullet)={\Delta}_{\overline{Y}_\bullet}(D)$. Note that $D^n=\operatorname{vol}_{\overline{X}}(D)=\operatorname{vol}_X(W_\bullet)$. By applying [@AKL Proposition 4] to ${\Delta}_{\overline{Y}_\bullet}(D)$, we obtain the assertion.
We now show the rational polyhedrality of ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$.
\[ratsimval\] Let $D$ be an effective ${\mathbb{Q}}$-divisor on $X$ with finitely generated section ring $R(X,D)$. Then there exists an admissible flag $Y_\bullet$ on $X$ containing a Nakayama subvariety of $D$ such that ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ is a rational simplex in $\{0\}^{n-\kappa(D)}\times{\mathbb{R}}^{\kappa(D)}$ of dimension $\kappa(D)$.
Let $m>0$ be a sufficiently divisible and large integer such that $mD$ is a ${\mathbb{Z}}$-divisor and the section ring $R(X, mD)$ is generated by $H^0(X, mD)$. We take a log resolution $f \colon \widetilde{X} \to X$ of the base ideal $\frak{b}(|mD|)$ so that we obtain a decomposition $f^*(mD)=M+F$ into a base point free divisor $M$ and the fixed part $F$ of $|f^*(mD)|$. Note that the morphism $\phi \colon \tilde{X} \to Z$ given by $|M|$ is the Iitaka fibration of $f^*D$. Let $A_1, \ldots, A_{n-\kappa(D)}$ be sufficiently general ample divisors on $\tilde{X}$ such that each $Y_i':=A_1 \cap \cdots \cap A_i$ for $1 \leq i \leq n-\kappa(D)$ is a smooth irreducible subvariety of dimension $n-i$. By Remark \[gensub\], $U:=Y_{n-\kappa(D)}'$ is a Nakayama subvariety of $f^*D$. Let $W_k$ be the image of the natural injective map $H^0(\widetilde{X}, kf^*(mD)) \to H^0(U, kf^*(mD)|_U)$ for any integer $k \geq 0$. Then $W_\bullet$ is a graded linear series on $U$ generated by $W_1$. Note that $\phi|_U \colon U \to Z$ is a surjective morphism of projective varieties of the same dimension $\kappa(D)$ defined by $W_1$. Now take sufficiently general members $E_1, \ldots, E_{\kappa(D)}$ of $W_1$ such that $Y_{n-\kappa(D)+i}':=E_1 \cap \cdots \cap E_i$ for $1 \leq i \leq \kappa(D)-1$ is a smooth irreducible subvariety of $X$ (and $U$) of dimension $\kappa(D)-i$, and $Y_n'=\{x\}$ where $x$ is a general point in $U$. In particular, $Y_\bullet': Y'_0\supseteq \cdots \supseteq Y'_n$ is an admissible flag on $\widetilde{X}$ and the partial flag $Y'_{n-\kappa(D)\bullet}$ is an admissible flag on $U$. Then by Lemma \[simplex\], ${\Delta}_{Y'_{n-\kappa(D)\bullet}}(W_\bullet)$ is a $\kappa(D)$-dimensional simplex. Recall from [@CHPW1 Remark 3.11] that ${\Delta^{\operatorname{val}}}_{Y'_\bullet}(f^*D)={\Delta}_{Y'_{n-\kappa(D)\bullet}}(W_\bullet)$. Furthermore, by the genericity assumption on $Y_\bullet'$, we can assume that $Y_\bullet : f(Y'_0)\supseteq \cdots\supseteq f(Y'_n)$ is an admissible flag on $X$ and $f(Y_{n-\kappa(D)}')$ is a Nakayama subvariety of $D$. By Lemma \[okbdbir\], ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)={\Delta^{\operatorname{val}}}_{Y'_\bullet}(f^*D)$, and hence, ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ is a rational simplex. Finally, by Theorem \[chpwmain\] (1), ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ is contained in $\{0\}^{n-\kappa(D)}\times{\mathbb{R}}^{\kappa(D)}$ and is of dimension $\kappa(D)$.
\[ratpolval\] Let $D$ be an effective ${\mathbb{Q}}$-divisor on $X$ which admits the birational good Zariski decomposition. Then there exists an admissible flag $Y_\bullet$ on $X$ containing a Nakayama subvariety of $D$ such that ${\Delta^{\operatorname{val}}}_{Y_\bullet}(D)$ is a rational simplex in $\{0\}^{n-\kappa(D)}\times{\mathbb{R}}^{\kappa(D)}$ of dimension $\kappa(D)$.
By Proposition \[zdabfg\], $D$ has a finitely generated section ring. Then the assertion now follows from Theorem \[ratsimval\].
We now turn to the limiting Okounkov body case.
\[oklimnef\] Let $P$ be a nef divisor on $X$, and consider an admissible flag $Y_\bullet$ on $X$ containing a smooth positive volume subvariety $V=Y_{n-{\kappa_\nu}(D)}$ of $P$. Then ${\Delta^{\lim}}_{Y_\bullet}(P)={\Delta}_{Y_{n-{\kappa_\nu}(P)\bullet}}(P|_V)$.
By definition, it is clear that ${\Delta^{\lim}}_{Y_\bullet}(P) \supseteq {\Delta}_{Y_{n-{\kappa_\nu}(P)\bullet}}(P|_V)$. Thus it is sufficient to show that their Euclidean volumes in ${\mathbb{R}}^{{\kappa_\nu}(P)}$ are equal, i.e.,$ \operatorname{vol}_{{\mathbb{R}}^{{\kappa_\nu}(P)}}({\Delta^{\lim}}_{Y_\bullet}(P))=\operatorname{vol}_{{\mathbb{R}}^{{\kappa_\nu}(P)}}({\Delta}_{Y_{n-{\kappa_\nu}(P)\bullet}}(P|_V))$, or equivalently, $\operatorname{vol}_{X|V}^+(P)=\operatorname{vol}_{V}(P|_V)$ by Theorem \[chpwmain\]. Fix an ample divisor $A$ on $X$. Since $P+{\varepsilon}A$ is ample for any ${\varepsilon}>0$, it follows that $\operatorname{vol}_{X|V}(P+{\varepsilon}A)=\operatorname{vol}_V((P+{\varepsilon}A)|_V)$. By the continuity of the volume function, we obtain $$\operatorname{vol}_{X|V}^+(P)=\lim_{{\varepsilon}\to 0+}\operatorname{vol}_{X|V}(P+{\varepsilon}A)=\lim_{{\varepsilon}\to 0+} \operatorname{vol}_V((P+{\varepsilon}A)|_V)=\operatorname{vol}_V(P|_V),$$ so we complete the proof.
We next obtain an analogous result on the rational polyhedrality of ${\Delta^{\lim}}_{Y_\bullet}(D)$.
\[ratsimlim\] Let $D$ be a pseudoeffective ${\mathbb{Q}}$-divisor on $X$ which admits the birational good Zariski decomposition. Then there exists an admissible flag $Y_\bullet$ on $X$ containing a positive volume subvariety of $D$ such that ${\Delta^{\lim}}_{Y_\bullet}(D)$ is a rational simplex in $\{0\}^{n-{\kappa_\nu}(D)}\times{\mathbb{R}}^{{\kappa_\nu}(D)}$ of dimension ${\kappa_\nu}(D)$.
Let $f \colon \widetilde{X} \to X$ be a birational morphism of smooth projective varieties of dimension $n$ such that $f^*D=P+N$ is the good Zariski decomposition. Let $A_1, \ldots, A_{n-{\kappa_\nu}(D)}$ be sufficiently general ample divisors on $\tilde{X}$ such that each $Y_i':=A_1 \cap \cdots \cap A_i$ for $1 \leq i \leq n-{\kappa_\nu}(D)$ is a smooth irreducible subvariety of dimension $n-i$. By Remark \[gensub\], $V:=Y_{n-{\kappa_\nu}(D)}'$ is a positive volume subvariety of $f^*D$. By [@CHPW1 Theorem 2.18], $P|_V$ is big, and $mP|_V$ on $V$ is base point free for a sufficiently divisible and large integer $m>0$. Let $E_1, \ldots, E_{{\kappa_\nu}(D)-1} \in |mP|_V|$ be general members such that each $Y_{n-{\kappa_\nu}(D)+i}':=E_1 \cap \cdots \cap E_i$ for $1 \leq i \leq {\kappa_\nu}(D)-1$ is a smooth irreducible subvariety of $X$ of dimension ${\kappa_\nu}(D)-i$, and $Y'_n:=\{ x\}$ where $x$ is a general point in $V$. Then $Y'_\bullet : \widetilde{X}=Y'_0\supseteq \cdots\supseteq Y'_n$ is an admissible flag on $\widetilde{X}$. By [@AKL Theorem 7], ${\Delta}_{Y'_{n-{\kappa_\nu}(D)\bullet}}(P|_V)$ is a ${\kappa_\nu}(D)$-dimensional simplex. By Lemma \[oklimnef\], ${\Delta^{\lim}}_{Y'_\bullet}(P)={\Delta}_{Y'_{n-{\kappa_\nu}(D)\bullet}}(P|_V)$, and by Lemma \[okbdzd\], ${\Delta^{\lim}}_{Y'_\bullet}(f^*D)={\Delta^{\lim}}_{Y'_\bullet}(P)$. By the genericity assumption on $Y'_\bullet$, we can assume that $Y_\bullet : f(Y'_0) \supseteq \cdots\supseteq f(Y'_n)$ is an admissible flag on $X$ and $f(Y_{n-{\kappa_\nu}(D)}')$ is a positive volume subvariety of $D$. By Lemma \[okbdbir\], we obtain ${\Delta^{\lim}}_{Y_\bullet}(D)={\Delta^{\lim}}_{Y'_\bullet}(f^*D)$, and hence, ${\Delta^{\lim}}_{Y_\bullet}(D)$ is a rational simplex. Finally, by Theorem \[chpwmain\], ${\Delta^{\lim}}_{Y_\bullet}(D)$ is in $\{0\}^{n-{\kappa_\nu}(D)}\times{\mathbb{R}}^{{\kappa_\nu}(D)}$ and of dimension ${\kappa_\nu}(D)$.
\[ratrem\] The problem of the rational polyhedrality of Okounkov body is not yet fully understood. It was shown in [@AKL Corollary 13] and [@CPW Theorems 1.1 and 4.17] that on a smooth projective surface, there always exists an admissible flag with respect to which the Okounkov body of any ${\mathbb{Q}}$-divisor is a rational polytope. Thus, in particular, even if a pseudoeffective ${\mathbb{Q}}$-divisor is not abundant or does not have finitely generated section ring, the associated Okounkov body can still be a rational polytope with respect to some admissible flag. On the other hand, even when the given variety is a Mori dream space, the Okounkov body can be non-polyhedral for some admissible flag (see [@KLM Section 3]).
[ELMNP1]{}
D. Anderson, A. Küronya, and V. Lozovanu, *Okounkov bodies of finitely generated divisors*, Int. Math. Res. Not. **2014** no.9, 2343-2355.
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S. Boucksom, *Corps D’Okounkov (d’après Okounkov, Lazarsfeld- Mustaţă et Kaveh-Khovanskii)*, Séminaire Bourbaki, 65ème année, 2012-2013, no. 1059.
S. Choi, Y. Hyun, J. Park, and J. Won, *Okounkov bodies associated to pseudoeffective divisors*, preprint, arXiv:1508.03922v4.
S. Choi, Y. Hyun, J. Park, and J. Won, *Asymptotic base loci via Okounkov bodies*, preprint, arXiv:1507.00817v2.
S. Choi, J. Park, and J. Won, *Okounkov bodies and Zariski decompositions on surfaces*, to appear in Bull. Korean. Math. Soc. (special volume for Magadan Conference).
T. Eckl, *Numerical analogues of the Kodaira dimension and the abundance conjecture*, Manuscripta Math. **150** (2016), 337-356.
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L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, and M. Popa, *Restricted volumes and base loci of linear series*, Amer. J. Math. **131** (2009), 607-651.
S.-Y. Jow, *Okounkov bodies and restricted volumes along very general curves*, Adv. Math. **223** (2010), 1356-1371.
K. Kaveh and A. G. Khovanskii, *Newton convex bodies, semigroups of integral points, graded algebras and intersection theory*, Ann. of Math. (2) **176** (2012), 925-978.
A. Küronya, V. Lozovanu, and C. Maclean, *Convex bodies appearing as Okounkov bodies of divisors*, Adv. Math. **229** (2012), 2622-2639.
R. Lazarsfeld, *Positivity in algebraic geometry I and II*, Ergeb. Math. Grenzgeb., **48** and **49** (2004), Springer-Verlag, Berlin.
R. Lazarsfeld and M. Mustaţă, *Convex bodies associated to linear series.* Ann. Sci. Ec. Norm. Super. (4) **42** (2009), 783-835.
B. Lehmann, *Comparing numerical dimensions*, Alg. and Num. Theory. **7** (2013), 1065-1100.
B. Lehmann, *On Eckl’s pesudo-effective reduction map*, Trans. Amer. Math. Soc. **366** (2014), 1525-1549.
C. Mourougane and F. Russo, *Some remarks on nef and good divisors on an algebraic variety*, C. R. Acad. Sci. Paris **325** (1997), 499-504.
N. Nakayama, *Zariski-decomposition and abundance*, MSJ Memoirs **14**. Mathematical Society of Japan, Tokyo, 2004.
A. Okounkov, *Brunn-Minkowski inequality for multiplicities*, Invent. Math. **125** (1996) 405-411.
A. Okounkov, *Why would multiplicities be log-concave?* in *The Orbit Method in Geometry and Physics*, Progr. Math. **213** (2003), Birkhauser Boston, Boston, MA, 329-347.
Y. Prokhorov, *On the Zariski decomposition problem*, Tr. Mat. Inst. Steklova **240** (2003) no. Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 43-72 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. **1 (240)** (2003), 37-65.
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[^1]: S. Choi and J. Park were partially supported by NRF-2016R1C1B2011446. J. Won was partially supported by IBS-R003-D1, Institute for Basic Science in Korea.
|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'This paper is a sequel to our previous work where we introduced the [[[MapDE]{}]{}]{}algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system ([[DPS]{}]{}) to a specific target [DPS]{}, and sometimes by heuristic integration an explicit form of the mapping. A particular feature was to exploit the Lie symmetry invariance algebra of the source without integrating its equations, to facilitate [[[[MapDE]{}]{}]{}]{}, making algorithmic an approach initiated by Bluman and Kumei. In applications, however, the explicit form of a target [[DPS]{}]{} is not available, and a more important question is, can the source be mapped to a more tractable class? We extend [[[[MapDE]{}]{}]{}]{} to determine if a source nonlinear [[DPS]{}]{} can be mapped to a linear differential system. [[[MapDE]{}]{}]{}applies differential-elimination completion algorithms to the various over-determined [[DPS]{}]{} by applying a finite number of differentiations and eliminations to complete them to a form for which an existence-uniqueness theorem is available, enabling the existence of the linearization to be determined among other applications. The methods combine aspects of the Bluman-Kumei mapping approach with techniques introduced by Lyakhov, Gerdt and Michels for the determination of exact linearizations of . The Bluman-Kumei approach for focuses on the fact that such linearizable systems must admit a usually infinite Lie sub-pseudogroup corresponding to the linear superposition of solutions in the target. In contrast, Lyakhov et al. focus on [[ODE]{}]{} and properties of the so-called derived sub-algebra of the (finite) dimensional Lie algebra of symmetries of the [[ODE]{}]{}. Examples are given to illustrate the approach, and a heuristic integration method sometimes gives explicit forms of the maps. We also illustrate the powerful maximal symmetry groups facility as a natural tool to be used in conjunction with [[[[MapDE]{}]{}]{}]{}.'
author:
- Zahra Mohammadi
- 'Gregory J. Reid'
- 'S.-L. Tracy Huang'
date: 'Received: date / Accepted: date'
title: 'Symmetry-based algorithms for invertible mappings of polynomially nonlinear PDE to linear PDE '
---
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Introduction {#sec:Intro}
============
This paper is a sequel to [@MohReiHua19:Intro] and is part of a series in which we explore algorithmic aspects of exact and approximate mappings of differential equations. We are interested in mapping less tractable differential equations into more tractable ones, in particular in this article focusing on mapping nonlinear systems to linear systems. This builds on progress in [@MohReiHua19:Intro] where we considered mappings from a specific differential system to a specific target system and mappings from a linear to a linear constant coefficient differential equation.
As in [@MohReiHua19:Intro] we consider systems of (partial or ordinary) differential equations with $n$ independent variables and $m$ dependent variables which are local analytic functions of their arguments. Suppose ${R}$ has independent variables $x = (x^1, \ldots ,x^n )$ and dependent variables $u = (u^1, \ldots , u^m)$ and ${\hat R}$ has independent variables $\hat{x} = (\hat{x}^1, \ldots , \hat{x}^n )$ and dependent variables $\hat{u} = (\hat{u}^1, \ldots , \hat{u}^m)$. In particular, we consider local analytic mappings $\Psi$: $(\hat{x}, \hat{u} ) = \Psi (x,u) = (\psi (x,u), \phi(x,u) )$ so that ${R}$ is locally and invertibly mapped to ${\hat R}$: $$\label{eq:MapTrans}
\hat{x}^j = \psi^j (x,u), \qquad
\hat{u}^k = \phi^k (x,u)$$ where $j = 1,\ldots, n$ and $k = 1,\ldots, m$. The mapping is locally invertible, so the determinant of the Jacobian of the mapping is nonzero: $$\label{eq:Jac}
\mbox{Det} \mbox{Jac}(\Psi) = \mbox{Det} \frac{\partial (\psi, \phi )}{\partial (x, u)} \not = 0,$$ where $\frac{\partial (\psi, \phi )}{\partial (x, u)} $ is the usual Jacobian $(n+m) \times (n+m)$ matrix of first order derivatives of the $(n+m)$ functions $(\psi, \phi)$ with respect to the $(n+m)$ variables $(x, u)$. Note throughout this paper: we will call ${\hat R}$ the [[[*Target*]{}]{}]{}system of the mapping, which will generally have some more desirable features than ${R}$, which we call the [[[*Source*]{}]{}]{}system. A well-known direct approach to forming the equations satisfied by $\Psi$ is roughly to substitute the general change of variables into ${\hat R}$, evaluate the result modulo ${R}$, appending equations that express the independence of $\Psi$ on derivative jet variables (or equivalently decomposing in independent expressions in the jet variables). The resulting equations for $\Psi$ are generally nonlinear overdetermined systems. Algorithmic manipulation of these and other over-determined systems of [[PDE]{}]{} are at the core of the algorithms used in this paper. We make prolific use of differential-elimination completion ([[[[dec]{}]{}]{}]{}) algorithms, which apply a finite number of differentiations and eliminations to complete such over-determined systems to a form including their integrability conditions, for which an existence uniqueness theorem is available. [[[[Maple]{}]{}]{}]{} is fortunate to have several such differential elimination packages. Currently we use the [[[[rif]{}]{}]{}]{} algorithm via the [[[[Maple]{}]{}]{}]{} command ${{\tt rifsimp}\xspace}$ [@Rus99:Exi] in our implementation, but other [[[Maple]{}]{}]{}packages could be used [@Robertz106:Tdec; @BLOP:Diffalg]. To be algorithmic we restrict to systems $R$ and $\hat{R}$ that are polynomially nonlinear (i.e. differential polynomial systems, DPS). In this paper [[[dec]{}]{}]{}refers to a Differential Elimination Completion algorithm to emphasize that a number of algorithms are available.
A very general approach to such problems, concerning maps $\Psi$ from ${R}$ to ${\hat R}$, is Cartan’s famous Method of Equivalence which finds invariants that label the classes of systems equivalent under the pseudogroup of such mappings. See especially texts [@PetOlv107:Sym] and [@Man10:Pra]. The fundamental importance and computational difficulty of such equivalence questions has attracted attention from the symbolic computation community [@NeutPetitotDridi2009]. For recent developments and extensions of Cartan’s moving frames for equivalence problems see [@Fel99:Mov], [@Valiquette13:LocEquiv] and [@Arnaldsson17:InvolMovingFrames]. The [[DifferentialGeometry]{}]{} package [@And12:New] is available in [[[[Maple]{}]{}]{}]{} and has been applied to equivalence problems [@KruglikovThe18]. Underlying these calculations is that overdetermined [PDE]{}systems with some non-linearity must be reduced to forms that enable the statement of a local existence and uniqueness theorem [@Hub09:Dif; @GolKonOvcSza09:complexityDiffElim; @Sei10:Inv; @Rus99:Exi; @Bou95:Rep].
Our methods here and in [@MohReiHua19:Intro] are based on the mapping approach initiated by Bluman and Kumei [@BluKu109:DiEq] which focuses on the interaction between such mappings and Lie symmetries via their infinitesimal form on the source and target. In particular, let ${\mathcal{G}}$ be the Lie group of transformations leaving ${R}$ invariant. Also, let ${\hat{\mathcal{G}}}$ be the Lie group of transformations leaving ${\hat R}$ invariant. Locally, such Lie groups are characterized by their linearizations in a neighborhood of their identity, that is by their Lie algebras ${\mathcal{L}}$, ${\hat{\mathcal{L}}}$. If an invertible map $\Psi$ exists then ${\mathcal{G}}\simeq {\hat{\mathcal{G}}}$ and ${\mathcal{L}}\simeq {\hat{\mathcal{L}}}$. This yields a subsystem of linear equations for $\Psi$ which we call the Bluman-Kumei equations. It is a significant challenge to translate the methods of Bluman and Kumei into procedures that are algorithmic (i.e., guaranteed to succeed on a defined class of inputs in finitely many steps). Please see [@AnBlWo110:Mapp; @BLuYang119:SysAlg; @Wolf116:SymSof; @TWolf108:ConLa] for progress in their approach and some (heuristic) integration-based computer implemented methods. Our methods are also inspired by remarkable recent progress on this question for [ODE]{}by Lyakhov et al.[@LGM101:LG] who presented an algorithm for determining when an [ODE]{}is linearizable. It was also stimulated by their use of an early method by one of us (see Reid [@Rei91:Fin]), which has been dramatically improved and extended [@RLB92:Alg; @LisleReidInfinite98] with the latest improvements in the [[[[LAVF]{}]{}]{}]{} package [@LisH:Alg; @Huang:Thesis].
In our previous work [@MohReiHua19:Intro] we provided an algorithm to determine the existence of a mapping of a linear differential equation to the class of constant coefficient linear homogeneous differential equations. Key for this application was the exploitation of a commutative sub-algebra of symmetries of ${\hat{\mathcal{L}}}$ corresponding to translations of the independent variables in the target. The main contribution of this paper is to present an algorithmic method for determining the mapping of a nonlinear system to a linear system when it exists. Using a technique of Bluman and Kumei, we exploit the fact that ${\hat R}$ must admit a sub-pseudo group corresponding to the superposition property that linear systems by definition must satisfy. Once existence is established, a second stage can determine features of the map and sometimes by integration, explicit forms of the mapping. For an algorithmic treatment using differential elimination (differential algebra), we limit our treatment to systems of differential polynomials, with coefficients from $\mathbb{Q}$ or some computable extension of $\mathbb{Q}$ in $\mathbb{C}$. Thus our input system ${R}$ should be a system of [[DPS]{}]{}. Some non-polynomial systems can be converted to differential polynomial form by using the Maple command, [[dpolyform]{}]{}.
In §\[sec:GeomDPSID\] we provide some introductory material on differential-elimination algorithms, initial data and Hilbert dimensions. In §\[sec:PreMapEqs\] we give an introduction to symmetries and mapping equations. In §\[sec:Algorithms\], we introduce the [[[MapDE]{}]{}]{}algorithm. Examples of application [[[MapDE]{}]{}]{}are given in §\[sec:Examples\], and we conclude with a discussion in §\[sec:Discussion\]. Our [[[MapDE]{}]{}]{}program and a demo file are publicly available on GitHub at: <https://github.com/GregGitHub57/MapDETools>.
Differential-elimination algorithms, initial data and Hilbert functions {#sec:GeomDPSID}
=======================================================================
The geometric approach to [[DPS]{}]{} centers on the jet locus, the solution set of the equations obtained by replacing derivatives with formal variables, yielding systems of polynomial equations and inequations and differences of varieties (solution sets of polynomial equations). In this way the algorithmic tools of algebraic geometry can be applied to systems of [[DPS]{}]{}. The union of prolonged graphs of local solutions of a [[DPS]{}]{} is a subset of the jet locus in $J(\mathbb{C}^n, \mathbb{C}^m)$, the jet space, with $n$ independent variables, and $m$ dependent variables. For details concerning Jet geometry see [@Olv93:App; @Sei10:Inv].
Throughout this paper we make prolific use of differential-elimination algorithms which apply a finite number of differentiations and eliminations to an input [[DPS]{}]{} to yield in a form that yields information about its properties and solutions. For example, consider $$\label{EzSys}
u_{xyy} - u_{yy} = 0, \; \; u_{xx} + u_{xy} - u_x - u_y = 0, \; \; u_{xx} - u_{xy} - u_x + u_y = 0$$ Simply eliminating using the ordering $u_{xx} \succ u_{xy} \succ u_{yy} \succ u_{x} \succ u_{y}$ gives the equivalent system $u_{xyy} = u_{yy} , u_{xx} = u_x , u_{xy} = u_y$. The first equation can be omitted since it is a derivative of the third, yielding $$\label{EzSys2}
u_{xx} = u_x , \; \; \; u_{xy} = u_y$$ The operations to reduce the example above mirror those to reduce a related polynomial system via $\frac{\partial}{\partial x} \leftrightarrow X$, $\frac{\partial}{\partial y} \leftrightarrow Y$, $X Y^2 - Y^2 = 0$, $X^2 + X Y - X - Y = 0$, $X^2 - X Y - X + Y = 0$ to a Gröbner Basis. Indeed a natural generalization of Gröbner Bases exists for linear homogeneous [[PDE]{}]{}. However [[DPS]{}]{} are much tougher theoretically and computationally, with straightforward generalizations yielding infinite bases, and undecidable problems. Currently we use the [[[[rif]{}]{}]{}]{} algorithm via the [[[[Maple]{}]{}]{}]{} command ${{\tt rifsimp}\xspace}$ [@Rus99:Exi] in our implementation, but other [[[Maple]{}]{}]{}packages could be used such as [[DifferentialThomas]{}]{} Package [@Thomas106:Tdec; @Robertz106:Tdec], and the [[DifferentialAlgebra]{}]{} package[@Bou95:Rep; @BLOP:Diffalg; @lemaire:tel] or [[casesplit]{}]{} which offers a uniform interface to such packages.
A key aspect of these [[[[dec]{}]{}]{}]{} packages for [[DPS]{}]{} is that they split on cases where certain leading polynomial quantities are zero or nonzero. This leads to systems of differential polynomial equations and inequations. In particular a system $R$ of equations $\{ p_1 = 0, p_2 = 0, \cdots , p_b =0 \} $ and inequations $ \{ q_1 \not = 0, \cdots , q_c \not = 0 \}$ has solution locus $$\label{Z=E-I}
Z(R) = V^= (R) \setminus V^{\not =} (R)$$ where $V^= (R)$ are the solutions satisfying $\{ p_1 = 0, p_2 = 0, \cdots , p_c =0 \} $ and $V^{\not =}(R)$ is the set of solutions of $\Pi_{i = 1}^{i = c} \: q_i = 0$.
Moreover, a central input in such algorithms are rankings of derivatives [@Rus97:Ran]. Indeed let $\Omega({R})$ be all the derivatives of dependent variables for ${R}$. Throughout this paper the set of derivatives also includes $0$-order derivatives (i.e. dependent variables). A ranking on $\Omega({R})$ is a total order $\prec$ that satisfies the axioms in [@Rus97:Ran]. Given a ranking and algorithms in [@Rus99:Exi] determine initial data and the existence and uniqueness of formal power series solutions. Additionally, if the ranking is orderly and of Riquier type (i.e. ordered first by total order of derivative, with a ranking specified by a Riquier ranking matrix) analytic initial data yields local analytic solutions. See [@Riquier:diffalg] for a proof of this result. We will need some block elimination rankings that eliminate groups of dependent variables in favor of others. Enforcing the block order via the first row of the Riquier Matrix and then enforcing total order of the derivative as the next criterion for each block enables analytic data to yield analytic solutions that is sufficient for this paper.
The differential-elimination algorithms used in this article enable the algorithmic posing of initial data for the determination of unique formal power series of differential systems. We will exploit a powerful measure of solution dimension information given by Differential Hilbert Series and its related Differential Hilbert Function [@MLPK13:Hilbert; @MLH:DCP]. Indeed given a ranking algorithms such as the [[[rif]{}]{}]{}algorithm, Rosenfeld Groebner and Thomas Decomposition partition the set of derivatives of the unknown function at a regular point $x_0$ into a set of parametric derivatives $\mathcal{P}$ and a set of principal derivatives. Principal derivatives are derivatives of the leading derivatives and parametric derivatives are its complement, the ones that can be ascribed arbitrary values at $x_0$. Then the Hilbert Series is defined as: $$\label{HS1}
\mbox{HS}({R}, x_0) := \sum_{G \in \mathcal{P} } s^{\mbox{dord}(G)} = \sum_{n=0}^\infty a_n s^n$$ where $\mbox{dord}(G)$ is the differential order of $G$, $a_n$ is the number of parametric derivatives of differential order $n$. To algorithmically compute such a series exploits the fact that the parametric data can be partitioned into subsets.
For example consider ${R}= \{ u_{xx} = u_x, u_{xy} = u_y\}$, which is already in [[[rif]{}]{}]{}-form with respect to an orderly ranking. Then $\mathcal{P} = \{ u_x \} \cup \{ u , u_y, u_{yy}, u_{yyy}, \cdots \}$, and the associated set of initial data is $ \{ u_x (x_0, y_0) = c_0 \} \cup \{ u (x_0, y_0) = c_1 , u_y (x_0, y_0) = c_2 , u_{yy} (x_0, y_0) = c_3 , \cdots \}$. In what follows it is helpful to associate these derivatives with corresponding points in $\mathbb{N}^2$ via $ \frac{\partial^{i+j}}{\partial x^i \partial y^j} u \leftrightarrow (i, j) \in \mathbb{N}^2$. See Fig. \[fig:HilbertPic\] for a graphical depiction of $\mathcal{P}$.
\[fig:HilbertPic\]
(0,0) grid (4.5,4.5); (0,0) – (4.5,0) node\[anchor=north west\] [x]{}; (0,0) – (0,4.5) node\[anchor=south east\] [y]{}; in [0,1,2,3,4]{} (cm,1pt) – (cm,-1pt) node\[anchor=north\] [$\x$]{}; in [0,1,2,3,4]{} (1pt,cm) – (-1pt,cm) node\[anchor=east\] [$\y$]{}; (1,1) circle\[radius= 0.3 em\]; (1,2) circle\[radius= 0.3 em\]; (1,3) circle\[radius= 0.3 em\]; (1,4) circle\[radius= 0.3 em\]; (2,0) circle\[radius= 0.3 em\]; (2,1) circle\[radius= 0.3 em\]; (2,2) circle\[radius= 0.3 em\]; (2,3) circle\[radius= 0.3 em\]; (2,4) circle\[radius= 0.3 em\]; (3,0) circle\[radius= 0.3 em\]; (3,1) circle\[radius= 0.3 em\]; (3,2) circle\[radius= 0.3 em\]; (3,3) circle\[radius= 0.3 em\]; (3,4) circle\[radius= 0.3 em\]; (4,0) circle\[radius= 0.3 em\]; (4,1) circle\[radius= 0.3 em\]; (4,2) circle\[radius= 0.3 em\]; (4,3) circle\[radius= 0.3 em\]; (4,4) circle\[radius= 0.3 em\]; (0,0) circle\[radius= 0.3 em\]; (0,1) circle\[radius= 0.3 em\]; (0,2) circle\[radius= 0.3 em\]; (0,3) circle\[radius= 0.3 em\]; (0,4) circle\[radius= 0.3 em\]; (1,0) circle\[radius= 0.3 em\];
Then $$\label{HS2}
\mbox{HS}({R}, (x_0, y_0)) = 1 + 2s + s^2 + s^3 + s^4 + \cdots$$ A crucial condition to check in our algorithms will be that two Hilbert series are the same, which is complicated since no finite algorithm exists for checking equality of series. For our example, however, the series can be expressed finitely $\mbox{HS}({R}, (x_0, y_0)) = 1 + 2s + s^2 + s^3 + s^4 + \cdots = s + \frac{1}{1 - s}$. This collapsing of the series into a rational function can be accomplished in general. In our implementation, we exploited the output of the [[initialdata]{}]{} algorithm in the [[[[rif]{}]{}]{}]{} package. For example this returns data as a partition of two sets: a finite set of initial data $\mathcal{F}$ and an set that represents an infinite set of data $\mathcal{I}$ by compressing them into arbitrary functions. For our example this compression is: $$\label{FuI}
\mathcal{F} \cup \mathcal{I} = \{ u_x (x_0, y_0) = c_0 \} \cup \{ u (x_0, y) = f(y) \}$$ This approach is easily extended to yield the Differential Hilbert Function by applying a function that acts on each piece of initial data in $\mathcal{F}$ and $\mathcal{I}$ $$\label{HF(FuI)}
\mbox{HF} (\mathcal{F}, \mathcal{I} ) :=
\sum_{G \in \mathcal{F} } s^{\mbox{dord}(G)} + \sum_{G \in \mathcal{I}} \frac{s^{\mbox{dord}(G)}}{ ( \mbox{free}(G) - 1)! } \left( \frac{d}{d s} \right)^{\mbox{free}(G) - 1} (1 - s)^{-1}$$ In the above formula $\mbox{free}(G)$ is the number of free variables in the right hand side of the infinite data set $\mathcal{I}$. So for our example with $G = u (x_0, y) = f(y)$ we get $\mbox{free}( u (x_0, y) = f(y) )=1$ and $\mbox{dord}(G) = 0$. Then the formula yields as before $$\label{HFcalc}
\mbox{HF} (\mathcal{F}, \mathcal{I} ) := s + \frac{1}{1 - s}$$ For leading linear [[DPS]{}]{}, in [[[[rif]{}]{}]{}]{}-form with respect to an orderly Riquier ranking, the Differential Hilbert Series gives coordinate independent dimension information. If a ranking is not orderly then the Differential Hilbert Series is no longer invariant. For example just consider the initial data for $v_{xx} = v_t$ in an orderly ranking compared to $v_t = v_{xx}$ in a non-orderly ranking.
The output of the initial data which partitions the parametric derivatives into disjoint cones of various dimensions $\leq n$, can express this in the rational function form $\mbox{HS}(s) = \frac{P(s)}{(1 - s)^{\textbf{d}}}$ where $\textbf{d} = \textbf{d}({R})$ is the differential dimension of ${R}$. It corresponds to the maximum number of free independent variables appearing in the functions for the initial data. For further information on differential Hilbert Series see [@MLPK13:Hilbert]. The algorithms are simple modifications of those for Gröbner bases for modules.
Symmetries & Mapping Equations {#sec:PreMapEqs}
==============================
Symmetries
----------
Infinitesimal Lie [*point*]{} symmetries for ${R}$ are found by seeking vector fields $$\label{eq:symmOp}
V = \sum_{i=1}^n \xi^i(x,u) {\frac{\partial}{\partial x^i}} + \sum_{j=1}^m \eta^j(x,u){\frac{\partial}{\partial u^j}}$$ whose associated one-parameter group of transformations $$\begin{aligned}
\label{eq:LieInfTrans}
{x^*} &= x + \xi(x,u) \epsilon + O(\epsilon^2) \nonumber\\
{u^*} &= u + \eta(x,u) \epsilon + O(\epsilon^2)\end{aligned}$$ away from exceptional points preserve the jet locus of such systems, mapping solutions to solutions. See [@BluKu111:Sym; @BluKu112:Sym] for applications. The [*infinitesimals*]{} $(\xi^i,\eta^j)$ of a symmetry vector field for a system of [DEs]{}are found by solving an associated system of linear homogeneous defining equations $S$ (or determining equations) for the infinitesimals. The defining system $S$ is derived by a prolongation formula for which numerous computer implementations exist [@Car00:Sym; @Che07:GeM; @Roc11:SAD]. Lie’s classical theory of groups and their algebras requires local analyticity in its defining equations. Such local analyticity will be a key assumption throughout our paper.
The resulting vector space of vector fields is closed under its commutator. The commutator of two vector fields for vector fields $X = \sum_{i=1}^{m+n} \nu^i {\frac{\partial}{\partial z^i}}$, $Y = \sum_{i=1}^{m+n} \mu^i {\frac{\partial}{\partial z^i}} $ in a Lie algebra $\mathcal{L}
$ and $z = (x,u)$, is: $$\label{eq:com}
\left[X , Y \right] = X Y - Y X = \sum_{i=1}^{m+n} \omega^i {\frac{\partial}{\partial z^k}}$$ where $\omega^k = \sum_{i=1}^{m+n} \left( \nu^i \mu^k_{z^i} - \mu^i \nu^k_{z^i} \right)$.
Similarly, we suppose that the [[[*Target*]{}]{}]{}admits symmetry vector fields
$$\label{eq:vf}
\hat{V} = \sum_{i=1}^n \hat{\xi}^i(\hat{x},\hat{u}) {\frac{\partial}{\partial \hat{x}^i}} + \sum_{j=1}^m \hat{\eta}^j(\hat{x},\hat{u}){\frac{\partial}{\partial \hat{u}^j}}$$
in the [[[*Target*]{}]{}]{}infinitesimals $(\hat{\xi}, \hat{\eta})$ that satisfies a linear homogeneous defining system $\hat{S}$ generating a Lie algebra $\hat{\mathcal{L}}$. Computations with defining systems will be essential in our approach and are implemented using Huang and Lisle’s powerful object oriented [[[[LAVF]{}]{}]{}]{}, [[[[Maple]{}]{}]{}]{} package [@LisH:Alg].
\[ex:ODE3(a)\] Consider as a simple example the third order nonlinear [[ODE]{}]{} which is in [[[rif]{}]{}]{}-form with respect to an orderly ranking $$\label{ODE3}
u_{{xxx}} = {\frac { 3 \left( uu_{{xx}}+{u_{{x}}}^{2}+1 \right) ^{2}}{u(uu_{{x}}+x)}}
- {\frac{3 u_{{x}}u_{{xx}}}{u}}
+ {\frac {8 x \left( uu_{{x}}+x \right) ^{4} \left( {u}^{2}+{x}^{2}+1 \right) }{u({u}^{2}+{x}^{2})}}$$ at points $u \not = 0, uu_{{x}}+x \not = 0, {u}^{2}+{x}^{2} \not = 0$. When [[[[Maple]{}]{}]{}]{}’s [[dsolve]{}]{} is applied to (\[ODE3\]) it yields no result. Later in this section, we will discover important information about (\[ODE3\]) using symmetry aided mappings. It will be used as a simple running example to illustrate the techniques of the article.
The defining system for Lie point symmetries of form $\xi(x,u) {\frac{\partial}{\partial x}} + \eta(x,u) {\frac{\partial}{\partial u}}$ of (\[ODE3\]) has [[[[rif]{}]{}]{}]{} form with respect to an orderly ranking given by: $$\begin{aligned}
\label{eq:ODE3DetSys}
S = [& \xi = -{\frac {\eta\,u}{x}}, \quad
\eta_{{x,u}}={\frac { \left( u-x \right)
\left( u+x \right) \eta}{{u}^{3}x}}+{\frac { \left( {u}^{2}+{x}^{2}
\right) \eta_{{u}}}{x{u}^{2}}}-{\frac {\eta_{{x}}}{u}}+{\frac {x\eta_
{{u,u}}}{u}}, \nonumber \\ \qquad
& \eta_{{x,x}}=-{\frac { \left( 2\,{u}^{4}-{x}^{2}{u}^{2}+{
x}^{4} \right) \eta}{{u}^{4}{x}^{2}}}+{\frac { \left( {u}^{2}+{x}^{2}
\right) \eta_{{u}}}{{u}^{3}}}+2\,{\frac {\eta_{{x}}}{x}}+{\frac {{x}^
{2}\eta_{{u,u}}}{{u}^{2}}}, \\
& \eta_{{u,u,u}}=-{\frac { \left( 16\,{u}^{8}
+24\,{u}^{6}{x}^{2}+8\,{u}^{4}{x}^{4}+16\,{u}^{6}+8\,{u}^{4}{x}^{2}+3
\,{u}^{2}+3\,{x}^{2} \right) \eta}{ \left( {u}^{2}+{x}^{2} \right) {u}
^{3}}} \nonumber \\
& -{\frac { \left( 8\,{u}^{6}{x}^{2}+8\,{u}^{4}{x}^{4}+8\,{u}^{4}{
x}^{2}-3\,{u}^{2}-3\,{x}^{2} \right) \eta_{{u}}}{{u}^{2} \left( {u}^{2
}+{x}^{2} \right) }} +8\,{\frac {{u}^{3}x \left( {u}^{2}+{x}^{2}+1
\right) \eta_{{x}}}{{u}^{2}+{x}^{2}}} \nonumber ]
\end{aligned}$$ Its corresponding initial data is $$\begin{aligned}
\hspace{-0.5cm} \mbox{{\sc{ID}}} (S) = [\eta \left( x_{{0}},u_{{0}} \right) =c_{{1}}, \eta_{{x}} \left( x_{{0}
},u_{{0}} \right) =c_{{2}}, \eta_{{u}} \left( x_{{0}},u_{{0}} \right) =
c_{{3}}, \eta_{{u,u}} \left( x_{{0}},u_{{0}} \right) =
c_{{4}}]
\end{aligned}$$ There are $4$ arbitrary constants in the initial data at regular points $(x_0, u_0)$, so (\[ODE3\]) has a $4$ dimensional local Lie algebra of symmetries ${\mathcal{L}}$ in a neighborhood of such points: $\dim \mathcal{L} = 4$. The structure of ${\mathcal{L}}$ of (\[eq:ODE3DetSys\]) can be algorithmically determined without integrating the defining system [@RLB92:Alg; @LisleReidInfinite98; @LisH:Alg; @Huang:Thesis]: $$\begin{aligned}
[Y_{{1}},Y_{{2}}] &=-Y_{{1}}-2\,Y_{{2}}, \quad [Y_{{1}},Y_{{3
}}]=Y_{{1}}-2\,Y_{{3}},\quad [Y_{{1}}, Y_{{4}}]=-2\,Y_{{4}}, \nonumber \\
[Y_{{2}},Y_{{3}}] &=Y_{{2}}+Y_{{3}}, \quad [Y_{{2}},Y_{{4}}]=Y_{{4}},\quad [Y_{{3}},Y_{{4}}]=-Y_{{4}}
\end{aligned}$$ where a regular point ($x_0 = 1, u_0 = 1$) was substituted into the relations.
But what can such symmetry information tell us about nonlinear systems ${R}$ such as the above [[ODE]{}]{} using mappings? In particular, in this paper we focus on the question of when a system ${R}$ can be mapped to a linear system ${\hat R}$. Throughout this paper we maintain blanket local analyticity assumptions. So the case of a single differential equation ${\hat R}$ has the form $\mathcal{H} \hat{u} = f(\hat{x})$ where $\mathcal{H}$ is a linear differential operator with coefficients that are analytic functions of $\hat{x}$ and $f$ is also analytic. Lewy’s famous counterexample of a single linear differential equation in 3 variables, of order 1, where $\mathcal{H}$ is analytic with smooth inhomogeneous term, without smooth solutions, provides a counterexample in the smooth case. Then supposing we have the existence of a local analytic solution $\tilde{u}$ in a neighborhood of $\hat{x}_0$, in this neighborhood the point transformation $\hat{u} \rightarrow \hat{u} - \tilde{u}$ implies that without loss we can consider ${\hat R}$ to be a homogeneous linear differential equation $\mathcal{H} \hat{u} = 0$ where $\hat{u} \in \mathcal{A}(\hat{x}_0, \delta)$, the set of analytic functions on some sufficiently small disk $| \hat{x} - \hat{x}_0 | < \delta$. Then solutions of ${\hat R}$ satisfy the superposition property $\mathcal{H} (\hat{v} + \hat{w}) = \mathcal{H} \hat{v} + \mathcal{H} \hat{w} = 0$. This corresponds to point symmetries generated by the Lie algebra of vectorfields $$\label{L*}
\hat{\mathcal{L}}^* := \left\{ \hat{v}(\hat{x}) \frac{\partial}{\partial \hat{u}}: \mathcal{H} \hat{v}(\hat{x}) = 0 \; and \;
\hat{v} \in \mathcal{A}(\hat{x}_0, \delta) \right\}$$ Consequently, assuming the existence of a local analytic map, $\dim \hat{\mathcal{L}}^* = \dim {\hat R}= \dim {R}$. If ${R}$ is an [[ODE]{}]{} of order $d \geq 2$ then $\dim \hat{\mathcal{L}}^* = \dim {\hat R}= \dim {R}= d$. Similarly the superposition property $\mathcal{H} (c\hat{v}) = c \mathcal{H} \hat{v} = 0$ corresponds to a $1$ parameter family of scalings with symmetry vectorfield $\hat{u} \frac{\partial}{\partial \hat{u}}$. So we get the well-known and obvious result that an [[ODE]{}]{} of order $d$ that can be mapped to a linear [[ODE]{}]{} must have $\dim {\mathcal{L}} = \dim \hat{\mathcal{L}} \geq d + 1$. Similarly if ${R}$ is linearizable and $\dim {R}= \infty$ then $\dim {\mathcal{L}} = \dim \hat{\mathcal{L}} = \infty$ with similar properties for systems. The Lie sub-algebra $\hat{\mathcal{L}}^*$ is easily shown to be ‘abelian‘ by direct computation of commutator [@PetOlv107:Sym] in both the finite and infinite dimensional case. Indeed consider the so-called derived algebra $\mathcal{L}' = \mbox{DerivedAlgebra}(\mathcal{L})$, which is the Lie subalgebra of $\mathcal{L}$ generated by commutators of members of $\mathcal{L}$ and similarly for $\hat{\mathcal{L}}'$. By direct computation of commutators $\hat{\mathcal{L}}^*$ is a sub-algebra of $\hat{\mathcal{L}}'$ (e.g. $[ \hat{v}(\hat{x}) \frac{\partial}{\partial \hat{u}},
\hat{u} \frac{\partial}{\partial \hat{u}} ] = \hat{v}(\hat{x}) \frac{\partial}{\partial \hat{u}} \in \hat{\mathcal{L}}$). Thus, a necessary condition for the existence of a map $\Psi$ to a linear target is that $\hat{\mathcal{L}}'$ has a $d$ dimensional abelian subalgebra in the finite and infinite dimensional cases (see Olver [@PetOlv107:Sym]). In the preceding paragraph we considered the case of a single dependent variable, which is easily extended to the multivariate case. For example in equation (\[L\*\]) the symmetry generator $\hat{v}(\hat{x}) \frac{\partial}{\partial \hat{u}}$ can be replaced by $ \sum_{i = 1}^{m} \hat{v}^i (\hat{x}) \frac{\partial}{\partial \hat{u}^i} $ for the case of a system.
Lyakhov, Gerdt and Michels [@LGM101:LG] use this to implement a remarkable algorithm to determine the existence of a linearization for a single [[ODE]{}]{} of order $d$. See Algorithm \[alg:LGMLinTest\] and [@LGM101:LG; @ML102:LSA] for further background. There are two main cases. The first is when the nonlinear [[ODE]{}]{} has a Lie symmetry algebra of maximal dimension, as shown in Step 5 of Algorithm \[alg:LGMLinTest\]. Such maximal cases are always linearizable. These occur for $d = 1, 2$ where the maximal dimensions of $\mathcal{L}$ are $\infty$ and $8$ respectively, and for $d > 2$ where the maximal dimension of $\mathcal{L}$ is $d+4$. The second main case is sub-maximal and occurs for $d > 2$ when $\dim \mathcal{L} = d+1$ or $\dim \mathcal{L} = d+2$.
: a leading linear [[ODE]{}]{} ${R}$ solved for its highest derivative of order $\geq 1$ : Lin = true if $R$ linearizable otherwise Lin = false Lin:= false Compute $S := \mbox{ThomasDecomposition}(\mbox{DetSys}(R))$ Find $\dim\mathcal{L} := \dim S$, $d := \mbox{difforder}(R)$ ComRels := Structure$(S)$ **if** $d=1$ or ($d=2$ and $\dim \mathcal{L}=8$) or ($d>2$ and $\dim \mathcal{L}= d+4$) **then** Lin := true **else if** $d>2$ and ($\dim \mathcal{L}= d+1$ or $\dim \mathcal{L}= d+2$) **then** $\mathcal{L}' := \mbox{DerivedAlgebra}(\mbox{ComRels})$ **if** IsAbelian$(\mathcal{L}')$ and $d = \dim(\mathcal{L}')$ **then** Lin := true **end if** **end if** **return** Lin
\[alg:LGMLinTest\]
\[ex:ODE3(b)\] We illustrate the above discussion and Algorithm \[alg:LGMLinTest\] by a continuation of Example \[ex:ODE3(a)\]. For that example $d = 3$ and $\dim \mathcal{L} = 4$. Then from the commutation relations the derived algebra $\mathcal{L}'$ is generated by $$[Z_{{1}}=Y_{{1}}-2\,Y_{{3}},\; Z_{{2}}=Y_{{2}}+Y_{{3}},\;Z_{{3}}=Y_{{4}}]$$ Thus $\dim \mbox{DerivedAlgebra}( \mathcal{L}) = \dim \mathcal{L}' = 3$. Also its structure is easily found as $$[Z_{{1}},Z_{{2}}] = [Z_{{1}},Z_{{3}}] = [Z_{{2}},Z_{{3}}] = 0$$ so $ \mathcal{L}'$ is abelian and by Algorithm \[alg:LGMLinTest\], (\[ODE3\]) is exactly linearizable.
The Algorithms introduced by Lyakhov et al. [@LGM101:LG] have two stages: the first given above is to determine whether the system is lineaizable. The second stage is to attempt to construct an explicit form for the mapping by integration. A fundamental algorithmic tool for both stages is the [[ThomasDecomposition]{}]{} algorithm which is a differential elimination algorithm which outputs a disjoint decomposition of a [[DPS]{}]{} finer than that of [@Bou95:Rep] or [@Rus99:Exi]. It is based on the work of Thomas [@Thomas106:Tdec; @Robertz106:Tdec]. The algorithm is available in distributed Maple 18 and later versions. We note that the construction step involves heuristic integration. The algorithm that they use to construct a system for the mapping to a linear [[ODE]{}]{}, before it is reduced using [[ThomasDecomposition]{}]{}, is related to the algorithm [[EquivDetSys]{}]{} given in our introductory paper [@MohReiHua19:Intro]. It is expensive as we illustrate later with examples. One of the contributions of our paper is to find a potentially more efficient algorithm that avoids the application of the full nonlinear equivalence equations generated by [[EquivDetSys]{}]{} in [@MohReiHua19:Intro].
Bluman-Kumei Mapping Equations
------------------------------
Assume the existence of a local analytic invertible map $\Psi = (\psi, \phi)$ between the [[[*Source*]{}]{}]{}system ${R}$ and the [[[*Target*]{}]{}]{}system ${\hat R}$, with Lie symmetry algebras $\mathcal{L}$, $\hat{\mathcal{L}}$ respectively. Applying $\Psi$ to the infinitesimals $(\hat{\xi}, \hat{\eta})$ of a vectorfield in $\hat{\mathcal{L}}$ yields what we will call the *Bluman-Kumei (BK) mapping equations*: $$\label{eq:BK}
{\small{M_{\mbox{\sc{BK}}}(\mathcal{L}, \hat{\mathcal{L}}) }} =
\left\{ \begin{array}{ccl}
\hat{\xi}^k (\hat{x}, \hat{u}) & = & \sum_{i=1}^n \xi^i(x,u) \frac{\partial \psi^k}{\partial x^i} + \sum_{j=1}^m \eta^j(x,u)\frac{\partial \psi^k}{\partial u^j} \vspace{0.5cm} \\
\hat{\eta}^\ell (\hat{x}, \hat{u}) & = & \sum_{i=1}^n \xi^i(x,u) \frac{\partial \phi^\ell}{\partial x^i}+\sum_{j=1}^m \eta^j(x,u)\frac{\partial \phi^\ell}{\partial u^j} \\
\end{array}\right.$$ where $1 \leq k \leq n$ and $1 \leq \ell \leq m$, and $(\xi, \eta)$ are infinitesimals of Lie symmetry vectorfields in $\mathcal{L}$. See Bluman and Kumei [@BK105:BluKu; @Blu10:App] for details and generalizations (e.g. to contact transformations). Note that all quantities on the RHS of the BK mapping equations are functions of $(x,u)$ including $\phi$ and $\psi$. See [@MohReiHua19:Intro Example 1] for an introductory example of mappings and the examples in [@Blu10:App].
When considered together with $\hat{x} = \psi(x,u), \hat{u} = \phi(x,u)$ the BK mapping equations are a change of variables from $(x,u)$ to $(\hat{x}, \hat{u})$ coordinates. Simply interchanging target and source variables then yields the inverse of the BK mapping equations below. Considered together with $x = \hat{\psi}(\hat{x}, \hat{u}), u = \hat{\phi}(\hat{x}, \hat{u})$ these are a change of variables from $(\hat{x}, \hat{u})$ to $(x,u)$ coordinates. \[rem:InvBK\] $$\label{eq:InvBK}
{\small{M_{\mbox{\sc{BK}}}( \hat{\mathcal{L}}, \mathcal{L}) }} =
\left\{ \begin{array}{ccl}
\xi^k(x,u) & = & \sum_{i=1}^n \hat{\xi}^i (\hat{x}, \hat{u}) \frac{\partial \hat{\psi}^k}{\partial \hat{x}^i} + \sum_{j=1}^m \hat{\eta}^j (\hat{x}, \hat{u}) \frac{\partial \hat{\psi}^k}{\partial \hat{u}^j} \vspace{0.5cm} \\
\eta^\ell (x,u) & = & \sum_{i=1}^n \hat{\xi}^i (\hat{x}, \hat{u}) \frac{\partial \hat{\phi}^\ell}{\partial \hat{x}^i}+\sum_{j=1}^m\hat{\eta}^j (\hat{x}, \hat{u}) \frac{\partial \hat{\phi}^\ell}{\partial \hat{u}^j} \\
\end{array}\right.$$ We note the following relation between Jacobians in $(x,u)$ and $(\hat{x},\hat{u})$ coordinate systems $$\label{eq:PsiPsihat}
\frac{\partial(\hat{\psi},\hat{\phi})}{\partial(\hat{x}, \hat{u})} = \left[ \frac{\partial(\psi,\phi)}{\partial(x, u)} \right]^{-1}$$
If an invertible map $\Psi$ exists mapping ${R}$ to ${\hat R}$ then it most generally depends on $\dim(\mathcal{L}) = \dim(\hat{\mathcal{L}})$ parameters. But we only need one such $\Psi$. So reducing the number of such parameters, e.g., by restricting to a Lie subalgebra $\mathcal{L}^{'}$ of $\mathcal{L}$ with corresponding Lie subalgebra $\hat{\mathcal{L}}^{'}$ of $\hat{\mathcal{L}}$ that still enables the existence of such a $\Psi$, is important in reducing the computational difficulty of such methods. We will use the notation $S',\hat{S}'$ to denote the symmetry defining systems of Lie sub-algebras $\mathcal{L}^{'} $, $\hat{\mathcal{L}}^{'}$ respectively. See [@Blu10:App; @PetOlv107:Sym] for discussion on this matter.
For mapping from nonlinear to linear systems, a natural candidate for $\mathcal{L}^{'}$ is the $ \mbox{DerivedAlgebra} (\mathcal{L}) $, and the natural target Lie symmetry algebra is $\hat{\mathcal{L}}^{*}$ corresponding to the superposition defined in (\[L\*\]).
\[ex:ODE3(c)\] This is a continuation of Examples \[ex:ODE3(a)\] and \[ex:ODE3(b)\] concerning (\[ODE3\]). Here we will use the BK mapping equations where $\mathcal{L}^{'} = \mbox{DerivedAlgebra} (\mathcal{L}) $. For the construction of $\Psi$ we actually need differential equations for $ \mathcal{L}'$, in addition its structure which are provided by the algorithm ${{\tt DerivedAlgebra}\xspace}$ in the [[[[LAVF]{}]{}]{}]{} package which for (\[ODE3\]) yields its [[[[rif]{}]{}]{}]{}-form: $$\begin{aligned}
\label{eq:ODE3_L'}
S' = [\xi = &-{\frac {\eta\,u}{x}},\eta_{{x}}={\frac { \left( {u}^{2}+{x}^{2}
\right) \eta}{x{u}^{2}}}+{\frac {\eta_{{u}}x}{u}}, \\
\eta_{{u,u,u}}&=-{
\frac { \left( 8\,{u}^{8}+8\,{u}^{6}{x}^{2}+8\,{u}^{6}+3\,{u}^{2}+3\,{
x}^{2} \right) \eta}{ \left( {u}^{2}+{x}^{2} \right) {u}^{3}}}+3\,{
\frac {\eta_{{u}}}{{u}^{2}}}] \nonumber
\end{aligned}$$ The derived algebra is then shown by [[[[LAVF]{}]{}]{}]{} commands to be both $3$ dimensional and abelian. Moreover, its determining system (\[eq:ODE3\_L’\]) is much simpler than the determining system of $\mathcal{L}$ given in (\[eq:ODE3DetSys\]). Crucially it means we can exploit this determining system using the BK mapping equations. Since the target infinitesimal generator is $\hat{\xi} {\frac{\partial}{\partial \hat{x}}} + \hat{\eta} {\frac{\partial}{\partial \hat{u}}} = 0 \cdot {\frac{\partial}{\partial \hat{x}}} + \hat{\eta} (\hat{x}) {\frac{\partial}{\partial \hat{u}}}$ where $\mathcal{H} \hat{u} = 0$. So $ \hat{\xi} = 0$ and $\mathcal{H} \hat{u} = 0$ and the BK equations are: $$\label{eq:BKODE3}
{\small{M_{\mbox{\sc{BK}}}(\mathcal{L}^{'}, \hat{\mathcal{L}}^{*})}} =
\left\{ \begin{array}{ccl}
0 & = & \xi (x,u) \frac{\partial \psi}{\partial x} + \eta(x,u)\frac{\partial \psi}{\partial u} \vspace{0.5cm} \\
\hat{\eta} (\hat{x}, \hat{u}) & = & \xi(x,u) \frac{\partial \phi}{\partial x}+\eta(x,u)\frac{\partial \phi}{\partial u} \\
\end{array}\right.$$ where $\hat{\eta}_{\hat{u}} = 0$ and $\mathcal{H} \hat{u} = 0$. These equations are an important necessary condition for the linearization of (\[ODE3\]) and this will be exploited in Section §\[sec:Algorithms\] in the computation of the mapping.
Algorithms and Preliminaries for the MapDE Algorithm {#sec:Algorithms}
====================================================
The [[[MapDE]{}]{}]{}algorithm introduced in [@MohReiHua19:Intro] is extended to determine if there exists a mapping of a nonlinear source ${R}$ to some linear target ${\hat R}$, using the target input option ${{\tt {{{\it Target}\xspace}}}\xspace}= {{\tt LinearDE}\xspace}$.
Symmetries of the linear target and the derived algebra {#sec:SymDer}
-------------------------------------------------------
We summarize and generalize some aspects of the discussion in §\[sec:Intro\] and §\[sec:PreMapEqs\]. The following theorem is a straightforward consequence of the necessary conditions in [@Blu10:App] where we have also required that the target system is in [[[[rif]{}]{}]{}]{}-form.
\[thm:SupSymLinSys\] Suppose that the analytic system ${R}$ is exactly linearizable by a local holomorphic diffeomorphism $\hat{x} = \psi(x,u), \hat{u} = \phi(x,u)$ to yield a linear target system. Then ${\hat R}$ locally takes the form $$\label{eq:Hsy}
{\hat R}: \mathcal{H} \hat{u}(\hat{x}) = 0$$ where $\mathcal{H}$ is a vector partial differential operator, with coefficients that are local analytic functions of $\hat{x}$ and the system (\[eq:Hsy\]) is in [[[[rif]{}]{}]{}]{}-form with respect to an orderly ranking. Moreover ${\hat R}$ admits the symmetry vector field $\sum_{j=1}^m \hat{\eta}^j (\hat{x}) \frac{\partial}{\partial \hat{u}^j}$: $$\label{eq:Hsym}
\hat{S}^* := \left\{ \hat{\xi}^i = 0, \; \mathcal{H} \hat{\eta} = 0, \; \hat{\eta}^j_{\hat{u}^k} = 0: 1 \leq i \leq n, \; 1 \leq j,k \leq m \right \}.$$
From the previous discussion, computation of determining systems for derived algebras is important in both the finite and infinite cases. In the Remark below we sketch what appears to be the first algorithm to compute such systems in the infinite case.
\[rem:DerivedAlgebraAlgorithm\] A simple consequence of the commutator formula (\[eq:com\]) is that the commutators generate a Lie algebra which is called the derived algebra. Lisle and Huang [@LisH:Alg] implement efficient algorithm in the [[[[LAVF]{}]{}]{}]{} package to compute the determining system for the derived algebra for finite dimensional Lie algebras of vectorfields. We have made a first implementation in the infinite dimensional case, together with the Lie pseudogroup structure relations [@RLB92:Alg; @LisleReidInfinite98]. First each of the $\nu$, $\mu$, $\omega$ in the commutation relations (\[eq:com\]) must satisfy the determining system of $\mathcal{L}$ so we enter three copies of those determining systems. We then reduce the combined system using a block elimination ranking which ranks any derivative of $\omega$ strictly less than those of $\mu, \nu$. The resulting block elimination system for $\omega$ generates the derived algebra in the infinite case.
The commutator between any superposition generator and the scaling symmetry admitted by linear systems yields $$\label{eq:DerCom}
\left[ \sum_{i = 1}^{m} \hat{v}^i(\hat{x}) \frac{\partial}{\partial \hat{u}^i} \; , \; \sum_{i = 1}^{m} \hat{u}^i \frac{\partial}{\partial \hat{u}^i} \right] = \sum_{i = 1}^{m} \hat{v}^i (\hat{x}) \frac{\partial}{\partial \hat{u}^i}$$ So we have the following result as an easy consequence (See Olver [@PetOlv107:Sym] for related discussion in both the finite and infinite case).
\[thm:DerivedAlgebraThm\] Suppose that the analytic system ${R}$ is exactly linearizable by a local holomorphic diffeomorphism $\hat{x} = \psi(x,u), \hat{u} = \phi(x,u)$, to yield a linear target system (\[eq:Hsym\]) and $\mathcal{L}$, $\mathcal{L}'$ are the Lie symmetry algebra and its derived algebra for ${R}$. Also, let $\hat{\mathcal{L}}$, $\mathcal{\hat{L}}'$ be the corresponding algebras for ${{\hat R}}$. Let $\mathcal{L}^*$, $\mathcal{\hat{L}}^*$ be the superposition algebras under $\Psi$. Then $\mathcal{\hat{L}}^*$ is a subalgebra of $\mathcal{\hat{L}}'$ and $\mathcal{L}^*$ is a subalgebra of $\mathcal{L}'$. Moreover $\mathcal{L}^*$ and $\mathcal{\hat{L}}^*$ are abelian.
We wish to determine if a system ${R}$ is linearizable and if so, characterize the target ${\hat R}$, i.e $\mathcal{H} \hat{\eta}= 0$. But initially we don’t know $\mathcal{H}$. One approach is to write a general form for this system that specifies $\hat{S}^*$ with undetermined coefficient functions whose form is established in further computation. See for example, [@LGM101:LG] use this approach in the case of a single [[ODE]{}]{}, but don’t consider the Bluman-Kumei mapping system. We will apply our method to a test set of [[ODE]{}]{} (\[LGMTestODE\]) given in [@LGM101:LG]. Instead, we only include $\xi^i = 0, \hat{\eta}^j_{\hat{u}^k} = 0$ and don’t include $\mathcal{H} \hat{\eta} = 0$. Thus, we only include a subset $\hat{S}^\star$ of $\hat{S}^*$, denoting the truncated system as $$\label{eq:Sstar}
\hat{S}^\star := \left\{ \hat{\xi}^i = 0, \hat{\eta}^j_{\hat{u}^k} = 0: 1 \leq i \leq n, \; 1 \leq j,k \leq m \right\} $$ and allow $\mathcal{H} \hat{\eta} = 0$ to be found naturally later in the algorithm. We note that $\hat{S}^\star$ are the defining equations of a (usually infinite) Lie pseudogroup.
Algorithm PreEquivTest for excluding obvious nonlinearizable cases {#sec:PreEquivTest}
------------------------------------------------------------------
As discussed in §\[sec:PreMapEqs\], ${R}$ being linearizable implies that the superposition is in its Lie symmetry algebra and is a coordinate change of ${R}$. This implies some fairly well-known efficient tests for screening out obvious non-linearizable cases. For linearization necessarily $\dim{S} \geq d+1$ and $\dim{S'} \geq d$ for finite $d$. For $d = \infty$, necessarily $\dim{S} = \infty = \dim{S'}$ and in terms of differential dimensions $\textbf{d}(S) \geq \textbf{d}(S') \geq \textbf{d}(R)$. Note that Algorithm \[Alg:PreEquivTest\] returns null, if all its tests are true. The most well-known of the above tests occur when $d = \infty$ and $\dim{S} = \infty$ and are given for example in Bluman and Kumei [@BluKu112:Sym]. Also see Theorem 6.46 in Chapter 6 of Olver [@PetOlv107:Sym].
${\mbox{{{{\tt PreEquivTest}\xspace}}}}({R}, \mbox{IDR}, \mbox{\sc{IDS}}, \mbox{IDS}')$ : ${R}$ is a leading linear [[DPS]{}]{} system in [[[[dec]{}]{}]{}]{}-form ${R}$ with no leaders of order $0$ with respect to an orderly Riquier ranking $\mbox{IDR}$, $\mbox{IDS}$, $\mbox{IDS}'$ are respectively the initial data for ${R}$, $S$ and $S'$ where $S$ and $S'$ are the symmetry determining systems for $\mathcal{L}$ and $\mathcal{L}'$ : \[IsLinearizable, DimInfo\] IsLinearizable = false if one of the necessary conditions $T_j$ tests false DimInfo is the dimension info (dimension, differential dimension and Differential Hilbert Function, computed for each of $R, S, S'$. Set IsLinearizable := null. Apply DifferentialHilbertFunction to IDR, IDS, IDS’ to get DimInfo := \[ $\dim{{R}}$, $\textbf{d}(R)$, HF(R), $ \dim{S}$, $\textbf{d}(S)$, HF(S) $, \dim{S'}$, $\textbf{d}(S')$, HF$(S')$\] **if** $d = \infty$ **then** $T_1$ := evalb($\dim{S} < \infty$) $T_2$ := evalb($\dim{S'}< \infty$) $T_3$ := evalb($\textbf{d}(S) < \textbf{d}(R)$) $T_4$ := evalb($\textbf{d}(S') < \textbf{d}(R)$) **else if** $d < \infty$ **then** $T_5$ := evalb($\dim{S} < d+1$) $T_6$ := evalb($\dim{S'} < d$) **end if** **if** $\land_{i = 1}^{i = 6} T_i $ = false **then** IsLinearizable := false; **end if** **return** \[ IsLinearizable, DimInfo \]
\[Alg:PreEquivTest\]
Algorithm ExtractTarget for extracting the linear target system
---------------------------------------------------------------
\[Alg:ExtractTarget\]
When a system ${R}$ is determined to be linearizable by Algorithm \[Alg:MapDE2Linear\], the conditions for linearizability will yield a list of cases $\bigcup_{c \in C} Q_c$ where each $Q_c$ is in [[[[rif]{}]{}]{}]{}-form. To implicitly determine the target linear system ${\hat R}$ for a case $Q_c$, Algorithm $\text{ExtractTarget}$ is applied to $Q_c$. It first selects from $Q_c$ the linear homogeneous differential sub-system $ R_c^*$ in $\xi(x,u)$, $\eta(x,u)$ with coefficients depending on $(x,u,\psi, \phi)$ in the $(x,u)$ coordinates. Algorithm $\text{ExtractTarget}$ then applies the inverse BK transformations (\[eq:InvBK\]) to convert the system $ R_c^*$ to $(\hat{x},\hat{u})$ coordinates, after which $\hat{\xi} = 0$ and also $\hat{\eta}(\hat{x},\hat{u}) =\hat{ \eta}( \hat{x} )$ is imposed. This yields $ R_c^*$ as a system $\hat{ R}_c^*$ which is a linear homogeneous differential system in $\hat{\eta}(\hat{x})$ with coefficients in $\hat{x}, \hat{u},\hat{\psi}, \hat{\phi}$. Though $ R_c^*$ is in [[[[rif]{}]{}]{}]{}-form, $\hat{ R}_c^*$ is not usually in [[[[rif]{}]{}]{}]{}-form so another application of [[[[rif]{}]{}]{}]{} is applied, to yield $\hat{ R}_c^*$ in [[[[rif]{}]{}]{}]{}-form for $\hat{\eta}(\hat{x})$; case splitting is not required here. As shown in the proof of Algorithm \[Alg:MapDE2Linear\], the coefficients of $\hat{ R}_c^*$ depend only on $\hat{x}, \hat{\psi}, \hat{\phi}$ and not on $\hat{u}$. The proof of Algorithm \[Alg:MapDE2Linear\] also shows that $\hat{\eta}(\hat{x})$ can be replaced in $\hat{ R}_c^*$ with $\hat{u}(\hat{x})$ yielding the target linear homogeneous differential equation ${\hat R}_c$.
Heuristic integration for the Mapping functions in MapDE using PDSolve {#Method:PDSolve}
-----------------------------------------------------------------------
This routine is still in the early stages of its development. The heuristic integration routine [[PDSolve]{}]{}, basically a simple interface to [[[[Maple]{}]{}]{}]{}’s [[pdsolve]{}]{} which is applied to the $\Psi$ sub-system (the sub-system with highest derivatives in $\psi, \phi$) together with its inequations in $Q_c$ to attempt to find an explicit form of the mapping $\Psi$. We naturally use a block elimination ranking in $\Psi$ sub-system where all derivatives of $\phi$ are higher than all derivatives of $\psi$. Then we attempt to solve uncoupled subsystem for $\psi$ using the [[[LAVF]{}]{}]{}routine Invariants, which depends on integration, and subsequently solve the substituted system for $\phi$ by [[pdsolve]{}]{}.
The geometric idea is that in the $(\hat{x}, \hat{u})$ coordinates the Lie symmetry generator corresponding to linear superposition has the form $\sum_k \hat{\eta}^k(\hat{x}) \frac{\partial}{\partial \hat{u}^k }$ and generates an abelian Lie algebra with obvious invariants $\hat{x}$. So the independent variables for the target linear equation are invariants of this vector field acting on the base space of variables $(\hat{x}, \hat{u})$, and thus on $(x,u)$ space via the map $\Psi$. The process of integrating the mapping equations first starts with the determination of these invariants in terms of $(x,u)$ using the [[[[LAVF]{}]{}]{}]{} command [[Invariants]{}]{}. If the integration is successful this yields $\hat{x}^j = \psi^j = I^j(x,u)$, for $j = 1 , \cdots , n$. Then substitution into the $\Psi$ sub-system, yields a system with dependence only on the $\hat{\phi}$ mapping functions, which we attempt to integrate using Maple’s [[pdsolve]{}]{}.
The MapDE Algorithm {#sec:MapDE2Linear}
===================
The main subject here is Algorithm \[Alg:MapDE2Linear\] which makes heavy use of differential-elimination completion ([[[[dec]{}]{}]{}]{}) algorithms, which in our current implementation is the [[[[rif]{}]{}]{}]{} algorithm accessed via [[[[Maple]{}]{}]{}]{}’s ${{\tt rifsimp}\xspace}$. Other [[[[dec]{}]{}]{}]{} algorithms could be used such as [[ThomasDecomposition]{}]{}, [[RosenfeldGroebner]{}]{} or [[casesplit]{}]{}.
In §\[sec:MapDEcode\] we will describe pseudo-code for [[[MapDE]{}]{}]{}. In §\[sec:NotesMapDE2Linear\] we will give notes about the steps of [[[MapDE]{}]{}]{}and in §\[sec:ProofMapDE\] we will a proof of correctness of [[[MapDE]{}]{}]{}.
Pseudo-code for the MapDE algorithm {#sec:MapDEcode}
-----------------------------------
Here we describe the pseudo-code for [[[MapDE]{}]{}]{}.
${\mbox{{{{\tt MapDE}\xspace}}}}({{{\it Source}\xspace}}, {{{\it Target}\xspace}}, {{{\it Map}\xspace}}, Options)$ : [[[*Source*]{}]{}]{}: A leading linear [[DPS]{}]{} system in [[[[dec]{}]{}]{}]{}-form ${R}$ with no leaders of order $0$ with respect to an orderly Riquier ranking; vars $[x,u]$, $[\xi, \eta]$ [[[*Target*]{}]{}]{}: [[Target]{}]{}=[[LinearDE]{}]{} [[[*Map*]{}]{}]{}: $\Psi$ $Options$: Additional options (for strategies, outputs, etc). : IsLinearizable = false if there $\nexists$ an invertible local linearization $\Psi$ IsLinearizable = true if there $\exists$ an invertible local linearization $\Psi$ and — Collection of cases in [[[[rif]{}]{}]{}]{}-form yielding such linearizations — Implicit form of the target linear equation for ${\hat R}$ (see \[Alg:ExtractTarget\] ) — Explicit form of $\Psi$ if the heuristic method [[PDSolve]{}]{} is successful (see §\[Method:PDSolve\]) Set IsLinearizable := null. Compute $ \mbox{IDR} := \mbox{ID}({R})$ Let $\mathcal{L}' = {{\tt DerivedAlgebra}\xspace}(\mathcal{L})$ and compute: $S:= \mbox{{{{\sc dec}\xspace}}}(\mbox{DetSys}(\mathcal{L}))$, $S' := \mbox{{{{\sc dec}\xspace}}}(\mbox{DetSys}(\mathcal{L}'))$ $ \mbox{IDS} :=\mbox{ID}(S )$, $ \mbox{IDS}' :=\mbox{ID}(S')$
:= $ {\mbox{{{{\tt PreEquivTest}\xspace}}}}({R}, \mbox{IDR}, \mbox{\sc{IDS}}, \mbox{IDS}')$ **if** IsLinearizable = false **then** **return** \[ IsLinearizable, DimInfo \] **end if** When ${R}$ is an [[ODE]{}]{} also calculate $\mbox{LGMLin} := {\mbox{LGMLinTest}}({R})$.
Set $\hat{S}^\star := \left\{ \hat{\xi}^i= 0, \hat{\eta}^j_{\hat{u}^k} = 0: 1 \leq i \leq n, \; 1 \leq j,k \leq m \right\}$ $M := S' \cup \; \hat{S}^\star|_{\Psi} \; \cup \; M_{{\sc{BK}}}(\mathcal{L}', \hat{\mathcal{L}}^\star ) \; \cup \{\mbox{Det} \mbox{Jac}(\Psi) \not = 0\} $ Compute list of consistent cases $P = [P_1, \cdots , P_{\mbox{nc}}]$ with dim $\geq d$: $P := \text{{{{\sc dec}\xspace}}}(M, \prec, casesplit, mindim=d)$ **if** $P=\emptyset$ **then** IsLinearizable := false **return** \[ IsLinearizable, DimInfo \] **end if** $Q$ := \[ \] **for** $k = 1$ [**to**]{} $\text{nc}$ **do** **if** $\mbox{HF}({R}) = \mbox{HF}(P_k)$ **then** $ Q := Q \cup P_k$ **end if** **end do** **if** $Q=$ \[ \] **then** IsLinearizable := false **return** \[ IsLinearizable, DimInfo \] **else if** $Q \not =$ \[ \] **then** IsLinearizable := true **end if** **if** CaseSelect $\not \in$ Options **then** $C := [ 1 ]$ **else** Assign $C$ using Options **end if** **for** $c \in C$ **do** ${\hat R}_c := \text{ExtractTarget}(Q_c)$ **end do** (See \[Alg:ExtractTarget\]) **for** $c \in C$ **do** Attempt heuristic integration $ \Psi^c_{\mbox{sol}} := {{\tt PDSolve}\xspace}(Q_c)$ **end do** (See \[Method:PDSolve\])
**return** $\bigcup_{c \in C}$ \[$Q_c$, ${\hat R}_c$, ${\hat R}_c|_{{\Psi}^c_{\mbox{sol}}}$, $\Psi^c_{\mbox{sol}}$\] $\hrulefill$ **Abbreviations used above**: [[[dec]{}]{}]{}: Differential Elimination Completion, $M_{{\sc{BK}}}$ BK system, ID: InitialData
\[Alg:MapDE2Linear\]
Notes on the MapDE Algorithm with Target = LinearDE {#sec:NotesMapDE2Linear}
---------------------------------------------------
We briefly list some main aspects of Algorithm \[Alg:MapDE2Linear\].
- Due to current limitations of [[[[Maple]{}]{}]{}]{}’s [[DeterminingPDE]{}]{} we restrict to input a single system ${R}$, in [[[[dec]{}]{}]{}]{} (i.e. [[[[rif]{}]{}]{}]{}-form) leading linear equations with leading derivatives of differential order $\geq 1$, together with inequations and no leading nonlinear equations. This form is more general than Cauchy–Kowalevski form, and includes over and under-determined systems, but not systems with $0$ order (algebraic) constraints.
[[Options]{}]{} refers to additional options for strategies and outputs. For example including OutputDetails in Options yields more detailed outputs.
- See Remark \[rem:DerivedAlgebraAlgorithm\], where we briefly describe our new algorithm for computing determining systems for infinite dimensional derived algebras.
- As discussed in §\[sec:PreMapEqs\], ${R}$ being linearizable means that linear superposition generates a symmetry sub-algebra of $\mathcal{L}$, yielding some fairly well-known efficient tests for rejecting many non-linearizable systems. See Algorithm \[Alg:PreEquivTest\] for details. We also apply Algorithm \[alg:LGMLinTest\] for the LGMLinTest [@LGM101:LG] when ${R}$ is an [[ODE]{}]{} in order to compare and test our Hilbert linearization test which occurs later in the algorithm.
- $\hat{S}^\star|_{\Psi}$ is $\hat{S}^\star$ evaluated in $(x,u)$ coordinates via $\Psi$ using differential reduction. $\hat{S}^\star := \left\{ \hat{\xi}^i= 0, \hat{\eta}^j_{\hat{u}^k} = 0: 1 \leq i \leq n, \; 1 \leq j,k \leq m \right\}$
- Here ${{\tt mindim}\xspace}= \dim({R}) = d$, as computed by the [[[Maple]{}]{}]{}command [[initialdata]{}]{} of the [[[dec]{}]{}]{}form of ${R}$. The mindim option avoids computing cases of dimension $< d$ by monitoring an upper bound based on initial data of such cases. The block elimination ranking $\prec$ ranks all infinitesimals and their derivatives for the first block $[\xi, \eta, \hat{\xi}, \hat{\eta}]$ strictly greater than the second block of the $\phi$ map variables, which are strictly greater than all derivatives of the third block of $\psi$ variables. This maintains linearity in the variables $[\xi, \eta, \hat{\xi}, \hat{\eta}]$. The mindim dimension is computed with respect to these variables, and not the degrees of freedom in the map variables $(\psi, \phi)$. The block structure also facilitates the later integration phase. Each case $P_k$ consists of equations and inequations.
- The Hilbert Functions of $R$ and $P_k$, disregarding the equations that don’t involve infinitesimals, should be equal if the system is linearizable.
- Note that $Q$ can consist of several systems. If *CaseSelect = all* is included in Options, then all cases leading to linearization are returned. By default, [[[[MapDE]{}]{}]{}]{} returns only one such case: $C = [1]$.
Proof of correctness of the MapDE Algorithm {#sec:ProofMapDE}
-------------------------------------------
\[thm:MapDEThm\] Let ${R}$ be a single input system in [[[[rif]{}]{}]{}]{}-form) consisting of leading linear equations with leaders of differential order $\geq 1$, inequations, and with no leading nonlinear equations. Then Algorithm \[Alg:MapDE2Linear\] converges in finitely many steps, and determines whether there exists a local holomorphic diffeomorphism $\hat{x} = \psi(x,u), \hat{u} = \phi(x,u)$ transforming ${R}$ to a linear homogeneous target system $$\label{eq:Hsys}
{\hat R}: \mathcal{H} \hat{u}(\hat{x}) = 0$$ In the case of existence the output [[[[rif]{}]{}]{}]{}-form consists of [[DPS]{}]{} of equations and inequations including those for the mapping function $(\psi, \phi )$.
.
We first note that Algorithm \[Alg:MapDE2Linear\] converges in finitely many steps due to finiteness of each of the sub-algorithms used [@Rus99:Exi; @LGM101:LG; @Bou95:Rep; @Robertz106:Tdec]. To complete the proof we need to establish correctness of the two possible outcomes:\
*Case I:* IsLinearizable = true and *Case II:* IsLinearizable = false.
***Case I: IsLinearizable = true***\
Our task here is to show that given consistent input ${R}$, IsLinearizable = true and output $Q$ then there exists a local holomorphc diffeomorphism $\Psi$ to some linear system ${\hat R}$. To do this we build initial data for a solution of ${R}$, and initial data for solutions of $Q$. A complication is that these spaces have different independent variables $x$ and $(x,u)$. The assumption that all leaders for the [[[[rif]{}]{}]{}]{}-form of $R$ are of order $\geq 1$ enables us to regard $(x,u)$ as independent variables for $Q$. The inequations for $Q$ include those for ${R}$ together with the invertibility condition $\mbox{DetJac}(\Psi) \not =0$.
Suppose that the input [[[[rif]{}]{}]{}]{}-form of $R$ has differential order $d_R$ and consists of equations and inequations with associated varieties $ V^{=}(R)$, $ V^{\not=}(R)$ in Jet space $J^{d_R} (\mathbb{C}^m, \mathbb{C}^{n})$. So any point on the jet locus satisfies $(x, u^{(\leq d_R)}) \in V^=(R) \setminus V^{\not =}(R)$ in jet space $J^{d_R} (\mathbb{C}^m, \mathbb{C}^{n})$ where $u^{(\leq d_R)}$ denotes the jet variables of total differential order $\leq d_R$. Let $\pi_0^{d_R}: J^{d_R} \rightarrow X $ be the projection of points in $J^{d_R}$, the jet space of order $d_R$, to the base space of independent variables $X \simeq \mathbb{C}^m$ where $x \in X$. The assumption that all leaders for $R$ are of order $\geq 1$ implies that $\pi_0^{d_R} (V^=(R) \setminus V^{\not =} (R) )
= \mathbb{C}^m \setminus \pi_0^{d_R} (V^{\not =} (R)) $.
When IsLinearizable = true there will be several systems $P_k$ in the list of systems $Q$ at Step 11 of Algorithm \[Alg:MapDE2Linear\]. We consider the case where there is only one such system, and without loss denote it by $Q$. For the case of several systems in $Q$ we simply repeat the argument below for each such system. Suppose the system has differential order $d_Q$, and consists of equations and inequations for $v = (\xi, \eta, \psi, \phi, \hat{\eta})$ with associated varieties $V^=(Q)$, $V^{\not=}(Q)$ in $J^{d_Q}(\mathbb{C}^{m+n}, \mathbb{C}^{(2m+3n)})$, so that $((x,u), v^{(\leq d_Q)}) \in V^=(Q) \setminus V^{\not =}(Q)$ in $J^{d_Q}(\mathbb{C}^{m+n}, \mathbb{C}^{(2m+3n)})$. Then $ \pi_0^{d_Q} (V^=(Q) \setminus V^{\not =} (Q) ) = \mathbb{C}^{m+n} \setminus \pi_0^{d_Q} (V^{\not =} (Q) )$.
Consider points $x_0 \in \mathbb{C}^m \setminus \pi_0^{d_R} (V^{\not =} (R)) $ and $(x_0, u_0) \in \mathbb{C}^{m+n} \setminus \pi_0^{d_Q} (V^{\not =} (Q) )$ belonging to the projections of $R$ and $Q$ onto their base spaces $X \simeq \mathbb{C}^m$ and $X \times U \simeq \mathbb{C}^{m+n}$. Then a family of initial data corresponding to all local analytic solutions $u$ in a neighborhood of $x_0$ exists, and similarly for $v$. For $R$ there exists a neighborhood $\mathcal{N}(x_0, u^{(\leq d_R)}_0) \subseteq V^=(R) \setminus V^{\not =}(R)$ in $J^{d_R}$ and from $Q$ there exists a neighborhood $\mathcal{N}((x_0,u_0), v^{(\leq d_Q)}_0) \subseteq V^=(Q) \setminus V^{\not =}(Q)$ in $ J^{d_Q}$. The existence and uniqueness Theorems associated with [[[[rif]{}]{}]{}]{}-form implies that for such analytic initial data there corresponds unique local analytic solutions and implies that there exists a local holomorphic diffeomorphism $\Psi$ between neighborhoods mapping ${R}$ to ${\hat R}$, and similarly between neighborhoods mapping $Q$ to to $\hat{Q}$. Under this diffeomorphism the images of $\mathcal{N}(x_0, u^{(\leq d_R)}_0) $ and $\mathcal{N}((x_0, u_0), v^{(\leq d_Q)}_0)$ are $\hat{\mathcal{N}}(\hat{x}_0, \hat{u}^{(\leq d_R)}_0) $ and $\hat{\mathcal{N}}((\hat{x}_0,\hat{u}_0), \hat{v}^{(\leq d_Q)}_0)$ respectively.
To show that ${\hat R}$ is linear we consider the subsystem of $Q$ for $\hat{\eta}$: $$\label{targetLin}
\mathcal{L}^{**} = \left\{ \sum_{\ell=1}^m \hat{\eta}^\ell (\hat{x}) {\frac{\partial}{\partial \hat{u}^\ell}}:
\hat{\eta}^j_{\hat{u}^k} = 0, {\mathcal{H}} (\hat{\eta}) = 0 \right\}$$ where the linear system for $\hat{\eta}$ is in [[[[rif]{}]{}]{}]{}-form and ultimately we will show that ${\mathcal{H}} (\hat{\eta})$ can be taken as ${\hat R}$. First we note the linear operator $\mathcal{H}$ cannot have any coefficients depending on $\hat{u}$. If not, and a coefficient did depend on a particular $\hat{u}^\ell$, then differentiating (\[targetLin\]) with respect to $\hat{u}^\ell$ would yield a relation between parametric quantities, violating the freedom to assign values independently to these parametric quantities. This would violate the [[[[rif]{}]{}]{}]{}-form and its existence and uniqueness Theorem. So $\mathcal{H}$ only has coefficients depending on $\hat{x}$. As a remark we note that it is now easily verified that $\mathcal{L}^{**} $ generates an abelian Lie pseudogroup.
Exponentiating the infinitesimal symmetry (\[targetLin\]) and applying its prolongation to a solution $ \hat{u}(\hat{x})$ in $\hat{\mathcal{N}}(\hat{x}_0, \hat{u}^{(\leq d_R)}_0) $ yields another solution in $\hat{\mathcal{N}}(\hat{x}_0, \hat{u}^{(\leq d_R)}_0) $ given by $ \tilde{u}(\hat{x}) = \hat{u}(\hat{x}) + \hat{\eta}$ where ${\mathcal{H}} (\hat{\eta}) = 0$. We have assumed in Step 9 of Algorithm \[Alg:MapDE2Linear\] that $\mbox{HF}({R}) = \mbox{HF}(Q)$, which implies that all local analytic solutions in $\hat{\mathcal{N}}(\hat{x}_0, \hat{u}^{(\leq d_R)}_0) $ are of form $ \tilde{u}(\hat{x}) = \hat{u}(\hat{x}) + \hat{\eta}$ in $\hat{\mathcal{N}}(\hat{x}_0, \hat{u}^{(\leq d_R)}_0) $. Consequently by a point change $\tilde{u}(\hat{x}) \rightarrow \tilde{u}(\hat{x}) - \hat{u}(\hat{x})$, ${\hat R}$ is equivalent to the linear homogeneous system $ {\mathcal{H}} (\hat{\eta}(\hat{x})) = 0$. The [[[[rif]{}]{}]{}]{}-form of $Q$ includes the system for $\Psi$ that determines mappings to ${\hat R}$ given by $ {\mathcal{H}} (\hat{\eta}(\hat{x})) = 0$.
***Case II: IsLinearizable = false***\
Suppose to the contrary that Algorithm \[Alg:MapDE2Linear\] returns IsLinearizable = false, yet a local analytic linearization exists. Since the tests in Algorithm [[PreEquivTest]{}]{} in Step 3 of Algorithm \[Alg:MapDE2Linear\] are all necessary conditions for a linearization to exist, they all test true.
Step 5 of Algorithm \[Alg:MapDE2Linear\] applies [[[[rif]{}]{}]{}]{} using binary splitting, partitioning the jet locus into disjoint cases; and the linearization must belong to some of these cases. By assumption, and the discussion above, there is diffeomorphism $\Psi$ of ${R}$ in some neighborhood $\mathcal{N}(x_0, u^{(\leq d_R)}_0)$ in $J^{d_R}$ to a linear system ${\hat R}$. Further the equations of the cases corresponding to linearization in terms of $v = (\xi, \eta, \psi, \phi, \hat{\eta})$ must have dimension $d$.
It cannot belong to a case of dimension $< d$, the ones discarded by the $\mbox{mindim} = d$ option. Therefore by disjointness it must belong to one of the $\mbox{nc}$ cases in $P$, say $P_s$. Therefore this case must fail the condition that $\mbox{HF}({R}) = \mbox{HF}(P_s)$ which is contrary to our assumption that such a linearization exist, completing our proof of correctness.
Examples {#sec:Examples}
========
To illustrate the [[[MapDE]{}]{}]{}Algorithm \[Alg:MapDE2Linear\] we consider some examples.
(Continuation and conclusion for Examples \[ex:ODE3(a)\],\[ex:ODE3(b)\] and \[ex:ODE3(c)\] using Algorithm \[Alg:MapDE2Linear\].) \[ex:ODE3(d)\]
The input is (\[ODE3\]) which is in [[[[rif]{}]{}]{}]{}-form with respect to the orderly ranking $u \prec u_x \prec u_{xx} \prec \cdots $, together with the inequations $u \not = 0, uu_{{x}}+x \not = 0, {u}^{2}+{x}^{2} \not = 0$ or equivalently $u (uu_{{x}}+x)({u}^{2}+{x}^{2}) \not = 0$. This can be regarded as being derived from the leading linear [[DPS]{}]{} which results from multiplication by factors in its denominators. **Step 1:** Set IsLinearizable := null. Here ${\mbox{ID}}({R}) = [u(x_0)= c_1, u_x(x_0)= c_2,u_{xx}(x_0)= c_3]$ and $\dim {R}= 3$, subject to $ u (uu_{{x}}+x)({u}^{2}+{x}^{2}) \not = 0$.
**Step 2:** See Example \[ex:ODE3(a)\] for $S:= \mbox{{{{\sc rif}\xspace}}}(\mbox{DetSys}(\mathcal{L}))$ in (\[eq:ODE3DetSys\]), together with its $\mbox{ID}(S)$ and $\dim \mathcal{L} = \dim(S) = 4$. See Example \[ex:ODE3(c)\] and in particular (\[eq:ODE3\_L’\]) for $S' := \mbox{{{{\sc rif}\xspace}}}(\mbox{DetSys}(\mathcal{L}'))$ which yields $\dim \mathcal{L}' = \dim(S') = 3$.
**Step 3:** Since $\dim{S} = 4 \geq d+1 = 4$ and $\dim{S'} = 3 \geq d=3$, the simplest necessary conditions for linearizability hold. Also $\textbf{d} (S) = \textbf{d}(S') = \textbf{d}(R) = 0$. Application of Algorithm \[alg:LGMLinTest\] for the LGMLinTest in Example \[ex:ODE3(b)\] shows that ${R}$ is linearizable, subject to $ u (uu_{{x}}+x)({u}^{2}+{x}^{2}) \not = 0$.
**Step 4:** $\mbox{Det} \mbox{Jac}(\Psi) = \psi_x \phi_u - \psi_u \phi_x \not = 0$, $\hat{S}^\star := \left\{ \hat{\xi}= 0, \hat{\eta}_{\hat{u}} = 0 \right\}$ and $\hat{\mathcal{L}}^* $ is replaced with $\hat{\mathcal{L}}^\star$ in (\[eq:BKODE3\]) to yield: $$\label{eq:BKODE3repeat}
\hspace{-1.0cm} M_{{\sc{BK}}}(\mathcal{L}', \hat{\mathcal{L}}^\star ) = \left\{
\hat{\xi} (\hat{x}, \hat{u}) = 0 = \xi \psi_x + \eta \psi_u,
\hat{\eta} (\hat{x}, \hat{u}) = \xi \phi_x +\eta \phi_u \right\}$$ Evaluate $ \hat{S}^\star$ modulo $\Psi: \hat{x} = \psi(x,u), \hat{u} = \phi(x,u)$ to obtain $ \hat{S}^\star|_{\Psi}$. This yields $ \hat{S}^\star|_{\Psi} = \left\{ \hat{\xi} = 0, \psi_u \hat{\eta}_x - \psi_x \hat{\eta}_u = 0 \right\} $. Note that for brevity of notation we have replaced $\hat{\xi}(\hat{x},\hat{u})$ with $\hat{\xi}(x,u)$ and $\hat{\eta}(\hat{x},\hat{u})$ with $\hat{\eta}(x,u)$. Thus, the mapping system $M = S' \cup \; \hat{S}^\star|_{\Psi} \; \cup \; M_{{\sc{BK}}}(\mathcal{L}', \hat{\mathcal{L}}^\star ) \; \cup \{\mbox{Det} \mbox{Jac}(\Psi) \not = 0\} $ is: $$\begin{aligned}
\label{eq:Msys'}
M = [\xi &=-{\frac {\eta\,u}{x}},\quad \eta_{{x}}={\frac { \left( {u}^{2}+{x}^{2}
\right) \eta}{x{u}^{2}}}+{\frac {\eta_{{u}}x}{u}}, \nonumber \\
& \eta_{{u,u,u}}=-{
\frac { \left( 8\,{u}^{8}+8\,{u}^{6}{x}^{2}+8\,{u}^{6}+3\,{u}^{2}+3\,{x}^{2} \right) \eta}{ \left( {u}^{2}+{x}^{2} \right) {u}^{3}}}+3\,{
\frac {\eta_{{u}}}{{u}^{2}}}, \\
& \hat{\xi} = 0,\quad \psi_u \hat{\eta}_x - \psi_x \hat{\eta}_u = 0, \quad
\hat{\xi} = \xi \psi_x + \eta \psi_u, \quad \nonumber \\
& \hat{\eta} = \xi \phi_x +\eta \phi_u , \quad
\psi_x \phi_u - \psi_u \phi_x \not = 0 ] \nonumber
\end{aligned}$$
**Step 5:** Compute $P := \text{{{{\sc rif}\xspace}}}(M, \prec, casesplit, mindim=d)$ where $d = 3$. This results in $3$ cases, two of which are rejected before their complete calculation since an upper bound in the computation drops below $mindim = d = 3$. The output for the single consistent case $P_1$ found is: $$\begin{aligned}
\label{eq:rifMsys'}
P_1 = [\xi &=-{\frac {\eta\,u}{x}},\eta_{{x}}={\frac { \left( {u}^{2}+{x}^{2}
\right) \eta}{x{u}^{2}}}+{\frac {\eta_{{u}}x}{u}}, \nonumber \\
& \eta_{{u,u,u}}=-{
\frac { \left( 8\,{u}^{8}+8\,{u}^{6}{x}^{2}+8\,{u}^{6}+3\,{u}^{2}+3\,{
x}^{2} \right) \eta}{ \left( {u}^{2}+{x}^{2} \right) {u}^{3}}}+3\,{
\frac {\eta_{{u}}}{{u}^{2}}}, \nonumber \\
& \phi_{{x,x}}={\frac {2\,\phi_{{x,u}}x{u}^{2}-\phi_{{u,u}}{x}^{2}u+\phi
_{{u}}{u}^{2}+\phi_{{u}}{x}^{2}}{{u}^{3}}} , \;
\psi_{{x}}={\frac {\psi_{{u}}x}{u}} , \\
& \hat{\eta} =-{\frac { \left(u\phi_{{x}}-x\phi_{{u}
} \right) \eta}{x}} , \;
\hat{\xi} = 0, \; x\phi_{{u}}-u\phi_{{x}} \neq 0 , \; \psi_u \neq 0 ] \nonumber
\end{aligned}$$
**Step 6:** $P\neq \emptyset$ contains $1$ case.
**Step 7:** Initialize $Q := [ \; ] $
**Step 8:** $k = \text{nc} = 1$
**Step 9:** Also $\mbox{HF}({R}) = 1 + s + s^2$ and the for $P_1$ yields $ \mbox{HF}(P_1 ) = 1 + s + s^2$. So $\mbox{HF}({R}) = \mbox{HF}(P_1)$ and the system is linearizable.
**Step 12:** We set $Q := [P_1 ]$. To extract the target we apply Algorithm \[Alg:ExtractTarget\] so that ${\hat R}_1 := \text{ExtractTarget}(Q_1)$ which yields ${\hat R}_1$ in the form: $$\label{TargetODE3a}
\left({\frac{\partial}{\partial \hat{x}}}\right)^3 \hat{u} (\hat{x}) = a_2 (\hat{x}) \left( {\frac{\partial}{\partial \hat{x}}}\right)^2 \hat{u} (\hat{x})
+ a_1 (\hat{x}) {\frac{\partial}{\partial \hat{x}}} \hat{u} (\hat{x})
+ a_0(\hat{x}) \hat{u} (\hat{x})$$ where $a_2 (\hat{x})$, $a_1 (\hat{x})$, $a_0 (\hat{x})$ are explicit expressions in $(\hat{x},\hat{u},\hat{\psi}(\hat{x},\hat{u}), \hat{\phi}(\hat{x},\hat{u}))$ and derivatives of $\hat{\psi}(\hat{x},\hat{u}), \hat{\phi}(\hat{x},\hat{u})$. **Step 13:** The $\psi$ system here is $\psi_{{x}}={\frac {\psi_{{u}}x}{u}} $. Using ${{\tt Invariants}\xspace}$ from the [[[[LAVF]{}]{}]{}]{} package yields a single invariant $x^2 + u^2$ and so $\psi = x^2 + u^2$. Here and elsewhere the ${{\tt Invariants}\xspace}$ removes the need for us to specify arbitrary functions which would be the case if we started from the general solution of the $\psi$ equation which is in this case $\psi = F(x^2+u^2)$. Then substitution and solution of the $\phi$ equation then yields $\phi = G(x^2 + u^2) x + H(x^2 + u^2)$. The program specializes the arbitrary functions and constants to satisfy the inequations including the Jacobian condition, and in this case yields $$\begin{aligned}
\label{phipsiSol1}
\hat{x} & = \psi = x^2 + u^2 \\
\hat{u} & = \phi = x \nonumber
\end{aligned}$$ Substitution of (\[phipsiSol1\]) into the target (\[TargetODE3a\]) requires first inverting (\[phipsiSol1\]) using [[[[Maple]{}]{}]{}]{}’s [[solve]{}]{} to give $x = \hat{\psi} = \hat{u}, u = \hat{\phi} = \left( \hat{x} - \hat{u}^2 \right)^{1/2}$ yields it explicitly as: $$\label{TargetODE31}
\left({\frac{\partial}{\partial \hat{x}}}\right)^3 \hat{u} (\hat{x}) =
- \frac{ (\hat{x}+1)}{\hat{x}} \hat{u} (\hat{x})$$ So far our work on the integration of the mapping equations to determine the transformations is preliminary and experimental. We have shown that the basic structure of the linear target can be determined implicitly. It remains to be seen how useful this would be in applications, where the mapping cannot be determined explicitly. Heuristic methods appear to be useful here, and we encourage the reader to try explore their own approaches.
From the output we also subsequently explored how far we could make the Target explicit before the integration of the map equations. In particular we exploited the transformation (as do [@LGM101:LG]) that any such [[ODE]{}]{} is point equivalent to one with its highest coefficients (here $a_2$, $a_1$) being zero. This yields additional equations on $\psi$, $\phi$ and the target takes the very simple form: $$\label{TargetODE3b}
\left({\frac{\partial}{\partial \hat{x}}}\right)^3 \hat{u} (\hat{x}) =
- \frac{8u^3 (u^2+x^2+1)}{(u^2+x^2) \psi_u^3} \hat{u} (\hat{x})$$ The [[[rif]{}]{}]{}-form of the system for $\phi, \psi$ is: $$\begin{aligned}
\label{rifpsiphisys}
\psi_{{x}}&={\frac {\psi_{{u}}x}{u}} \nonumber \\
\psi_{{u,u,u}} &=-1/2\,{\frac {-3\,{\psi_{{u,u}}}^{2}{u}^{2}+3\,{\psi_{
{u}}}^{2}}{\psi_{{u}}{u}^{2}}} \nonumber \\
\phi_{{x,x}}&=2\,{\frac {x\psi_{{u,u}} \left( \phi_{{x}}u-\phi_{{u}}x
\right) }{\psi_{{u}}{u}^{2}}}+{\frac {\phi_{{u,u}}{x}^{2}u+\phi_{{u}}
{u}^{2}-2\,\phi_{{x}}ux+\phi_{{u}}{x}^{2}}{{u}^{3}}} \nonumber \\
\phi_{{x,u}}&={\frac {
\psi_{{u,u}}\phi_{{x}}u-\psi_{{u,u}}\phi_{{u}}x+\psi_{{u}}\phi_{{u,u}}
x-\psi_{{u}}\phi_{{x}}}{\psi_{{u}}u}} \nonumber \\
\end{aligned}$$ The general solution of the system is found by [[[[Maple]{}]{}]{}]{} and yields the same particular solution as before for $\psi, \phi$. It seems to have a made a straightforward problem, more difficult!
(Lyakhov, Gerdt and Michels Test Set) Lyakhov, Gerdt and Michels [@LGM101:LG] consider the following test set of [ODE]{}of order $d$, for $3 \leq d \leq 15$: $$\label{LGMTestODE}
\left( \frac{d}{dx}\right)^d (u(x)^{2})+ u(x)^{2}=0$$ By inspection this has the linearization for any $d$: $$\label{LGMTestODEtrans}
\Psi=\{\hat{x} = x, \, \hat{u} = u^2 \}$$
They report times on an Intel(R)Xeon(R) X5680 CPU clocked at 3.33 GHz and 48GB RAM. All the following times are measured from the entry to the program. The times for detecting the existence of the linearization by the LGM Test in [@LGM101:LG] range from 0.2 secs for $d = 3$ to about 150 secs for $d = 15$. For comparison, we run MapDE with our own implementation of the LGMLinTest using [[[[LAVF]{}]{}]{}]{}. Our runs of the same tests to detect the existence of the linearization using on a 2.61 GHz I7-6600U processor with 16 GB of RAM range from 0.422 secs when $d = 3$ to 11.5 secs when $d= 15$.
Their method from start to existence and then construction of the linearization (existence and construction), takes 7512.9 secs for $d=9$ and is out of memory for $d \geq 10$. In contrast, we report times for existence and construction that are only slightly longer than our existence times for $3 \leq d \leq 15$. For $d=3$ to $d=15$ we also report the time for our Hilbert test for existence of linearization and the total time for the existence and construction of the explicit linearization. Thus LGMTest time $<$ Hilbert Test time $<$ Existence and Construction. These results are displayed in Fig. \[fig:testTime\] on a $\log_{10} $ axis.
![The graph represents the CPU times for $\left( \frac{d}{dx}\right)^d (u(x)^{2})+ u(x)^{2}=0$. Timings from $t = 0$ (start of [[[[MapDE]{}]{}]{}]{}) to the time to LGMLin linearization Existence confirmation (Red), time to Hilbert Existence confirmation (Yellow), time from $t = 0$ to existence and construction of the linearizing transformations (Green).[]{data-label="fig:testTime"}](LGMtestTime.png){width="6cm"}
On this test, our approach appears to have more favorable memory behavior, which possibly is due to our equations being less nonlinear those of Lyakhov et al. [@LGM101:LG]. However, more testing and analysis are needed to make a reasonable comparison.
\[ex:Burgers\] Consider Burger’s equation, modeling the simplest nonlinear combination of convection and diffusion: $$u_{{x,x}}= u_{{t}} -u u_x$$ Using our algorithm [[[MapDE]{}]{}]{}with ${{\tt TargetClass}\xspace} = {{\tt LinearDE}\xspace}$ shows that it has finite dimensional Lie symmetry algebra with $\dim \mathcal{L} = 5 < \infty$. Thus by the preliminary equivalence test [[PreEquivTest]{}]{}, it is not linearizable by point transformation. However rewriting this equation in conserved form ${\frac{\partial}{\partial x}} (u_x + \frac{1}{2} u^2 ) = {\frac{\partial}{\partial t}} u$ implies that there exists $v$: $$\label{potBurgers}
v_x = u, \; \; \; v_t = u_x + \frac{1}{2} u^2$$ Applying [[[MapDE]{}]{}]{}with ${{\tt TargetClass}\xspace} = {{\tt LinearDE}\xspace}$ to (\[potBurgers\]) shows that this new system is linearizable, with the [[[[rif]{}]{}]{}]{}-form of the $\Psi$ system given by: $$\begin{aligned}
\label{mapsysBurgers}
\phi_{{u,u}}&=0,\; \varphi_{{u,u}}=0, \; \phi_{{u,v}}=-1/2\,\phi_{{u}}, \nonumber \\
\varphi_{{u,v}}&=-1/2\,\varphi_{{u}}, \; \phi_{{v,v}}=-1/2\,\phi_{{v}}, \nonumber \\
\varphi_{{v,v}}&=-1/2\,\varphi_{{v}},
\Upsilon_{{u}}=0,\psi_{{u}}=0,
\Upsilon_{{v}}=0,\psi_{{v}}=0
\end{aligned}$$ After integration this yields $\Psi$: $${\it {\hat{x}}}= \psi = x ,\, {\it {\hat{t}}}= \Upsilon = t,\, {\it {\hat{u}}}= \varphi = u \exp\left(-\frac{v}{2} \right), \, {\it {\hat{v}}}= \phi = \exp\left(-\frac{v}{2}\right)$$ and the Target system ${\hat R}$ is $[{\it {\hat{u}}}_{{{\it {\hat{x}}}}}=-2 {\it {\hat{v}}}_{{{\it \hat{t}}}}, {\it {\hat{v}}}_{{{\it {\hat{x}}}}}=-{\it {\hat{u}}}/2]$ so that ${\it {\hat{v}}}_{\it \hat{t}} = {\it {\hat{v}}}_{{{\it {\hat{x}}{\hat{x}}}}}$, ${\it {\hat{u}}}_{\it \hat{t}} = {\it {\hat{u}}}_{{{\it {\hat{x}}{\hat{x}}}}}$. This implies that the original Burger’s equation is apparently also linearizable, through the introduction of the auxiliary nonlocal variable $v$. This paradox is resolved in that the resulting very useful transformation is not a point transformation, since it effectively involves an integral. For extensive developments regarding such nonlocally related systems see [@Blu10:App].
(Nonlinearizable examples with infinite groups) \[ex:KPLiouville\] Consider the KP equation $$\label{KP}
u_{{x,x,x,x}}=-6\,uu_{{x,x}}-6\,{u_{{x}}}^{2}-4\,u_{{x,t}}-3\,u_{{y,y
}}$$ which has $$\begin{aligned}
\mbox{ID}(R) = \{ & u(x_0, y, t) = F_1(y, t),\; u_x(x_0, y, t) = F_2(y, t), \nonumber \\
&u_{x,x}(x_0, y, t) = F_3(y, t),\; u_{x,x,x} (u)(x_0, y, t) = F_4(y, t)\}.
\end{aligned}$$
Applying [[[[MapDE]{}]{}]{}]{} shows that the defining system $S$ for symmetries $\xi {\frac{\partial}{\partial x}} + \eta {\frac{\partial}{\partial y}} + \tau {\frac{\partial}{\partial t}} + \beta {\frac{\partial}{\partial u}}$, has initial data which is the union of infinite data along the Hyperplane $p = (x_0, y_0, t, u_0)$ and finite initial data at the point $z_0 = (x_0, y_0, t_0, u_0)$: $$\begin{aligned}
\mbox{ID}(S) &= \{ \beta \left( p \right) =H_{{1}} \left( t
\right) , \beta_y \left( p \right) =H_{{2}} \left( t \right) , \beta_{y,y} \left( p \right) =H_{{3
}} \left( t \right) \} \nonumber \\
\phantom{XX} & \hspace{3cm} \cup \\
\{ \beta_x (z_0 ) &=c_{{1}}, \beta_u (z_0 ) =c_{{2}},\eta (z_0 ) =c_{{3}},\eta_t ( z_{{0}}) =c_{{4}},\tau (z_0 ) =c_{{5}},\xi (z_0 ) =c_{{6}} \} \nonumber
\end{aligned}$$ So both the KP equation and its symmetry system have infinite dimensional solution spaces: $\dim R = \dim S = \infty$ since both have arbitrary functions in their data. However the KP equation has differential dimension $\textbf{d} (R) = 2$ since there are a max of $2$ free variables ($y,t$) in its initial data while its symmetry system has $\textbf{d} (S) = 1$ since it has a max of one free variable in its data. Thus $\textbf{d} (S) = 1 < 2 = \textbf{d} (R)$ and by the [[PreEquivTest]{}]{} the KP equation is not linearizable by point transformation.
Consider Liouville’s equation $u_{{x,x}} + u_{{y,y}} = \mbox{e}^u$ which we rewrite as a [[DPS]{}]{} using [[[[Maple]{}]{}]{}]{}’s function [[dpolyform]{}]{}. That yields $v = \mbox{e}^u$ and the Liouville equation in the form $v_{{x,x}} = - v_{{y,y}} + v_x^2/v + v_y^2/v + v^2$. [[[[MapDE]{}]{}]{}]{} determines that $\dim \mathcal{L} = \infty = \dim {R}$, and also that the Liouville equation is not linearizable by point transformation. Interestingly it is known that Liouville’s equation is linearizable by contact transformation (a more general transformation involving derivatives). For extensive developments regarding such contact related systems see [@Blu10:App].
\[ex:NLTelegraph\] Given that exactly linearizable systems are not generic among the class of nonlinear systems, a natural question is how to identify such linearizable models. Since linearizability requires large (e.g. $\infty$ dimensional) symmetry groups a natural approach is to embed a model in a large class of systems and seek the members of the class with the largest symmetry groups. Indeed in Wittkopf and Reid [@ReWit113:rif] developed such as approach. We now illustrate how this approach can be used here. For a nonlinear telegraph equation one might embed it in the general class of spatially dependent nonlinear telegraph systems $$\label{polTel}
v_x = u_t, \; \; \; v_{t} = C(x,u)u_x + B(x,u)$$ where $B_u \neq 0, C_u \neq 0, B_x \neq 0, C_x \neq 0$. Then applying the [[maxdimsystems]{}]{} algorithm available in [[[[Maple]{}]{}]{}]{} with $dim = \infty$ a quick calculation yields $11$ cases, only $4$ of which satisfy the dimension restriction which we further narrow by requiring restriction to those that have the greatest freedom in $C(x,u)$, $B(x,u)$. Integration yields the linearizable class: $$\label{TelLin}
v_x = u_t, \; \; \; v_{t} = \frac{1}{q_x u} f\left( \frac{u}{q_x}\right) u_x - \frac{q_{xx} } {q_x^2 } f\left( \frac{u}{q_x}\right)$$ and the linearizing transformation $$\label{TelLinTran}
\hat{x} = \frac{u}{q_x}, \hskip6pt \hat{t} = v, \hskip6pt \hat{u} = q(x), \hskip6pt \hat{v} = t$$ Similarity, we can seek the maximal dimensional symmetry group for the normalized linear Schrödinger Equation $$\label{ex:ConsCoeffR}
i \hslash \varphi_t = -\frac{\hslash^2}{2 m}
\nabla^2 \varphi +V(x,y,t) \varphi$$ Restricting to $2$ space plus one time yields $V(x,y,t) = \omega(t) (x^2 + y^2) + b(t)x + c(t)y + d(t)$, and satisfies the conditions for mapping to a constant coefficient [[DE]{}]{} via the methods of our previous paper [@MohReiHua19:Intro].
Discussion {#sec:Discussion}
==========
In this paper we give an algorithmic extension of [[[[MapDE]{}]{}]{}]{} introduced in [@MohReiHua19:Intro], that decides whether an input [[DPS]{}]{} can be mapped by local holomorphic diffeomorphism to a linear system, returning equations for the mapping in [[[[rif]{}]{}]{}]{}-form, useful for further applications. This work is based on creating algorithms that exploit results due to Bluman and Kumei [@BK105:BluKu; @Blu10:App] and some aspects of [@LGM101:LG]. This is a natural partner to the algorithm for deciding the existence of an invertible map of a linear [[DPS]{}]{} to a constant coefficient linear [[DE]{}]{} given in our previous paper [@MohReiHua19:Intro].
The mapping approach [@LGM101:LG] for [[ODE]{}]{} explicitly introduces a target linear system ${\hat R}$ with undetermined coefficients, then uses the full nonlinear determining equations for the mapping and applies the [[ThomasDecomposition]{}]{} Algorithm [@Thomas106:Tdec; @Robertz106:Tdec]. In contrast, like Bluman and Kumei, we exploit the fact that the target appears implicitly as a subalgebra of the Lie symmetry algebra $\mathcal{L}$ of ${R}$ and avoid using the full nonlinear determining equations for the transformations. Unlike Bluman and Kumei, who depend on extracting this subalgebra by explicit non-algorithmic integration, we use algorithmic differential algebra. We exploit the fact that the subalgebra corresponding to linear super-position appears as a subalgebra of the derived algebra $\mathcal{L}'$ of $\mathcal{L}$, generalizing the technique for [[ODE]{}]{} in [@LGM101:LG]. Instead of using the mapping equations , [@LGM101:LG] apply the transformations directly to the [ODE]{}. In contrast, our method works at the linearized Lie algebra level instead of the nonlinear Lie Group level used in [@LGM101:LG] which may be a factor in the increased space and time usage for their test set compared to our timings given in Fig. \[fig:testTime\].
Bluman and Kumei give necessary conditions in [@Blu10:App Theorem 2.4.1] and sufficient conditions [@Blu10:App Theorem 2.4.2] for linearization of nonlinear [[PDE]{}]{} systems with $m \geq 2$. Their requirement $\dim{\mathcal{L}} = \infty$ is dropped in our approach allowing us to deal with [[ODE]{}]{} and also linearization of overdetermined [[PDE]{}]{} systems. They also use the Jacobian condition to introduce a solved form of the BK equations with coefficients $\alpha^i_\sigma(x,u), \beta^\nu_\sigma(x,u)$ (see [@Blu10:App Eq 2.69]), and further decompose the resulting system with respect to their $f^\sigma (\phi)$’s (our $\hat{\eta}$’s). This decomposition results for input [[PDE]{}]{} having no zero-order (i.e. algebraic) relations among the input systems of [[PDE]{}]{}, a condition that is not explicitly given in the hypotheses of their theorems. We are planning to take advantage of this decomposition in future work, as an option to [[[[MapDE]{}]{}]{}]{}, since it can improve efficiency, when applicable. For the more general case of contact transformations for $m=1$, not considered here, see [@Blu10:App Theorems 2.4.3-2.4.4].
An important aspect of Theorem \[thm:MapDEThm\] concerning the correctness of [[[[MapDE]{}]{}]{}]{}, is to show that the existence of an infinitesimal symmetry $\sum_\ell \hat{\eta}^\ell (\hat{x}) {\frac{\partial}{\partial \hat{u}^\ell}}$ where ${\mathcal{H}} (\hat{\eta}) = 0$, when exponentiated to act on a local analytic solution of ${\hat R}$ produces all local analytic solutions in a neighborhood. Showing this depends on showing $\mbox{HF}({R}) = \mbox{HF}(P_k)$ in Step 9 of Algorithm \[Alg:MapDE2Linear\] or in intuitive terms, the size of solution space of the input system is the same as the size of the symmetry subgroup corresponding to $P_k$. In contrast the statement and proof of [@Blu10:App Theorem 2.4.2] appears to miss this crucial hypothesis about the size of the solution space of the input system being equal to the size of the solution space of the symmetry sub-group for linearization (as measured by Hilbert Series).
It is important to develop simple, efficient tests to reject the existence of mappings, based on structural and dimensional information. In addition to existing tests [@LGM101:LG], [@Blu10:App; @AnBlWo110:Mapp; @TWolf108:ConLa] we introduced a refined dimension test based on Hilbert Series. We will extend these tests in future work. We note that the potentially expensive change of rankings needed by our algorithms (for example to determine the derived algebra when it is infinite dimensional) could be more efficiently accomplished by the change of rankings approach given in [@BouLemMazChangeOrder:2010].
Mapping problems such as those considered in this paper are theoretically and computationally challenging. Given that nonlinear systems are usually not linearizable, a fundamental problem is to identify such linearizable models. For example [@AnBlWo110:Mapp; @TWolf108:ConLa] use multipliers for conservation laws to facilitate the determination of linearization mappings. Wolf’s approach [@TWolf108:ConLa] enables the determination of partially linearizable systems. Setting up such problems by finding an appropriate space to define the relevant mappings is important for discovering new non-trivial mappings. See Example \[ex:Burgers\] and [@Blu10:App] for such embedding approaches where the model is embedded in spaces that have a natural relation to the original space in terms of solutions but are not related by invertible point transformation. Another method is to embed a given model in a class of models and then efficiently seek the members of the class with the largest symmetry groups and most freedom in the functions/parameters of the class. Example \[ex:NLTelegraph\] illustrates this strategy.
We provide a further integration phase to attempt to find the mappings explicitly, based on [[[[Maple]{}]{}]{}]{}’s [[pdsolve]{}]{} which will be developed in further work. Even if the transformations can’t be determined explicitly, they can implicitly identify important features. Linearizable systems have a rich geometry that we are only beginning to exploit, such as the availability group action on the source and target. This offers interesting opportunities to use invariantized methods, such as invariant differential operators, and also moving frames [@Man10:Pra; @Hub09:Dif; @Fel99:Mov; @Arnaldsson17:InvolMovingFrames; @LisleReid2006]. Furthermore, they are available for the application of symbolic and symbolic-numeric approximation methods, a possibility that we will also explore. Finally a model that is not exactly linearizable may be *close* to a linearizable model or other attractive target, providing motivation for our future work on approximate mapping methods.
Acknowledgments {#acknowledgments .unnumbered}
===============
One of us (GR) acknowledges his debt to Ian Lisle who was the initial inspiration behind this work, especially his vision for Lie pseudogroups and an algorithmic calculus for symmetries of differential systems. GR and ZM acknowledge support from GR’s NSERC discovery grant.
Our program and a demo file included some part of our computations are publicly available on GitHub at: <https://github.com/GregGitHub57/MapDETools>
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\[ind:index\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called *terminals* in ${\ensuremath{\mathbb{R}}}^d$, find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair’s Manhattan (that is, $L_1$-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless ${\cal P}\!=\!{\cal NP}$). Approximation algorithms are known for 2D, but not for 3D.
We present, for any fixed dimension $d$ and any ${\ensuremath{\varepsilon}\xspace}>0$, an $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation algorithm. For 3D, we also give a $4(k-1)$-approximation algorithm for the case that the terminals are contained in the union of $k \ge 2$ parallel planes. 0.5em **Keywords:** Approximation Algorithms, Computational Geometry, Minimum Manhattan Network
author:
- Aparna Das
- 'Emden R. Gansner'
- Michael Kaufmann
- Stephen Kobourov
- Joachim Spoerhase
- Alexander Wolff
title: Approximating Minimum Manhattan Networks in Higher Dimensions
---
Introduction
============
In a typical network construction problem, one is given a set of objects to be interconnected such that some constraints regarding the connections are fulfilled. Additionally, the network must be of little cost. For example, if the objects are points in Euclidean space and the constraints say that, for some fixed $t>1$, each pair of points must be connected by a path whose length is bounded by $t$ times the Euclidean distance of the points, then the solution is a so-called *Euclidean $t$-spanner*. Concerning cost, one usually requires that the total length of the network is proportional to the length of a Euclidean minimum spanning tree of the points. Such low-cost spanners can be constructed efficiently [@admss-esstl-95].
In this paper, we are interested in constructing 1-spanners, with respect to the Manhattan (or $L_1$-) metric. Rather than requiring that the total length of the network is proportional to the minimum spanning tree of the points, our aim is to minimize the total length (or *weight*) of the network. Note that the Euclidean 1-spanner of a set of points is simply the complete graph (if no three points are collinear) and hence, its weight is completely determined. Manhattan 1-spanners, in contrast, have many degrees of freedom and vastly different weights.
More formally, given two points $p$ and $q$ in $d$-dimensional space ${\ensuremath{\mathbb{R}}}^d$, a *Manhattan path* connecting $p$ and $q$ (a $p$–$q$ *M-path*, for short) is a sequence of axis-parallel line segments connecting $p$ and $q$ whose total length equals the Manhattan distance between $p$ and $q$. Thus an M-path is a monotone rectilinear path. For our purposes, a set of axis-parallel line segments is a *network*. Given a network $N$, its *weight* $\|N\|$ is the sum over the lengths of its line segments. A network $N$ *Manhattan-connects* (or *M-connects*) two given points $p$ and $q$ if it “contains” a $p$–$q$ M-path $\pi$. Note that we slightly abuse the notation here: we mean pointwise containment, that is, we require $\bigcup \pi \subseteq \bigcup N$. Given a set $T$ of points—called *terminals*—in ${\ensuremath{\mathbb{R}}}^d$, a network $N$ is a *Manhattan network* (or *M-network*) for $T$ if $N$ M-connects every pair of terminals in $T$. The *minimum Manhattan network problem* (MMN) consists of finding, for a given set $T$ of terminals, a minimum-weight M-network. For examples, see Fig. \[fig:examples\]. M-networks have important applications in several areas such as VLSI layout and computational biology. For example, Lam [et al.]{} [@lap-pafst-03] used them in gene alignment in order to reduce the size of the search space of the Viterbi algorithm for pair hidden Markov models.
Previous work
-------------
The 2D-version of the problem, 2D-MMN, was introduced by Gudmundsson [et al.]{} [@gln-ammn-01]. They gave an 8- and a 4-approximation algorithm. Later, the approximation ratio was improved to 3 [@bwws-mmnpa-06; @fs-s3amm-08t] and then to 2, which is currently the best possible. It was achieved in three different ways: via linear programming [@cnv-raamm-08], using the primal–dual scheme [@n-eprmc-05] and with purely geometric arguments [@gsz-yaa2a-08]. The last two algorithms run in $O(n
\log n)$ time, given a set of $n$ points in the plane. A ratio of 1.5 was claimed [@su-15amm-05], but apparently the proof is incomplete [@fs-s3amm-08t]. Chin [et al.]{} [@cgs-mmnnp-11] finally settled the complexity of 2D-MMN by proving it NP-hard.
A little earlier, Mu[ñ]{}oz [et al.]{} [@msu-mmnp3-09] considered 3D-MMN. They showed that the problem is NP-hard and that it is NP-hard to approximate beyond a factor of 1.00002. For the special case of 3D-MMN, where any cuboid spanned by two terminals contains other terminals or is a rectangle, they gave a $2\alpha$-approximation algorithm, where $\alpha$ denotes the best approximation ratio for 2D-MMN. They posed the design of approximation algorithms for general 3D-MMN as an open problem.
Related problems
----------------
As we observe in Section \[sec:relat-stein-type\], MMN is a special case of the *directed Steiner forest problem* (DSF). More precisely, an instance of MMN can be decomposed into a constant number of DSF instances. The input of DSF is an edge-weighted directed graph $G$ and a set of vertex pairs. The goal is to find a minimum-cost subgraph of $G$ (not necessarily a forest) that connects all given vertex pairs. Recently, Feldman [et al.]{} [@fkn-iaadsf-09] reported, for any ${\ensuremath{\varepsilon}\xspace}>0$, an $O(n^{4/5+{\ensuremath{\varepsilon}\xspace}})$-approximation algorithm for DSF, where $n$ is the number of vertices of the given graph. This bound carries over to $d$D-MMN.
An important special case of DSF is the *directed Steiner *tree* problem* (DST). Here, the input instance specifies an edge-weighted digraph $G$, a *root* vertex $r$, and a subset $S$ of the vertices of $G$ to which $r$ must connect. An optimum solution for DST is a minimum-weight $r$-rooted subtree of $G$ spanning $S$. DST admits an $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation for any ${\ensuremath{\varepsilon}\xspace}>0$ [@cccdggl-aadsp-98].
A *geometric* optimization problem that resembles MMN is the *rectilinear Steiner arborescence problem* (RSA). Given a set of points in ${\ensuremath{\mathbb{R}}}^d$ with non-negative coordinates, a rectilinear Steiner arborescence is a spanning tree that connects all points with M-paths to the origin. As in MMN, the aim is to find a minimum-weight network. For 2D-RSA, there is a polynomial-time approximation scheme (PTAS) [@lr-ptasrsap-00] based on Arora’s technique for approximating geometric optimization problems such as TSP [@a-asnph-03]. It is not known whether 2D-MMN admits a PTAS. Arora’s technique does not directly apply here as M-paths between terminals forbid detours and thus may not respect portals.
Our contribution
----------------
We first present a $4(k-1)$-approximation algorithm for the special case of 3D-MMN where the given terminals are contained in $k \ge 2$ planes parallel to the $x$–$y$ plane; see Section \[sec:k-planes\].
Our main result is an $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation algorithm for $d$D-MMN, for any ${\ensuremath{\varepsilon}\xspace}>0$. We first present the algorithm in detail for three dimensions; see Section \[sec:general-case\]. Since the algorithm for arbitrary dimensions is a straightforward generalization of the algorithm for 3D but less intuitive, we describe it in the appendix.
Our $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation algorithm for $d$D-MMN constitutes a significant improvement upon the best known ratio of $O(n^{4/5+{\ensuremath{\varepsilon}\xspace}})$ for (general) directed Steiner forest [@fkn-iaadsf-09]. We obtain this result by exploiting the geometric structure of the problem. To underline the relevance of our result, we remark that the bound of $O(n^{\ensuremath{\varepsilon}\xspace})$ is the best known result also for other directed Steiner-type problems such as DST [@cccdggl-aadsp-98] or even acyclic DST [@zelikovsky-adstapprox-97].
Our $O(k)$-approximation algorithm for the $k$-planes case relies on recent work by Soto and Telha [@st-2dorg-11]. They show that, given a set of red and blue points in the plane, one can determine efficiently a minimum-cardinality set of points that together *pierce* all rectangles having a red point in the lower left corner and a blue point in the upper right corner. Combining this result with an approximation algorithm for 2D-MMN, yields an approximation algorithm for the 2-planes case. We show how to generalize this idea to $k$ planes.
Some Basic Observations {#sec:basic-observations}
=======================
We begin with some notation. Given a point $p \in {\ensuremath{\mathbb{R}}}^3$, we denote the $x$-, $y$- and $z$-coordinate of $p$ by $x(p)$, $y(p)$, and $z(p)$, respectively. Given two points $a$ and $c$ in ${\ensuremath{\mathbb{R}}}^2$, let $R(a,c)=\{ b \in {\ensuremath{\mathbb{R}}}^2 \mid x(a) \le x(b) \le x(c), \, y(a) \le y(b)
\le y(c) \}$ be the *rectangle spanned by $a$ and $c$*. If a line segment is parallel to the $x$-, $y$-, or $z$-axis, we say that it is $x$-, $y$-, or $z$-*aligned*. In what follows, we consider the 3-dimensional case of the MMN problem, unless otherwise stated.
Quadratic Lower Bound for Generating Sets in 3D {#sec:generating}
-----------------------------------------------
Intuitively, what makes 3D-MMN more difficult than 2D-MMN is the following: in 2D, if the bounding box of terminals $s$ and $s'$ and the bounding box of $t$ and $t'$ cross (as in Fig. \[sfg:2D-MMN\]), then any $s$–$s'$ M-path will intersect any $t$–$t'$ M-path, which yields $s$–$t'$ and $t$–$s'$ M-paths for free (if $s$ and $t$ are the lower left corners of their respective boxes). A similar statement for 3D does not hold; M-paths can “miss” each other—even if their bounding cuboids cross; see Fig. \[sfg:miss\].
Let us formalize this observation. Given a set $T$ of terminals, a set $Z$ of pairs of terminals is a *generating set* [@kia-iammn-02] if any network that M-connects the pairs in $Z$ in fact M-connects *all* pairs of terminals. In 2D, any MMN instance has a generating set of linear size [@kia-iammn-02]. Unfortunately this result does not extend to 3D. Below, we construct an instance that requires a generating set of size $\Omega(n^2)$. The idea of using linear-size generating sets is exploited by several algorithms for 2D-MMN [@cnv-raamm-08; @kia-iammn-02]. The following theorem shows that these approaches do not easily carry over to 3D.
There exists an instance of 3D-MMN with $n$ terminals that requires a generating set of size $\Omega(n^2)$.
We construct an instance that requires a generating set of size at least $n^2/4$. The main idea of the construction is to ensure that $n^2/4$ of the terminal pairs must use an edge segment unique to that specific pair. The input consists of two sets $T$ and $T'$, each with $n/2$ terminals, with the following coordinates: for $0\le
i < {n}/{2}$, terminal $t_i \in T$ is at $(i, {n}/{2} -i,
{n}/{2} -i)$ and terminal $t'_i \in T'$ is at $({n}/{2}+i,
n-i, n-i)$. Figure \[fig:genpairex\] shows the instance for $n=6$.
![The constructed network for $n=6$.[]{data-label="fig:genpairex"}](genpairex)
Consider any given generating set $Z \subset T \times T'$ such that there is a pair $(\tilde{t}, \tilde{t'})$, $\tilde{t}\in T$ and $\tilde{t}'\in T'$ that is not in $Z$. We now construct a specific network that contains M-paths between all terminal pairs in $Z$ but no M-path between $(\tilde{t}, \tilde{t'})$.
Consider any pair $(t,t')\in Z$ such that $t=(i,j,k) \in T$ and $t'=(i',j',k') \in T'$. The M-path from $t$ to $t'$ has three segments: an $x$-aligned segment from $(i,j,k)$ to $(i',j,k)$, a $y$-aligned segment from $(i',j,k)$ to $(i',j',k)$, and a $z$-aligned segment from $(i',j', k)$ to $(i',j',k')$. To ensure an M-path between each generating pair $t_i, t_j \in T$ (similarly between $t_i', t_j' \in T'$), we add M-paths between each pair of consecutive terminals in $T$ (similarly for $T'$) as follows: we connect $t_i, t_{i+1} \in T$ by adding a $z$-aligned segment from $t_i=(i,j,k)$ to $(i,j,k-1)$, a $y$-aligned segment to $(i,j-1,k-1)$, and an $x$-aligned segment to $t_{i+1}=(i+1,j-1,k-1)$; see Fig. \[fig:genpairex\].
It is easy to verify that, in this construction, the M-path between terminals $t = (i,j,k) \in T$ and $t' = (i',j',k') \in T'$ must use the $y$-aligned segment between $(i',j,k)$ and $(i',j',k)$. Since this segment is added only between terminal pairs that are present in the generating set $Z$, there is no M-path between terminals $\tilde{t}\in T$ and $\tilde{t}'\in T'$ which are not in $Z$. Thus, in order to obtain M-paths between all pairs of terminals in $T \cup T'$, we need at least all of the ${n^2}/{4}$ pairs in $T \times T'$.
Hanan Grid and Directional Subproblems {#sec:hann-grid-direct}
--------------------------------------
First, we note that any instance of MMN has a solution that is contained in the *Hanan grid*, the grid induced by the terminals; see Fig. \[fig:examples\](a). Gudmundsson [et al.]{} [@gln-ammn-01] showed this for 2D; their proof generalizes to higher dimensions. In what follows, we restrict ourselves to finding feasible solutions that are contained in the Hanan grid.
Second, to simplify our proofs, we consider the *directional* subproblem of 3D-MMN which consists of connecting all terminal pairs $(t,t')$ such that $t$ *dominates* $t'$, that is, $x(t) \le
x(t')$, $y(t) \le y(t')$, $z(t) \le z(t')$, and $t \ne t'$. We call such terminal pairs *relevant*.
The idea behind our reduction to the directional subproblem is that any instance of 3D-MMN can be decomposed into four subproblems of this type. One may think of the above-defined directional subproblem as connecting the terminals which are oriented in a north-east (NE) configuration in the $x$–$y$ plane (with increasing $z$-coordinates). Analogous subproblems exist for the directions NW, SE, and SW. Note that any terminal pair belongs to one of these four categories (if seen from the terminal with smaller $z$-coordinate).
The decomposition extends to higher dimensions $d$, by fixing the relationship between $(t,t')$ for one dimension (for example, $z$), and enumerating over all possible relationships for the remaining $d-1$ dimensions. This decomposes $d$D-MMN into $2^{d-1}$ subproblems, which is a constant number of subproblems as we consider $d$ to be a fixed constant.
This means that any $\rho$-approximation algorithm for the directional subproblem leads to an $O(\rho)$-approximation algorithm for the general case. Thus we can focus on designing algorithms for the directional subproblem.
\[obs:dir-sub\] Any instance of MMN can be decomposed into a constant number of directional subproblems. Thus a $\rho$-approximation algorithm for the directional subproblem leads to an $O(\rho)$-approximation algorithm for MMN.
Relation to Steiner Problems {#sec:relat-stein-type}
----------------------------
We next show that there is an approximation-preserving reduction from directional 3D-MMN to the directed Steiner forest (DSF) problem, which by Observation \[obs:dir-sub\], carries over up to a constant factor, to general 3D-MMN.
Let $T$ be a set of $n$ points in $\mathbb{R}^3$. Let $H$ be the Hanan grid induced by $T$. We consider $H$ as an undirected graph where the length of each edge equals the Euclidean distance between its endpoints. We orient each edge in $H$ so that, for any edge $(p,p')$ in the resulting digraph $H'$, the start node $p$ dominates the end node $p'$. We call $H'$ the *oriented Hanan grid* of $T$. Now let $(t,t')$ be a relevant pair of points in $T$, that is, $t$ dominates $t'$. Any M-path in $H$ connecting $t$ to $t'$ corresponds to a directed path in $H'$ from $t$ to $t'$. The converse also holds: every directed path in $H'$ corresponds to an M-path in $H$.
Let $I$ be an instance of directional 3D-MMN and let $I'$ be an instance of DSF where the input graph is $H'$ and where every relevant terminal pair of $I$ has to be connected. Then, each feasible solution $N$ of $I$ contained in $H$ corresponds to a sub-graph $N'$ of $H'$ that connects every relevant terminal pair, and is therefore a feasible solution to $I'$. It is easy to see that $N'$ has the same cost as $N$, as $N'$ uses the oriented version of each edge of $N$. Conversely, every feasible solution $N'$ for $I'$ corresponds to a subgraph $N$ of $H$ that M-connects every relevant terminal pair. Therefore, $N$ is a feasible solution to $I'$ with the same cost as $N'$. This establishes an efficiently computable one-to-one correspondence between feasible solutions to $I$ that are contained in $H$ and feasible solutions to $I'$. Since there is an optimum solution to $I$ contained in $H$ [@gln-ammn-01], this is an approximation-preserving reduction from directional 3D-MMN to DSF.
By means of the above transformation of the Hanan grid into a digraph, we also obtain an approximation-preserving reduction from 3D-RSA to DST. We use this later in Section \[sec:general-case\] to develop an approximation algorithm for 3D-MMN. Let $I$ be an instance of 3D-RSA given by a set $T$ of terminals with non-negative coordinates that are to be M-connected to the origin $o$. We construct an instance $I'$ of DSF as above where $\{o\}\times T$ is the set of node pairs to be connected. Note that any feasible solution to $I'$ is, without loss of generality, a tree. Hence, $I'$ is an instance of DST with root $o$. All in all, we have an approximation-preserving reduction from 3D-RSA to DST.
The $k$-Plane Case {#sec:k-planes}
==================
In this section we consider 3D-MMN, under the assumption that the set $T$ of terminals is contained in the union of $k \ge 2$ planes $E_1,\dots, E_k$ that are parallel to the $x$–$y$ plane. Of course, this assumption always holds for some $k \le n$. We present a $4(k-1)$-approximation algorithm, which outperforms our algorithm for the general case in Section \[sec:general-case\] if $k \in
o(n^{\ensuremath{\varepsilon}\xspace})$.
Let [$N_\mathrm{opt}$]{}be some fixed minimum M-network for $T$, let [${\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{hor}$]{}be the set of all $x$-aligned and all $y$-aligned segments in [$N_\mathrm{opt}$]{}, and let [${\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}$]{}be the set of all $z$-aligned segments in [$N_\mathrm{opt}$]{}. Let [$\mathrm{OPT}$]{}denote the weight of [$N_\mathrm{opt}$]{}. Clearly, [$\mathrm{OPT}$]{}does not depend on the specific choice of [$N_\mathrm{opt}$]{}; the weights of [${\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{hor}$]{}and [${\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}$]{}, however, may depend on [$N_\mathrm{opt}$]{}. For $i \in \{1,\dots,k\}$, let $T_i =
T \cap E_i$ be the set of terminals in plane $E_i$. Further, let [$T_{xy}$]{}be the projection of $T$ onto the $x$–$y$ plane.
Our algorithm consists of two phases. Phase I computes a set [$N^\mathrm{hor}$]{}of horizontal (that is, $x$- and $y$-aligned) line segments, phase II computes a set [$N^\mathrm{ver}$]{}of vertical (that is, $z$-aligned) line segments. Finally, the algorithm returns the set $N = {\ensuremath{N^\mathrm{hor}}\xspace}\cup {\ensuremath{N^\mathrm{ver}}\xspace}$.
Phase I is simple; we compute a 2-approximate M-network [$N_{xy}$]{}for [$T_{xy}$]{}(using the algorithm of Guo [et al.]{} [@gsz-yaa2a-08]) and project [$N_{xy}$]{}onto each of the planes $E_1,\dots, E_k$. Let [$N^\mathrm{hor}$]{}be the union of these projections. Note that [$N^\mathrm{hor}$]{}M-connects any pair of terminals that lie in the same plane.
\[obs:Nhor\] $\| {\ensuremath{N^\mathrm{hor}}\xspace}\| \le 2k\|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{hor}}\xspace}\|$.
The projection of [${\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{hor}$]{}to the $x$–$y$ plane is an M-network for [$T_{xy}$]{}. Hence, $\|{\ensuremath{N_{xy}}\xspace}\| \le 2 {\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{hor}}\xspace}$. Adding up over the $k$ planes yields the claim.
In Phase II, we construct a [*pillar network*]{} by computing a set [$N^\mathrm{ver}$]{}of vertical line segments, so-called *pillars*, of total cost at most $4(k-1)\|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\|$. This yields an overall approximation factor of $4(k-1)$ since $\|{\ensuremath{N^\mathrm{hor}}\xspace}\cup {\ensuremath{N^\mathrm{ver}}\xspace}\| \le 2k\|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{hor}}\xspace}\| +
4(k-1)\|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\| \le 4(k-1) (\|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{hor}}\xspace}\| + \|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\|) \le 4(k-1)
{\ensuremath{\mathrm{OPT}}\xspace}$.
Below we describe Phase II of our algorithm for the directional subproblem that runs in direction north-east (NE) in the $x$–$y$ plane (with increasing $z$-coordinates). For this directional subproblem, we construct a pillar network ${\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}$ of weight at most $(k-1)\|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\|$ that, together with [$N^\mathrm{hor}$]{}, M-connects all relevant pairs. We solve the analogous subproblems for the directions NW, SE, and SW in the same fashion. Then [$N^\mathrm{ver}$]{}is the union of the four partial solutions and has weight at most $4(k-1)\|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\|$, as desired. Our directional subproblem is closely linked to the *(directional) bichromatic rectangle piercing problem* ([$\mathrm{BRP}$]{}), which is defined as follows. Let [$R$]{}and [$B$]{}be sets of red and blue points in ${\ensuremath{\mathbb{R}}}^2$, respectively, and let ${\cal R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$ denote the set of axis-aligned rectangles each of which is spanned by a red point in its SW-corner and a blue point in its NE-corner. Then the aim of [$\mathrm{BRP}$]{}is to find a minimum-cardinality set $P \subset {\ensuremath{\mathbb{R}}}^2$ such that every rectangle in ${\cal R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$ is *pierced*, that is, contains at least one point in $P$. The points in $P$ are called *piercing points*. The problem dual to BRP is the *(directional) bichromatic independent set of rectangles problem* (BIS) where the goal is to find the maximum number of pairwise disjoint rectangles in ${\cal
R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$, given the sets [$R$]{}and [$B$]{}.
Recently, Soto and Telha [@st-2dorg-11] proved a beautiful min–max theorem saying that, for ${\cal R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$, the minimum number of piercing points always *equals* the maximum number of independent rectangles. This enabled them to give efficient exact algorithms for [$\mathrm{BRP}$]{}and BIS running in $\tilde{O}(n^{2.5})$ worst-case time or $\tilde{O}(n^{\gamma})$ expected time, where the $\tilde{O}$-notation ignores polylogarithmic factors, $\gamma <
2.4$ is the exponent for fast matrix multiplication, and $n=|{\ensuremath{R}\xspace}|+|{\ensuremath{B}\xspace}|$ is the input size.
The details of Phase II appear, for $k=2$ planes, in Section \[sec:pillar2\], and, for $k>2$ planes, in Section \[sec:pillark\]. Algorithm \[alg:kplanes\] summarizes of our $k$-planes algorithm.
\
Let $T_{xy}$ be the projection of $T$ onto the $x$–$y$ plane Phase I:\
Compute ${\ensuremath{N_{xy}}\xspace}$, a 2-approximate M-network for $T_{xy}$ using the algorithm of Guo [et al.]{} [@gsz-yaa2a-08].\
Let ${\ensuremath{N^\mathrm{hor}}\xspace}$ be the union of the projections of $N_{xy}$ onto each of the planes $E_1, \ldots, E_k$. Phase II:\
If $k=2$, construct a pillar network ${\ensuremath{N^\mathrm{ver}}\xspace}$ by Algorithm \[alg:2pillars\]; see Section \[sec:pillar2\].\
Otherwise, construct a pillar network ${\ensuremath{N^\mathrm{ver}}\xspace}$ by Algorithm \[alg:kpillars\]; see Section \[sec:pillark\].
Output: ${\ensuremath{N^\mathrm{hor}}\xspace}\cup {\ensuremath{N^\mathrm{ver}}\xspace}$.
Pillar Network for Two Planes {#sec:pillar2}
-----------------------------
Our phase-II algorithm for two planes is very simple. We sketch it first in order to provide some intuition for the $k$-planes case. Let the terminals in $T_1$ be red and those in $T_2$ be blue. Ignore the $z$-coordinates of the terminals. Then the relevant red–blue point pairs span exactly the rectangles in ${\cal
R}(T_1,T_2)$, which we call relevant, too.
Input: Sets $T_1 \subset E_1$ and $T_2 \subset E_2$ of terminals. \[alg:2pillars\]
Color $T_1$ red and $T_2$ blue. Ignoring $z$-coordinates of terminals, let ${\cal R}(T_1,T_2)$ be the set of rectangles spanned by relevant red–blue pairs. Compute a minimum piercing $\hat{P}$ of ${\cal R}(T_1,T_2)$ such that for each relevant red–blue pair $(r,b) \in T_1 \times T_2$ the piercing point for $(r,b)$ lies on an $r$–$b$ M-path in $N_{xy}$, as described in Lemma \[lem:paths\]. Erect pillars from $E_1$ to $E_2$ at each piercing point $\hat{p} \in
\hat{P}$; let ${\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}$ be the resulting set of pillars.
Output: ${\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}$.
Our algorithm (Algorithm \[alg:2pillars\]) consists of two steps. First, we compute a minimum piercing $P$ of ${\cal R}(T_1,T_2)$ using the algorithm of Soto and Telha [@st-2dorg-11]. Second, we move each piercing point $p \in
P$ to a new position $\hat{p}$—a nearby junction of [$N_{xy}$]{}—and erect, at $\hat{p}$, a pillar connecting the two planes. Let $\hat{P}$ be the set of piercing points after the move, and let ${\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}$ be the corresponding set of pillars.
\[lem:4approx\] It holds that $\|{\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}\| \le \|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\|$.
It is easy to see that $|\hat{P}|=|P|$. Integrating over the distance $d$ of the two planes yields $\|{\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}\| = |\hat{P}| \cdot d = |P| \cdot d
\le \|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\|$. The last inequality is due to the fact that $P$ is a *minimum* piercing of ${\cal R}(T_1,T_2)$ and that the pillars in [${\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}$]{}pierce ${\cal R}(T_1,T_2)$—otherwise [$N_\mathrm{opt}$]{}would not be feasible.
Now we turn to feasibility. We first detail how we move each piercing point $p$ to its new position $\hat{p}$. For the sake of brevity, we identify terminals with their projections to the $x$–$y$ plane. Our description assumes that we have at our disposal some network $M$ (such as [$N_{xy}$]{}) connecting the relevant pairs in [$T_{xy}$]{}.
For a piercing point $p \in P$, let $A_p$ be the intersection of the relevant rectangles pierced by $p$; see Fig. \[fig:piercing\]. Clearly, $p \in A_p$. Note that the bottom and left sides of $A_p$ are determined by terminals [$t_\mathrm{W}$]{}and [$t_\mathrm{S}$]{}to the west and south of $A_p$, respectively. Symmetrically, the top and right sides of $A_p$ are determined by terminals [$t_\mathrm{E}$]{}and [$t_\mathrm{N}$]{}to the east and north of $A_p$, respectively. Terminals [$t_\mathrm{W}$]{}and [$t_\mathrm{S}$]{}may coincide, and so may [$t_\mathrm{E}$]{}and [$t_\mathrm{N}$]{}. It is easy to see that the network $M$ contains an M-path [$\pi_\mathrm{SN}$]{} connecting [$t_\mathrm{S}$]{}and [$t_\mathrm{N}$]{}and an M-path [$\pi_\mathrm{WE}$]{}connecting [$t_\mathrm{W}$]{}and [$t_\mathrm{E}$]{}. The path [$\pi_\mathrm{SN}$]{}goes through the bottom and top sides of $A_p$ and [$\pi_\mathrm{WE}$]{}goes through the left and right sides. Hence, the two paths intersect in a point $\hat{p} \in A_p$. This is where we move the original piercing point $p$.
![Paths [$\pi_\mathrm{SN}$]{}and [$\pi_\mathrm{WE}$]{}meet in a point $\hat{p}$ in $A_p$.[]{data-label="fig:piercing"}](pillar)
Since $\hat{p} \in A_p$, the point $\hat{p}$ pierces the same relevant rectangles as $p$, and the set $\hat{P} = \{ \hat{p} \mid p \in P \}$ is a (minimum) piercing for the set of relevant rectangles.
\[lem:paths\] Let ${\cal R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$ be an instance of [$\mathrm{BRP}$]{}and let $M$ be a network that M-connects every relevant red–blue point pair. Then we can efficiently compute a minimum piercing of ${\cal
R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$ such that $M$ contains, for every relevant red–blue point pair $(r,b)$ in ${\ensuremath{R}\xspace}\times {\ensuremath{B}\xspace}$, an $r$–$b$ M-path that contains a piercing point.
We use the algorithm of Soto and Telha [@st-2dorg-11] to compute a minimum piercing $P$ of ${\cal R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$. Then, as we have seen above, $\hat{P}$ is a minimum piercing of ${\cal
R}({\ensuremath{R}\xspace},{\ensuremath{B}\xspace})$, too. Now let $(r,b)$ be a relevant red–blue pair in ${\ensuremath{R}\xspace}\times{\ensuremath{B}\xspace}$, and let $p \in P$ be a point that pierces $R(r,b)$. Clearly, $\hat{p}$ pierces $R(r,b)$, too. As we have observed before, both $p$ and $\hat{p}$ lie in $A_p$.
Since $(r,b)$ is a relevant pair, $r$ lies to the [$\mathrm{SW}$]{}of $A_p$ and $b$ to the [$\mathrm{NE}$]{}; see Fig. \[sfg:rin\]. We prove that $M$ contains an $r$–$\hat{p}$ M-path; a symmetric argument proves that $M$ also contains a $\hat{p}$–$b$ M-path. Concatenating these two M-paths yields the desired $r$–$b$ M-path since $r$ lies to the [$\mathrm{SW}$]{}of $\hat{p}$ and $\hat{p}$ lies to the [$\mathrm{SW}$]{}of $b$. Recall that $\hat{p}$ lies on the intersection of the [$t_\mathrm{W}$]{}–[$t_\mathrm{E}$]{}M-path [$\pi_\mathrm{WE}$]{}and the [$t_\mathrm{S}$]{}-[$t_\mathrm{N}$]{} M-path [$\pi_\mathrm{SN}$]{}, where [$t_\mathrm{W}$]{}, [$t_\mathrm{E}$]{}, [$t_\mathrm{S}$]{}, [$t_\mathrm{N}$]{}are the terminals that determine the extensions of $A_p$; see Fig. \[fig:piercing\]. To show that $M$ M-connects $r$ and $\hat{p}$, we consider two cases.
*Case I:* $r \in R({\ensuremath{t_\mathrm{W}}\xspace},{\ensuremath{t_\mathrm{S}}\xspace})$; see Fig. \[sfg:rin\]. According to our assumption, $M$ contains *some* $r$–$b$ M-path $\pi$. Then $\pi$ must intersect [$\pi_\mathrm{WE}$]{} or [$\pi_\mathrm{SN}$]{}at some point $x$ to the [$\mathrm{SW}$]{}of $\hat{p}$. Thus, we can go, in a monotone fashion, along $\pi$ from $r$ to $x$ and then along [$\pi_\mathrm{WE}$]{} or [$\pi_\mathrm{SN}$]{}from $x$ to $\hat{p}$. This is the desired $r$–$\hat{p}$ M-path.
[![Sketches for the proof of Lemma \[lem:paths\].[]{data-label="fig:paths"}](rin "fig:")]{} [![Sketches for the proof of Lemma \[lem:paths\].[]{data-label="fig:paths"}](rout "fig:")]{}
*Case II:* $r$ lies to the [$\mathrm{SW}$]{}of [$t_\mathrm{W}$]{}or [$t_\mathrm{S}$]{}; see Fig. \[sfg:rout\]. In this case $M$ contains M-paths from $r$ to [$t_\mathrm{W}$]{}and to [$t_\mathrm{S}$]{}. If $r$ lies to the [$\mathrm{SW}$]{}of [$t_\mathrm{W}$]{}, we can go, again in a monotone fashion, from $r$ to [$t_\mathrm{W}$]{}and then along [$\pi_\mathrm{WE}$]{}from [$t_\mathrm{W}$]{}to $\hat{p}$. Otherwise, if $r$ lies to the [$\mathrm{SW}$]{}of [$t_\mathrm{S}$]{}, we can go from $r$ to [$t_\mathrm{S}$]{}and then on [$\pi_\mathrm{SN}$]{}from [$t_\mathrm{S}$]{}to $\hat{p}$.
Since these are the only two possibilities, this concludes the proof.
Lemmas \[lem:4approx\] and \[lem:paths\] (with ${\ensuremath{R}\xspace}=T_1$, ${\ensuremath{B}\xspace}=T_2$, and $M={\ensuremath{N_{xy}}\xspace}$) yield the following.
\[thm:select-pill-locat\] We can efficiently compute a 4-approximation for the 2-plane case.
Pillar Network for $k$ Planes {#sec:pillark}
-----------------------------
Now we show how our phase-II algorithm generalizes to $k$ planes. As in the 2-planes case, we restrict ourselves to the directional subproblem and construct a pillar network ${\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}$ of weight at most $(k-1) \|{\ensuremath{{\ensuremath{N_\mathrm{opt}}\xspace}^\mathrm{ver}}\xspace}\|$. As we have argued at the beginning of Section \[sec:k-planes\], this suffices to prove Theorem \[thm:k-planes\].
\[thm:k-planes\] There exists a $4(k-1)$-approximation algorithm for 3D-MMN where the terminals lie in the union of $k \ge 2$ planes parallel to the $x$–$y$ plane.
Our pillar-placement algorithm (Algorithm \[alg:kpillars\]) is as follows. Let $i \in \{1,\dots,k-1\}$. We construct an instance ${\cal I}_i$ of [$\mathrm{BRP}$]{}where we two-color [$T_{xy}$]{}such that each point corresponding to a terminal of some plane $E_j$ with $j\leq i$ is colored red and each point corresponding to a terminal of some plane $E_{j'}$ with $j'\geq i+1$ is colored blue. For ${\cal I}_i$, we compute a minimum piercing $\hat{P}_i$ according to Lemma \[lem:paths\] with $M={\ensuremath{N_{xy}}\xspace}$. In other words, for any relevant pair $(t_j,t_{j'})$, there is some M-path in $N_{xy}$ that contains a piercing point of $\hat{P}_i$. We choose $i^\star \in \{1,\dots,k-1\}$ such that $\hat{P}_{i^\star}$ has minimum cardinality. This is crucial for our analysis. At the piercing points of $\hat{P}_{i^\star}$, we erect pillars spanning all planes $E_1,\ldots,E_k$. Let $\hat{N}_{i^\star}$ be the set of these pillars. We now show that $\hat{N}_{i^\star}$, along with [$N^\mathrm{hor}$]{}, creates a feasible network for any relevant terminal pair $(t_j, t_j')$ such that $j\le i^{\star}$ and $j'\ge i^{\star}+1$.
Input: Sets $T_s \subset E_s, \dots, T_t \subset E_t$ of terminals with $s\le t$ (initially $s=1$ and $t=k$).
Let $T'$ be the projection of $T_s \cup \dots \cup T_t$ onto the $x$–$y$ plane. Let ${\cal I}_i$ be an instance of [$\mathrm{BRP}$]{}where each point in $T'$, corresponding to a terminal in $T_j$ with $j\leq i$, is colored red and each point in $T'$, corresponding to a terminal in $T_{j'}$ with $j'\geq i+1$, is colored blue. Compute a minimum piercing $\hat{P}_i$ according to Lemma \[lem:paths\] with $M={\ensuremath{N_{xy}}\xspace}$. Choose $i^\star \in \{s,\dots,t\}$ such that $\hat{P}_{i^\star}$ has minimum cardinality. Let $\hat{N}_{i^\star}$ be the set of pillars erected at each piercing point of $\hat{P}_{i^\star}$, spanning planes $E_s, \dots, E_t$. Let $\hat{N}_{\le i^\star}$ be the output of this algorithm applied recursively to $T_s, \dots, T_{i^{\star}}$. Let $\hat{N}_{> i^\star}$ be the output of this algorithm applied recursively to $T_{i^{\star}+1}, \dots, T_t$.
Output: $\hat{N}_{i^\star} \cup \hat{N}_{\le i^\star} \cup \hat{N}_{> i^\star}$
\[lem:feasibility\] The network ${\ensuremath{N^\mathrm{hor}}\xspace}\cup \hat{N}_{i^\star}$ M-connects any relevant terminal pair in $T_j \times T_{j'}$ with $j \le i^\star$ and $j' \ge i^\star+1$.
Consider a pair $(t_j,t_{j'})$ in $T_j \times T_{j'}$ as in the statement. We construct an M-path from $t_j$ to $t_{j'}$ as follows. We know that there exists an M-path $\pi$ that connects the projections of $t_j$ and $t_{j'}$ in [$N_{xy}$]{}and contains a piercing point $p$ of $\hat{P}_{i^\star}$. Therefore, we can start at $t_j$ and follow the projection of $\pi$ onto plane $E_j$ until we arrive at $p$. Then we use the corresponding pillar in $\hat{N}_{i^\star}$ to reach the plane $E_{j'}$, where we follow the projection of $\pi$ (onto that plane) until we reach $t_{j'}$.
In order to also M-connect relevant terminal pairs in $T_j \times T_{j'}$, where either ($j \le i^\star$ and $j' \le i^\star$) or ($j \ge i^\star+1$ and $j' \ge i^\star+1$), we simply apply the pillar-placement algorithm recursively to the sets $T_1,\dots,T_{i^\star}$ and $T_{i^\star+1},\dots,T_k$. This yields the desired pillar network ${\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}$. By Lemma \[lem:feasibility\], ${\ensuremath{N^\mathrm{ver}}\xspace}_{\ensuremath{\mathrm{dir}}}\cup {\ensuremath{N^\mathrm{hor}}\xspace}$ is feasible. Next, we bound $\|\hat{N}_{i^\star}\|$.
\[lem:piercing-split\] Let $M$ be an arbitrary directional Manhattan network for $T$, and let [$M^\mathrm{ver}$]{}be the set of vertical segments in $M$. Then the pillar network $\hat{N}_{i^\star}$ has weight at most $\|{\ensuremath{M^\mathrm{ver}}\xspace}\|$.
Without loss of generality, we assume that $M$ is a subnetwork of the Hanan grid [@gln-ammn-01]. We may also assume that any segment of [$M^\mathrm{ver}$]{}spans only consecutive planes. For $1\leq i\leq j\leq k$, let $M_{i,j}$ denote the subnetwork of [$M^\mathrm{ver}$]{}lying between planes $E_i$ and $
E_j$. Let $d_{i,j}$ be the vertical distance between planes $E_i$ and $E_j$.
We start with the observation that, for any $j=1,\ldots,k-1$, the network $M_{j,j+1}$ is a set of pillars that forms a valid piercing of the piercing instance ${\cal I}_j$ (defined right after Theorem \[thm:k-planes\]). Hence, $|M_{j,j+1}| \ge |\hat{P}_j|
\ge |\hat{P}_{i^\star}|$, which implies the claim of the lemma as follows: $$\|{\ensuremath{M^\mathrm{ver}}\xspace}\|=\sum_{j=1}^{k-1}\|M_{j,j+1}\|=\sum_{j=1}^{k-1}|M_{j,j+1}|
\cdot d_{j,j+1} \ge \sum_{j=1}^{k-1}|P_{i^\star}| \cdot d_{j,j+1} =
|P_{i^\star}| \cdot d_{1,k} =
\|P_{i^\star}\|.$$
It is crucial for our construction that the pillars constructed recursively span either $E_1,\dots,E_{i^\star}$ or $E_{i^\star+1},\dots,E_k$, but not all planes. For $1\leq j\leq j'\leq k$, let ${\ensuremath{\mathrm{weight}}\xspace}_z(j,j')$ denote the weight of the vertical part of the network produced by the above pillar-placement algorithm, when applied to planes $E_j,\dots,E_{j'}$ recursively. For technical reasons we set ${{\ensuremath{\mathrm{weight}}\xspace}_z(j,j)=0}$. Now assume that $j<j'$ and that the algorithm makes the partition at plane $E_{i'}$ with $j\leq i'< j'$ when planes $E_j,\dots,E_{j'}$ are processed. By means of Lemma \[lem:piercing-split\], we derive the recursion $$\label{eqn:weight}
{\ensuremath{\mathrm{weight}}\xspace}_z(j,j')\leq \|M_{j,j'}\|+{\ensuremath{\mathrm{weight}}\xspace}_z(j,i')+{\ensuremath{\mathrm{weight}}\xspace}_z(i'+1,j')\, ,$$ which holds for any M-network $M$ for $T$. We now claim that $${\ensuremath{\mathrm{weight}}\xspace}_z(j,j')\leq (j'-j)\|M_{j,j'}\|.$$ Our proof is by induction on the number of planes processed by the algorithm. By the inductive hypothesis, we have that ${\ensuremath{\mathrm{weight}}\xspace}_z(j,i')\leq(i'-j)\|M_{j,i'}\|$ and ${\ensuremath{\mathrm{weight}}\xspace}_z(i'+1,j')\leq(j'-i'-1)\|M_{i'+1,j'}\|$. We plug these expressions into the recursion \[eqn:weight\]. Since $\|M_{j,i'}\|+\|M_{i'+1,j'}\|\leq \|M_{j,j'}\|$ and ${\ensuremath{\mathrm{weight}}\xspace}_z(l,l)=0$ for any $l\in\{1,\dots,k\}$, the claim follows.
We conclude that the weight of the solution produced by the algorithm, when applied to all planes $E_1,\dots,E_k$, is bounded by ${\ensuremath{\mathrm{weight}}\xspace}_z(1,k)\leq
(k-1)\|M_{1,k}\|=(k-1)\|{\ensuremath{M^\mathrm{ver}}\xspace}\|$. This completes the proof of Theorem \[thm:k-planes\].
The General Case {#sec:general-case}
================
In this section, we present an approximation algorithm, which we call the *grid algorithm*, for the general 3D-MMN problem. Our main result is the following.
\[thm:3dmain\] For any ${\ensuremath{\varepsilon}\xspace}>0$, there exists an $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation algorithm for 3D-MMN that, given a set of $n$ terminals, runs in $n^{O(1/{\ensuremath{\varepsilon}\xspace})}$ time.
This result is better than the one in the previous section if the given set of terminals is distributed over $\omega(n^{\ensuremath{\varepsilon}\xspace})$ horizontal planes. Moreover, the approach in this section extends to higher dimensions; see appendix.
For technical reasons, we assume that the terminals are in general position, that is, any two terminals differ in all three coordinates. By Observation \[obs:dir-sub\] it suffices to describe and analyze the algorithm for the directional subproblem.
The 3D Grid Algorithm {#sec:gridalg}
---------------------
We begin the description with a high-level summary. To solve the directional subproblem, we place a 3D grid that partitions the instance into a constant number of cuboids; see Fig. \[sfg:gridedges\]. Cuboids that differ in only two coordinates form *slabs*. We connect terminals from different slabs by M-connecting each terminal to the corners of its cuboid and by using the edges of the grid to connect the corners. We connect terminals from the same slab by recursively applying our algorithm to the slabs.
[*Step 1: Partitioning into cuboids and slabs.*]{} Consider the bounding cuboid $C$ of $T$ and set $c=3^{1/{\ensuremath{\varepsilon}\xspace}}$. Partition $C$ by $3(c-1)$ separating planes into $c\times c\times c$ axis-aligned subcuboids $C_{ijk}$ with $i,j,k \in \{1,\ldots,c\}$. The indices are such that larger indices mean larger coordinates. Place the separating planes such that the number of terminals between two consecutive planes is at most $n/c$. This can be accomplished by executing a simple plane-sweep for each direction $x, y, z,$ and by placing separating planes after every $n/c$ terminals. Here we exploit our general-position assumption. The edges of the resulting subcuboids—except the edges on the boundary of $C$, which we do not need—induce a three-dimensional grid $\mathcal{G}$ of axis-aligned line segments. We insert $\mathcal{G}$ into the solution.
For each $i \in \{1,\dots,c\}$, define the $x$-aligned *slab*, $C_i^x$, to be the union of all cuboids $C_{ijk}$ with $j,k \in
\{1,\dots,c\}$. Define $y$-aligned and $z$-aligned slabs $C_j^y$, $C_k^z$ analogously; see Fig. \[sfg:slabs\].
[*Step 2: Add M-paths between different slabs.*]{} Consider two cuboids $C_{ijk}$ and $C_{i'j'k'}$ with $i<i'$, $j<j'$, and $k<k'$. Any terminal pair $(t,t') \in C_{ijk} \times C_{i'j'k'}$ can be M-connected using the edges of $\mathcal G$ as long as $t$ and $t'$ are connected to the appropriate corners of their cuboids; see Fig. \[sfg:patchingpath\]. To this end, we use the following *patching* procedure.
Call a cuboid $C_{ijk}$ *relevant* if there is a non-empty cuboid $C_{i'j'k'}$ with $i<i'$, $j<j'$, and $k<k'$. For each relevant cuboid $C_{ijk}$, let $\hat{p}_{ijk}$ denote a corner that is dominated by all terminals inside $C_{ijk}$. We define *up-patching* $C_{ijk}$ to mean M-connecting every terminal in $C_{ijk}$ to $\hat{p}_{ijk}$. We up-patch $C_{ijk}$ by solving (approximately) an instance of 3D-RSA with the terminals in $C_{ijk}$ as points and $\hat{p}_{ijk}$ as origin. We define *down-patching* analogously; cuboid $C_{ijk}$ is relevant if there is a non-empty cuboid $C_{i'j'k'}$ with $i>i'$, $j>j'$, $k>k'$; we let $\check{p}_{ijk}$ be the corner that dominates all terminals in $C_{ijk}$.
We complete this step by inserting the up-patches and the down-patches of all relevant cuboids into the solution.
[*Step 3: Add M-paths within slabs.*]{} To M-connect relevant terminal pairs that lie in the same slab, we apply the grid algorithm (steps 1–3) recursively to each slab $C_i^x$, $C_j^y$, and $C_k^z$ with $i,j,k \in \{1,\dots,c\}$.
Analysis
--------
We first show that the output of the algorithm presented in Section \[sec:gridalg\] is feasible, then we establish its approximation ratio of $O(n^{\ensuremath{\varepsilon}\xspace})$ and its running time of $n^{O(1/{\ensuremath{\varepsilon}\xspace})}$ for any ${\ensuremath{\varepsilon}\xspace}> 0$. In this section, [$\mathrm{OPT}$]{}denotes the weight of a minimum M-network (*not* the cost of an optimal solution to the directional subproblem).
\[lem:mpaths\] The grid algorithm M-connects all relevant terminal pairs.
Let $(t,t')$ be a relevant terminal pair. First, suppose that $t$ and $t'$ lie in cuboids of different slabs. Thus, there are $i<i',j<j',k<k'$ such that $t\in C_{ijk}$ and $t'\in
C_{i'j'k'}$. Furthermore, $C_{ijk}$ and $C_{i'j'k'}$ are relevant for up- and down-patching, respectively. When up-patching, we solve an instance of RSA connecting all terminals in $C_{ijk}$ to $\hat{p}_{ijk}$. Similarly, down-patching M-connects $t'$ to $\check{p}_{i'j'k'}$. The claim follows as $\cal G$ M-connects $\hat{p}_{ijk}$ and $\check{p}_{i'j'k'}$.
Now, suppose that $t$ and $t'$ lie in the same slab. As the algorithm is applied recursively to each slab, there will be a recursion step where $t$ and $t'$ lie in cuboids in different slabs. Here, we need our general-position assumption. Applying the argument above to that particular recursive step completes the proof.
#### Approximation ratio.
Next, we turn to the performance of our algorithm. Let $r(n)$ be its approximation ratio, where $n$ is the number of terminals in $T$. The total weight of the output is the sum of $\|{\cal G}\|$, the cost of patching, and the cost for the recursive treatment of the slabs. We analyze each of the three costs separately.
The grid $\mathcal G$ consists of all edges induced by the $c^3$ subcuboids except the edges on the boundary of $C$. Let $\ell$ denote the length of the longest side of $C$. The weight of $\cal G$ is at most $3(c-1)^2 \ell$, which is bounded by $3c^2{\ensuremath{\mathrm{OPT}}\xspace}$ as $\ell \le {\ensuremath{\mathrm{OPT}}\xspace}$.
Let ${\ensuremath{r_\mathrm{patch}}}(n)$ denote the cost of patching all relevant cuboids in step 2. Lemma \[lem:dstpatch\] (given below) proves that ${\ensuremath{r_\mathrm{patch}}}(n) = O(n^{{\ensuremath{\varepsilon}\xspace}}){\ensuremath{\mathrm{OPT}}\xspace}$.
Now consider the recursive application of the algorithm to all slabs. Recall that ${\ensuremath{N_\mathrm{opt}}\xspace}$ is a fixed minimum M-network for $T$. For $i \in
{1,\ldots,c}$, let ${\ensuremath{\mathrm{OPT}}\xspace}_i^x$ be the optimum cost for M-connecting *all* (not only relevant) terminal pairs in slab $C_i^x$. Define ${\ensuremath{\mathrm{OPT}}\xspace}_i^y$ and ${\ensuremath{\mathrm{OPT}}\xspace}_i^z$ analogously.
Slightly abusing of notation, we write ${\ensuremath{N_\mathrm{opt}}\xspace}\cap C_i^x$ for the set $\{ s \cap C_i^x \mid s \in {\ensuremath{N_\mathrm{opt}}\xspace}\}$ of line segments of [$N_\mathrm{opt}$]{}intersected with slab $C_i^x$. Observe that ${\ensuremath{N_\mathrm{opt}}\xspace}\cap C_i^x$ forms a feasible solution for $C_i^x$. Thus, ${\ensuremath{\mathrm{OPT}}\xspace}_i^x\leq \|{\ensuremath{N_\mathrm{opt}}\xspace}\cap C_i^x\|$. By construction, any slab contains at most $n/c$ terminals. Hence, the total cost of the solutions for slabs $C_1^x,\dots,C_c^x$ is at most $$\sum_{i=1}^cr\left(\frac{n}{c}\right){\ensuremath{\mathrm{OPT}}\xspace}_i^x\leq
r\left(\frac{n}{c}\right)\sum_{i=1}^c\|{\ensuremath{N_\mathrm{opt}}\xspace}\cap C_i^x\|\leq
r\left(\frac{n}{c}\right){\ensuremath{\mathrm{OPT}}\xspace}\,.$$ Clearly, the solutions for the $y$- and $z$-slabs have the same bound.
Summing up all three types of costs, we obtain the recursive equation $$r(n){\ensuremath{\mathrm{OPT}}\xspace}\le 3c^2{\ensuremath{\mathrm{OPT}}\xspace}+{\ensuremath{r_\mathrm{patch}}}(n){\ensuremath{\mathrm{OPT}}\xspace}+3r\left(\frac{n}{c}\right){\ensuremath{\mathrm{OPT}}\xspace}\,.$$ Hence, $r(n)=O(n^{\max\{{\ensuremath{\varepsilon}\xspace},\log_c3\}})$. Plugging in $c =
3^{1/{\ensuremath{\varepsilon}\xspace}}$ yields $r(n)=O(n^{\ensuremath{\varepsilon}\xspace})$, which proves the approximation ratio claimed in Theorem \[thm:3dmain\].
\[lem:dstpatch\] Patching all relevant cuboids costs ${\ensuremath{r_\mathrm{patch}}}(n) \in O(n^{{\ensuremath{\varepsilon}\xspace}}){\ensuremath{\mathrm{OPT}}\xspace}$.
First note that it suffices to consider up-patching; the down-patching case can be argued analogously.
Lemma \[lem:patching-cost\] shows the existence of a near-optimal M-network that up-patches all relevant cuboids. Lemma \[lem:patching-algorithm\] shows that by reducing the patching problem to 3D-RSA, we can find such a network of cost $O(\rho){\ensuremath{\mathrm{OPT}}\xspace}$, where $\rho$ is the approximation factor of 3D-RSA.
We argued in Section \[sec:relat-stein-type\] that there exists an approximation-preserving reduction from 3D-RSA to DST. DST, in turn, admits an $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation for any ${\ensuremath{\varepsilon}\xspace}>0$ [@cccdggl-aadsp-98]. Hence, the cost of up-patching is indeed bounded by $O(n^{\ensuremath{\varepsilon}\xspace}){\ensuremath{\mathrm{OPT}}\xspace}$.
We now turn to the two lemmas that we just used in the proof of Lemma \[lem:dstpatch\]. For our analysis, we need the network $N'$ that is the union of $\mathcal G$ with [$N_\mathrm{opt}$]{}and the projections of [$N_\mathrm{opt}$]{}onto every separating plane of $\mathcal G$. Since there are $3(c-1)$ separating planes and, as we have seen above, $\|{\cal G}\|
\le 3c^2{\ensuremath{\mathrm{OPT}}\xspace}$, it holds that $\|N'\| \le 3(c^2+c){\ensuremath{\mathrm{OPT}}\xspace}= O({\ensuremath{\mathrm{OPT}}\xspace})$.
\[lem:patching-cost\] There exists an M-network of total cost at most $3(c^2+c){\ensuremath{\mathrm{OPT}}\xspace}$ that up-patches all relevant cuboids.
We claim that $N'$ up-patches all relevant cuboids. To this end, let $t\in C_{ijk}$ and let $t'\in C_{i'j'k'}$ with $i<i',j<j',k<k'$. Follow the M-path connecting $t$ and $t'$, starting from $t$. This path must leave $C_{ijk}$ at a certain point $\bar p$, which lies on some face $F$ of $C_{ijk}$. Face $F$, in turn, lies on some separating plane $S$ of the grid $\mathcal G$. From now on follow the projection of the M-path from $\bar p$ to $t'$ on plane $S$. This projected path must leave the face $F$, since $t'$ lies in $C_{i'j'k'}$ with $i<i',j<j',k<k'$, and the projection of $t'$ onto $S$ must therefore lie outside of $F$. Moreover, the point $\bar p'$ where this path leaves $F$ must lie on an edge of $C_{ijk}$ incident to $p_{ijk}$. Hence, we obtain a $t$–$p_{ijk}$ M-path by going from $t$ to $\bar p$, from $\bar p$ to $\bar p'$ and then from $\bar
p'$ to $p_{ijk}$.
\[lem:patching-algorithm\] Given a number $\rho \ge 1$ and an efficient $\rho$-approximation of 3D-RSA, we can efficiently up-patch all relevant cuboids at cost no more than $12(c^2+c) \rho{\ensuremath{\mathrm{OPT}}\xspace}$.
In Lemma \[lem:patching-cost\], we showed the existence of a network $N'$ that up-patches all relevant cuboids at low cost. Now consider an arbitrary relevant cuboid $C_{ijk}$. Clearly $N'\cap
C_{ijk}$ up-patches $C_{ijk}$. Hence ${\ensuremath{{\ensuremath{\mathrm{OPT}}\xspace}^\mathrm{up}}\xspace}_{ijk} \le \|N'\cap
C_{ijk}\|$, where ${\ensuremath{{\ensuremath{\mathrm{OPT}}\xspace}^\mathrm{up}}\xspace}_{ijk}$ denotes the cost of a minimum up-patching of $C_{ijk}$. The problem of optimally up-patching $C_{ijk}$ is just an instance $I_{ijk}$ of 3D-RSA in which all terminals in $C_{ijk}$ have to be connected by an M-path to $\hat{p}_{ijk}$. Applying the factor-$\rho$ approximation algorithm for 3D-RSA to each instance $I_{ijk}$ with $C_{ijk}$ relevant, we patch at total cost at most $$\rho\sum_{C_{ijk}\text{
relevant}}{\ensuremath{{\ensuremath{\mathrm{OPT}}\xspace}^\mathrm{up}}\xspace}_{ijk} \quad\le\quad
\rho\sum_{C_{ijk}\text{ relevant}}\|N'\cap
C_{ijk}\| \quad\le\quad 4\rho \|N'\|\,.$$ The last inequality follows from the fact that each edge of $N'$ occurs in at most *four* cuboids. The lemma follows since $\|N'\|\leq 3(c^2+c){\ensuremath{\mathrm{OPT}}\xspace}$.
#### Running time.
Finally, we analyze the running time. Let $T(n)$ denote the running time of the algorithm applied to a set of $n$ terminals. The running time is dominated by patching and the recursive slab treatment. Using the DST algorithm of Charikar [et al.]{} [@cccdggl-aadsp-98], patching cuboid $C_i$ requires time $n_i^{O(1/{\ensuremath{\varepsilon}\xspace})}$, where $n_i$ is the number of terminals in $C_i$. As each cuboid is patched at most twice and there are $c^3$ cuboids, patching takes $O(c^3) n^{O(1/{\ensuremath{\varepsilon}\xspace})} =
n^{O(1/{\ensuremath{\varepsilon}\xspace})}$ time. The algorithm is applied recursively to $3c$ slabs. This yields the recurrence $T(n) = 3c T(n/c) + n^{O(1/{\ensuremath{\varepsilon}\xspace})}$, which leads to the claimed running time.
This completes the proof of Theorem \[thm:3dmain\].
Open Problems
=============
We have presented, for any ${\ensuremath{\varepsilon}\xspace}>0$, a grid-based $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation algorithm for $d$D-MMN. This is a significant improvement over the ratio of $O(n^{4/5+{\ensuremath{\varepsilon}\xspace}})$ which is achieved by reducing the problem to DSF. For 3D, we have described a $4(k-1)$-approximation algorithm for the case when the terminals lie on $k \ge 2$ horizontal planes. This outperforms our grid-based algorithm when $k \in o(n^{\ensuremath{\varepsilon}\xspace})$. Whereas 2D-MMN admits a 2-approximation [@cnv-raamm-08; @gsz-yaa2a-08; @n-eprmc-05], it remains open whether $O(1)$- or $O(\log n)$-approximation algorithms exist for higher dimensions.
Our $O(n^{{\ensuremath{\varepsilon}\xspace}})$-approximation algorithm for $d$D-MMN solves instances of $d$D-RSA for the subproblem of patching. We conjecture that $d$D-RSA admits better approximation ratios. While this is an interesting open question, a positive result would still not be enough to improve our approximation ratio, which is dominated by the cost of finding M-paths inside slabs.
The complexity of the *undirectional* bichromatic rectangle piercing problem (see Section \[sec:k-planes\]) is still unknown. Currently, the best approximation has a ratio of 4, which is (trivially) implied by the result of Soto and Telha [@st-2dorg-11]. Any progress would immediately improve the approximation ratio of our algorithm for the $k$-plane case of 3D-MMN (for any $k>2$).
#### Acknowledgments. {#acknowledgments. .unnumbered}
This work was started at the 2009 Bertinoro Workshop on Graph Drawing. We thank the organizers Beppe Liotta and Walter Didimo for creating an inspiring atmosphere. We also thank Steve Wismath, Henk Meijer, Jan Kratochv[í]{}l, and Pankaj Agarwal for discussions. We are indebted to Stefan Felsner for pointing us to Soto and Telha’s work [@st-2dorg-11].
Appendix: Extension to Higher Dimensions {#sec:dd .unnumbered}
========================================
We now describe the approximation algorithm for $d$D-MMN, for $d>3$, as a generalization of the 3D-MMN idea from Section \[sec:general-case\].
For any fixed dimension $d$ and for any ${\ensuremath{\varepsilon}\xspace}>0$, there exists an $O(n^{\ensuremath{\varepsilon}\xspace})$-approximation algorithm for $d$D-MMN.
Large parts of the algorithm and the analysis are straightforward generalizations of the algorithm for 3D-MMN. The presentation of both follows closely the 3D case. However, Lemma \[lem:dd-patching-cost\], where the cost of the patching procedure is bounded, requires non-trivial additional insights.
As in the 3D case we decompose the overall problem into a constant number of instances of the directional subproblem. The directional subproblem consists in M-connecting any terminal pair $(t,t')$ such that $x^i(t)\leq x^i(t')$ for any $i\in\{1,\dots,d\}$. Here, we use $x^i(p)$ to denote the $i$-th coordinate of point $p\in\mathbb{R}^d$. We can decompose the general problem into $2^{d-1}$ directional subproblems. Once again, we assume that the terminals are in general position.
The $d$D Grid Algorithm {#the-dd-grid-algorithm .unnumbered}
-----------------------
We begin the description with a high-level summary. To solve the directional $d$D-MMN problem we place a $d$D-grid which partitions the instance into cuboids and slabs. Terminal pairs lying in different slabs are handled by M-connecting each terminal to the corner of its cuboid and then using the edges of grid. Terminal pairs from the same slab are M-connected by applying the algorithm recursively to all slabs. Each slab contains only a constant fraction of the terminals.
[*Step 1: Partitioning into cuboids and slabs.*]{} Consider the bounding cuboid $C$ for the set $T$ of terminals and choose a large constant $c=d^{1/{\ensuremath{\varepsilon}\xspace}}$. For each dimension $i\in\{1,\dots,d\}$ we choose $c+1$ separating planes determined by values $x_1^i<\dots<x_{c+1}^i$. Planes $x_1^i$ and $x_{c+1}^i$ coincide with the boundary of $C$ in dimension $i$. The separating planes for dimension $i$ partition $C$ into $c$ slabs $C_j^i$, where $j\in\{1,\dots,c\}$. Slab $C_j^i$ is the set of points $p\in C$ such that $x_i^j\leq x_i(p)\leq x_i^{j+1}$. We place the separating planes so that each slab contains at most $n/c$ terminals. Altogether we have $d(c+1)$ separating planes.
Let $j_1,\dots,j_d\in\{1,\dots,c\}$. The *subcuboids* $C_{j_1,\dots,j_d}$ is the set of points $p\in C$ such that $x_{j_i}^i\leq x^i(p)\leq x_{j_i+1}^i$ for each $i\in \{1,\dots,d\}$.
Consider the $i^{th}$ dimension, $i\in\{1,\dots,d\}$ and integers $j_k\in\{2,\dots,c\}$ for each $k\in\{1,\dots,d\}-\{i\}$. Let $s$ be the axis-parallel line segment that contains all points $p\in C$ such that $x^k(p)=x_{j_k}$ for each $k\neq i$. We call $s$ a *grid segment for dimension $i$*. The *grid $\cal G$* is the set of all grid segments and there are $d(c-1)^{d-1}$ grid segments in $\cal
G$.
Given $j_k\in\{2,\dots,c\}$ for each $k\in\{1,\dots,d\}$ we call $(x_{j_1}^1,\dots,x_{j_d}^d)$ a *grid point* of $\mathcal G$ and there are $(c-1)^d$ grid points in total.
[*Step 2: Add M-paths between different slabs.*]{} Consider two cuboids $C_{j_1,\dots,j_d}$ and $C_{j_1',\dots,j_d'}$ with $j_i<j_i'$ for each $i\in\{1,\dots,d\}$. Any pair of terminals $t\in C_{j_1',\dots,j_d'} $ and $t'\in C_{j_1',\dots,j_d'}$ can be M-connected using the segments of $\mathcal G$ as long as $t$ and $t'$ are suitably connected to the corners (grid points) of their cuboids. We use [*patching*]{} (described below) to connect all terminal to the corners of their cuboid.
Call a cuboid $C_{j_1,\dots,j_d}$ *relevant* if there is a cuboid $C_{j_1',\dots,j_d'}$ that contains at least one terminal and satisfies $j_i<j_i'$ for each $i\in\{1,\dots,d\}$. For each relevant cuboid $C_{j_1,\dots,j_d}$, let $p_{j_1,\dots,j_d}$ denote the grid point $(x_{j_1+1}^i,\dots,x_{j_d+1}^d)$. *Up-patching* $C_{j_1,\dots,j_d}$ means to M-connect every terminal in $t\in
C_{j_1,\dots,j_d}$ to $p_{j_1,\dots,j_d}$. Up-patch $C_{j_1,\dots,j_d}$ by solving the $d$D-RSA problem with the terminals inside $C_{j_1,\dots,j_d}$ as the terminals and $p_{j_1,\dots,j_d}$ as the origin.
[*Down-patching*]{} is defined analogously; cuboid $C_{j_1,\dots,j_d}$ is relevant if there is a non-empty cuboid $C_{j_1',\dots,j_d'}$ with $j_i'<j_i$, $i=1,\dots,d$ and using grid point $p_{j_1,\dots,j_d}':=(x_{j_1+1}^i,\dots,x_{j_d+1}^d)$ as origin instead of $p_{j_1,\dots,j_d}$.
The output of this step is the union of grid $\mathcal G$ with a network that up-patches and down-patches all relevant cuboids. This produces M-paths between all terminal pairs in different slabs.
[*Step 3: Add M-paths within slabs.*]{} To also connect terminal pairs that lie in a common slab we apply the algorithm (Steps 1–3) recursively to each slab $C_j^i$ with $i\in\{1,\dots,d\}$ and any $j\in\{1,\dots,c\}$.
Analysis {#analysis-1 .unnumbered}
--------
We now show that the algorithm presented above yields a feasible solution to directional $d$D-MMN, with cost at most $O(n^{\ensuremath{\varepsilon}\xspace}){\ensuremath{\mathrm{OPT}}\xspace}$, for any ${\ensuremath{\varepsilon}\xspace}>0$. Here, [$\mathrm{OPT}$]{}denotes the cost of an optimum solution to the *general* $d$D-MMN instance rather than the minimum cost ${\ensuremath{\mathrm{OPT}}\xspace}'$ achievable for the directional subproblem. The reason is that the cost of the grid $\cal
G$ is generally not related to ${\ensuremath{\mathrm{OPT}}\xspace}'$ but to [$\mathrm{OPT}$]{}. We finish the section by arguing that the running time of the algorithm is $n^{O(1/{\ensuremath{\varepsilon}\xspace})}$.
\[lem:dd-mpaths\] The dD grid algorithm M-connects all relevant terminal pairs.
Let $(t,t')$ be a relevant terminal pair. First suppose that $t\in
C_{j_1,\dots,j_d}$ and $t'\in C_{j_1',\dots,j_d'}$ where $j_i<j_i'$ for all $i\in\{1,\dots,d\}$. Hence, $C_{j_1,\dots,j_d}$ and $C_{j_1',\dots,j_d'}$ are relevant for up-patching and down-patching, respectively. Consider the corners $p_{j_1,\dots,j_d}$ of $C_{j_1,\dots,j_d}$ and the corner $p_{j_1',\dots,j_d'}'$ of $C_{j_1',\dots,j_d'}$. In the up-patching step of our algorithm we solve an RSA problem with terminals of $C_{j_1,\dots,j_d}$ as the input points, and corner $p_{j_1,\dots,j_d}$ as the origin. By definition, an RSA solution M-connects $t$ to $p_{j_1,\dots,j_d}$. Similarly, down-patching M-connects $t'$ to $p_{j_1',\dots,j_d'}'$. It follows that $t$ and $t'$ are connected, since $p_{j_1,\dots,j_d}$ and $p_{j_1',\dots,j_d'}'$ are M-connected via grid $\mathcal G$. Both terminals are even M-connected since additionally $t\leq
p_{j_1,\dots,j_d}\leq p_{j_1',\dots,j_d'}'\leq t'$, where $\leq$ denotes the domination relation between points.
Now suppose $t$ and $t'$ lie in the same slab. As the algorithm is applied recursively to each slab there will be a recursion step where $t$ and $t'$ will lie in cuboids in different slabs. Here, we need our assumption of general position. Applying the argument above to that particular recursive step completes the proof.
#### Approximation ratio.
Let $r(n)$ denote the approximation ratio of our algorithm where $n$ is the number of terminals in $T$. The total cost of our solution consists of the cost for the grid $\mathcal G$, the cost of up-patching and down-patching all relevant cuboids, and the cost for the recursive treatment of the slabs in all $d$ dimensions. We analyze each of these costs separately.
The grid $\mathcal G$ consists of the $d(c-1)^{d-1}$ grid segments. The length of any grid segment $s$ is a lower bound on ${\ensuremath{\mathrm{OPT}}\xspace}$. This holds because there are two terminals on the boundary of $C$ whose $L_1$-distance is at least the length of $s$. It follows that the cost of the grid is bounded by $d(c-1)^{d-1}{\ensuremath{\mathrm{OPT}}\xspace}$.
Let $r_{\text{patch}}(n)$ denote the cost of patching all relevant cuboids as is done in Step 2. Lemma \[lem:dd-dstpatch\] (given below) proves that $r_{\text{patch}}(n) = O(n^{{\ensuremath{\varepsilon}\xspace}}){\ensuremath{\mathrm{OPT}}\xspace}$.
Now consider the recursive application of the algorithm to all slabs $C_j^i$, where $i\in\{1,\dots,d\}$ and $j\in\{1,\dots,c\}$. First recall that we placed the separating planes so that $|C_j^i|\leq n/c$ for any $i\in\{1,\dots,d\}$ and any $j\in\{1,\dots,c\}$.
Consider dimension $i\in\{1,\dots,d\}$. Let ${\ensuremath{\mathrm{OPT}}\xspace}_j^i$ be the optimum cost for M-connecting *all* (not only relevant) terminal pairs in slab $C_j^i$, where $j\in\{1,\dots,c\}$. Slightly abusing notation, we write ${\ensuremath{N_\mathrm{opt}}\xspace}\cap C_j^i$ for the set of line segments of ${\ensuremath{N_\mathrm{opt}}\xspace}$ that are completely contained in the slab $C_j^i$. Observe that ${\ensuremath{N_\mathrm{opt}}\xspace}\cap
C_j^i$, forms a feasible solution for $C_j^i$. Thus ${\ensuremath{\mathrm{OPT}}\xspace}_j^i\leq
\|{\ensuremath{N_\mathrm{opt}}\xspace}\cap C_j^i\|$. Each such $C_j^i$ contains at most $n/c$ terminals, and therefore the total cost of the solutions for the all slabs $C_j^i$ of dimension $i$ is at most $$\sum_{j=1}^cr\left(\frac{n}{c}\right){\ensuremath{\mathrm{OPT}}\xspace}_j^i\leq
r\left(\frac{n}{c}\right)\sum_{j=1}^c\|{\ensuremath{N_\mathrm{opt}}\xspace}\cap C_j^i\|\leq
r\left(\frac{n}{c}\right){\ensuremath{\mathrm{OPT}}\xspace}\,.$$ Summing all costs, we obtain the following recursive equation for $r(n)$ $$r(n){\ensuremath{\mathrm{OPT}}\xspace}\leq dc^{d-1}\cdot{\ensuremath{\mathrm{OPT}}\xspace}+d\cdot
r\left(\frac{n}{c}\right){\ensuremath{\mathrm{OPT}}\xspace}+r_{\text{patch}}(n){\ensuremath{\mathrm{OPT}}\xspace}\,.$$ Hence $r(n)=O(n^{\max\{{\ensuremath{\varepsilon}\xspace},\log_cd\}})$. Choosing $c \ge d^{1/{\ensuremath{\varepsilon}\xspace}}$, as in Step 1, yields $O(n^{\ensuremath{\varepsilon}\xspace})$ proving the approximation ratio claimed in Theorem \[thm:3dmain\].
\[lem:dd-dstpatch\] The cost of patching all relevant cuboids, ${\ensuremath{r_\mathrm{patch}}}(n)$, is $O(n^{{\ensuremath{\varepsilon}\xspace}}){\ensuremath{\mathrm{OPT}}\xspace}$.
First consider up-patching. Lemma \[lem:dd-patching-cost\] (below) shows the existence of a near optimal network that up-patches all relevant cuboids. Lemma \[lem:dd-patching-algorithm\] shows that by reducing the patching problem to $d$D-RSA, we can find such a network of cost $O(\rho){\ensuremath{\mathrm{OPT}}\xspace}$, where $\rho$ is the approximation factor of $d$D-RSA.
Analogously to the 3D-case there is a approximation-preserving reduction from $d$D-RSA to DST (see Section \[sec:relat-stein-type\]), which implies that $d$D-RSA is approximable within a factor $O(n^{\ensuremath{\varepsilon}\xspace})$ for any ${\ensuremath{\varepsilon}\xspace}>0$. Hence the same approximation factor can be achieved for $d$D-RSA by choosing $\epsilon$ sufficiently small.
The lemma follows as the analysis holds analogously for down-patching.
\[lem:dd-patching-cost\] There exists an M-network of total cost at most $(c+1)^d{\ensuremath{\mathrm{OPT}}\xspace}$ that up-patches all relevant cuboids.
Let $I\subseteq\{1,\dots,d\}$ be a set of dimensions. For every $i\in I$ we choose a separating plane $x_{j_i}^i$ where $j_i\in\{1,\dots,c\}$. Let $J$ be the set of these separating planes and let $C(J)$ be the intersection of $C$ with all separating planes in $J$. We call $C(J)$ a *$C$-face*. There are most $(c+1)^d$ such $C$-faces. Project ${\ensuremath{N_\mathrm{opt}}\xspace}$ onto each $C$-face. Let $N'$ be the union of all these projections. Clearly, the cost of $N'$ is at most $(c+1)^d{\ensuremath{\mathrm{OPT}}\xspace}$.
We claim that $N'$ up-patches all relevant cuboids. To this end, let $(t,t')$ be a relevant terminal pair such that $t\in
C_{j_1,\dots,j_d}$, $t'\in C_{j_1',\dots,j_d'}$ and $j_i<j_i'$ for all $i\in\{1,\dots,d\}$.
We claim that there is an M-path $\pi_t$ from $t$ to $p_{j_1,\dots,j_d}$ in $N'$. To see this, traverse the M-path $\pi_{tt'}$ in ${\ensuremath{N_\mathrm{opt}}\xspace}$ connecting $t$ and $t'$, starting from $t$. We assume w.l.o.g. that the separating planes that bound cuboid $C_{j_1,\dots,j_d}$ are entered by $\pi_{tt'}$ in the order $(x_{j_1+1}^1,\dots,x_{j_d+1}^d)$. The desired path $\pi_t$ starts at $t$ and follows $\pi$ until the separating plane $x_{j_1+1}^1$ is entered. From this point on we follow the projection $\pi_{tt'}(x_{j_1+1}^1)$ of $\pi_{tt'}$ onto $C$-face $C(x_{j_1+1}^1)$. If $\pi_{tt'}(x_{j_1+1}^1)$ enters $x_{j_2+1}^2$ we follow the projection $\pi_{tt'}(x_{j_1+1}^1,x_{j_2+1}^2)$ of $\pi_{tt'}$ onto $C(x_{j_1+1}^1,x_{j_2+1}^2))$. We proceed in this fashion until we reach the $C$-face $C(x_{j_1+1}^1,\dots,x_{j_d+1}^d)$, which is just the corner $p_{j_1,\dots,j_d}$. Since $N'$ contains the projection of $\pi_{tt'}$ onto each $C$-face, the path $\pi_t$ described above is contained in $N'$. This reasoning remains valid if the separating planes that bound $C_{j_1,\dots,j_d}$ are entered in an arbitrary order, as we projected ${\ensuremath{N_\mathrm{opt}}\xspace}$ onto each $C$-face.
\[lem:dd-patching-algorithm\] Given an efficient algorithm that approximates $d$D-RSA within a factor of $\rho$, we can efficiently up-patch all relevant cuboids at cost at most $(2(c+1))^d \rho{\ensuremath{\mathrm{OPT}}\xspace}$.
In Lemma \[lem:dd-patching-cost\], we showed the existence of a network $N'$ that up-patches all relevant cuboids at low cost. Now consider an arbitrary relevant cuboid $C_{j_1,\dots,j_d}$. Clearly $N'\cap C_{j_1,\dots,j_d}$ up-patches $C_{j_1,\dots,j_d}$. Hence ${\ensuremath{{\ensuremath{\mathrm{OPT}}\xspace}^\mathrm{up}}\xspace}_{j_1,\dots,j_d} \le \|N'\cap C_{j_1,\dots,j_d}\|$, where ${\ensuremath{{\ensuremath{\mathrm{OPT}}\xspace}^\mathrm{up}}\xspace}_{j_1,\dots,j_d}$ denotes the cost of a minimum up-patching of $C_{j_1,\dots,j_d}$. The problem of optimally up-patching $C_{j_1,\dots,j_d}$ is just an instance $I_{j_1,\dots,j_d}$ of $d$D-RSA, in which all terminals in $C_{j_1,\dots,j_d}$ have to be connected by an M-path to $p_{j_1,\dots,j_d}$. Applying the factor-$\rho$ approximation algorithm for $d$D-RSA to each instance $I_{j_1,\dots,j_d}$ with $C_{j_1,\dots,j_d}$ relevant, we patch at total cost at most $$\rho\sum_{C_{j_1,\dots,j_d}\text{
relevant}}{\ensuremath{{\ensuremath{\mathrm{OPT}}\xspace}^\mathrm{up}}\xspace}_{j_1,\dots,j_d} \quad\le\quad
\rho\sum_{C_{j_1,\dots,j_d}\text{ relevant}}\|N'\cap
C_{j_1,\dots,j_d}\| \quad\le\quad 2^d\rho \|N'\|\,.$$ The last inequality follows from the fact that each segment of $N'$ occurs in at most $2^d$ cuboids. The lemma follows since $\|N'\|\leq (c+1)^d{\ensuremath{\mathrm{OPT}}\xspace}$.
#### Running time.
Let $T(n)$ denote the running time of the algorithm for $n$ terminals. The running time is dominated by patching and the recursive treatment of slabs. Using the DST algorithm of Charikar [et al.]{} [@cccdggl-aadsp-98], patching cuboid $C_j^i$ requires time $(n_j^i)^{O(1/{\ensuremath{\varepsilon}\xspace})}$, where $n_j^i$ is the number of terminals in $C_j^i$. As each cuboid is patched at most twice and there are $c^d$ cuboids, patching requires total time $O(c^d) n^{O(1/{\ensuremath{\varepsilon}\xspace})} =
n^{O(1/{\ensuremath{\varepsilon}\xspace})}$. The algorithm is applied recursively to $dc$ slabs. This yields the recurrence $T(n) = dc T(n/c) + n^{O(1/{\ensuremath{\varepsilon}\xspace})}$, which leads to the claimed running time.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The ultraviolet background (UVB) emitted by quasars and galaxies governs the ionization and thermal state of the intergalactic medium (IGM), regulates the formation of high-redshift galaxies, and is thus a key quantity for modeling cosmic reionization. The vast majority of cosmological hydrodynamical simulations implement the UVB via a set of spatially uniform photoionization and photoheating rates derived from UVB synthesis models. We show that simulations using canonical UVB rates reionize and, perhaps more importantly, spuriously heat the IGM, much earlier $z \sim 15$ than they should. This problem arises because at $z > 6$, where observational constraints are nonexistent, the UVB amplitude is far too high. We introduce a new methodology to remedy this issue, and we generate self-consistent photoionization and photoheating rates to model any chosen reionization history. Following this approach, we run a suite of hydrodynamical simulations of different reionization scenarios and explore the impact of the timing of reionization and its concomitant heat injection on the the thermal state of the IGM. We present a comprehensive study of the pressure smoothing scale of IGM gas, illustrating its dependence on the details of both hydrogen and helium reionization, and argue that it plays a fundamental role in interpreting [Lyman-$\alpha$]{} forest statistics and the thermal evolution of the IGM. The premature IGM heating we have uncovered implies that previous work has likely dramatically overestimated the impact of photoionization feedback on galaxy formation, which sets the minimum halo mass able to form stars at high redshifts. We make our new UVB photoionization and photoheating rates publicly available for use in future simulations.'
author:
- 'Jose Oñorbe, Joseph F. Hennawi'
- Zarija Lukić
bibliography:
- 'apj-jour.bib'
- 'uvmodeling.bib'
title: 'Self-Consistent Modeling of Reionization in Cosmological Hydrodynamical Simulations'
---
Introduction {#sec:intro}
============
In our current standard model of the universe, hydrogen and helium account for 99% of the baryonic mass density [@Planck:2015]. After the recombination epoch, these elements remain neutral until ultraviolet radiation from star-forming galaxies and active galactic nuclei reionizes them. Therefore, this ultraviolet background (UVB) governs the ionization state of intergalactic gas and plays a key role in its thermal evolution through photoheating. During the reionization of ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and later ${\ensuremath{\textrm{He} \, \textsc{ii}}}$, ionization fronts propagate supersonically through the intergalactic medium (IGM), impulsively heating gas to $\sim10^{4}$ K [see, e.g., @Abel:1999; @McQuinn:2012; @Davies:2016]. As the universe evolves, it is well known that the balance between cooling due to Hubble expansion and inverse-Compton scattering of cosmic microwave background (CMB) photons and heating due to the gravitational collapse and photoionization heating give rise to a well-defined temperature-density relationship in the IGM [@Hui:1997; @McQuinn:2012]: $$T=T_{0}\times \Delta^{\gamma-1}
\label{eq:T0gamma}$$ where $\Delta=\rho/\bar{\rho}$ is the overdensity with respect to the mean and $T_{0}$ is the temperature at the mean density. Immediately after the reionization of ${\ensuremath{\textrm{H} \, \textsc{i}}}$ ($z \lesssim 6$) or ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ ($z \lesssim 3$), $T_{0}$ is likely to be around $\sim2\times 10^{4}$ K and $\gamma\sim1$ [@Bolton:2009; @McQuinn:2009]; at lower redshifts, $T_{0}$ decreases as the universe expands, while $\gamma$ is expected to increase and asymptotically approach a value of $1.62$ [@Hui:1997].
Another important physical ingredient to describe the thermal state of the IGM is the gas pressure support. At small scales and high densities, baryons experience pressure forces that prevent them from tracing the collisionless dark matter. This pressure results in an effective 3D smoothing of the baryon distribution relative to the dark matter, at a characteristic scale. known as the Jeans pressure smoothing scale, ${\lambda_{\rm P}}$. In an expanding universe with an evolving thermal state, this scale at a given epoch is expected to depend on the entire thermal history, because fluctuations at earlier times expand or fail to collapse depending on the IGM temperature at that epoch [@Gnedin:1998; @Kulkarni:2015]. Recently, @Rorai:2013 and @Rorai:2015 have shown that an independent measurement of the pressure smoothing scale can be obtained using the coherence of [Lyman-$\alpha$]{} forest absorption in close quasar pairs [@Hennawi:2006; @Hennawi:2010].
[Lyman-$\alpha$]{} forest observations between $2<z<6$ probe the moderate overdensities characteristic of the IGM and therefore are a crucial tool to understand the properties of the UVB. In the last decade, the precision of these measurements has continued to grow both in terms of their numbers (BOSS[^1] survey) and in quality [high signal-to-noise ratio spectrum from, e.g. @OMeara:2015]. However, while it seems that we keep learning more and more about the ionization history of the universe, for both ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionizations [e.g. @Becker:2013; @Syphers:2014; @Worseck:2014; @Becker:2015] the thermal history of the universe is still far from certain. The statistical properties of the [Lyman-$\alpha$]{} forest are sensitive to the thermal state of the gas, trough both thermal broadening of lines and pressure support. When constraints on the thermal history are reviewed, they yield very puzzling results. Measurements of $T_{0}$ from different groups utilizing different methodology are in poor agreement [@Schaye:2000; @Bolton:2008; @Lidz:2010; @Becker:2011; @Rudie:2012; @Garzilli:2012; @Boera:2014; @Bolton:2014]. A similar problem appears when measurements of the slope of the temperature–density relation, $\gamma$, are compared. At $z\simeq3$ some authors have even found that $\gamma$ is either close to isothermal ($\gamma=1$) or even inverted [$\gamma<1$; @Bolton:2008; @Viel:2009 but see @Lee:2015]. Most studies of the thermal state of the IGM ignore uncertainties resulting from the unknown pressure smoothing scale [but see @Becker:2011; @Puchwein:2015], which produces a 3D smoothing that is difficult to disentangle from the the similar but 1D smoothing resulting from thermal broadening [@Peeples:2010a; @Peeples:2010b; @Rorai:2013]. Therefore, ignoring this effect has probably contributed to the confusing and sometimes contradictory published constraints on $T_{0}$ and $\gamma$ [@Puchwein:2015].
With the help of accurate models of the IGM, the statistics of the [Lyman-$\alpha$]{} forest can be used to constrain its thermal parameters and ultimately cosmic reionization. Ideally one will run coupled radiative transfer hydrodynamical simulations that include extra physics governing the sources of ionizing photons (stars, quasars, etc.). Despite significant progress on this front [@Wise:2014; @So:2014; @Gnedin:2014; @Pawlik:2015; @Norman:2015; @Ocvirk:2015] these simulations are still too costly for sensible exploration of the parameter space. For this reason, the dominant approach, implemented in the vast majority of hydrodynamical codes, is to assume that all gas elements are optically thin to ionizing photons, such that their ionization state can be fully described by a uniform and isotropic UV+X-ray background radiation field. Thus, the radiation field is encapsulated by a set of photoionization and photoheating rates that evolve with redshift for each relevant ion. The minimal set of ions are ${\ensuremath{\textrm{H} \, \textsc{i}}}$, ${\ensuremath{\textrm{He} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ in order to track the most relevant ionization events, as well as the thermal heating associated with them. Of course, although this optically thin approximation is a valid assumption once the mean free path of ionization photons, $\lambda_{\rm mfp,\nu}$, is large enough, it is certainly not true during cosmic reionization events. As such, this optically thin approach is not meant to provide an accurate description of reionization itself, but it should at least provide a reasonable description of the heat injection associated with reionization.
This is important since galaxies forming during the reionization epoch are sensitive to the thermal state of the gas, and even well after reionization gas elements can retain thermal memory of reionization heating [@Gnedin:1998; @Kulkarni:2015]. It is important to remark here that these UVB models have relevant consequences for galaxy formation and evolution models and hydrodynamical simulations. Several groups have already shown how important the UVB model is to determine the star formation of the first galaxies and their evolution by not only setting the minimum halo mass able to form stars [i.e., halos massive enough to overcome gas pressure forces; @Rees:1986; @Sobacchi:2013] but also regulating the gas accretion from the IGM into the more massive halos [@Quinn:1996; @Simpson:2013; @BenitezLlambay:2015; @Wheeler:2015a].
The standard approach is to adopt photoionization and photoheating rates from semianalytical synthesis models of the UVB [@Haardt:1996; @Haardt:2001; @FaucherGiguere:2009; @Haardt:2012]. However, these UVB synthesis models surely break down during reionization events, and the validity of using them in optically thin simulations (during reionization) is questionable. Moreover, as we will show, these models are fundamentally inconsistent during reionization, leading to different reionization histories in the simulations than the ones given by the authors. Specifically, they reionize the universe too early, and as a result they produce spurious heating of the IGM at early times (see Section \[sec:typmodels\] and Figure \[fig:Qhistgas0\]). In this paper, we improve on the limitations of current UVB models to provide reliable ionization and thermal histories during reionization by developing a new method to model ionization and heating during reionization in hydrodynamical simulations. In the context of this method, we demonstrate how to run simulations with self-consistent ionization and thermal histories that agree with constraints from the CMB and IGM measurements. Moreover, we make these new tables publicly available in the default format used by most cosmological codes.
{width="\textwidth"}
The outline of the paper is as follows. In Section \[sec:typmodels\] we discuss in detail current standard methods that include the effect of the UVB in optically thin hydrodynamical simulations. We show that these models have problems reproducing the desired ionization and thermal histories. In Section \[sec:newmodel\] we present a new method to improve the current models of the UVB during reionization events. The different reionization models considered in this work, based on current observational constraints, are motivated in Section \[sec:reionmodels\]. We describe the basic details of the hydrodynamical cosmological code that we have used in this work, the analysis pipeline, and the properties of the simulations in Section \[sec:code\]. The ionization and thermal histories of the simulations using the new UVB models are shown and examined in Section \[sec:results\]. We explore the possibility of reproducing observational constraints on the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ transmission in Section \[sec:meanflux\]. In Section \[sec:discuss\] we discuss the limitations of our new approach, provide a comparison to previous work, and discuss previous work using incorrect UVB models in galaxy formation simulations that likely overestimate the impact of photoionization feedback. We conclude in Section \[sec:conc\]. In Appendix \[app:volave\] we provide details on how the new photoionization and photoheating rates are derived in our method. In Appendix \[app:HMold\] we present the ionization and thermal histories of several widely used UVB models. The effects of cosmology on the new models are discussed in Appendix \[app:cosmo\]. Finally, in Appendix \[app:tables\] we present the photoionization and photoheating rates of the new models.
Ionization and Thermal Histories of Common UVB Models {#sec:typmodels}
=====================================================
Current UVB models used in optically thin hydrodynamical simulations give the evolution of both photoionization, $\Gamma_{\gamma}$ (${\ensuremath{\textrm{H} \, \textsc{i}}},{\ensuremath{\textrm{He} \, \textsc{i}}},{\ensuremath{\textrm{He} \, \textsc{ii}}}$), and photoheating $\dot{q}$ (${\ensuremath{\textrm{H} \, \textsc{i}}},{\ensuremath{\textrm{He} \, \textsc{i}}},{\ensuremath{\textrm{He} \, \textsc{ii}}}$) rates with redshift. Therefore, during reionizations the simulated IGM is photoheated everywhere by the same spectrum using a homogeneous value of $\dot{q}$. It is important to remark that the photoheating rates are given as energy per second per ion. Then, for example, in the case of ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, the photoheating rate per volume for each resolution element is ${n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$.
@Haardt:1996 were the first to try to develop self-consistent UVB models in a cosmological context using radiative transfer methods and taking into account observations of the ionizing sources (namely, quasars and galaxy luminosity functions) and the absorption of the ionizing photons (column density distribution of neutral hydrogen, $N_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$, absorbers, and ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux). An ionization front heats up the gas behind it [see, e.g., @Abel:1999; @McQuinn:2012; @Davies:2016] and therefore it is crucial to include radiative transfer effects in the UVB modeling. Self-consistent methods use photoionization modeling codes that implement 1D radiative transfer (e.g. <span style="font-variant:small-caps;">cloudy</span>) to try to take this effect into account. Subsequent efforts have developed these models further [@FaucherGiguere:2009; @Haardt:2012]. These UVB models are adopted in essentially all nonadiabatic cosmological hydrodynamic simulations to compute the ionization state and photoheating rates of intergalactic gas [@Somerville:2015]. These include simulations focusing on the properties of the IGM [e.g. @Katz:1996; @MiraldaEscude:1996; @Lukic:2015], but also simulations modeling galaxy formation and evolution [e.g. @Vogelsberger:2013; @Hopkins:2014; @Shen:2014; @Governato:2015; @Dave:2016].
We first want to present the ionization and thermal histories obtained when one of the most widely used UVB models (e.g. tabulated photoionization and photoheating rates) is used [@Haardt:2012 hereafter HM12]. The upper panel of Figure \[fig:Qhistgas0\] shows the ${\ensuremath{\textrm{H} \, \textsc{ii}}}$ (solid black line) and ${\ensuremath{\textrm{He} \, \textsc{iii}}}$ (dot-dashed black line) ionization history calculated by the HM12 model (black lines), which indicates that ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization should finish at $z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}=6.7$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ at $z_{\rm reion,{\ensuremath{\textrm{He} \, \textsc{ii}}}}=2.75$. These are given by the volume filling fraction evolution, $Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}(z)$ which can be thought of the probability that the hydrogen in a given region is ionized [@Madau:1999a], and $Q_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}(z)$ the analogous quantity for a doubly ionized helium region. In this panel we also show the ionization history of an hydrodynamical cosmological simulation using the HM12 UVB model (green lines). This simulation uses the standard methodology employed in other optically thin simulations that we describe it in detail in Section \[sec:code\]. To obtain the ionization history from the simulations we computed the ionization fraction of each volume element in the simulation, $x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}={n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}/{n_{{\ensuremath{\textrm{H}}}}}$ and $x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}={n_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}/{n_{{\ensuremath{\textrm{He}}}}}$, and then calculate the average weighting in volume (i.e. averaging all of the cells)[^2]. The first striking thing that we learn from this comparison is that using the HM12 photoionization rates effectively reionizes the universe much earlier than the reionization redshift reported by HM12. It appears that some aspect of the HM12 calculation is not internally consistent.
The middle panel of Figure \[fig:Qhistgas0\] shows the integrated electron scattering optical depth, defined as $$\begin{aligned}
& {\tau_{\rm e}}(z)=c\sigma_T \int_0^{z}{n_{\rm e}}{(1+z')^2dz'\over H(z')}\\
\end{aligned}
\label{eq:tauCMB}$$ where $c$ is the velocity of light, $\sigma_T$ is the Thomson cross section, $H(z)$ is the Hubble parameter, and ${n_{\rm e}}$ is the proper electron density. We have computed the electron density in our simulation as ${n_{\rm e}}={\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}(1+\chi){\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle} +\chi {\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}$ where $\chi=Y_{\rm p}/(4X_{\rm p})$ and $X_{\rm p}$ and $Y_{\rm p}$ are the hydrogen and helium mass abundances, respectively (see Appendix \[ssec:volave\] for a detailed derivation of this equation). We also make the standard assumption that the reionization of ${\ensuremath{\textrm{He} \, \textsc{i}}}$ is perfectly coupled with that of ${\ensuremath{\textrm{H} \, \textsc{i}}}$. The observational constraints on ${\tau_{\rm e}}$ coming from the CMB [${\tau_{\rm e}}=0.078\pm 0.019$ @Planck:2015] are indicated by a gray band[^3]. The results of the simulation (green) not only differ from the expected results of the model (black) but also they are in strong disagreement with the observational constraints.
Moreover, the lower panel of Figure \[fig:Qhistgas0\] shows the thermal history of this simulation (green line) via the evolution of the temperature at mean density, $T_{0}$ defined in eqn. (\[eq:T0gamma\]). See Section \[ssec:samplesum\] for a discussion of the procedure used to fit the $\rho$-$T$ relation. We can see that, not only do reionization events occur too early, but also that the heating associated with them starts at much earlier times, $z\sim15$. We have confirmed that this result is not due to resolution, assumed cosmological model, atomic rates considered, or some particular code characteristic [see Figures 3 and A1 in @Puchwein:2015 for the same effect but using a SPH Lagrangian code, GADGET-3, and using both an equilibrium and non-equilibrium ionization solver]. Thus, why do the reionization histories in the simulation and the one calculated by these authors differ so much?
UVB semianalytic synthesis models — such as the one used by HM12 — rely on two main assumptions: (1) the photoionizing background is everywhere uniform, and (2) radiative transfer is optically thin. Clearly, both assumptions break down during reionization. While a full solution requires a radiative transfer simulation, a large number of IGM studies are insensitive to the reionization details. Thus, the goal is to nevertheless have an approximate UVB background representing the mean UVB to adopt in an optically thin simulation. Therefore, it is important to state upfront that the rates obtained under these sets of assumptions and the ones obtained in a patchy reionization model (e.g., a radiation transfer simulation) can differ significantly during reionization. @Lidz:2007 illustrate this point in a simple way by considering two toy models. In the first case, representing patchy reionization, we imagine equal-sized ionized bubbles each with an interior neutral fraction $x_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm IN}<<1$, filling a fraction $Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}$ of the volume of the IGM, which is otherwise completely neutral. For simplicity, we neglect helium and consider an IGM with a uniform density and temperature. In the second model, representing uniform ionization the neutral fraction is identical at each location within the IGM with ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}=1-Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}$. In each case, photoionization equilibrium tells us that ${\langle \Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}\sim{\langle \alpha{n_{{\ensuremath{\textrm{H}}}}}\rangle}{\langle (1-x_{{\ensuremath{\textrm{H} \, \textsc{i}}}})^{2}/x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}$, where $\alpha$ here is the recombination factor. Now, in a uniformly ionized IGM we get ${\langle \Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}_{\rm uniform}\sim {\langle \alpha{n_{{\ensuremath{\textrm{H}}}}}\rangle} Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}^{2}/(1-Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}})$. In the toy patchy model, we have $(1-x_{{\ensuremath{\textrm{H} \, \textsc{i}}}})^{2}/x_{{\ensuremath{\textrm{H} \, \textsc{i}}}}\sim 1/x_{{\ensuremath{\textrm{H} \, \textsc{i}}}, \rm IN}$ inside each ionized bubble and $\sim 0$ outside of ionized regions. Hence, ${\langle \Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}_{\rm patchy}\sim Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} {\langle \alpha{n_{{\ensuremath{\textrm{H}}}}}\rangle}/x_{{\ensuremath{\textrm{H} \, \textsc{i}}}, \rm IN}$. The ratio of the volume-averaged photoionization rates is just ${\langle \Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}_{\rm patchy}/{\langle \Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}_{\rm uniform}\sim (1-Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}})/Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}x_{{\ensuremath{\textrm{H} \, \textsc{i}}}, \rm IN}$. This will typically be a very large number: for example, if $50$% of the volume is filled by ionized bubbles $Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}=0.5$ each with an interior neutral fraction of $x_{{\ensuremath{\textrm{H} \, \textsc{i}}}, \rm IN}=10^{-4}$, the volume-averaged photoionization rate is a factor of $10^{4}$ times larger in the patchy reionization model than in the uniform model.
This is also relevant to understanding how the ionization history given by the volume filling factor can be compared with the one given by the volume-averaged ionization fractions and the intrinsic differences between the two. The $Q$ formalism only knows about sources and sinks of ionizing photons and does not tell us anything about the value of the neutral fraction in highly ionized regions. In what follows we focus on hydrogen reionization, but analogous considerations also apply to helium. The volume filling factor evolution in UVB synthesis models is computed as $$\frac{dQ_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}{dt}=\frac{\dot{n}_{\rm photon,{\ensuremath{\textrm{H} \, \textsc{ii}}}}}{{\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}}-\frac{Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}{{\langle t_{rec,H} \rangle}}
\label{eq:Q}$$ where $t_{\rm rec,{\ensuremath{\textrm{H}}}}$ is the hydrogen recombination time and $\dot{n}_{\rm photon,{\ensuremath{\textrm{H} \, \textsc{i}}}}$ is the mean number of ionizing photons emitted by all radiation sources available per second[^4]. In the context of these models, $\dot{n}_{\rm photon,i}=\int_{\nu_{L,i}}^{\infty} \frac{\epsilon_{\nu}}{h\nu}d\nu$, is considered to be the number of ionizing photons emitted into the IGM by all radiation sources [HM12; @So:2014] where $\epsilon_{\nu}$ is the total emissivity as a function of frequency obtained by the assumption of the sources (i.e. galaxies and quasars luminosity functions)[^5] This model uses observational constraints of the distribution of absorbers along the line of sight, and it is able to reproduce available measurements of the mean free path at $1$ Ryd and the [Lyman-$\alpha$]{} effective opacity.
To clarify what is happening at these high redshifts we can use the equation of cosmological radiative transfer in its “source function” approximation, which allows to write the following relation between emissivity, radiation intensity and mean free path, $\lambda_{\nu}$ by ignoring photon redshifting effects (minimal at high redshifts): $4\pi J_{\nu}= \lambda_{\nu}\epsilon_{\nu}$ if only local radiation sources contribute to the ionizing background intensity (HM12). From this relation it is now much more easy to see what went wrong in the UVB model. The ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rates, $\Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ given by HM12 are calculated as $$\Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=\int 4\pi \frac{J_{\nu}}{h\nu} \sigma_{\nu,{\ensuremath{\textrm{H} \, \textsc{i}}}} d\nu$$ where $J_{\nu}$ is the radiation intensity. The ionization history, $Q$, was computed using the emissivity alone, using an analytical approximation that does not know anything about the mean free path assumed in the model. On the other hand, the photoionization rates depend on both the emissivity and the mean free path, indicating that the mean free path extrapolation done at high redshift was wrong and yields values that are systematically too high.. To illustrate this, we show in Figure \[fig:mfp\] the mean free path assumed in the model (green line) and observational constraints on the mean free path at $1$ Ryd (black symbols). Notice that at high redshifts $z>5$, there are no available constraints and the model thus corresponds to a blind extrapolation.
![ Evolution of the mean free path, $\lambda_{\rm mfp,\nu}$, at 1 Ryd (solid lines) and 4 Ryd (dot-dashed lines). Green symbols stand for a compilation of observational measurements of the mean free path at 1 Ryd [@FaucherGiguere:2008; @Prochaska:2009; @Songaila:2010; @OMeara:2013; @Fumagalli:2013; @Worseck:2014]. Red lines: fit from [@Worseck:2014] to observations only until $z=5.5$. Orange lines: tabulated values of the mean free path given by HM12.Black lines: $\lambda_{\rm mfp,\nu}$ obtained from the tabulated emissivities $\epsilon_{\nu}$ and intensities $J_{\nu}$ given by HM12 assuming the source function approximation. Blue line: $\lambda_{\rm mfp,\nu}$ obtained from one new reionization model. See text for more details. \[fig:mfp\]](meanfreepath){width="48.00000%"}
The ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoheating rate (energy per time per ion), $\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$, given by these models is calculated as $$\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=\int 4\pi \frac{J_{\nu}}{\nu} (\nu-\nu_{{\ensuremath{\textrm{H} \, \textsc{i}}}}) \sigma_{\nu,{\ensuremath{\textrm{H} \, \textsc{i}}}} d\nu$$ so the photoionization rates are overestimated for the same reason that the photoheating rates are. As noted above, the total heating rate does not depend on the the amplitude of $J_{\nu}$ but only on its shape [@Theuns:2002a; @McQuinn:2009]. However, as we argued above, in the standard UVB semianalytic synthesis modeling approach, $Q$ and $J_{\nu}$ are not required to be internally consistent, so in practice the actual heating rate per volume approaches a constant at early times ($z>z_{\rm reion}$), whereas it should go to zero. Therefore, the total heat produced during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization in the simulation was also applied earlier than when it should be.
We have also tested other widely used UVB models in the literature [@Haardt:1996; @Haardt:2001; @FaucherGiguere:2009 hereafter HM96, HM01, FG09, respectively]. We present the full results for these other models in Appendix \[app:HMold\] showing that they all share a similar problem, i.e., for ${\ensuremath{\textrm{H} \, \textsc{i}}}$ or ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ (or both) reionization the gas heating associated with the reionization event occurs much earlier than one will naively expect from these models. Although it was known that these UVB models do not properly model the reionization process, it is important that this problem has not been directly confronted in the literature [see, however, @Puchwein:2015]. This oversight is likely due to the fact that, for the ionization and thermal histories of the IGM, most comparisons between simulations and observations have been performed in the context of the [Lyman-$\alpha$]{} forest and focused on the redshift range for which such observations are available, i.e. between $2<z<6$, or even lower redshifts. The thermal parameters studied so far in the literature, $T_{0}$ and $\gamma$, depend on the instantaneous values of these rates once we are far enough in time from reionization. So they quickly forget about early heating Therefore, what happens at higher redshifts is not that relevant for these values. However, it is expected that this problem will have more important consequences for correctly modeling the pressure smoothing scale, as it depends on the full thermal history. This is because at IGM densities, the dynamical time that it takes the gas to respond to temperature changes at the Jeans scale (i.e., the sound-crossing time) is the Hubble time [@Gnedin:1998].
To summarize, current UVB models result in different ionization and thermal histories than those quoted by the authors. This is because the low-$z$ mean free path values have been blindly extrapolated to high-$z$ (see Figure \[fig:mfp\]), where they result in a photoionization rate far too high. Motivated by these results, we have developed a new method to build self-consistent effective ionization and thermal histories during reionization epochs in optically thin simulations, which we describe in the next section.
Improved UVB Models {#sec:newmodel}
===================
As stated in the previous section, current photoionization rates widely used in optically thin codes do not properly track the desired ionization histories and lead to incorrect thermal histories for the gas in these simulations, where the gas is heated much earlier than it should be. We present here a new way of creating self-consistent UVB models during reionization events (${\ensuremath{\textrm{H} \, \textsc{i}}}$, ${\ensuremath{\textrm{He} \, \textsc{i}}}$, and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$) to be used in optically thin hydrodynamical simulations. Different groups in the field have modified these tables, both ionization and photoheating rates (especially photoheating, and more in the context of ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization where more observations are available), with different justifications: accounting for different physical effects as nonequilibrium ionization or radiative transfer effects, matching specific observables, or just exploring the parameter space [e.g., @Haehnelt:1998; @Theuns:2002; @Bolton:2005; @Jena:2005; @Wiersma:2009; @Pawlik:2009; @Puchwein:2015; @Lukic:2015]. However, this has generally been done by just applying a multiplying factor to the standard models assumed or by applying different simple cutoffs[^6].
Our approach will be based on building effective values of the photoionization and photoheating rates that can be substituted into the standard optically thin equations to yield the desired results (see FG09 for an early motivation of this approach). The main goal is to make sure that the heating due to reionization in the simulation is consistent with the reionization model itself. To enforce this, we have calculated the volume-averaged values of both the photoionization and photoheating rates that give us the desired ionization and total heat injection for an input reionization model. We give here a global overview of the method and the different assumptions made in our models. We explain in full detail how we derive the photoionization rates in Appendix \[ssec:volave\] and the photoheating rates in Appendix \[ssec:dTdz\]. We will discuss the different caveats and limitations of our model in Section \[sec:discuss\].
Each of our reionization models is defined by one free parameter, the total heat input $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ during reionization, and one free function, the reionization history, which in this context we define as the volumen-averaged ionization fraction evolution, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}(z)$. The reionization of ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ is analogously treated. Using these parameters, we will derive effective photoionization and photoheating rates that can be used in hydrodynamical simulations using the following assumptions:
1. That all species are in ionization equilibrium at all times. This is done to be fully consistent with (most of) the codes that will be using these UVB models but it could be changed in the future.
2. That the gas composition can be approximated as primordial. Therefore, the evolution of the number density of electrons, ${n_{\rm e}}$, is given as ${n_{\rm e}}={n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}+{n_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}}+2{n_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}$.
3. That ${\ensuremath{\textrm{He} \, \textsc{i}}}$ reionization is perfectly coupled with ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization.
4. That ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization is not relevant during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization and vice versa.
5. That the heating due to reionization is perfectly coupled to the reionization process. Therefore, the heating can be written as a function of the total heat injection and the ionization history, i.e., $$\frac{dT_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{dt} \propto \Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \frac{d{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}}{dt}$$
Finally, the new effective rates are only used during reionization, i.e., while ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}<1.0$. Once the reionization redshift, defined as when the input ionization history is one, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}=1$, is reached, we can simply use the photoionization and photoheating rates of common UVB models (in our case HM12).
Let us first focus on how we obtain the new photoionization rates to be applied during reionization. We obtain the new effective photoionization rates by volume averaging the ionization equilibrium equations and using the assumptions enumerated above. In particular, for the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rates we get: $${\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)= C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} {\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}(z)
\alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}}({\langle T \rangle})(1+\chi) \frac{{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}^{2}(z)}{{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)} \\[1.5mm]
\label{eq:invphoto}$$ where $C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}$ is a volumen-averaged correction factor linked with the well known clumping factor and $\alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}}({\langle T \rangle})$ is the recombination coefficient for which a specific volumen-averaged temperature of the IGM, ${\langle T \rangle}$, has to be assumed. We refer the reader to Appendix \[ssec:volave\] for all the details[^7].
In addition to affecting abundances, photoionization injects energy into the gas when a high-energy ($h\nu> h\nu_{\rm T}$) photon transfers more energy to an electron than what is necessary to unbind it from the atom. In order to include this heat transfer to the IGM by the UVB, models also provide photoheating rates that are included in simulations as an effective energy source. The temperature of the IGM is much more affected by this photoheating and different atomic cooling processes than by the gravitational collapse of cosmic structure. These processes are accounted for as a global heating and cooling term that is added to the equation of energy for the gas: $$\begin{aligned}
& \frac{\partial E}{\partial t} =
- \frac{1}{a}\vec{v}\cdot \nabla E
-\frac{\dot{a}}{a}(3\frac{p}{\rho}+\vec{v}^{2})\\
& \qquad -\frac{1}{\rho a}\nabla\cdot (p\vec{v})
+\frac{1}{a}\vec{v}\cdot\nabla\Phi
+\frac{\Lambda_{\rm HC}}{a \rho}\\
\end{aligned}
\label{eq:energy}$$ where $\Lambda_{\rm HC}$ represents the combined heating and cooling terms,[^8] which can be expanded as $$\begin{aligned}
& \Lambda_{\rm HC}= a^{4} {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}+ a^{4} {n_{{\ensuremath{\textrm{He} \, \textsc{i}}}}}\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{i}}}} + a^{4} {n_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}}\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}\\
& -a^{4}{n_{\rm e}}f_{\rm c}({n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}},{n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}},{n_{{\ensuremath{\textrm{He} \, \textsc{i}}}}},...)-a^{4}{n_{\rm e}}f_{\rm Compton}(T,T_{\rm CMB})+...\\
\end{aligned}
\label{eq:interenergy}$$ where the first three terms represent the photoheating rates of ${\ensuremath{\textrm{H} \, \textsc{i}}}$, ${\ensuremath{\textrm{He} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ respectively[^9]. We have also included an atomic cooling term ($f_{\rm c}$) and an inverse-Compton scattering term ($f_{\rm Compton}$).
We can obtain effective photoheating rates for the reionization heating by volume-averaging the relation between the heat per unit of time produced by a certain reionization model, $d\Delta T/dt$, and the one produced by a photoheating rate. This will allow the use of our new models in standard hydrodynamical codes. For the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoheating rate, and using the set of assumptions enumerated above, we obtain $$\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=C_{\dot{q},{\ensuremath{\textrm{H} \, \textsc{i}}}}\frac{ 3 k_{B}}{2 {\langle \mu \rangle} X_{\rm p} {\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)} \frac{d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{dt} \label{eq:qdot}$$ where $X_{\rm p}$ is the hydrogen mass abundance, ${\langle \mu \rangle}$ is the volumen-averaged molecular weight, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)$ the assumed reionization history of the model, $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ is its total heat input, and $C_{\dot{q},{\ensuremath{\textrm{H} \, \textsc{i}}}}$ is a volumen-averaged correction factor that we set to one at all redshifts. We refer the reader to Appendix \[ssec:dTdz\] for more details on how this equation was obtained.
{width="39.00000%"} {width="60.00000%"}
Figure \[fig:dTdz\] shows an example of how a UVB model is built. In the left panel we show a model that should match a specific late ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization history that assumes a reionization redshift of $z_{\rm rei}=6.55$ (late reionization model; see more details in Section \[sec:reionmodels\]) with a total heat input due to ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization of $\Delta
T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=2\times10^{4}$ K. The upper left panel shows the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ ionization history assumed to build the model. The middle panel shows the assumed temperature evolution of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization event. As explained above, we have assumed that the temperature evolution follows the ionization history. The lower left panel shows the actual instantaneous heat input, $d \Delta
T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}/dz$, derived using eqn. (\[eq:dTdz\]) for this model. The integral of this line gives the total heat input, $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$.
The upper right panel gives the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rate (blue line) obtained using eqn. (\[eq:invphoto\]) and the method described above. This rate is compared with the equivalent of the HM12 model (green line). We also plot observational constraints on the photoionization rates from different studies [@Bolton:2007; @FaucherGiguere:2008; @Wyithe:2011; @Calverley:2011; @Becker:2013]. Notice that all these constraints are below $z=6$. The middle right panel shows the photoheating rates derived using eqn. (\[eq:qdot\]) for the ${\ensuremath{\textrm{H} \, \textsc{i}}}$. Notice that for the new model, below its reionization redshift ($z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}=6.55$) we apply the same photoionization and photoheating rates used in HM12 as we think that they are reasonable after reionization. The rates of our model raise abruptly at around the reionization redshift because in the input ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization model the transition from ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}\sim0.1$ to ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}\sim10^{-4}$ is very fast. The lower right panel shows the reionization function input for the model (solid blue line; defined to be $1.0$ at $z=z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}$). It also shows the volumen-averaged ${\ensuremath{\textrm{H} \, \textsc{i}}}$ ionization fraction obtained running a cosmological hydrodynamical simulation that uses this new photoionization rate (dashed blue line). The solid black line shows the reionization evolution of the HM12 model as calculated by their authors, while the dashed green line stands for the outcome of a hydrodynamical simulation using the HM12 photoionization rate. Current best observational constraints on the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ fraction at different redshifts are also plotted using different symbols and errorbars [@Bolton:2005; @Fan:2006; @Bolton:2007; @McGreer:2015]. Notice again that all available constraints are below $z=6$ so that both models are in agreement with these observations.
We can derive the evolution of the mean free path at 1 Rydberg for the new model, making some simple assumptions, and compare it with the ones obtained by HM12. Our modeling has no shape information on the intensity, $J_{\nu}$; however, one can assume the HM12 shape and that all intensities differ by the same constant: $J_{\rm \nu,new}=C\times J_{\rm \nu,HM12}$. With this we can approximate the new radiation intensity at 1 Ryd as the ratio between the two photoionization rates $J_{\rm 912,new}=(\Gamma_{\rm {\ensuremath{\textrm{H} \, \textsc{i}}},new}/\Gamma_{\rm {\ensuremath{\textrm{H} \, \textsc{i}}},HM12})\times J_{\rm 912,HM12}$. Then we obtain the mean free path using the source function approximation that has the same emissivity assumed by HM12: $\lambda_{\rm mfp,912,new}=4\pi J_{\rm 912,new}/\epsilon_{\rm 912,HM12}$. Figure \[fig:mfp\] shows the evolution of the mean free path derived for this new model (blue line). The green solid line stands for the mean free path used in the HM12 model. This plot illustrates how the volumen-averaged mean free path drops significantly above the reionization redshift.
Reionization Models {#sec:reionmodels}
===================
We will now discuss the different reionization models that we want to simulate by using our new approach to compute photoionization and photoheating during reionization. In order to define a reionization model, we need to set the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization history via the volumen-averaged ionization fractions[^10].
First, we need to define the shape of our reionization histories. We use the lower incomplete gamma function, $g$: $${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}=\begin{cases}
0.5+0.5\times g(1/n_{1},|z-z_0|^{n_{1}}), & z <= z_0 \\
0.5-0.5\times g(1/n_{2},|z-z_0|^{n_{2}}), & z > z_0 \\
\end{cases}
\label{eq:Qana}$$ where $n_{1}=50$, $n_{2}=1$ and $z_0$ is a free parameter that sets the redshift where $x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}(z_0)=0.5$. This function defines a slower start for the reionization function but a fast finish. This specific shape was motivated by radiative transfer simulation results [@Ahn:2012; @Park:2013; @Pawlik:2015] which seem to favor rapid and sudden ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization histories in $\sim 20^3$ (Mpc/h)$^3$ cosmological volumes. In particular, $n_{1}$ and $n_{2}$ were fitted to mimic @Pawlik:2015 results. In this work we use only this shape and experiment with the redshift of reionization, but our function could be easily modified in the future to explore a wider range of models. We explore the relevant range of reionization parameters taking into account the CMB constraints on the integrated electron scattering optical depth, ${\tau_{\rm e}}=0.078\pm 0.019$ [@Planck:2015][^11]. We consider three models: an early, middle, and late ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization history, which have reionization redshifts[^12] of $z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}=9.70$, $8.30$, and $6.55$, respectively, all of which give ${\tau_{\rm e}}$ are within $1\sigma$ of the CMB measurements. These ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization models are plotted as solid lines in the upper panel of Figure \[fig:Qsims\] and their associated ${\tau_{\rm e}}$ values are shown in the lower panel.
For ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization we consider a history based on FG09 (last column of their Table $2$) results which sets the HeII reionization redshift to finish at $z_{\rm reion,{\ensuremath{\textrm{He} \, \textsc{ii}}}}=3.0$ (hereafter He\_A). The specific evolution of the full ionized fraction can be well described by ${\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}=1.0-arctan(z-z_{\rm reion})$. This is in agreement with standard models of ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization, which, as a result of the high energy requirement to double ionize helium ($E_{\nu}>54.4$ eV), assume that quasars must be the main drivers of this process [e.g., @MiraldaEscude:2000; @Compostella:2014; @Worseck:2015]. This ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization model is plotted as a dot-dashed line in the upper panel of Figure \[fig:Qsims\].
Total Heat Input due to ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ Reionization
------------------------------------------------------------------------------------------------------------------------------
The other two parameters that define the reionization events in our models are the total heat input that happens during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionizations. There have been several efforts to calculate the specific cumulative energy increase or total heat input ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$,$\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$) due to reionization events. Approximate analytical estimates of this energy can be made given some specific assumptions for the intrinsic spectral slopes of the sources responsible for reionization, and in general neglecting redshifting effects [@Efstathiou:1992; @MiraldaEscude:1994]. One-dimensional radiative transfer codes have also been used to obtain a better understanding of the radiative transfer effects on the energy input into the IGM due to the reionization process [e.g. HM96; @Abel:1999]. In recent years better radiative transfer codes have been used to study this problem, and current simple analytic models are motivated by by these radiative transfer calculations [@Tittley:2007; @McQuinn:2009; @McQuinn:2012]. Most efforts have focused on ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization [@Bolton:2004; @McQuinn:2009], but the same set of assumptions have also been applied to ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization [@Bolton:2009]. The range of values discussed in these works for ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization center around $\sim2\times10^{4}$ K with up to a factor of two or three difference depending on the exact assumptions. For ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization typical values have been around $1.5\times 10^{4}$ K with a similar range of uncertainty. In the context of their calculation of self-consistent UVB synthesis models, FG09 computed the total heat input of their model ($\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=14269$ K) and its evolution (i.e. $dT/dz$), finding good agreement with the heat input determined from detailed radiative transfer simulations [@McQuinn:2009].
We have treated the total heat input of ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization as free parameters in our models and have chosen values based on the aforementioned literature. In particular, for our default models we will use the standard values assumed for both ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization: $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=2\times 10^{4}$ K and $\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=1.5\times 10^{4}$ K. We also consider other models with a range of heat input for both ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=1.5\times 10^{4}-4\times 10^{4}$ K) and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=1\times 10^{4}-3\times 10^{4}$ K), respectively.
Simulations {#sec:code}
===========
![Reionization history obtained in simulations using different UVB models compared with the input models. Upper panel: the ${\ensuremath{\textrm{H} \, \textsc{ii}}}$ and ${\ensuremath{\textrm{He} \, \textsc{iii}}}$ volume filling factor evolution calculated in the simulations (dashed and dotted lines, respectively) compared with the input models (full and dot-dashed lines respectively). Lower panel: integrated electron scattering optical depth, ${\tau_{\rm e}}$ computed from the above volume filling factors. Dashed lines show the results from the simulations, and solid lines show those for the input models. Results from a simulation using the HM12 model are also shown for direct comparison (green lines, see also Figure \[fig:Qhistgas0\]). The gray band stands for the last constraints on ${\tau_{\rm e}}$ coming from @Planck:2015 data. \[fig:Qsims\]](Qtau_hist2){width="45.00000%"}
The simulations used in this work were performed with the Nyx code [@Almgren:2013]. Nyx follows the evolution of dark matter simulated as self-gravitating Lagrangian particles, and baryons modeled as an ideal gas on a uniform Cartesian grid. The Eulerian gas dynamics equations are solved using a second-order-accurate piecewise parabolic method (PPM) to accurately capture shock waves. We do not make use of adaptive mesh refinement (AMR) capabilities of Nyx in the current work, as the [Lyman-$\alpha$]{} forest signal spans nearly the entire simulation domain rather than isolated concentrations of matter, where AMR is more effective. For more details of these numerical methods and scaling behavior tests, see @Almgren:2013.
Besides solving for gravity and the Euler equations, we also include the main physical processes fundamental to modeling the [Lyman-$\alpha$]{} forest. First, we consider the chemistry of the gas as having a primordial composition with hydrogen and helium mass abundances of $X_{\rm p}$, and $Y_{\rm p}$, respectively. In addition, we include inverse-Compton cooling off the microwave background and keep track of the net loss of thermal energy resulting from atomic collisional processes. We used the updated recombination, collision ionization, dielectric recombination rates, and cooling rates given in @Lukic:2015. All cells are assumed to be optically thin, and radiative feedback is accounted for via a spatially uniform but time-varying UVB radiation field given to the code as a list of photoionization and photoheating rates that vary with redshift following the method described in Section \[sec:newmodel\].
[lccccc]{} Sim & ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}(z)$ & $z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & ${\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}(z)$ & $\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$\
& & & (K) & & (K)\
HM12 & HM12 & ... & ... & ... & ...\
FG09 & FG09 & ... & ... & ... & ...\
HM01 & HM01 & ... & ... & ... & ...\
HM96 & HM96 & ... & ... & ... & ...\
LateR & Late reionization & $6.55$ & $2\times10^{4}$ & He\_A & $1.5\times10^{4}$\
MiddleR & Middle reionization & $8.30$ & $2\times10^{4}$ & He\_A & $1.5\times10^{4}$\
EarlyR & Early reionization & $9.70$ & $2\times10^{4}$ & He\_A & $1.5\times10^{4}$\
MiddleR-Hcold & Middle reionization & $8.30$ & $1.5\times10^{4}$& He\_A & $1.5\times10^{4}$\
MiddleR-Hwarm & Middle reionization & $8.30$ & $3\times10^{4}$ & He\_A & $1.5\times10^{4}$\
MiddleR-Hhot & Middle reionization & $8.30$ & $4\times10^{4}$ & He\_A & $1.5\times10^{4}$\
MiddleR-noHe & Middle reionization & $8.30$ & $2\times10^{4}$ & None & ...\
MiddleR-Hecold & Middle reionization & $8.30$ & $2\times10^{4}$ & He\_A & $1\times10^{4}$\
MiddleR-Hewarm & Middle reionization & $8.30$ & $2\times10^{4}$ & He\_A & $2\times10^{4}$\
MiddleR-Hehot & Middle reionization & $8.30$ & $2\times10^{4}$ & He\_A & $3\times10^{4}$\
In order to generate the initial conditions, we have used the <span style="font-variant:small-caps;">music</span> code [@Hahn:2011] and a <span style="font-variant:small-caps;">camb</span> [@Lewis:2000; @Howlett:2012] transfer function. All simulations started at $z_{\rm ini}=159$ to be sure that nonlinear evolution is not compromised [see, e.g., @Onorbe:2014 for a detailed discussion on this issue]. Unless otherwise stated, all the simulations discussed in this paper assumed a $\Lambda$CDM cosmology with the following fundamental parameters: $\Omega_{\rm m}=0.3192$, $\Omega_{\Lambda}=0.6808$, $\Omega_{\rm b}=0.04964$, $h=0.67038$, $\sigma_{8}=0.826$ and $n_{\rm s}=0.9655$. These values are within $1\sigma$ agreement with last cosmological constraints from the CMB [@Planck:2015]. The choice of hydrogen and helium mass abundances ($X_{\rm p}=0.76$ and $Y_{\rm p}=0.24$, therefore $\chi=0.0789$) is in agreement with the recent CMB observations and Big Bang nucleosynthesis [@Coc:2013]. Simulations were run down to $z=0.2$, saving 32 snapshots[^13] from $z=20$. Unless otherwise stated, all simulations presented here have a box size of length $L_{\rm box}=20$ ${\rm Mpc}/h$ and $1024^{3}$ resolution elements. This dynamical range guarantees that all the different physical parameters analyzed in this paper are converged with enough accuracy ($<5\%$ errors). We will discuss this issue in more detail in Section \[ssec:convergence\].
We first run one simulation using photoionization and photoheating values from the most widely used models (HM96, HM01, FG09 and HM12). We have already presented the reionization and thermal histories of the HM12 model in Section \[sec:typmodels\] but below we will further explore other properties of this simulation.[^14] We also run the three ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization histories presented above, an early, middle, and late reionization model (EarlyR, MiddleR and LateR; see above and Figure \[fig:Qsims\]). All of them share the same heat input during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, $\Delta T_{HI}=2\times10^4$ K, and ${\ensuremath{\textrm{He} \, \textsc{iii}}}$ reionization model: ${\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}(z)$ and $\Delta T_{HeIII}=1.5\times10^{4}$ K.
In order to study the effect of different total heat input during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$, we run three more simulations that share all parameters with the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ middle reionization run, MiddleR, but varying this parameter: MiddleR-Hcold ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=1.5\times10^{4}$ K), MiddleR-Hwarm ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=3\times10^{4}$ K) and MiddleR-Hhot ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=4\times10^{4}$ K). We also explored the effects of different global net heating during ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization by running a set of four more ${\ensuremath{\textrm{H} \, \textsc{i}}}$ middle reionization simulations (MiddleR, $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=2\times10^{4}$) in which we just changed the heat input during ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization, $\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$: MiddleR-noHe (no ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization), MiddleR-Hecold ($\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=1\times10^{4}$ K), MiddleR-Hewarm ($\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=2\times10^{4}$ K) and MiddleR-Hehot ($\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=3\times10^{4}$ K). A summary of all the relevant parameters used in the runs presented in this work is shown in Table \[tab:sims\] along with the naming conventions we have adopted.
Analysis of the Simulations {#ssec:samplesum}
---------------------------
Whenever [Lyman-$\alpha$]{} forest spectra are created from the simulation, we compute the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ optical depth at a fixed redshift, which can then be easily converted into a transmitted flux fraction, $F=e^{-\tau}$. That is, we do not account for the speed of light when we cast rays in the simulation; we use the gas state at a single cosmic time. The simulated spectra are not meant to look like full [Lyman-$\alpha$]{} forest spectra, but just recover the statistics of the flux in a small redshift window. Our calculation of the spectra accounts for Doppler shifts due to bulk flows of the gas, as well as for thermal broadening of the [Lyman-$\alpha$]{} line. We refer to @Lukic:2015 for specific details of these calculations. This procedure results in the [Lyman-$\alpha$]{} flux as a function of wavelength or equivalently time or distance. Following the standard approach, we then rescale the UV background intensity so that the mean flux of all the extracted spectra from the simulation matches the observed mean flux at the respective redshift (see Section \[sec:meanflux\] for more details on the specific value that we have chosen). We therefore have neglected noise and metal contamination in our skewers so far, but this will not be relevant in this paper.
Using these skewers we have also calculated the curvature flux statistics, ${\langle|\kappa|\rangle}$, where, $$\kappa=\frac{F''}{[1+(F')^{2}]^{3/2}}$$ $F'$ is the first derivative of the flux with respect to the velocity separation between pixels, and $F''$ is the second derivative. We have done this for each simulation following the method described in @Becker:2011[^15].
We measured the thermal parameters of the simulation at each snapshot by fitting the $\rho_{\rm b}-T$ relation with linear least squares in $\log_{10} \Delta$ and $\log_{10} T$, fitting the range $-0.7< \log_{10} \Delta < 0.0$ and $\log_{10} T/K< 4.5$ [^16].
To characterize the gas pressure support in all our simulations, we have followed the recent work by @Kulkarni:2015 and use the real-space [Lyman-$\alpha$]{} flux, $F_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm real}$. This quantity is defined as $F_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm real}=\exp(-\tau_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm real})$, where $\tau_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm real}$ is the real-space [Lyman-$\alpha$]{} optical depth which is identical to the observed [Lyman-$\alpha$]{} optical depth except that the convolution integral that accounts for the redshift-space effects of the peculiar velocity field and thermal line broadening has not been included. This field naturally suppresses dense gas, and is thus robust against the poorly understood physics of galaxy formation, revealing pressure smoothing in the diffuse IGM. The $F_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm real}$ 3D power spectrum is accurately described by a simple fitting function with a gaussian cutoff at ${\lambda_{\rm P}}$, which is then defined as the pressure smoothing scale. This statistic has the added advantage that it directly relates to observations of correlated [Lyman-$\alpha$]{} forest absorption in close quasar pairs, proposed as a method to measure this scale, and enables one to quantify it in simulations [@Rorai:2013; @Rorai:2015].
The Ionization and Thermal History of the IGM {#sec:results}
=============================================
We now present thermal properties of the IGM in LateR, MiddleR, and EarlyR simulations, which only differ in their redshift of ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization. We first focus on the the evolution with redshift of the temperature at mean density, $T_{0}$, and the slope of the temperature-density relation, $\gamma$. The left panel of Figure \[fig:Thistgas1\] shows the evolution of these parameters for the HM12 run (green line). It also shows the thermal history of the LateR (blue), MiddleR (magenta) and EarlyR (orange) simulations. In the upper left panel we plot the evolution of $\gamma$, which exhibits the expected convergence to a value close to $\sim 1.6$ after all reionization events for all models, resulting from the balance of photoheating with adiabatic cooling[@Hui:1997; @McQuinn:2016]. The larger decrease of $\gamma$ during ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization in the EarlyR, MiddleR, and LateR runs seems to indicate a temperature increase more independent of density than in the HM12 run. @Puchwein:2015 have also shown that, for a fixed UVB model, using a nonequilibrium approach will tend to create a more pronounced feature (we will further discuss this in Section \[sec:discuss\]). In our modeling we use equilibrium photoionization; hence, the flattening occurs for different reasons. In fact, this is just because our ionization model injects more heat to the IGM than in the HM12 model. One expects this type of effect during reionization when applying a uniform UVB model through the whole volume, as in that case we are applying a constant temperature increase at each resolution element. At higher initial temperature this corresponds to a lower increase in the logarithm of the temperature, so that the temperature-density relation flattens in log-log space. This effect gets magnified as we increase the amount of heat applied to the whole box. Therefore, our results show that the exact evolution of $\gamma$ in simulations depends on the assumed shape for the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization histories and their total heat input. In this panel we also plot the value of $\gamma$ of @Bolton:2014 at $z=2.4$, derived from absorption-line profiles in the [Lyman-$\alpha$]{} forest. [^17]
{width="48.00000%"} {width="48.00000%"}
The lower left panel of Figure \[fig:Thistgas1\] shows the evolution of $T_{0}$ for the same set of simulations. As expected, in the new models ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization produces a much later heating than in the HM12 run. In fact, it can be clearly seen that the temperature at mean density of the new runs rises following the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization of each model. The heating during ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization also shows significant differences with the HM12 run. Although the heating happens at basically the same time as in the HM12 run, the new models also exhibit a larger and sharper temperature increase at lower redshifts ($3\lesssim z\lesssim4$), along with the expected decrease of the $\gamma$ value. This is due to the different ionization history assumed for ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ which rises steeply at these redshifts. We compare these models to the measurements of the temperature at mean density, $T_{0}$, from @Lidz:2010, based on the wavelet technique [@Meiksin:2000; @Theuns:2000]. We also plot constraints obtained by combining the $\gamma$ measurement by @Bolton:2014 plotted in the upper panel with measurements of the temperature at the optimal density, $T({\Delta_{\bigstar}})$, by @Becker:2011 and @Boera:2014 derived from the observed curvature of the [Lyman-$\alpha$]{} forest transmitted flux. In this case we propagated errors from both measurements. Since observations of the [Lyman-$\alpha$]{} forest are only available at $z\lesssim6$, Figure \[fig:Thistgas1\] illustrates that it will be challenging to constrain ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization from measurements of $T_{0}$ and $\gamma$ at these redshifts because the IGM quickly loses thermal memory of HI reionization. Our new UVB models provide, by construction, a sharper temperature rise due to ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization than the HM12 run. This again illustrates that the exact evolution of the IGM thermal parameters depends on the the assumed shape for the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization histories, as well as their total heat input.
In the upper right panel of Figure \[fig:Thistgas1\] we plot another interesting property defining the thermal state of the IGM, which is the temperature at the optimal overdensity, $T({\Delta_{\bigstar}})$. This “optimal” overdensity at each redshift is defined as the one for which curvature measurements of the [Lyman-$\alpha$]{} forest are more sensitive [see @Becker:2011; @Boera:2014]. The curvature measurements allow one to determine this parameter because they are not able to break the degeneracy between $T_{0}$ and $\gamma$. To calculate the $T({\Delta_{\bigstar}})$ of our simulations, we have used the function fit to these optimal densities by @Becker:2011 from a suite of hydrodynamical simulations ($\log_{10} {\Delta_{\bigstar}}=A\times z + B$ where $A=-0.24596$ and $B=1.22218$)[^18]. Inspecting Figure \[fig:Thistgas1\], we see that all simulations give the same temperature at the optimal density at $z<6$, which is not surprising as we have already shown that at these redshifts all of them have roughly the same temperature-density relation (see the left panel of Figure \[fig:Thistgas1\]). We also see the more pronounced rise in temperature due to ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization in the new models compared with HM12 between $3\lesssim z\lesssim4$.
The middle right panel of Figure \[fig:Thistgas1\] shows the curvature flux statistics, ${\langle|\kappa|\rangle}$, for all these simulations, and it is clear that they do not match at these redshifts. This is because, as explained above, the flux statistics depends not only on the temperature density relation of the IGM ($\gamma$, $T_{0}$), but also on the pressure smoothing scale of the IGM, ${\lambda_{\rm P}}$. In fact, the lower right panel of Figure \[fig:Thistgas1\] shows the evolution of the pressure smoothing scale, ${\lambda_{\rm P}}$, with redshift for all the simulations. This panel clearly illustrates the dependence of the pressure smoothing scale on the full thermal history of the universe and not just on the instantaneous temperatures. That is, whereas the temperatures of all models agree at $z<7$, differences in the pressure smoothing scale persist to much lower redshifts. This explains the differences between the curvature statistics for all the simulations and indicates that for fixed values of $T_{0}$ and $\gamma$, the value of the optimal density, ${\Delta_{\bigstar}}$, will be degenerate with the pressure smoothing scale, ${\lambda_{\rm P}}$.
We have confirmed this by recreating the same study done by @Becker:2011 to obtain the optimal densities using hydrodynamical simulations. With a similar set of simulations to that of these authors, we found almost identical results for the values of the optimal densities[^19]. However, by including simulations that have identical values of $T_0$ and $\gamma$ but different values of ${\lambda_{\rm P}}$, complicates the simple unique definition of ${\Delta_{\bigstar}}$ by @Becker:2011, and instead adds scatter to the relationship between curvature and $T({\Delta_{\bigstar}})$. The upper panel of Figure \[fig:Thistgas1\] also shows the determinations of the temperature at the optimal density using the curvature of the [Lyman-$\alpha$]{} forest transmitted flux [@Becker:2011; @Boera:2014] compared with our simulations. Given the strong dependence of the curvature on the pressure smoothing scale ${\lambda_{\rm P}}$ resulting from the different reionization histories, it is clear that the error bars on $T({\Delta_{\bigstar}})$ are likely underestimated. These issues pertaining to the thermal history are discussed in @Becker:2011 [see also @Puchwein:2015] but were not included in the error budget. The difference in $\log {\langle|\kappa|\rangle}$ between our late reionization model (LateR) and early reionization model (EarlyR) at $z\sim4$ is $\sim0.05$. From the results presented by @Becker:2011 [Figure 1 and 10] this difference implies already $20-25\%$ error in temperature.
The aforementioned issues related to the pressure smoothing scale and thermal history can ease the $2\sigma$ level of disagreement between @Lidz:2010 measurements of $T_{0}$ and the @Becker:2011 measurements of $T({\Delta_{\bigstar}})$ at $z > 4$ (Figure \[fig:Thistgas1\]) although this does not seem to be enough to explain it fully. At lower redshifts a comparison of the two measurements is more challenging because it depends on what one assumes for the the temperature density relation slope, $\gamma$. Based on our results, it is clear that these conflicting measurements lead to different interpretations of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization events. On the one hand, the @Becker:2011 results point toward a temperature of the IGM at mean density of $T_{0}\sim 1\times 10^{4}$ K by $z\sim 4.7$, and a clear heating event later at $z\sim3$, which they associated with ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization. On the other hand, @Lidz:2010 higher measured temperatures require a higher energy injection ($\Delta_{T}$) than we assumed in our models ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=2\times10^{4}$ K) and an earlier injection of heat that could be associated with ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization at higher redshift than inferred by @Becker:2011 and @Boera:2014. Based on their measurements, @Lidz:2010 claim that the ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization event should be completed by $z=3.4$ and that the temperature at lower redshifts is consistent with the fall-off expected from adiabatic cooling. Although one can argue about the statistical significance of these discrepancies, especially given that the error bars are underestimated because neither study marginalized out the pressure smoothing scale ${\lambda_{\rm P}}$, we see no reason to prefer one set of measurements over the other. For this reason we will not attempt any further interpretation of these measurements with our numerical simulations, and we defer detailed data-to-model comparisons to future work.
We now want to discuss the evolution of the pressure smoothing scale in the different simulations, which, as can clearly be seen from Figure \[fig:Thistgas1\], retains memory of the reionization events. As we mentioned in the introduction, this is because, at IGM densities, the dynamical time that it takes the gas to respond to temperature changes at the Jeans scale (i.e., the sound-crossing time) is the Hubble time. The first thing to notice is that the HM12 model results in a much larger pressure smoothing scale than that of any our models, even the early reionization one. This is a direct result of the premature reionization of HI and the spurious associated heating (starting at $z\sim15$) produced by this model. Our new models correct this issue, properly tying reionization heating to reionization history, resulting in later heating and a smaller overall pressure scale. In addition, significant differences are also found below $z=6$ among the simulations with different ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization histories (LateR, MiddleR and EarlyR) even though these simulations share exactly the same photoionization and photoheating values at these redshifts and therefore have very similar thermal parameters, i.e. $T_{0}$, $\gamma$, $T(\Delta)$. These differences in ${\lambda_{\rm P}}$ arise because the IGM has had more time to respond to its hotter temperature when reionization occurs earlier, resulting in a larger pressure scale. The sensitivity of the pressure smoothing scale to the reionization history highlights the importance of constructing self-consistent models of reionization and applying them to optically thin simulations to better understand the thermal evolution of the IGM.
It is interesting to discuss the redshift evolution that we find for the pressure smoothing scale, using ${\lambda_{\rm P}}$. This parameter can be seen as the gas pressure scale at the density most sensitive for [Lyman-$\alpha$]{} observations. Therefore, there are two physical effects that contribute to the value of this parameter. First, the IGM is heated as the universe evolves, so we expect the pressure smoothing scale to increase with time. On the other hand, there is not just one pressure smoothing scale in the IGM, but one for each density and, just from linear theory, we expect it to be higher at lower densities, ${\lambda_{\rm P}}\propto {n_{{\ensuremath{\textrm{H}}}}}^{-1/2}$ [@Schaye:2001]. As we go to lower redshifts, the neutral hydrogen density is being further diluted by the expansion of the universe, and observations start to be more sensitive to higher densities that have a smaller pressure smoothing scale. The combination of both processes can produce the somewhat surprising behavior of flattening of ${\lambda_{\rm P}}$ at $z\lesssim4$ (decrease in the case of the HM12 model due to an ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization with a lower heat injection). We also found that since all these models share the same UVB after reionization, they tend to converge to the same ${\lambda_{\rm P}}$ values at lower redshifts. Although the pressure smoothing scale of the IGM depends on the full thermal history, the thermal memory of past reionization events eventually fades as the gas evolves toward lower redshifts.
Heating during Hydrogen Reionization {#ssec:hehy}
------------------------------------
{width="48.00000%"} {width="48.00000%"}
Figure \[fig:hehymodels\] shows the results from simulations (MiddleR-Hcold, MiddleR, MiddleR-Hwarm and MiddleR-Hhot) using different total input heat during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, i.e., different $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ ($1\times 10^{4}$, $2\times 10^{4}$, $3\times 10^{4}$ and $4\times 10^{4}$ K, respectively). All these simulations share the same ${\ensuremath{\textrm{H} \, \textsc{i}}}$ ionization history (middle reionization, $z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}=8.3$) and exactly the same ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ ionization history and heating. The left panel of Figure \[fig:hehymodels\] shows the evolution of $\gamma$ and $T_{0}$ for these simulations. As expected, they differ significantly during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, due to the different heat input applied in them. At lower redshifts, $z<z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}$, all these simulations share the same HM12 photoionization and photoheating rates; hence, eventually $T_{0}$ and $\gamma$ thermal parameters tend to converge to the same values. This shows again that these thermal parameters depend more strongly on the instantaneous value of these rates. However, it is very interesting to remark that this convergence is not immediate, but that it takes some time for each simulation to converge after reaching the redshift at which they all have exactly the same rates [@McQuinn:2016]. In any case, our simulations show that $\gamma$ and $T_{0}$ have little sensitivity to the details of ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization after $z<5$.
In the right panels of Figure \[fig:hehymodels\] we show the evolution of the temperature at optimal density ($T({\Delta_{\bigstar}})$, upper panel), the curvature (${\langle|\kappa|\rangle}$, middle panel), and the pressure smoothing scale (${\lambda_{\rm P}}$, lower panel). As expected, $T({\Delta_{\bigstar}})$, follows the same trend as $T_{0}$ and $\gamma$ (see discussion above). However, the curvature statistics and the pressure smoothing scale for these simulations clearly show a different behavior at redshifts above $z\gtrsim3$, while $T_{0}$, $\gamma$ and $T({\Delta_{\bigstar}})$ have already forgotten reionization at $z\sim 5$. Simulations with a higher heat input at high redshift show higher pressure smoothing scale values even at lower redshift, due to the dependence of this parameter on the full thermal history. Comparing this result with Figure \[fig:Thistgas1\] we can see that the ionization history and the total heat input are degenerate in terms of the pressure smoothing scale. That is, the earlier that ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization injects heat into the IGM, the larger the gas pressure scale, $\lambda_{P}$. However, a later but hotter (larger heat input) ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization also results in a larger pressure smoothing scale. This cautions one about the interpretation of curvature-based measurements of the IGM temperature [@Becker:2011] at $z>3.5$, since the middle right panel of Figure \[fig:hehymodels\] clearly illustrates that the curvature has a strong dependence on thermal history, even when the instantaneous temperature is the same in all models.
Heating during Helium Reionization {#ssec:hehe}
----------------------------------
{width="48.00000%"} {width="48.00000%"}
Finally, we discuss simulations for which we modified only the total heat input during ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization. These are MiddleR-noHe, MiddleR-Hecold, MiddleR, MiddleR-Hewarm, MiddleR-Hehot, and the specific values of $\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$ used in them were $0$ (no ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization), $1\times 10^{4}$, $1.5\times 10^{4}$, $2\times
10^{4}$, and $3\times 10^{4}$ K, respectively. Apart from this, these simulations share exactly the same ${\ensuremath{\textrm{He} \, \textsc{i}}}$ ionization history and also the same ${\ensuremath{\textrm{H} \, \textsc{i}}}$ ionization and photoheating rates (middle reionization, $z_{\rm reion,{\ensuremath{\textrm{H} \, \textsc{i}}}}=8.30$). Therefore, it is not surprising that the evolution of $\gamma$, $T_{0}$, and $T_{{\Delta_{\bigstar}}}$ thermal parameters at redshift above $z\gtrsim 5$ is exactly the same for all simulations (see upper left, lower left, and upper right panels of Figure \[fig:hehemodels\]). It is only when ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization starts heating the IGM that these simulations differ. In particular, runs with a higher total heat input result in a much steeper rise when ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization commences in $T_{0}$, accompanied by a commensurate fall when ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization is completed. With the slope of the temperature-density relation, $\gamma$, the effect is the opposite. The curvature statistic and the pressure smoothing scale also illustrate the effect of the different heating during ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization. Simulations with a larger late heat input due to ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization give rise to a larger pressure smoothing scale. Notice that the pressure smoothing scales of these models begin to diverge at $z<3$, once ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization has already completed, as there is a delay before the effect propagates. This delay is due to the dynamical time that it takes the gas to respond to temperature changes at the Jeans scale (i.e., the sound crossing time), which, as discussed above for IGM densities, is close to the Hubble time.
Calibrating the UVB to Yield the Correct Mean Flux {#sec:meanflux}
==================================================
The simplest possible [Lyman-$\alpha$]{} flux statistic is the mean transmitted flux ${\langle F_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}$, or equivalently, the effective optical depth $\tau_{{\ensuremath{\textrm{H} \, \textsc{i}}}}= − \log {\langle F_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}$. It is commonly the case that simulations do not recover the observed mean flux, but that simulated fluxes are rescaled to match the observed mean. This rescaling is often understood as equivalent to adjusting the specific intensity of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rate used in the simulation and is justified based on how poorly constrained the ionizing background is. Notice, however, that this rescaling is generally done directly in redshift space. @Lukic:2015 conducted a detailed study of the effect that this rescaling can have on different [Lyman-$\alpha$]{} statistics. They found that for the large rescalings — those where optical depth has to be rescaled by a factor of 2 or more — the error on flux power spectrum is a few percent. Also, the larger the rescaling is, the larger the error that is introduced. The rescaling error is therefore small, but not negligible, and most importantly, it is puzzling why one should continue to run simulations that systematically produce mean flux values excluded by observations at the few sigma level, and continue to compensate by rescaling the optical depth by a factor of few as is currently required with the HM12 or FG09 UVB tables. For this reason we wish to correct for this error in our new UVB models by renormalizing the input ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rate so that the post-processing correction will be minimal at all relevant redshifts. We want to emphasize here that the goal of this step is not to remove the need for future rescaling of the optical depth in simulations, but only to ensure that mean fluxes obtained by simulations are roughly consistent with current observations, therefore removing the need for large rescalings. Trying to do better than that would be pointless exercise, as the change in cosmological parameters, as well as having different resolution or box size, will anyway change the mean flux at a few percent level. The resolution of the simulations discussed here is thus sufficient for obtaining mean flux converged at a few percent level [@Lukic:2015]. In fact, we have run simulations of our LateR, MiddleR and EarlyR UVB models using a larger box size, $L_{box}=40$ $Mpc/h$ and $2048^3$ resolution elements to confirm that this is indeed the case.
Observational constraints coming from quasar absorption lines [@Fan:2006; @Becker:2007; @Kirkman:2007; @FaucherGiguere:2008; @Becker:2013] show that the mean flux smoothly evolves from about $0.2$ at $z=5$, to about $0.9$ at $z=2$, as expansion gradually lowers the density and the UVB intensity slowly increases. Figure \[fig:meanflux\_fit\] shows a compilation of these observations using different symbols with $1\sigma$ error bars. We also plot suggested fits to the mean flux evolution by various authors [@Fan:2006; @Kim:2007; @Viel:2013b]. However, none of these fits do a particularly good job of describing the full evolution of the mean flux, and we therefore opt for our own fit using all observational data points between $0.2<z<5.85$. We found that the functional form $$\tau_{{\ensuremath{\textrm{H} \, \textsc{i}}}}= A \times e^{(B\times \sqrt{z})}$$ provides an optimal fit, with $A=0.00126$ and $B=3.294$ as the best-fit parameters. This fit is also shown in Figure \[fig:meanflux\_fit\] as a solid black line.
![Different observation sets of the [Lyman-$\alpha$]{} mean flux evolution from high resolution quasar spectra [@Fan:2006; @Kirkman:2007; @FaucherGiguere:2008; @Becker:2013]. The orange dotted line stands for the fit obtained using $\sim 13000$ quasar spectra from the BOSS collaboration [@PalanqueDelabrouille:2013]. We also plotted some suggested fits in the literature [@Fan:2006; @Kim:2007; @Viel:2013b] as well as our suggested fit the the whole high resolution data set $0.2<z<5.85$ (solid black line). See text for more details.[]{data-label="fig:meanflux_fit"}](meanflux_z_plot_fit){width="45.00000%"}
The left panel of Figure \[fig:meanfsims\] shows the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux evolution in simulations using different well-known UVB models. It is clear that several of them significantly underpredict the observed mean flux at all redshifts. It is also worth pointing out that, somewhat coincidentally, the HM01 UVB model is doing a good job in recovering the observed mean flux. Notice, however, that since our thermal history is different from theirs, we cannot just assume their photoionization rates. Hence, we have modified the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rate in our models after reionization so that the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux in the simulations matches the fit to current observational constraints. Before ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, when we apply our new methodology, this does not apply. We also modified the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{i}}}$ photoheating rate by the same factor so that the heat input at these redshifts is conserved, i.e., $n_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm old}\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm old}=n_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm new}\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}},\rm new}$ and therefore we get exactly the same thermal histories. We have confirmed that this is the case by comparing the evolution of the thermal parameters in the new models versus the old ones, and we show the mean flux evolution of the MiddleR simulation as a dashed line in Figure \[fig:meanfsims\].
Regarding the effect of changing the thermal history, we have compared the differences between the simulations in which we change the heat input due to ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization. We found differences in the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux up to $\sim 10\%$ and $\sim 15\%$ between our our fiducial MiddleR model and the two most extreme simulations MiddleR-Hehot and MiddleR-noHe, respectively. This is because through the recombination factor ($\alpha \propto T_{0}^{-0.7}$) the neutral fraction has a sensitivity to the gas temperature, not just the photoionization: $n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}\propto \Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}}^{-1} T_{0}^{-0.7} \Delta^{0.7*(\gamma-1)}$. This maximum difference corresponds to $z\sim 3$, the redshift at which the thermal parameters between these simulations are most different. Of course, simulations using thermal histories that deviate even further from these would increase these differences. We have confirmed that mean flux differences due to current uncertainties in the cosmological parameters have a much weaker effect on the mean flux than any of the above systematics in the UVB models described above (see Appendix \[app:cosmo\] for a full discussion on cosmological parameters).
{width="45.00000%"} {width="45.00000%"}
Finally, in the right panel of Figure \[fig:meanfsims\] we show the most recent observations of the ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ transmission [@Worseck:2015] and compare them again with the mean HeII flux derived from our hydrodynamical simulations using standard UVB models (HM96, HM01, FG09, HM12)[^20]. We show the mean fluxes, ${\langle F_{{\ensuremath{\textrm{He} \, \textsc{ii}}}} \rangle}=e^{-\tau_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}}$, and not the optical depths in order to focus on the low redshift results, $z\lesssim2.7$, where observations seem to indicate that ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization has already finished. Results at higher redshifts are not that conclusive and may indicate that this reionization happened much more slowly than has been assumed [see @Worseck:2015 for a detailed discussion]. Therefore, we want to focus here just on the low redshift values, where reionization is completed and our method should be valid. For these it seems that FG09 ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ photoionization rates are doing the best job in reproducing the observations. For this reason we decided to use the ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ photoionization and photoheating rates of this model in our new UVB models after ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization.
Discussion {#sec:discuss}
==========
In this section we elaborate on the convergence of the results presented in this work, we compare them with recent efforts done in the field, and finally discuss their implications for galaxy formation simulations.
Resolution and Convergence {#ssec:convergence}
--------------------------
We have also run a set of simulations to explore resolution and box size effects on the different methods and parameters discussed in this paper. Some of these results for the mean flux, $T_{0}$, $\gamma$ and flux power spectrum relevant for this work have already been presented in @Lukic:2015. We refer to this work for more details of the accuracy of these simulations. In this regard we are confident that the thermal parameters discussed in this paper, $T_{0}$, $\gamma$, and $T({\Delta_{\bigstar}})$ as well as the pressure smoothing scale, ${\lambda_{\rm P}}$, are converged at least at the 5% level for $z\lesssim6$. This is also the case for the curvature statistic, ${\langle|\kappa|\rangle}$. Although we have used Nyx, an Eulerian code, to run all the tests of the new models created in this work, they will produce the same ionization and thermal histories in any other optically thin hydrodynamical code available. We have explicitly confirmed that Nyx and Gadget (which uses the SPH method for the hydrodynamics) agree well in their values for the mean flux and $\rho$-$T$ relation. Some differences could arise at lower redshifts in some observables, depending on the specific galaxy formation sub-grid model implementation [see e.g., @Viel:2013a]. However, it is hard to think of a realistic galaxy formation feedback model that will significantly affect the global ionization and thermal histories of the IGM [see, e.g., @Kollmeier:2006; @Desjacques:2006; @Shull:2015 but see Figure 10 of @Meiksin:2014].
Comparison to Previous Work
---------------------------
Recently, @Puchwein:2015 tried to solve some of the discrepancies between the HM12 model and $2<z<4$ observations of thermal parameters by including a nonequilibrium ionization solver in their hydrodynamical simulations. This approach goes in the same direction as this work, in the sense that they both try to improve how things are currently done during reionization events. As was expected, for a fixed UVB model they showed that using a non-equilibrium solver will produce a bigger temperature increase of the IGM during reionization. There is no doubt that a nonequilibrium approach is more physically relevant, as during reionization events the equilibration timescale, which is the time it takes the ionized fraction to change in response to a change in the photoionization rate $\Gamma$, can be comparable to the Hubble time, $t_{\rm eq} \simeq \Gamma_{\rm UVB}^{-1} \simeq t_{\rm Hubble}$. In a time-dependent (nonequilibrium) ionization calculation the neutral fraction will thus be elevated relative to the equilibrium value, and this results in more photoionization heating, $\sim {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\times \dot{q}_{\ensuremath{\textrm{H} \, \textsc{i}}}$, i.e. ${n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}$ is higher in a nonequilibrium calculation. @Puchwein:2015 found that this effect brings the HM12 model much more in agreement with the @Becker:2011 curvature measurements.
@Puchwein:2015 also showed that the change of the slope in the temperature density relation of the IGM, $\gamma$, is in fact significantly smaller in the ionization equilibrium approximation by running the same UVB model using ionization equilibrium and nonequilibrium algorithms. However, the different thermal histories that we found using our new UVB models indicate that this in fact degenerated with the ionization history and total heat input of the reionization event assumed to build the UVB model. The @Puchwein:2015 calculations use the HM12 heating rates, which are based on <span style="font-variant:small-caps;">cloudy</span> 1D slab calculations. The validity of the various approximations is dubious, as the heating during reionization is a complicated physical process that depends not only on the shape of the spectrum but also on the local density field and how fast the ionizing front travels [@McQuinn:2012; @Davies:2016]. Due to the present lack of knowledge about how much and when the reionization heats the IGM, we prefer to simply parameterize our ignorance of the details of reionization heat injection with a free parameter $\Delta T$. The differences in the IGM thermal properties between equilibrium and nonequilibrium codes thus seem moot given the large uncertainty in this $\Delta T$ parameter. However, as observational constraints improve and begin to constrain $\Delta T$, an improvement of our calculation would be to implement a nonequilibrium calculation along the lines of @Puchwein:2015, but from the perspective of our thermal history. This would amount to modification of the value of the $\Delta T$ that we choose or infer from data. That said, most cosmological hydrodynamical and galaxy formation codes use equilibrium solvers, and thus our current tables have wider applicability in their present assumption of equilibrium.
Recently, @UptonSanderbeck:2015 have also analyzed the thermal histories of different reionization models in a similar spirit as our approach in this work. However, they adopt a fast semianalytical approach that allows them to study how intergalactic gas is heated and cooled during and after reionization processes in a multiple-zone scenario, as opposite to the one-zone model assumption currently used in hydrodynamical simulations. Using this approach, the authors also explored reionization models with different ionization histories and heat injection. This method allows them more freedom in the types of models and parameters that can be explored. It probes to be a very useful tool to build intuition on the possible effects of a wide variety of reionization scenarios. However, a full numerical hydrodynamical method is required to generate simulations of different reionization models that can be directly compared to the observations, and perhaps most importantly, it is not possible to simulate the pressure smoothing effects resulting from different thermal/reionization history analytically. So in fact, the authors must interpret their results on the thermal parameters derived from observations using hydrodynamical simulations.
Finally, we want to point out that both the @Puchwein:2015 and @UptonSanderbeck:2015 final conclusions on possible ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization scenarios are based only on @Becker:2011 measurements at high redshift ($z>4$), ignoring the constraints on $T_{0}$ given by @Lidz:2010, which point to a much hotter IGM. As discussed above, these two measurements seem to be in disagreement at the 2$\sigma$ level at high redshift, but there is no clear reason to us why any of them should be ignored.
Implications for Galaxy Formation
---------------------------------
As mentioned in the introduction, an ionizing UVB inhibits gas accretion and photoevaporates gas from the shallow potential wells of low-mass dwarf galaxies. This effect due to gas being heated up by photoionization can result in negative feedback: suppressing star formation inside reionized regions, thus impeding their continued growth [see, e.g. @Rees:1986; @Efstathiou:1992]. In our current picture of galaxy formation this feedback is considered to be a good candidate for resolving the “missing satellite” problem [@Moore:1999; @Klypin:1999] which arises because in the cold dark matter ($\Lambda$CDM) framework the number of simulated dark matter subhalos is much larger than the number of observed dwarf galaxies [@Babul:1992; @Bullock:2001]. This picture seems consistent with recent observations that have seen uniformly old stellar populations in ultrafaint galaxies [@Brown:2014].
This topic has been a very active area of research in the past years, and using cosmological simulations is key to trying to understand these effects in their full context [@Quinn:1996; @Thoul:1996; @Gnedin:2000; @Hoeft:2006; @McQuinn:2007; @Okamoto:2008; @Noh:2014; @BenitezLlambay:2015]. More recent simulations with much better resolution and a more complete feedback model also seem to point in this direction [@Onorbe:2015; @Wheeler:2015a]. In this context, the work by @Simpson:2013 is particularly relevant. Using an <span style="font-variant:small-caps;">enzo</span> high-resolution cosmological hydrodynamical simulation, they found that turning on the UVB at $z=7.0$ versus $z=8.9$ resulted in an order-of-magnitude change in the final stellar mass of a $10^{9}$ ${\ensuremath{\mathrm{M_{\odot}}}}$ dark matter halo. In addition, some very interesting constraints are starting to come from radiative transfer hydrodynamical simulations. @Wise:2014 find that very faint galaxies ($M_{UV}\sim-6$, $M_{*}\sim10^{3.59}$ in halos of $M_{h}=1\times 10^{7}$) will still form at high redshift and contribute a significant amount to the ionizing photon budget during cosmic reionization. However, in order to avoid overproducing the observed abundance of classical satellites of the Milky Way, studies based on dark-matter-only simulations argue for a critical mass closer to $\sim 10^9$ ${\ensuremath{\mathrm{M_{\odot}}}}$ [@Madau:2008; @BoylanKolchin:2014].
For all this it is clear that the use of current standard UVB models that reionize and, more importantly here, heat the IGM at a much higher redshift than was desired will have a strong impact on the results of galaxy formation hydrodynamical simulations. For example, simulations that use the HM12 UVB background start spuriously heating up the IGM at $z\sim15$ (see Figure \[fig:Qhistgas0\]). We can now perform a simple first analytical calculation to get an approximate idea of this effect. In linear theory, the instantaneous cosmological mass cooling threshold of the neutral IGM (sometimes also referred to as the Jeans mass) is given by [see, e.g., @Iliev:2007]: $$\begin{aligned}
& M_{\rm min}=3.9\times 10^{9} {\ensuremath{\mathrm{M_{\odot}}}}\left(\frac{T_{\rm IGM}}{10^{4}K}\right)^{3/2}
\left(\frac{1+z}{10}\right)^{3/2} \\
& \qquad \left(\frac{\Omega_{\rm b}h^{2}}{0.0223}\right)^{-3/5}
\left(\frac{\Omega_{\rm m}h^{2}}{0.15}\right)^{-1/2}
\label{eq:Mcool}
\end{aligned}$$ where $T_{\rm IGM}$ is the temperature of the IGM. As has been found in the simulations mentioned above, the actual cooling mass differs somewhat from this instantaneous Jeans mass since the mass scale on which baryons succeed in collapsing out of the IGM along with the dark matter must be determined, even in linear theory, by integrating the differential equation for perturbation growth over time for the evolving IGM [@Gnedin:1998]. In fact, there is no single mass above which a collapsing halo retains all its gas and below which the gas does not collapse with the dark matter. Instead, simulations show that the cooled gas fraction in halos decreases gradually with decreasing halo mass [e.g. @Okamoto:2008]. Using eqn. (\[eq:Mcool\]) can give us a first estimate of the possible effects of using different UVB models. Figure \[fig:MJ\] shows the results of this equation when we use the different thermal histories of the HM12, FG09 as well as LateR, MiddleR, and EarlyR models. The differences between the models are quite significant.
Different studies of the reionization redshift of collapse structures have shown that the median reionization redshift of halos moves to lower redshift and the scatter increases [@Weinmann:2007; @Alvarez:2009; @Li:2014]. Although both parameters depend substantially on the details of reionization [see, e.g. @Ocvirk:2013], massive halos, $>10^{15}$ ${\ensuremath{\mathrm{M_{\odot}}}}$, can be reionized significantly earlier than the average region in the universe [$\Delta z_{\rm reion,50\%}\sim 2$, see Figure 2 of @Li:2014]. For Milky Way halos ($\sim10^{12}$ ${\ensuremath{\mathrm{M_{\odot}}}}$, ), or dwarf galaxies ($\sim10^{10}$ ${\ensuremath{\mathrm{M_{\odot}}}}$) the typical reionization redshift is expected to be much lower ($\Delta z_{\rm reion,50\%} \sim 0.5$ and $\Delta z_{\rm reion,50\%} \lesssim 0$ respectively), making FG09 and HM12 models not optimal for these studies. This is even more true if one considers the last constraints on ${\tau_{\rm e}}$ from @Planck:2016a. By overestimating the heat at high $z$ we are not only overestimating the pressure smoothing scale at lower redshifts but also overestimating the effect of the UVB in galaxy formation and evolution[^21]. For all these reasons, a detailed review of the results of galaxy formation simulations using new consistent UVB models that fulfill all observational constraints, like the ones developed here, is certainly needed. This is another reason why we make the models presented in this work publicly available to the community (see Appendix \[app:tables\]) and encourage colleagues to adopt them in future galaxy formation work and revisit previous calculations.
![Expected cosmological Jeans mass of the neutral IGM for the different thermal histories of HM12, FG09, LateR, MiddleR, and EarlyR simulations. See text for details. \[fig:MJ\]](MJ_z01){width="48.00000%"}
Conclusions {#sec:conc}
===========
In this paper we have presented results from optically thin cosmological hydrodynamical simulations using the Nyx code [@Almgren:2013; @Lukic:2015]. As commonly done in multiple IGM and galaxy formation studies, the UV background is modeled as a uniform and isotropic field that evolves with redshift. Operationally, the UVB determines the photoioinization and photoheating rates of ${\ensuremath{\textrm{H} \, \textsc{i}}}$, ${\ensuremath{\textrm{He} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$, which are inputs to the code.
We have demonstrated that when canonical models of the UVB, like that of HM12, are used in hydrodynamical simulations, the ionization of the IGM and, more importantly, the concomitant heating occur far too early, inconsistent with the reionization histories calculated by the respective authors, and in violation of current observational constraints on reionization. We argue that this results from the fact that these models dramatically overestimated the mean free path of ionizing photons at root at $z > 5$, resulting from the blind extrapolation of a model fit to lower redshift ($z < 5$) measurements. As a result, the amplitudes of the photoionization and photoionization heating rates are far too high at $z > 6$. This premature heating spuriously heats the IGM to $\sim 10^4$ K by $z \sim 13$, and because the IGM gas pressure smoothing scale depends on the full thermal history, it produces an erroneously large pressure smoothing scale at nearly all redshifts. We argue that a correct and consistent model of the reionization and thermal history is crucial for obtaining the correct pressure scale in simulations, which is necessary for interpreting [Lyman-$\alpha$]{} forest statistics at $z<6$ – not doing so can bias estimates of the thermal state of the IGM. We also discussed the implications of this spurious early heating on galaxy formation simulations.
Motivated by these issues, we have developed a new method to generate UVB models for hydrodynamical simulations that allow one to self-consistently simulate different reionization models. We implement this by volume-averaging the photoionization and energy equations. In the model, each reionization event is defined by the ionization history with redshift and the total heat input of the reionization event. In this sense our new models provide a very promising tool to explore the parameter space of possible ionization and thermal histories. In this work, we investigated models in which we changed the redshift at which ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization ends and the amount of heat input associated with both ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization. We studied the effect of these changes on the thermal history of the IGM, in particular the temperature at mean density, $T_{0}$, the slope of the temperature-density relation, $\gamma$, the temperature at the optimal density probed by curvature measurements, $T({\Delta_{\bigstar}})$, and the gas pressure smoothing scale, $\lambda_{P}$. We have shown how important the degeneracies between these parameters can be in order to derive the thermal parameters of the IGM using the curvature [Lyman-$\alpha$]{} statistic. These rates have also been corrected to improve the agreement with measurements of the average ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ transmission after reionization.
The UVB plays a fundamental role in determining the star formation of the first galaxies and their evolution by not only setting the minimum halo mass able to form stars but also regulating the gas accretion from the IGM into more massive halos. Previous studies utilizing UVB models that suffer from the spurious early heating described in this paper have thus overestimated the effect of this photoheating feedback and the resulting suppression of star-formation at high redshift. We therefore argue that galaxy formation simulations should be revisited using our new UVB models.
We make our new UVB models publicly available so that the community can better explore the consequences and effects of different ionization and thermal histories in all types of hydrodynamical cosmological simulations. Tables with the photoionization and photoheating rates of the new models can be found in Appendix \[app:tables\], which are in the the standard “TREECOOL” file format, ready and easy to use with most cosmological codes, including <span style="font-variant:small-caps;">gadget</span>, <span style="font-variant:small-caps;">arepo</span>, and <span style="font-variant:small-caps;">gizmo</span>. We encourage anyone interested in implementing some other specific UVB model to contact the authors. We will also be happy to provide help incorporating these models into other codes by request.
Like all optically thin simulations, our approach misses UVB fluctuations that could produce scatter in reionization times and temperatures between different regions of the universe [e.g. @Abel:1999; @Meiksin:2004; @Pontzen:2014; @GontchoaGontcho:2014; @Davies:2014; @Malloy:2015; @DAloisio:2015; @Davies:2016b]. Of course, an immediate solution to this problem in optically thin simulations will be to add some extra dependence on a specific property of each resolution element in the simulation (e.g. density, distance from a halo that could host a galaxy/quasar, etc). However, the computational challenge is to try to do this without making the simulation prohibitively expensive. In this sense, an interesting solution that it is worth exploring will be to assign a specific reionization redshift to each resolution element of the simulation using, for example, an excursion set formalism [e.g. @Furlanetto:2004]. In this picture, a resolution element will not see the UVB background until its reionization redshift. We plan to pursue this idea in the near future. Another very valuable piece of information that would improve current optically thin hydrodynamical simulations would be if future radiative transfer simulations [e.g. @So:2014; @Gnedin:2014; @Pawlik:2015; @Norman:2015; @Ocvirk:2015] were to make the probability distribution function of their photoionization rates publicly available (and perhaps also its dependence on density), and not just the evolution of the mean and/or median values.
Finally, our new parameterization for the heating and ionization produced by the UVB allows us to explore a broader range of reionization models, as well as any other physical scenarios that could alter the thermal history of the IGM. This will allow us to better test the effect of such models in simulations of galaxy formation and the IGM. These models could include Population III stars [@Manrique:2015], X-ray pre-heating coming from from starburst galaxies, supernova remnants, or miniquasars [@Oh:2001; @Glover:2003; @Madau:2004; @Furlanetto:2006], dark matter annihilation or decay [@Liu:2016] or cosmic rays [@Samui:2005], from the intergalactic absorption of blazar TeV photons [@Chang:2012; @Puchwein:2012], or from broadband intergalactic dust absorption [@Inoue:2008]. We expect more detailed studies on these physically motivated models in the future.
We thank the members of the ENIGMA group at the Max Planck Institute for Astronomy (MPIA) for helpful discussions. J.F.H. acknowledges generous support from the Alexander von Humboldt foundation in the context of the Sofja Kovalevskaja Award. The Humboldt foundation is funded by the German Federal Ministry for Education and Research. Z.L. was supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy Office of Advanced Scientific Computing Research and the Office of High Energy Physics. Calculations presented in this paper used the <span style="font-variant:small-caps;">hydra</span> cluster of the Max Planck Computing and Data Facility (MPCDF, formerly known as RZG) MPCDF is a competence center of the Max Planck Society located in Garching (Germany). We also used resources of the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the U.S. Department of Energy under Contract no. DE-AC02-05CH11231. The ASCR Leadership Computing Challenge (ALCC) program has provided NERSC allocation under the project “Cosmic Frontier Computational End-Station”. This work made extensive use of the NASA Astrophysics Data System and of the astro-ph preprint archive at arXiv.org.
Volume-averaged Ionization and Heating Equations {#app:volave}
================================================
In this paper we have presented a new way of obtaining effective photoionization and photoheating rates for different reionization models. These can be used in optically thin hydrodynamical simulations to account for emission of the galaxies and quasars. Here we provide detailed derivation of these rates for future reference.
We remind the reader that the new effective rates are only computed during reionization, i.e., while ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}<1$. After reionization, our models use the rates from common UVB models (e.g. FG09, HM12, see Section \[sec:meanflux\]). New values of the photoionization rates during reionization are forced to never exceed the values of the model plugged after reionization. This is done to guarantee no numerical artifacts in the limit where reionization is almost complete, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}\sim1$, where our new rates are not well defined.
Volume Average Optically Thin Ionization Equations {#ssec:volave}
--------------------------------------------------
In the context of optically thin hydrodynamical codes, it is often assumed that the gas is of the primordial chemical composition, where the resulting reaction network includes six atomic species: ${\ensuremath{\textrm{H} \, \textsc{i}}}$, ${\ensuremath{\textrm{H} \, \textsc{ii}}}$, ${\ensuremath{\textrm{He} \, \textsc{i}}}$, ${\ensuremath{\textrm{He} \, \textsc{ii}}}$, ${\ensuremath{\textrm{He} \, \textsc{iii}}}$ and e$^-$. Codes generally evolve those species under the assumption of ionization equilibrium [see, however @Gnat:2007; @Vasiliev:2011; @Oppenheimer:2013; @Richings:2014a; @Richings:2014b; @Puchwein:2015 for nonequilibrium treatments]. The resulting system of algebraic equations is: $$\begin{aligned}
& \left( \Gamma_{\rm e, {\ensuremath{\textrm{H} \, \textsc{i}}}} {n_{\rm e}}+ \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \right) {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}= \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}} {n_{\rm e}}{n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\\[1.5mm]
& \left( \Gamma_{\rm e, {\ensuremath{\textrm{He} \, \textsc{i}}}} {n_{\rm e}}+ \Gamma_{\gamma, {\ensuremath{\textrm{He} \, \textsc{i}}}} \right) {n_{{\ensuremath{\textrm{He} \, \textsc{i}}}}}= \left( \alpha_{\rm r, {\ensuremath{\textrm{He} \, \textsc{ii}}}} + \alpha_{\rm d, {\ensuremath{\textrm{He} \, \textsc{ii}}}} \right)
{n_{\rm e}}{n_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}}\\[1.5mm]
& \left[ \Gamma_{\gamma, {\ensuremath{\textrm{He} \, \textsc{ii}}}} + \left(\Gamma_{\rm e, {\ensuremath{\textrm{He} \, \textsc{ii}}}}
+ \alpha_{\rm r, {\ensuremath{\textrm{He} \, \textsc{ii}}}} + \alpha_{\rm d, {\ensuremath{\textrm{He} \, \textsc{ii}}}} \right)
{n_{\rm e}}\right] {n_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}}\\[1.5mm]
& \qquad = \alpha_{\rm r, {\ensuremath{\textrm{He} \, \textsc{iii}}}} {n_{\rm e}}{n_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}+ \left( \Gamma_{\rm e, {\ensuremath{\textrm{He} \, \textsc{i}}}} {n_{\rm e}}+ \Gamma_{\gamma, {\ensuremath{\textrm{He} \, \textsc{i}}}} \right)
{n_{{\ensuremath{\textrm{He} \, \textsc{i}}}}}\end{aligned}
\label{eq:photo}$$ In addition, there are three closure equations for the conservation of charge and hydrogen and helium abundances. Radiative recombination ($\alpha_{\rm r, X}$), dielectronic recombination ($\alpha_{\rm d, X}$), and collisional ionization ($\Gamma_{\rm e, {\rm X}}$) rates are strongly dependent on the temperature, which itself depends on the ionization state through the mean mass per particle $\mu$ $$T = \frac{2}{3} \frac{m_p}{k_{\rm B}} \mu\ e_{\rm int}$$ where $m_p$ is the mass of a proton, $k_{\rm B}$ is the Boltzmann constant and $e_{\rm int}$ is the internal thermal energy per mass of the gas. For a gas composed of only hydrogen and helium, $\mu$ is related to the number density of free electrons relative to hydrogen by $\mu=(1+4\chi)/[1+\chi+({n_{\rm e}}/{n_{{\ensuremath{\textrm{H}}}}})]$. The reaction network equations are iteratively solved together with the ideal gas equation of state, $p = 2\rho e_{\rm int}/3$, to determine the temperature and equilibrium distribution of species. Above and throughout this paper we have assumed an adiabatic index of $5/3$.
In order to produce consistent UVB models that reliably reproduce different reionization histories, we have derived the volume-averaged version of the ionization equilibrium equations presented in eqn. (\[eq:photo\]). We start with the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization and derive here in detail the equation for ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization. We will address the ${\ensuremath{\textrm{He}}}$ single and double reionization afterward, as the method is very similar. We start by doing the volume-averages of eqn. (\[eq:photo\]) for ${\ensuremath{\textrm{H} \, \textsc{i}}}$, $$\begin{aligned}
& {\langle \Gamma_{\rm e, {\ensuremath{\textrm{H} \, \textsc{i}}}} {n_{\rm e}}{n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\rangle} + {\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\rangle}
= {\langle \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}} {n_{\rm e}}{n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}
\end{aligned}
\label{eq:photomean1}$$ The first thing is that collisional ionization terms, $\Gamma_{\rm e}$, are mainly relevant in shocks at high temperatures and densities. They will have a negligible effect on this volume-averaged calculation, so we can discard them. The important point here is that each term in this equation is nonlinear; hence, in principle, it is not possible to compute their volume averages unless the cross-correlation of the abundances of ${n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}$ and ${n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}$ with each other, as well as with the radiation and temperature fields are known (since $\Gamma_{\rm e,{\ensuremath{\textrm{H} \, \textsc{i}}}}$ and $\alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}}$ are temperature-dependent). A convenient way to encapsulate this unknown information is using correction factors. With all this we can write the volume-averaged equivalent of eqn. (\[eq:photo\]) as $$\begin{aligned}
& C_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}} {\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle} {\langle {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\rangle}
= C_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}}{\langle \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle} {\langle {n_{\rm e}}\rangle} {\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}
\end{aligned}
\label{eq:photomean2}$$ where the correction factors are defined as $$\begin{aligned}
& C_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}}=\frac{{\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\rangle}}{{\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle} {\langle {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\rangle} } \\[1.5mm]
& C_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}} =\frac{{\langle \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}} {n_{\rm e}}{n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}}{{\langle \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle} {\langle {n_{\rm e}}\rangle} {\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}}
\end{aligned}
\label{eq:clumping}$$ Then, we can rewrite eqn. (\[eq:photomean2\]) as $$\begin{aligned}
& {\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}
= \frac{C_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}}}{C_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}}}
\frac{{\langle \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle} {\langle {n_{\rm e}}\rangle} {\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}}{{\langle {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\rangle}} \\[1.5mm]
\end{aligned}
\label{eq:photomean3}$$
Now, in order to obtain an ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rate, we will make use of the following assumptions: 1) We assume that the ${\ensuremath{\textrm{He} \, \textsc{i}}}$ reionization occurs perfectly coupled with the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization process. This is a very common assumption in reionization models because ionization of hydrogen and the first ionization of helium require photons with similar energies (13.6 eV and 24.6 eV respectively). Therefore, the same physical process responsible for ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization must be also liable for the ${\ensuremath{\textrm{He} \, \textsc{i}}}$ reionization. We also consider that the the helium second ionized state number density is negligible during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{i}}}$ reionization, ${n_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}\sim 0$, so ${\langle x_{HeIII} \rangle}=0$. This is a correct assumption for all the models discussed in this work but could be easily modified in the future if needed. 2) The evolution of number density of free electrons, ${n_{\rm e}}$, can be approximated by ${n_{\rm e}}={n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}+{n_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}}+ 2{n_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}$. We can rewrite this equation as a function of the hydrogen density and the ionized fractions: ${n_{\rm e}}={n_{{\ensuremath{\textrm{H}}}}}\left[(1+\chi)x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} +\chi x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}\right]$ where $\chi=Y_{\rm p}/4X_{\rm p}$ and we have assumed again that ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization follows that of ${\ensuremath{\textrm{H} \, \textsc{ii}}}$ one. The volume-averaged value can be written as $$\begin{aligned}
{\langle {n_{\rm e}}\rangle}={\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}\left[C_{x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}} \left(1+\chi\right) {\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle} +
C_{x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}\chi {\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}\right]
\label{eq:elecden}
\end{aligned}$$ where $C_{x}$ factors stand for the correction factors defined as $C_{x_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}={\langle {n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\rangle}$, $C_{x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}{\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}={\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}$, etc. As we are assuming that ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ is not relevant during ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, we can discard the second term inside parentheses for our current derivation. Finally, putting together eqn. (\[eq:photomean3\]) with eqn. (\[eq:elecden\]), we arrive at $${\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)= C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} {\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}(z)
\alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}}({\langle T \rangle})(1+\chi) \frac{{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}^{2}(z)}{{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)} \\[1.5mm]
\label{eq:invphotoA}$$ where $C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}$ encapsulates all correction parameters described above and can be written as $$C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}=\frac{C_{\alpha_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\times C_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}}\times C_{x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}^{2}}{C_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}} \times C_{x_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}}
\label{eq:clumpingfinal}$$ where $C_{\alpha_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}}}$ is defined as $C_{\alpha_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}}}={\langle \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}/\alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}}({\langle T \rangle})$ to clarify that in general the volume-averaged recombination factor is redefined as the recombination factor at a certain mean temperature value. It is customary to use the temperature at mean density, $T_{0}$ as this value. In the case of optically thin hydrodynamical simulations a constant photoionization rate is used throughout the whole volume, so $C_{\gamma}=1$.
In general, the unknown ratio of the IGM’s true recombination rate to its hypothetical rate under the assumption of uniform density and temperature is often referred to as the IGM clumping factor ($C_{\rm IGM}=C_{\rm r,{\ensuremath{\textrm{H} \, \textsc{ii}}}}$). Notice also that, if for eqn. (\[eq:clumping\]) we assume that ${n_{\rm e}}={n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}$ and that $\alpha_{\rm r}$ is a constant, it is easy to redefine the IGM clumping in a much more familiar form: $C_{\rm IGM}={\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}^2 \rangle}/{\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}^{2}$ [@Kohler:2007; @Pawlik:2009; @Finlator:2012; @Kaurov:2014].
We can also obtain analogous equations for the volume-averaged helium photoionization rates from eqn. (\[eq:photo\]) to the one we obtained above, eqn. (\[eq:invphotoA\]), for the hydrogen. We have just made one extra assumption, which is that ${n_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}$ is negligible during the single reionization of helium and that during the double reionization of helium ${n_{{\ensuremath{\textrm{He} \, \textsc{i}}}}}$ is negligible. In the context of all the ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization models discussed in this paper this is a fair assumption. For the ${\ensuremath{\textrm{He}}}$ first ionization we obtain $${\langle \Gamma_{\gamma, {\ensuremath{\textrm{He} \, \textsc{i}}}} \rangle}(z)= C_{{\ensuremath{\textrm{He} \, \textsc{ii}}}} {\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}(z)
\alpha_{\rm r, {\ensuremath{\textrm{He} \, \textsc{ii}}}}({\langle T \rangle})(1+\chi) \frac{{\langle x_{{\ensuremath{\textrm{He} \, \textsc{ii}}}} \rangle}^{2}(z)}{{\langle x_{{\ensuremath{\textrm{He} \, \textsc{i}}}} \rangle}(z)} \\[1.5mm]
\label{eq:invphotoHeI}$$ and $C_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$ is analogous to $C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}$, $$C_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=\frac{C_{\alpha_{\rm r,{\ensuremath{\textrm{He} \, \textsc{ii}}}}}\times C_{\rm r,{\ensuremath{\textrm{He} \, \textsc{ii}}}}\times C_{x_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}}^{2}}{C_{\gamma,{\ensuremath{\textrm{He} \, \textsc{i}}}} \times C_{x_{{\ensuremath{\textrm{He} \, \textsc{i}}}}}}.
\label{eq:clumpingHeII}$$ For the ${\ensuremath{\textrm{He}}}$ double ionization, $$\begin{aligned}
& {\langle \Gamma_{\gamma, {\ensuremath{\textrm{He} \, \textsc{ii}}}} \rangle}(z)=\frac{1}{C_{\gamma,{\ensuremath{\textrm{He} \, \textsc{i}}}}} {\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}(z)
[C_{x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}(1+\chi) + \chi C_{{x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}}}{\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}(z)]\\
& \quad \left[C_{\alpha_{\rm r,{\ensuremath{\textrm{He} \, \textsc{iii}}}}}\times C_{\rm r,{\ensuremath{\textrm{He} \, \textsc{iii}}}}\frac{\alpha_{\rm r, {\ensuremath{\textrm{He} \, \textsc{iii}}}}({\langle T \rangle}){\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}(z)}{{\langle x_{{\ensuremath{\textrm{He} \, \textsc{ii}}}} \rangle}(z)}
-C_{\alpha_{\rm r,{\ensuremath{\textrm{He} \, \textsc{ii}}}}}\times C_{\rm r,{\ensuremath{\textrm{He} \, \textsc{ii}}}} \times \alpha_{\rm r, {\ensuremath{\textrm{He} \, \textsc{ii}}}}({\langle T \rangle})\right] \\[1.5mm]
\end{aligned}
\label{eq:invphotoHeII}$$ In order to describe the reionization history of one model,we will need to specify two functions, the evolution of ${\ensuremath{\textrm{H} \, \textsc{ii}}}$ and ${\ensuremath{\textrm{He} \, \textsc{iii}}}$ volumen-averaged ionization fractions, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}(z)$ and ${\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}(z)$. As ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle} + {\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}=1$ and ${\langle x_{{\ensuremath{\textrm{He} \, \textsc{i}}}} \rangle} + {\langle x_{{\ensuremath{\textrm{He} \, \textsc{ii}}}} \rangle} +{\langle x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}} \rangle}=1$ and we assume that ${\ensuremath{\textrm{He} \, \textsc{i}}}$ reionization is totally coupled with ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization.
To calculate the average values of the radiative recombination rates ($\alpha_{\rm r, i}$) which depend on temperature, we need to specify the evolution of the volume-averaged temperature, ${\langle T \rangle}(z)$ in our model. To do this, we first need to define the total heat input produced by each reionization event, $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ and $\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$ which will be two free parameters in our reionization models. We describe in the next paragraph the recent efforts to calculate these parameter from theoretical models. Thus, we will make another assumption in our model to describe the evolution of the volumen-averaged temperature. 3) That the evolution of the volumen-averaged temperature can be well approximated with the evolution of the reionization history times the total heat input, i.e, ${\langle T \rangle}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}(z)=\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}\times {\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}(z)$ and ${\langle T \rangle}_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}(z)=10^{4}+\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}\times x_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}(z)$. This is a very rough estimation of the thermal history, as we are neglecting all cooling, but it serves as a first-order approximation of the gas temperature during reionization in order to compute the different rates that we need to compute eqn. (\[eq:invphotoA\]). We will show below that this assumption is accurate enough for our purposes and for the range of models covered in this work. More elaborate assumptions, including analytic estimations of the Compton cooling, could be implemented in the future.
Finally, to generate the models of this work, we used the following correction factors. For the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{i}}}$ photoionization rates we used $C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}=C_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=1.5$ for $z\geq10$ and $C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}=C_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=2.0$ for $6<z<10$. Notice that we never go below redshift $z=6$ when we compute the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ rates, and that these values are in good agreement with results from radiative transfer simulations [clumping factor, $C_{100}$, of @Pawlik:2009] in this range of redshifts. For the ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ photoionization rates we used $C_{{\ensuremath{\textrm{He} \, \textsc{iii}}}}=1.5$ at all redshifts, which seems to allow us to accurately recover the input ionization models.
### First-order Estimation of the Ionization History in Hydrodynamical Simulations from the Photoionization Rates {#app:hifrac}
Using the relation between volume-averaged quantities derived above, one can obtain an analytical expectation of the volume-averaged ${\ensuremath{\textrm{H} \, \textsc{i}}}$ ionization fraction, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}$, evolution in an optically thin simulations once a certain photoionization rate is assumed. This can be useful to predict the approximate ionization history that a certain photoionization rate model will produce in a optically thin hydrodynamical simulation. In particular, for the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ photoionization rate we can rewrite eqn. (\[eq:invphotoA\]) above as $$C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} {\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}(z) \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}}({\langle T \rangle})(1+\chi) (1-C_{x,{\ensuremath{\textrm{H} \, \textsc{i}}}}{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z))^{2}
- C_{x,{\ensuremath{\textrm{H} \, \textsc{ii}}}}^{2}{\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z){\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)=0 \\[1.5mm]$$ where now we are using that $C_{x,{\ensuremath{\textrm{H} \, \textsc{ii}}}}{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}=1-C_{x,{\ensuremath{\textrm{H} \, \textsc{i}}}}{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}$. If we define $A=C_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} {\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}(z) \alpha_{\rm r, {\ensuremath{\textrm{H} \, \textsc{ii}}}}({\langle T \rangle})(1+\chi)$, we can rewrite this equation as $$A{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}^{2}(z) - [2A+{\langle \Gamma_{\gamma, {\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)]{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}+A=0$$ Solving for the volume-averaged ${\ensuremath{\textrm{H} \, \textsc{i}}}$ ionization fraction, we get $${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}=1-{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}=\frac{ 2A+\Gamma_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}} \pm \sqrt{\Gamma_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}}^{2}+4A\Gamma_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}}} } {2A}
\label{eq:hifrac}$$ which gives us an estimation of the expected hydrogen neutral fraction once we discard the nonphysical solution. It is expected that $C_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ changes between simulations that implement very different physical processes. Therefore, this formula could also be used to obtain the value of $C_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ in optically thin hydrodynamical simulations using one run and then compute analytically the expected results with different UVB models.
![Evolution of the volume-averaged ${\ensuremath{\textrm{H} \, \textsc{i}}}$ fraction, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}=1-{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}$, obtained in the hydrodynamical simulations (dashed lines) vs. the expected evolution using the volume-averaged analytical prediction given by eqn. (\[eq:hifrac\]). See text for more details. \[fig:xhiana\]](modelxHI_z){width="48.00000%"}
In Figure \[fig:xhiana\] we plot the result obtained from eqn. (\[eq:hifrac\]) using the photoionization rate from the HM12 model (green line) and the late reionization model (blue line). We assumed $T=2\times10^{4}$ K at all redshifts, and we used the clumping given by @Pawlik:2009 [their fit to $C_{100}$].
Volume-averaged Heating Equations {#ssec:dTdz}
---------------------------------
The next step is to include in a consistent way the heating occurring during different reionization epochs. To do this, we have implemented a variation of an the idea suggested by FG09.In this work the authors also raised the problems pointed out in the previous section of incorporating the effects of a prescribed UVB in cosmological hydrodynamical simulations under the assumption of an optically thin plasma. Their work was motivated by ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization, but the idea can be well applied for any reionization event. They proposed as a more physically motivated approach to increase the temperature of each gas element by an amount $d \Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}} (z) / dz$, (which is subsequently allowed to cool) at each time step $\Delta z$ in the simulation.[^22] Here $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}} (z)$ is the total heat input or cumulative temperature increase evolution owing to reionization and is precomputed given the desired ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization history. Both the total heat input from ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization and from ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization will be free parameters in our model. We discuss the specific values used in this work in Section \[sec:reionmodels\].
Now, we want to go one step further from the $d\Delta T/dt$ idea, and relate this change in temperature to a specific photoheating rate, $\dot{q}$. This will allow the use of our new models in standard hydrodynamical codes. To do this we can, again, volumen-averaged the relation between the heat per unit of time in one cell produced by the $dT/dz$ model and the one produced by a photoheating rate. The change of internal energy density due to a certain change in temperature of the gas can be related to a new photoheating rate in the following way: $$\frac{de_{int}}{dt}=\frac{3 k_{\rm B}}{2 m_{\rm p}\mu}\frac{d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{dt}=\frac{{n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{\rho_{\rm proper} }
\label{eq:dtdzq}$$ Using the same assumptions considered to obtain the new photoionization rates, we can get a volume-averaged value for the heating rate. We need also to assume that the volume-averaged molecular weight is ${\langle \mu \rangle}=(1+ 4\chi)/\left(1+\chi + {\langle {n_{\rm e}}/{n_{{\ensuremath{\textrm{H}}}}}\rangle}\right)$. Then we obtain $$\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=C_{\dot{q},{\ensuremath{\textrm{H} \, \textsc{i}}}}\frac{ 3 k_{\rm B}}{2 {\langle \mu \rangle} X_{\rm p} {\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}(z)} \frac{d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{dt} \label{eq:qdotA}$$ where $C_{\dot{q},{\ensuremath{\textrm{H} \, \textsc{i}}}}$ is a correction factor defined as $$\langle \frac{\mu}{x_{{\ensuremath{\textrm{H} \, \textsc{i}}}}} \frac{d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{dt} \rangle = C_{\dot{q},{\ensuremath{\textrm{H} \, \textsc{i}}}} \frac{{\langle \mu \rangle}}{{\langle x_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \rangle}}{\langle \frac{d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}{dt} \rangle}
\label{eq:Cqdot}$$ Note that this approach sets $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{i}}}}=0.0$ as we include the heating produced by ${\ensuremath{\textrm{He} \, \textsc{i}}}$ reionization in the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ heating rate. An identical approach is used to obtain the [$\textrm{He} \, \textsc{ii}$]{} heating rate, $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$. To compute these values in our models, we have used a correction factor, $C_{\dot{q}}=1$.
Therefore, in our model, once a total heat input due to reionization ($\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$) is chosen, the exact photoheating rates will depend on the assumption made on $d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}/dz$. In order to simplify this, we assume that the evolution of the total heat input can be well approximated by the volume-averaged ionization fraction evolution: $d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}/dt\sim d{\langle x_{HII} \rangle}/dt$. From here we can derive the derivative of the total heat input evolution with redshift, which is used in eqn. (\[eq:qdotA\]) to obtain the tabulated photoheating rates that will go into the code. For this reason in our reionization models, the thermal history is defined based on one free parameter, the total heat input $\Delta T$, and one free function, the ionization history, which in this context we define as the volumen-averaged ionization fraction evolution, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}(z)$. In the case of ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization the heating term is defined as $$\left|\frac{d\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}(z)}{dz}\right| = \frac{\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}} \left|
\frac{d{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}(z)}{dz} \right|}{\int_{z_{\rm reion}}^{z_{\rm recomb}} \left| \frac{d{\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}(z)}{dz} \right| dz}
\label{eq:dTdz}$$ The denominator factor is a normalization to guarantee that the integrated amount of heating injected at each time step corresponds to the total heat input.
Ionization and Thermal Histories for Other UVB Models {#app:HMold}
=====================================================
Here we present the ionization and thermal histories produced by other UVB models, widely used in the literature. We have run simulations using the following UVB models (HM96, HM01, FG09), in addition to the HM12 model largely discussed in this work. The left panel of the Figure \[fig:hmold\] shows the ionization history for all these models. It is particularly interesting to first focus on the results of the FG09 which use the same approximation as HM12 (Eq. \[eq:Q\]) to calculate the reionization redshift of their model that obtained complete ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization by $z\sim3$, and ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization by $z=6$. However, it is clear that this model is producing a much higher ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, more close to $z\sim10$, indicating that they have the same problems as the HM12 prescription. In fact, the method to compute their expected ionization history is the same as the one used by HM12. In this case the effect is not as extreme, only because tabulated values start at a lower redshift and, although they produce what seems to be a very high redshift ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization, the final outcome of the model is still within CMB observational constraints [@Planck:2015][^23]. We do not have any information on what were the expected ionization histories for the other two older Haardt & Madau models, but it is still very interesting to show their reionization histories, as they are still used in the literature. First of all, notice that in these models, as well as in the FG09 run, the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization is described as a simple step function. The HM01 model assumes a much more shallower ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization history than any other model considered in this work, starting as early as $z\sim10$. This will allow us to see the effect of these types of models on the thermal history. HM96 ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization happens very late, and therefore this model does not fulfill @Planck:2015 CMB constraints[^24].
The middle and right panels of Figure \[fig:hmold\] illustrate the thermal histories of these models. The first thing that we want to point out is the effect of a shallower ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ reionization model on the HM01 model. This basically produces a smoother evolution of the temperature at mean density, $T_{0}$, eliminating any sharp behavior at lower redshifts. At lower redshifts, the instantaneous thermal parameters $\gamma$ and $T(\Delta)$ are converging to the same values, but pressure smoothing scale and curvature are not.
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
Figure \[fig:hmcut\] shows results for hydrodynamical simulations in which a simple redshift cutoff was applied to the HM12 model. In this approach the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{i}}}$ UVB rates are set to zero above a certain redshift: $z=11$ (red), $z=9$ (blue), and $z=7$ (brown). We also show the original HM12 model (green lines) for comparison. All the runs share the same ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ rates. The left panel of Figure \[fig:hmcut\] shows the ionization histories of all these models. The middle and right panels present their thermal histories. As expected, by applying a cutoff to the UVB rates, the reionization redshift and its thermal signatures move down to the cutoff redshift. The gas pressure scale shows very clearly the effect of producing the heating due to ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization at lower redshift. As discussed in Section \[sec:results\], this is because the pressure smoothing scale depends on the full thermal history of the universe and not just on the instantaneous temperatures.
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
Cosmological Effects on the Ionization and Thermal Properties of the IGM {#app:cosmo}
========================================================================
In this appendix we present the results of simulations that only differ in the cosmological parameters and have the same UVB model, HM12. The random seeds in the initial condition are also the same. Table \[tab:cosmo\] summarizes the cosmological parameters used in each run. Cosmological model A is the default cosmological model used in this work. Models B and C were selected from the posterior distribution of the Planck results [@Planck:2015] in order to differ as much as possible in the expected matter power spectrum. These should maximize the difference between the models while keeping them within the limits allowed by CMB observations. Model D was used in @Lukic:2015 and is also in agreement with the last CMB results.
Figure \[fig:cosmology\] shows the ionization and thermal histories for these four simulations. Both ionization histories, as well as the evolution of thermal parameters, are almost identical for all the runs. This means that within current observational constraints, the cosmological structure formation does not change significantly the thermal evolution of the IGM, which is determined by the UVB model used. In fact, this result is expected, as from our analytical calculation of the photoionization and photoheating rates we can easily calculate how these rates will change with some cosmological parameters. The difference in photoionization rate values for different cosmologies will be the difference between $\Omega_{\rm b}h^{2}$ in the models. Difference in effective photoheating values for different cosmologies will be driven by the difference between $H(z)$[^25].
We want to emphasize here that even when two simulations/models share the same ionization and thermal evolution, that does not mean that the [Lyman-$\alpha$]{} observables (probability density function, flux power spectrum, etc.) from these runs will not show differences between them as the observables could have their own dependence on cosmological parameter or other parameters.
[lcccccc]{} Model & $\Omega_{\rm m}$ & $\Omega_{\rm b}$ & $\Omega_{\Lambda}$ & $h$ & $\sigma_{8}$ & $n_{\rm s}$\
A & 0.320 &0.0496&0.681&0.670&0.826&0.966\
B & 0.298 &0.0477&0.702&0.686&0.873&0.974\
C & 0.333 &0.0517&0.667&0.658&0.757&0.971\
D & 0.275 &0.0460&0.725&0.702&0.816&0.960\
![Cosmological dependence of the IGM properties from simulations using different cosmological parameters but the same UVB (HM12). Upper left panel: ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ ionization histories computed from the simulations. Upper right panel: evolution of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux. Lower left panel: evolution of thermal parameters. slope of the temperature density relation, $\gamma$ (upper), and the temperature at mean density, $T_{0}$ (lower). Lower right panel: evolution of the temperature at the characteristic density, $T(\Delta)$ (upper), the curvature statistics, ${\langle|\kappa|\rangle}$ (middle), and the pressure smoothing scale, ${\lambda_{\rm P}}$ (lower). \[fig:cosmology\]](Qtau_cosmo "fig:"){width="45.00000%"} ![Cosmological dependence of the IGM properties from simulations using different cosmological parameters but the same UVB (HM12). Upper left panel: ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ ionization histories computed from the simulations. Upper right panel: evolution of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux. Lower left panel: evolution of thermal parameters. slope of the temperature density relation, $\gamma$ (upper), and the temperature at mean density, $T_{0}$ (lower). Lower right panel: evolution of the temperature at the characteristic density, $T(\Delta)$ (upper), the curvature statistics, ${\langle|\kappa|\rangle}$ (middle), and the pressure smoothing scale, ${\lambda_{\rm P}}$ (lower). \[fig:cosmology\]](meanflux_z_cosmology "fig:"){width="45.00000%"}\
![Cosmological dependence of the IGM properties from simulations using different cosmological parameters but the same UVB (HM12). Upper left panel: ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ ionization histories computed from the simulations. Upper right panel: evolution of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux. Lower left panel: evolution of thermal parameters. slope of the temperature density relation, $\gamma$ (upper), and the temperature at mean density, $T_{0}$ (lower). Lower right panel: evolution of the temperature at the characteristic density, $T(\Delta)$ (upper), the curvature statistics, ${\langle|\kappa|\rangle}$ (middle), and the pressure smoothing scale, ${\lambda_{\rm P}}$ (lower). \[fig:cosmology\]](gammaT0_cosmo "fig:"){width="45.00000%"} ![Cosmological dependence of the IGM properties from simulations using different cosmological parameters but the same UVB (HM12). Upper left panel: ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{ii}}}$ ionization histories computed from the simulations. Upper right panel: evolution of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux. Lower left panel: evolution of thermal parameters. slope of the temperature density relation, $\gamma$ (upper), and the temperature at mean density, $T_{0}$ (lower). Lower right panel: evolution of the temperature at the characteristic density, $T(\Delta)$ (upper), the curvature statistics, ${\langle|\kappa|\rangle}$ (middle), and the pressure smoothing scale, ${\lambda_{\rm P}}$ (lower). \[fig:cosmology\]](kurvTjeans_z_cosmo "fig:"){width="45.00000%"}
New Optically Thin Photoionization and Photoheating Rates {#app:tables}
=========================================================
We are making our new UVB models freely available for public use, so that they can be used by the whole community in future hydrodynamical simulations. We provide here the photoionization and photoheating rates for the late ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization (table \[tab:uv1\]), middle ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization (table \[tab:uv2\]), and early ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization (table \[tab:uv3\]) models, assuming a total heat input during reionization of $\Delta T_{{\ensuremath{\textrm{H} \, \textsc{i}}}}=2\times10^{4}$ K for ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and $\Delta T_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}=1.5\times10^{4}$ K for ${\ensuremath{\textrm{He} \, \textsc{ii}}}$. The rates for these models are also shown in Figure \[fig:newuv\]. We refer to Section \[sec:newmodel\] for a careful explanation on how these rates were obtained. In our new models we have also applied a small correction to the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ and ${\ensuremath{\textrm{He} \, \textsc{i}}}$ photoionization rates of HM12 once reionization is completed to ensure that the simulations match current best observations of the ${\ensuremath{\textrm{H} \, \textsc{i}}}$ mean flux at different redshifts. The goal is to reduce the effect of the current standard post-process rescaling approach done with simulations that aim to reproduce [Lyman-$\alpha$]{} statistics. The photoheating rates have been corrected by the same factor in order to keep the heat input per volume element constant in the models and therefore keep exactly the same thermal histories. Section \[sec:discuss\] elaborates on the limitations and applicability of these models. Table \[tab:uv1\], table \[tab:uv2\] and table \[tab:uv3\] are published in their entirety in a machine-readable format. Just a portion is shown here, as a guidance to its format and content. We encourage anyone interested in running some other specific model to contact the authors.
![Evolution of photoionization and photoheating rates with redshift for our new late reionization, middle reionization, and early reionization UVB models. See text for more details on how these models were computed. The HM12 and FG09 models are also shown for comparison. \[fig:newuv\]](treecool_photoioHI_time "fig:"){width="33.00000%"} ![Evolution of photoionization and photoheating rates with redshift for our new late reionization, middle reionization, and early reionization UVB models. See text for more details on how these models were computed. The HM12 and FG09 models are also shown for comparison. \[fig:newuv\]](treecool_photoioHeI_time "fig:"){width="33.00000%"} ![Evolution of photoionization and photoheating rates with redshift for our new late reionization, middle reionization, and early reionization UVB models. See text for more details on how these models were computed. The HM12 and FG09 models are also shown for comparison. \[fig:newuv\]](treecool_photoioHeII_time "fig:"){width="33.00000%"}\
![Evolution of photoionization and photoheating rates with redshift for our new late reionization, middle reionization, and early reionization UVB models. See text for more details on how these models were computed. The HM12 and FG09 models are also shown for comparison. \[fig:newuv\]](treecool_photoheHI_time "fig:"){width="33.00000%"} ![Evolution of photoionization and photoheating rates with redshift for our new late reionization, middle reionization, and early reionization UVB models. See text for more details on how these models were computed. The HM12 and FG09 models are also shown for comparison. \[fig:newuv\]](treecool_photoheHeI_time "fig:"){width="33.00000%"} ![Evolution of photoionization and photoheating rates with redshift for our new late reionization, middle reionization, and early reionization UVB models. See text for more details on how these models were computed. The HM12 and FG09 models are also shown for comparison. \[fig:newuv\]](treecool_photoheHeII_time "fig:"){width="33.00000%"}
[lcccccc]{} $\log_{10}(z+1)$ & $\Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & $\Gamma_{{\ensuremath{\textrm{He} \, \textsc{i}}}}$ & $\Gamma_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$ & $\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{i}}}}$ & $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$\
& (s$^{-1}$) & (s$^{-1}$) & (s$^{-1}$) & (erg s$^{-1}$) & (erg s$^{-1}$) & (erg s$^{-1}$)\
0.0000 & 5.700e-14 & 3.100e-14 & 1.122e-16 & 3.561e-25 & 4.486e-25 & 5.008e-27\
0.0212 & 7.131e-14 & 3.942e-14 & 1.291e-16 & 4.466e-25 & 5.632e-25 & 5.729e-27\
0.0414 & 8.817e-14 & 4.882e-14 & 1.564e-16 & 5.546e-25 & 6.944e-25 & 6.874e-27\
0.0607 & 1.081e-13 & 6.037e-14 & 1.892e-16 & 6.806e-25 & 8.499e-25 & 8.215e-27\
...\
[lcccccc]{} $\log_{10}(z+1)$ & $\Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & $\Gamma_{{\ensuremath{\textrm{He} \, \textsc{i}}}}$ & $\Gamma_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$ & $\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{i}}}}$ & $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$\
& (s$^{-1}$) & s$^{-1}$) & (s$^{-1}$) & (erg s$^{-1}$) & (erg s$^{-1}$) & (erg s$^{-1}$)\
0.0000 & 5.700e-14 & 3.100e-14 & 1.122e-16 & 3.561e-25 & 4.486e-25 & 5.008e-27\
0.0212 & 7.131e-14 & 3.942e-14 & 1.291e-16 & 4.466e-25 & 5.632e-25 & 5.729e-27\
0.0414 & 8.817e-14 & 4.882e-14 & 1.564e-16 & 5.546e-25 & 6.944e-25 & 6.874e-27\
0.0607 & 1.081e-13 & 6.037e-14 & 1.892e-16 & 6.806e-25 & 8.499e-25 & 8.215e-27\
...\
[lcccccc]{} $\log_{10}(z+1)$ & $\Gamma_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & $\Gamma_{{\ensuremath{\textrm{He} \, \textsc{i}}}}$ & $\Gamma_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$ & $\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ & $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{i}}}}$ & $\dot{q}_{{\ensuremath{\textrm{He} \, \textsc{ii}}}}$\
& (s$^{-1}$) & (s$^{-1}$) & (s$^{-1}$) & (erg s$^{-1}$) & (erg s$^{-1}$) & (erg s$^{-1}$)\
0.0000 & 5.700e-14 & 3.100e-14 & 1.122e-16 & 3.561e-25 & 4.486e-25 & 5.008e-27\
0.0212 & 7.131e-14 & 3.942e-14 & 1.291e-16 & 4.466e-25 & 5.632e-25 & 5.729e-27\
0.0414 & 8.817e-14 & 4.882e-14 & 1.564e-16 & 5.546e-25 & 6.944e-25 & 6.874e-27\
0.0607 & 1.081e-13 & 6.037e-14 & 1.892e-16 & 6.806e-25 & 8.499e-25 & 8.215e-27\
...\
[^1]: Baryon Oscillation Spectroscopic Survey (BOSS): https://www.sdss3.org/surveys/boss.php
[^2]: Notice that calculating the volume filling factor, $Q_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}$, is not the optimal way to describe reionization in optically thin simulations. One needs to set up an ionization threshold (standard values are 0.999-0.9) and compute how many cells in the simulation have an ionization fraction above this level. It is easy to see that in optically thin simulations the filling factor evolution will be just a step function that jumps to 1 as soon as the volume-averaged ionization fraction, ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}$, reaches the chosen threshold. The temperature evolution shown in Figure \[fig:Qhistgas0\] illustrates why this approach will not be the correct description of how reionization took place in these simulations.
[^3]: During the making of this paper, new constraints on reionization from Planck were published [@Planck:2016a], moving these constraints to a lower value and reducing the errors: ${\tau_{\rm e}}=0.058\pm 0.012$. These results do no change any of the conclusions of this paper.
[^4]: The recombination time for ${\ensuremath{\textrm{H} \, \textsc{i}}}$ reionization is generally defined as $t_{\rm rec}=[(1+\chi)\alpha_{B} C_{IGM} {\langle {n_{{\ensuremath{\textrm{H}}}}}\rangle}]^{-1}$ where $\alpha_{B}$ is the recombination coefficient to the excited states of hydrogen, $\chi$ accounts also for the presence of photoelectrons from singly ionized helium, and $C_{IGM}\equiv{\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}^2 \rangle}/{\langle {n_{{\ensuremath{\textrm{H} \, \textsc{ii}}}}}\rangle}^2$ is the clumping factor of ionized hydrogen. A practical issue is how $t_{\rm rec}$ should be evaluated when $Q < 1$, and in particular when $Q \ll 1$. We refer to the nice and detailed discussion on this issue done by [@So:2014]. In any case, recombination rates are relatively unimportant at high redshifts, and this possibility cannot explain the big discrepancy between the model and the simulation.
[^5]: These models also need to assume some galaxy escape fraction at each redshift that is generally chosen by first iteratively solving the integrated cosmological radiative transfer equation
[^6]: Results of simulations using the cutoff approach can be found in Appendix \[app:HMold\].
[^7]: Based on the same idea used to derive the photoionization rates, in Appendix \[app:hifrac\], we introduce a numerical method to compute the expected volumen-averaged ionization history outcome in optically thin hydrodynamical simulations from a specific mean photoionization rate, $\Gamma_{\gamma,{\ensuremath{\textrm{H} \, \textsc{i}}}}(z)$. We recommend this approach to be used in the future when generating different UVB tabulated models.
[^8]: We have assumed comoving coordinates: $\vec{r}_{\rm proper}=a\vec{x}$, $\rho$ is the comoving baryon density ($\rho=a^{3}\rho_{\rm proper}$), $p$ comoving pressure ($p=a^{3}p_{\rm proper}$), $\vec{v}$ is the proper peculiar baryonic velocity ($\vec{v}=\vec{v}_{\rm proper}-\dot{a}\vec{x}$), $\Phi$ is the modified gravitational potential, and $E$ is the total comoving energy ($E=E_{\rm proper}-a\vec{x}\cdot\vec{v}-\frac{1}{2}\dot{a}^{2}\vec{x}^{2}$). Refer to @Almgren:2013 for details.
[^9]: Photoheating terms in eqn. (\[eq:interenergy\]) are written as they are usually defined in hydrodynamical simulations, a photoheating rate, $\dot{q}_{{\ensuremath{\textrm{H} \, \textsc{i}}}}$ (in units of energy per time per ion) multiplied by the ion density, ${n_{{\ensuremath{\textrm{H} \, \textsc{i}}}}}$, of that resolution element. Therefore, the global heating due to reionization will be determined also by the amount of ions present, and we cannot expect a constant temperature increase, independent of density. We will discuss this in detail in Section \[sec:discuss\].
[^10]: As eqn. (\[eq:invphoto\]) above shows, we also need to define the total heat input expected from each reionization process because there is also a weak dependence on temperature due to the recombination factor.
[^11]: In this work we used ${\tau_{\rm e}}$ less constraining results based just on temperature and polarization Planck data. Using more data reduces slightly the best value, but it is still in agreement with this best value. During the making of this paper new constraints on reionization from Planck were published [@Planck:2016a], moving this constraints to a lower value and reducing the errors: ${\tau_{\rm e}}=0.058\pm 0.012$. These results do no change any of the conclusions of this paper and in fact emphasize the disagreement between standard UVB models and these observational constraints.
[^12]: We define the reionization redshift of the models at the redshift when ${\langle x_{{\ensuremath{\textrm{H} \, \textsc{ii}}}} \rangle}=1$
[^13]: For all simulations we saved an snapshot at the following redshifts: $20$, $19$, $18$, $17$, $16$, $15$, $14$, $13$, $12$, $11$, $10$, $9$, $8$, $7$, $6$, $5$, $4$, $3$, $3.8$, $3.6$, $3.4$, $3.2$, $3.0$, $2.8$, $2.6$, $2.4$, $2.2$, $2.0$, $1.8$, $1.6$, $1.0$, $0.5$ and $0.2$.
[^14]: Results of the simulations using HM96, HM01 and FG09 models can be found in Appendix \[app:HMold\]
[^15]: i.e., we renormalize the fluxes of each skewer dividing them by its maximum flux value. Then we only used pixels where the renormalized fluxes are in the range $0.1\leq F_{{\ensuremath{\textrm{H} \, \textsc{i}}}}^{R20}\leq0.9$.
[^16]: We have tested that changing these thresholds within reasonable IGM densities produce differences just at a few per cent level [see @Lukic:2015 for similar conclusions] and in any case it does not affect the conclusions presented in this work. We also found no relevant effects in the main results of this paper if we employed a different fitting approach as the one used in @Puchwein:2015.
[^17]: We do not directly compare to other measurements of $\gamma$ [@Ricotti:2000; @Schaye:2000; @McDonald:2001; @Garzilli:2012] because they are either significantly less precise, employ outdated simulations, do not sufficiently treat degeneracies between $T_{0}$ and ${\lambda_{\rm P}}$, or have other differences in methodology that make direct comparisons between them challenging.
[^18]: Using optimal densities given by @Boera:2014 [$A=-0.21838$ and $B=1.05603$] does not change any of the conclusions of this paper.
[^19]: This grid of simulations was created by modifying HM12 heating rates using two factors, $A$ and $B$: $\dot{q}=A\Delta^{B}\dot{q}_{HM12}$
[^20]: Note that in this plot observations compute the mean flux averaging over a much smaller window than in the simulations; however this would only change the variance, but not the mean.
[^21]: Notice that as mentioned above, some authors have tried to solve for this issue by applying different simple redshift cutoffs to the standard UVB models. Results of simulations using the cutoff approach can be found in Appendix \[app:HMold\].
[^22]: This method injects the same amount of heat regarding of the density at each resolution element and in fact, we have also implemented it in our code and did several tests. This creates a very different evolution of the temperature-density relation slope, $\gamma$, with redshift. The evolution for the temperature at mean density, $T_{0}$, was the same regardless of the method. This method has the advantage of only requiring an additional heating term and adding negligible computational overhead while capturing the timescale and magnitude of the heat input more realistically. Although we might explore it in more detail in the future, we decided to change to our current approach because it implied changing the standard algorithm used in several state-of-the-art cosmological hydrodynamical codes. Our current method produces new UVB models that can be directly plugged into any of these codes.
[^23]: During the making of this paper, new constraints on reionization from Planck were published [@Planck:2016a], moving these constraints to a lower value and reducing the errors: ${\tau_{\rm e}}=0.058\pm 0.012$. The HM01 and FG09 models are in clear disagreement at $1\times\sigma$ with these new constraints.
[^24]: Notice, however, that it is in good agreement with the new @Planck:2016a constraints.
[^25]: In all cases we are assuming the same abundance values for hydrogen, $X_{\rm p}$, and helium, $Y_{\rm p}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have studied the reconstruction of Pt(111) theoretically using a two-dimensional Frenkel-Kontorova model for which all parameters have been obtained from [*ab initio*]{} calculations. We find that the unreconstructed surface lies right at the stability boundary, and thus it is relatively easy to induce the surface to reconstruct into a pattern of FCC and HCP domains, as has been shown experimentally. The top layer is very slightly rotated relative to the substrate, resulting in the formation of “rotors" at intersections of domain walls. The size and shape of domains is very sensitive to the density in the top layer, the chemical potential, and the angle of rotation, with a smooth and continuous transition from the honeycomb pattern to a Moiré pattern, via interlocking triangles and bright stars. Our results show clearly that the domain patterns found on several close-packed metal surfaces are related and topologically equivalent.'
author:
- Shobhana Narasimhan
- Raghani Pushpa
title: |
The Reconstruction of Pt(111) and\
Domain Patterns on Close-packed Metal Surfaces
---
Due to its enormous importance as a catalyst, Pt(111) is one of the most widely studied surfaces. Under “normal" conditions, it has the flat topography expected of a bulk-truncated face-centered-cubic (FCC) (111) surface. However, experiments have shown that one can induce the surface to reconstruct into either a honeycomb structure or a pattern of interlocking triangles – for example, by heating the surface above 1330 K [@sandy], or by placing it in a supersaturated Pt vapor [@bott]. The reconstructed structure is comprised of domains where the bulk FCC stacking sequence is retained, alternating with domains where the surface atoms instead occupy hexagonal-close-packed (HCP) sites, sitting directly above atoms two layers below.
This is one of a family of similar reconstructions, formed by a tessellation of FCC and HCP domains, seen on Au(111) as well as various heteroepitaxial systems on the (111) faces of FCC metals, and the structurally similar (0001) faces of HCP metals [@naau; @agpt; @curu; @niru]. These structures have attracted a great deal of interest, especially because they can be used as templates for growing ordered arrays of nanoparticles [@chambliss; @voigt; @brune]. Possible applications for such nanostructures include nanoelectronics, information storage, and nanoscale chemical reactors. To design such nanostructures, it is desirable to understand the factors controlling the geometry and spacing of the reconstruction patterns.
In this paper, we study the structure of the Pt(111) surface theoretically. We show that the unreconstructed Pt(111) surface is in fact teetering right at the brink of a domain of stability. Thus, slight changes in the environment can trigger a reconstruction, whose periodicity and geometry are very sensitive to various extrinsic and intrinsic parameters. In addition to obtaining excellent agreement with the structures reported experimentally for Pt(111), we also observe most of the structures reported experimentally for other systems.
The presence or absence of such reconstructions involves a very delicate balance between various contributions to the total energy, and it is therefore desirable to have as accurate a description of interatomic interactions as possible. Unfortunately, the very large unit cells of the reconstructed surfaces make it unfeasible to perform a fully [*ab initio*]{} calculation. However, one can use [*ab initio*]{} calculations to parametrize a model; this is the approach we will follow.
The driving force for the reconstruction of these close-packed surfaces is tensile surface stress, i.e., surface atoms would like to be closer together than the bulk nearest-neighbor (NN) spacing $a$ [^1]. This tendency is however opposed by the fact that (in general) it costs energy when: (i) surface atoms lose registry with the substrate (ii) extra atoms must be provided to increase the surface density. These factors are incorporated in the Frenkel-Kontorova (FK) model, which has been widely used to study commensurate-incommensurate transitions [@fk]. In its original form, the model consists of a one-dimensional chain of atoms connected by harmonic springs of equilibrium length $b \ne a$, sitting in a sinusoidal substrate potential. This has the advantage of being exactly solvable, and can serve as a useful guideline for studying structural stability.
However, a proper description requires a generalized version of the FK model, with a two-dimensional (2D) layer of atoms interacting via a more realistic anharmonic potential $V_{ss}$, and sitting in a 2D potential due to bulk atoms, $V_{sb}$, with competing minima at FCC and HCP sites. The Hamiltonian is given by [@mansfield]:
$$\label{eq:fk} H = \sum_i V_{ss}(l_i) + \sum_jV_{sb}({\bf r}_j) + \Gamma N,$$
where $i$ runs over all NN bonds of length $l_i$ between surface atoms, and $j$ runs over all atoms at positions ${\bf r}_j$ in the surface layer. $N$ is the total number of atoms, and $\Gamma$ is a chemical potential that contains information about the energy required to incorporate an atom into the surface layer; this depends on where the atom comes from (bulk, step edge, adatom, etc.)
To obtain $V_{ss}$, $V_{sb}$ and $\Gamma$, we have performed [*ab initio*]{} density functional theory calculations to study the energetics of compressing the surface layer, making surface stacking faults, and extracting atoms from various sites. We use the PWSCF package [@pwscf], with a plane wave basis with a cut-off of 20 Ry, ultrasoft pseudopotentials [@ultra], and the local density approximation to the exchange-correlation potential. Surface calculations have been carried out using supercells with 9 layers of atoms, separated by a vacuum of 6-layer thickness. A Monkhorst-Pack grid corresponding to 27 [**k**]{}-points in the irreducible part of the surface Brillouin zone is used to sample reciprocal space for calculations using a $(1\times1)$ surface cell; the grid is varied commensurately when using larger surface cells. We obtain the bulk lattice constant as $a_0=3.92 \AA$, and find that the first interlayer spacing $d_{12}$ is slightly expanded by 0.3% relative to the bulk interlayer spacing, and the next two interlayer spacings $d_{23}$ and $d_{34}$ are slightly contracted by 0.55% and 0.18% respectively. We obtain the (unreconstructed) surface energy $\gamma$ and surface stress $\sigma$ as 9.13 and 29.45 ${\rm mRy}/\AA^2$ respectively. These results are in good agreement with experiments and previous calculations [@needs; @feibelman; @boisvert].
The trickiest part of the parametrization concerns $V_{ss}$, the interaction between surface atoms. Earlier authors (who studied other systems), either used “physically reasonable" parameters [@hamiltonagpt], or did calculations to study the energetics of a monolayer of atoms on a jellium [@takeuchi]; the ambiguity in the latter approach arises from the uncertainty in the choice of jellium density. We have chosen instead to calculate the variation in the surface stress when compressing a slab of atoms [^2]. The distance $d_{12}$ is allowed to relax, and it is therefore a good approximation to assume that the variation in surface stress comes entirely from the surface-surface bonds that we are interested in. We use these results to parametrize a Morse potential $V_{ss}=A_0\{1-\exp[-A_1(a-b)]\}^2$; we obtain $A_0$ = 60.2 mRy, $A_1$ = 2.062 $\AA^{-1}$, and $b$ = 2.638 $\AA$, i.e., surface bonds would like to shorten their length by 4.7% from the bulk NN distance $a=a_0/\sqrt 2=2.77 \AA$.
To obtain $V_{sb}$, we study surface stacking faults. We find that the energy cost (per surface atom) of having the surface layer occupy HCP, top and bridge sites instead of the most favored FCC site is 5.05, 11.85 and 5.89 mRy respectively. These values are then used to expand $V_{sb}$ in a 2D Fourier series, using the first two shells of reciprocal lattice vectors for the 2D surface lattice [@takeuchi; @narasimhan]. Fig. 1 shows $V_{sb}$ for a line cutting through the surface along the $[11\bar2]$ direction. Note that though the HCP site is a local minimum, it lies considerably above the FCC site in energy.
![The inset shows a top view of the Pt(111) surface, while the main graph shows the surface-bulk potential $V_{sb}$ along the line PQRS marked in the inset.[]{data-label="F_fig1"}](fig1.eps){width="60mm"}
Finally, to obtain $\Gamma$, we use our [*ab initio*]{} results for the bulk cohesive energy, the surface energy, the adatom adsorption energy and the energy needed to detach an atom from a kink site at a step edge [@unpub] to obtain $\Gamma_b$ = 50.56 mRy for bulk atoms, $\Gamma_a$ = -67.12 mRy for adatoms, and $\Gamma_k$ = 50.4 mRy for kink atoms.
To see how stable the unreconstructed surface is, one can map the 2D problem onto a 1D FK model [@mansfield], and evaluate the dimensionless parameter $R=(3\pi a)({4\over3} \sigma -
\gamma)/8\sqrt{2kW}$, where $k$ is the spring constant for NN surface bonds and $W$ is the amplitude of $V_{sb}$ along the zigzag line connecting adjacent FCC and HCP sites. If $R >/< 1$, the surface will / will not reconstruct. Mansfield and Needs [@mansfield] have obtained $R=0.73$ for Pt(111) and $R=0.44$ for Au(111), i.e., they predict that unreconstructed Au(111) should be more stable than Pt(111), whereas it is well known that Au(111) reconstructs even at low temperatures. We believe that the deficiency lies not with their model but with the values they used to obtain $R$. Upon approximating $V_{ss}$ by a harmonic potential in the region of interest, we obtain $k$=370 mRy/$\AA^2$. Using $W$ = 5.89 mRy, we get $R$ = 0.998, i.e., we find that the unreconstructed surface is [*only just*]{} stable. This explains why it is relatively easy to induce the surface to reconstruct by increasing the temperature (reducing $W$) or by placing the surface in a supersaturated vapor (effectively increasing the numerator of $R$).
For a more precise determination of structural stability, we work with the full 2D Hamiltonian of Eq. 1. We study the variation in the surface energy $\gamma$ as a function of $\Delta\rho$, the excess density in the surface layer (relative to the unreconstructed case), and $\theta$, the angle between the unit vectors of the top layer and the substrate. (Though optimizing $\theta$ makes only a very small difference to $\gamma$, we will show below that allowing for $\theta
\ne 0$ is crucial to explain the “rotors" at intersections of domain walls.) In all cases, we start with a uniformly compressed layer of surface atoms, and obtain the atomic positions that minimize $\gamma$, using a conjugate gradient algorithm.
![Results from the 2D Frenkel-Kontorova model for isotropic reconstruction. (a) and (b) show how $\Delta\gamma$, the change in the surface energy (relative to the unreconstructed surface) and $\theta$, the angle of rotation of the top layer, depend on the excess density $\Delta\rho$; (c) and (d) show how the optimal excess density $\Delta\rho_{min}$ and the periodicity $l$ vary with the chemical potential $\Gamma$. []{data-label="F_fig2"}](fig2.eps){width="75mm"}
For Pt(111), we find that isotropically compressed structures, with threefold symmetry and sixfold atomic coordination everywhere, are always lower in energy than the uniaxial “stripe" patterns or structures with point dislocations. Fig. 2(a) shows how $\Delta\gamma$, the difference in $\gamma$ for the reconstructed and unreconstructed surfaces, varies with $\Delta\rho$. Each point is also optimized with respect to $\theta$; Fig. 2(b) shows that $\theta$ varies approximately linearly with $\Delta\rho$ and is very small. If the surface reconstructs, $\Delta \gamma$ will have a minimum at a non-zero value $\Delta\rho_{min}$. Under normal conditions, extra atoms are obtained from the bulk or step edges, and $\Gamma \simeq \Gamma_b$. It is clear from the figure that under these conditions, the surface will not reconstruct, which agrees with experiment. However, if adatoms are available for incorporation in the surface layer, $\Gamma$ is lowered, up to a minimum of $\Gamma_a$, and the surface then reconstructs. Fig. 2(c) shows how $\Delta\rho_{min}$ varies as $\Gamma$ varies between $\Gamma_a$ and $\Gamma_b$. The surface reconstructs only if $\Gamma < $ 36 mRy. As $\Gamma$ decreases and $\Delta\rho$ increases, the periodicity $l$ of the reconstruction decreases; Fig 2(d) shows how $l$ varies with $\Gamma$.
Our most striking results are obtained upon examining the domain patterns using a simple technique to obtain the surface corrugation and thus simulate STM images. The height of each atom in the unit cell is obtained using a 2D Fourier expansion similar to that used for $V_{sb}$, expanding about the heights at FCC, HCP, bridge and top sites. From our [*ab initio*]{} calculations on surface stacking faults, we obtain these heights as 0, 0.03, 0.29 and 0.06 $\AA$ respectively. These are smaller than the corrugations measured experimentally [@bott]. However, we find that [*ab initio*]{} simulations of the reconstructed structure (for small cell sizes accessible to computation) support our smaller values [@unpub]. As has been reported recently for Au(111), it appears that the STM exaggerates the value of the corrugation [@stm]; in any event, we are primarily interested in the domain patterns, rather than in the absolute heights.
![ Simulated STM images of the Pt(111) surface, as $\Delta\rho$ is varied from 2.9% to 21%. Individual atoms are shaded according to their height; lighter atoms are higher. In (a) the top layer is aligned with the substrate, in (b) to (f), the angle betwen the top layer and substrate has been optimized. (a) and (b) are the honeycomb, (c) and (d) are threefold whorls (e) is the bright star, and (f) is the Moiré pattern. The black/white line in each image has a length of 50 $\AA$[]{data-label="F_fig3"}](fig3.eps){width="65mm"}
A sequence of simulated STM images is shown in Fig. 3. In all six images, black, dark grey, light grey and white areas correspond to regions where atoms sit at FCC, HCP, bridge and top sites respectively. A non-linear grey scale has been used to show the domain walls (bridge sites) clearly. Fig. 3(a) shows the pattern obtained when $\Delta\rho$=2.9%, and $\theta=0$. Large hexagonal FCC domains are separated from narrow HCP domains by domain walls that form a honeycomb network. Alternate vertices of the hexagons are extremely bright; atoms here sit at top sites. Between these bright vertices, there is a “three-pointed star" similar to that obtained in an earlier simulation of the nucleation of the reconstruction [@jacobsen]. Fig. 3(b) shows the structure obtained for the same $\Delta\rho$, but with $\theta$ at its optimal value of $0.14^{\circ}$. It is evident from a comparison of the two images that breaking the symmetry by even this tiny amount has a big impact on the structure. The bright vertices are now transformed into “bright rotors", which are indeed observed experimentally [@hohage]. The similarity between Fig 3(b) and the experimental STM images [@bott; @hohage] is striking. Our finding that the rotors arise from a rotation of the top layer relative to the substrate explains why, in a given region, experiments tend to find either all clockwise or all anticlockwise rotors. The honeycomb pattern occurs in a region where $l$ is very sensitive to $\Gamma$; we believe this explains why STM images usually show irregular hexagons.
As $\Gamma$ is lowered, the surface layer densifies further. Figs. 3 (c) and (d) show the structures obtained when $\Delta\rho$ is increased to 4.0% and 6.8% respectively. The honeycomb transforms into a pattern of interlocking wavy triangles. Such structures have been observed on dense terraces of Pt(111) [@hohage]. This structure has also been seen for Na/Au(111) [@naau] and for 3 ML of Cu/Ru(0001) [@curu]. As $\Gamma$ is lowered and $\Delta\rho$ increased still further, the wavy domain walls straighten out, resulting in the “bright star" pattern shown in Fig. 3(e), for a value of $\Delta\rho$= 10.2%. This pattern has not been seen on Pt(111) as it requires a very low $\Gamma$ (high adatom coverage); however it is seen on other surfaces, e.g., Ni/Ru(0001) [@niru]. $\Delta\rho_{min}$ for Pt(111) has a maximum value of 12% (when $\Gamma=\Gamma_a$); it is nevertheless instructive to see what happens for larger $\Delta\rho$. The triangular domains of the bright star transform into hexagons, resulting in the Moiré pattern shown in Fig 3(f). The density distribution on the surface is now almost uniform. This pattern is also observed experimentally, e.g., for 4 ML of Cu/Ru(0001) [@curu].
The sequence of images shown in Fig. 3 makes it clear that all these structures “morph" smoothly into one another and are topologically equivalent. Progressively larger values of $\Delta\rho$ can be favored either by lowering $\Gamma$ \[as for Pt(111)\], or by increasing the surface stress – e.g., by depositing more overlayers in a heteroepitaxial system. Indeed, a similar progression of structures with overlayer thickness has been observed for the Cu/Ru(0001) system [@curu; @hamiltoncuru]. Our results show that the same physics is operating in all the systems mentioned above, and it is only slight changes in parameters that are responsible for the various domain patterns observed on different surfaces.
To summarize: we have shown that the unreconstructed Pt(111) surface is only just stable, and can thus be easily induced to reconstruct. We have confirmed that Pt(111) will reconstruct if the chemical potential is lowered by the presence of a large number of adatoms. In addition to the methods tried to date, we suggest that it should also be possible to induce the surface to reconstruct in an electrochemical environment or by depositing alkali metals on it. The domain patterns we obtain for Pt(111) are in excellent agreement with experiment. We have also shown that slight variations in conditions can lead to many of the other domain patterns seen on similar surfaces. We have shown that the periodicity of the reconstruction can be controlled by varying the chemical potential; this is important if one would like to use such surfaces as templates for growing ordered nanostructures.
We thank S. de Gironcoli for providing the Pt ultrasoft pseudopotential, and X. Gonze, I.K. Robinson, H. Brune and U.V. Waghmare for helpful discussions.
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[^1]: For heteroepitaxial systems, the driving force is sometimes compressive (not tensile) surface stress.
[^2]: In principle, only the top layer should be compressed; however for commensurate compressions of 2/3, 3/4 and 4/5 we have verified that the two procedures give essentially the same result.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The optical analog of vacuum triode with electron flow being replaced by photon flow — optical triode (OT) — is considered. Distinctions of such a device with respect to vacuum and semiconductor triodes are discussed. As an illustration and example of possible application of OT the design of RF generator without conventional active elements is experimentally demonstrated. The amplification and feedback are realised in optical channel by means of Pockels cell and vacuum photodetector. The application of suggested device as a sensitive photo receiver is discussed.'
author:
- 'Gleb G. Kozlov'
- 'Valentin G. Davydov'
title: 'To the abilities of optical triode: optical triode-based RF generator.'
---
Introduction {#introduction .unnumbered}
============
The fantastic development of radio technology and the use of electronic devices in almost all areas of human activity began with the invention by Lee de Forest in 1906 of a high-speed electric signal amplifier — a vacuum triode [@f1; @f2; @f3; @f4; @f5]. Despite the fact that currently vacuum triodes have mostly given way to solid state semiconductor amplifying elements, both of them have many common features, and many circuitry solutions found in the era of electro-vacuum amplifying devices still remain relevant. The simple design of a vacuum triode clearly illustrates the very principle of the amplifying element as a device in which relatively weak energy flow controls a stronger one. In the Lee de Forest vacuum triode, the electron flux arising due to thermionic emission from the cathode and the accelerating electric field of the anode is modulated by electric field of an electron-transparent grid placed between the anode and cathode. Changes in the grid potential leads to change (up to complete blocking) of the electron current from the cathode to the anode, and the electric power in the anode circuit can significantly exceed the power in the grid circuit (power gain).
In the proposed article, it is shown that a similar structure of the amplifying element can be completely reproduced in a device in which the electron beam is replaced by a light (photon) beam. The suggested device — an optical triode — is implemented with simple laboratory means and can serve to explain the principles of amplification and demonstration of elementary circuitry solutions in school and student classrooms.
The article is organized as follows. The first section describes the design of the optical triode, introduces the definition of its parameters, and compares the optical triode with conventional amplifying elements. The second section describes sample design of an optical-triode-based RF oscillator and provides its theoretical analysis. The third section describes the experimental implementation of the RF generator and discusses its usage as an active photodetector. The results of the paper are briefly summarized in Conclusion.
The optical triode
==================
Consider the electro-optical modulator (Pockels cell with associated polarization elements) and the photodetector (semiconductor or vacuum photodiode) sequentially arranged in a light beam. The variation of voltage $U$ on the electro-optical modulator will cause a change in transmitted light intensity and hence the photocurrent $I$. If $ U_ {\lambda / 2} $ is the electro-optical modulator half-wave voltage, then the following estimate holds $${dU\over dI}\equiv {1\over S}\sim {U_{\lambda /2}\over I_0}
\label{1}$$ where the photocurrent of “open" modulator $ I _0 $ can be derived from light beam intensity $ P $ and quantum efficiency of the photodetector $ \eta $ by the relation: $$I_0={\eta P e \over \hbar \omega},$$ where $e$ is electron charge and $\hbar \omega$ — photon energy. Described sequence of electro-optical elements (which we will call [*optical triode*]{}) works similar to electro-vacuum triode: light source plays role of the cathode, electro-optical modulator — grid, and photodetector — anode. By analogy with the vacuum triode, we consider transconductance $ S $ of the optical triode. Due to the negligible electrical conductance of the Pockels cell, the possibility of power amplification by an optical triode is quite obvious.
Let us estimate the possible transconductance of optical triode. Take a semiconductor laser with output power $ \sim $ 1.5 W as a light source, and a semiconductor photodiode with quantum efficiency $ \eta \lesssim 1 $ as a photodetector. Then current $I_0$ is $ \sim 0.5 $ A. To obtain a low half-wave voltage, the crystal in Pockels cell should be made as thin (in the direction of electric field) as possible. It is known that DKDP crystal thickness in off-the-shelf ML-103 cell is $ \sim 3 $ mm, the crystal length is $ \sim 50 $ mm and a half-wave voltage is $ \sim 200 $ V for visible light. It seems to be feasible to reduce the crystal thickness by 100 times (to 30 $\mu$m) at the cost of length reduction by 10 times (to 5 mm). Then transconductance of the optical triode is expected to be $ S \sim 250 $ ma/V, which is an order of magnitude greater than that of vacuum triodes with similar power consumption.
The input impedance of optical triode is expected to be greater than that of insulated gate field effect transistor (MOSFET) of comparable transconductance, mainly due to lower capacitance of the Pockels cell.
Transconductance $S$ of the optical triode can be linearly controlled by changing power of the light beam, e. g. by an additional optical modulator. Therefore one can create voltage controlled amplifiers.
Yet another valuable advantage of optical triode is its inherent galvanic isolation between input and output. Passthrough capacitance can be almost arbitrarely reduced by increasing the distance of light travel between optical modulator and photoreciver.
![Two-cascade electro-optical amplifier[]{data-label="fig1"}](2C)
Within the same general principle of electro-optical active element its design can be varied depending on the application. As an example, two cascade linear amplifier is presented in Fig. \[fig1\]. The polarisation beam splitters P1 and P2 are oriented so that when the input voltages of Pockels cells C1 and C2 are zero, photocurrents of D1, D2 are equal to each other (the same for D3, D4). Therefore net currents in load resistors R4 and R are zero (both pairs of photodiodes are balanced). If input voltage of Pockels cells becomes nonzero then intensities of horizontal and vertical (as depicted in Fig. \[fig1\]) light beams change in opposite directions giving rise to nonzero currents in load resistors R4 and R. If most of voltage amplification is provided by the first cascade, then intensity $I_1$ can be chosen relatively small. Required amplification coefficient $K$ can be obtained by choosing the value of resistor $R_4$ sufficiently large. For example, if $S_1=25$ ma/V ($I_1\sim 150$ mW) and $R_4=100$ k$\Omega$, we have $K_1\sim S_1R_4=2500$. To obtain large enough power in the load resistor $R$ the intensity $I_2$ should be sufficient to provide the required current of the second cascade photodetectors. For $I_2=0.25$ A, $S_2=100$ mA/V, $\pm U=15$ V and $R=50$ $\Omega$ the amplification coefficient of the second cascade is $K_2\sim S_2 R\sim 5$, and the maximum output power is $P_\text{out}=I_2^2R\sim 3$ W. The feedback circuit R3, R2 is used to linearise the amplifier’s amplitude response. Resulting amplification coefficient can be approximated as $$K={(R3+R2)\over R2}$$ provided that $K\ll K_1K_2$. Depending on the feedback loop coefficient $K_1K_2/K$, total optical length (and hence time delay) from Pockels cells to photodetectors and capacitances of Pockels cells and photodiodes, an amplifier with feedback could become unstable. To maintain stability, one can implement the Boucherot cell in parallel to (or even instead of) R4.
In conclusion let us derive the input noise voltage in a frequency band $\Delta\nu$ of the cascade with optical triode from the shot noise of the photocurrent. Calculation gives $$\sqrt{\langle \delta U^2\rangle}=U_{\lambda /2}\sqrt{\hbar\omega \Delta\nu\over P\eta}$$. The estimation of noise voltage in unit frequency band for the amplifier duscussed above gives the value $\sim $ 2 nV$/\sqrt{\mathrm Hz}$, which is comparable with best devices built upon bipolar transistors.
The RF oscillator on optical triode. Quantitative consideration.
================================================================
![LC oscillator based on optical triode. Polarization optics and parasitic impedances are not shown.[]{data-label="fig2"}](OSC)
Authors were limited in facilities to implement in hardware full-scale amplifier presented in the previous section. Nevertheless the possibility of obtaining the optical-triode-based amplification was demonstrated in a simple experiment described below. Let us consider the optical-triode-based RF generator (Fig. \[fig2\]). The light beam after passing through the electro-optical modulator M connected to the LC tank hits the photocathode of vacuum photocell D which is also connected to the same tank thus forming the feedback.
The DC components of currents and voltages in the schematics Fig. \[fig2\] affect only power consumption and will be omitted from further analysis. Hereinafter only AC components of all voltages and currents are considered. Then current $I$ of the photodetector can be expressed in terms of voltage $U$ on the electro optical modulator as $I=SU-S_3U^3$, were $S$ is defined by Eq. (\[1\]) and the cubic term represents general nonlinearity which is necessary for obtaining definite amplitude of steady state oscillations irrespective to the initial conditions. Denoting voltage on the capacitor as $U_C$, current of the photodetector as $I$, current in the coil as $I_L$, active resistance of the coil as $R$ (not shown in the Fig. \[fig2\]), one can write the following system of equations for these quantities:
$$\begin{aligned}
I&=SU_C-S_3U_C^3,
\label{2}\\
I&=C{dU_C\over dt}+I_L,
\label{3}\\
U_C&=L{dI_L\over dt}+RI_L.
\label{4}
\end{aligned}$$
By linearization of this system (i. e. ommiting the term $U_C^3$) one can analyse it’s stability and obtain the following necessary condition of unstability with respect to oscillations in the system: $${LS\over RC}={Z^2S\over R}=QZS>1.
\label{5}$$ Here $Z=\sqrt{L/C}$ is the characteristic impedance and $Q=Z/R$ is the Q-factor of the tank circuit. The estimation by this formula shows the possibility (not very strong) of generation.
The exact analytical solution of nonlinear system Eqs. (\[2\]–\[4\]) is not possible. Therefore, to obtain an amplitude of the steady state oscillations we have to use some [*ad hoc*]{} assumptions[@fn], namely i) the solution of the considered system at sufficiently large time is a periodic oscillations of constant amplitude, and ii) these oscillations are nearly harmonic with a frequency $\omega$ close to the resonant frequency $\omega_0=1/\sqrt{LC}$ of the tank circuit. Also we will consider the tank circuit Q-factor to be large: $Q\gg 1$. Solving (\[4\]) with respect to $I_L$ we obtain $$I_L(t)=I_L(0)\exp \left(-{t\over \tau}\right)+\int\limits_0^t\!\exp \left({t'-t\over \tau}\right)\,
{U_C(t')\over L}dt,'
\label{6}$$ where $\tau\equiv L/R$. In accordance with the above assumptions we set $$U_C(t)=A\cos(\omega t),
\label{7}$$ where $A$ denotes amplitude of oscillations and $\omega\approx \omega_0$. Then from Eq. (\[6\]) one can obtain the following equation for $I_L(t)$ at $t\gg\tau$: $$I_L(t)=A\left({\sin (\omega t)\over \omega L}+{\cos (\omega t)\over \omega^2 \tau L}\right).
\label{8}$$
Here we neglect terms of the order $\sim 1/\omega^2\tau^2\approx 1/Q^2\ll 1$. If considered circuit reached the regime when the assumptions i) and ii) are hold, then we can expect that nothing will substantially change when a narrow-band filter transmitting only frequencies in the viscinity of $\omega_0$ will be installed between photodetector and tank circuit. The LC circuit itself plays role of such a filter. It means that before substituting the current $I$ from Eq. (\[2\]) to (\[3\]) one can omit all harmonics in it with the exception of $I_\omega$. Substituting (\[7\]) to (\[2\]), we obtain: $$I_\omega= \cos(\omega t)\left(SA-{3S_3A^3\over 4}\right).
\label{9}$$ Then substituting to (\[3\]) $I_\omega$ obtained by (\[9\]), we get $$A\sin(\omega t)\left({1\over \omega L}-\omega C\right)=
A\cos(\omega t)\left({1\over \omega ^2\tau L}-S +{3\over 4}S_3A^2\right).$$ To satisfy this equation for any $t$ with nonzero $A$, both parenthesized differences must vanish. That can be written as two separate quadratic equations for $\omega$ $${1\over\omega L}=\omega C$$ and $A$ $${3\over4}S_3A^2=S-{1\over\omega^2\tau L},$$ which have their respective solutions $$\omega={1\over \sqrt{LC}}
\label{11}$$ and $$A=2\sqrt{S-RC/L\over 3S_3}.
\label{12}$$ Equation (\[11\]) shows that assumptions i) and ii) are consistent and (\[12\]) illustrates critical behaviour of the oscillations amplitude with excitation threshold exactly defined by relation (\[5\]).
Due to the fact that $S$ and $S_3$ are both proportional to the light intensity $P$ one can see that the sensitivity of the described oscillations amplitude to the small light intesity fluctuations $dA/dP$ diverges at the excitation threshold. This allows one to use near-critically excited oscillator as a sensitive photodetector.
Experiment
==========
In experiment we used cylindrical coil 30 mm in length and 25 mm in diameter comprising of two layers of 0.9 mm thick silver-plated copper wire. The tank circuit was formed by this coil together with the sum of input capacitance of Pockels cell, output capacitance of photocell, internal coil capacitance and other parasitic capacitances. We used linearly polarized 0.5 W argon laser as a light source, Pockels cell combined with a Glan-Taylor prism ML-103 as an electro-optical modulator and vacuum photoelectric cell F-22 with bulk photocathode as a photodetector. For the sake of better linearity we installed a matte scatterer in front of the photocell to uniformly illuminate the whole photocathode surface. The RF generation was observed by means of the auxiliary coil L’ (see Fig. \[fig2\]) located 5 mm apart from the tank circuit. Amplitude of the RF voltage observed on an 100 $\Omega$ load resistor reached $\sim 3$ V.
As it was already mentioned, if the generation threshold is barely exceeded, the amplitude of oscillations in the system strongly depends on light intensity (see Eq. (\[12\])). This fact allows one to use weakly excited generator as an “active" photodetector. In our experimental setup it was easy to obtain the regime when 5% fluctuations of the laser beam intensity produced 1 V modulation of the output RF voltage on 100 $\Omega$ load resistor. This is two orders of magnitude greater than voltage observed from the same intensity fluctuations with the same photocell on the same 100 $\Omega$ load resistor connected in conventional circuit (that is without coils and Pockels cell).
Conclusion
==========
The paper describes an optical analogue of the electro-vacuum triode, in which the electron flux is replaced by a photon flux, and the grid is replaced by an electro-optical modulator. The properties of the proposed amplifying element are considered. A scheme of an RF generator on an optical triode is proposed and its quantitative analysis is given. The possibility of using an optical triode generator as an active photodetector is demonstrated. According to the authors, the described RF generator on an optical triode can serve to demonstrate the principles of designing amplifying elements and simple radio circuits based on them in school and student audiences.
[99]{} *The Continuous Wave: Technology and American Radio, 1900 – 1932*, by Hugh G. J. Aitken, 1985. *De Forest and the Triode Detector*, by Robert A. Chipman, Scientific American, March 1965, pages 93 – 101. *Saga of the Vacuum Tube*, by Gerald E. J. Tyne (Indianapolis, IN: Howard W. Sams and Company, 1977). *Empire of the Air: The Men Who Made Radio*, by Ken Burns a PBS Documentary Video 1992. *Lee de Forest and the Invention of Sound Movies, 1918–1926* by Mike Adams, The AWA Review (vol. 26, 2013). A similar analysis method is described in the book: Andronov A.A, Vitt A.A. and Khaykin S.E., *Theory of Oscillations.* Translated from the Russian by F. Immirzi. Pergamon Press. Oxford-N.Y.-Toronto, Ont., 1966.
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abstract: 'This work develops a covariance function which allows for a stronger spatial correlation for pairs of points in the direction of a vector such as wind and weaker for pairs which are perpendicular to it. It derives a simple covariance function by stretching the space along the wind axes (upwind and across wind axes). It is shown that this covariance function is anisotropy in the original space and the functions is explicitly calculated.'
---
\
[Reza Hosseini]{}
Introduction
============
Many spatial processes show non-stationary behavior in the correlation function across space. Therefore various non-stationary models are proposed in the literature to deal with this non-stationarity. A classical work in this area is [@sampson-1992] which utilizes non-parametric [*space transformations*]{} of the space where the process is transformed into another space where the process is almost isotropy. In the presence of wind, we can expect a specific type of non-stationarity in which the correlation is larger between a pair of points along the wind direction (angle=0) and lower between a pair of points perpendicular to the wind direction (with the same distance as before). We can also expect that this non-stationarity varies smoothly as we vary the angle from 0 to $\pi/2$.
In this work using an intuitively appealing space transformation which satisfies the aforementioned expectation we derive the exact form of the covariance function for a wind of given velocity and direction (Section \[sect:covariance-derivation\]). It turns out that the covariance function is an anisotropic covariance function (see [@banerjee-2003]) of a specific form which we calculate here.
Covariance derivation {#sect:covariance-derivation}
=====================
We denote the wind speed vector by ${\bf v}$ and assume it is constant across the domain we consider. Here we only consider a 2-dimensional vector but it is possible to extend this method to 3 dimensions (and more). We denote the magnitude of the wind by $v$ and its angle with the $x$ axis by $\theta_0$. It is useful to consider a new set of coordinates with the same origin and x-axis with the same direction as wind and y-axis perpendicular to it. We denote the coordinates of a point $(x,y)$ in the original plane by $(x^*,y^*)$ and in the new coordinates which we call the [*wind coordinates*]{}. The idea of the method is depicted in Figure \[wind\_cov\_construct.pdf\] where the original space is considered to be the circle depicted in bold and the deformed space is given with in dashed. In the new space the transformed pair of points for which the connecting line segment is parallel to the wind direction are closer than the original space. We can find the transformation to achieve this result by stretching and rotation of the space in three steps: (1) rotate the space counterclockwise with the angle $\theta_0$ to find the coordinates of the points in the wind coordinates; (2) stretch the space along the wind axes; (3) rotate back the results to the get the value of the transformation in the original space.
Let us denote the counter-clockwise rotation matrix of angle $\theta_0$ by $R(\theta_0)$ which is given by $$R(\theta_0)=\left(\begin{array}{cc}\cos(\theta_0) & -\sin(\theta_0)\\
\sin(\theta_0) & \cos(\theta_0)\\ \end{array}\right)$$
Also let $a,b\geq 0$ denote the magnitude of the [*stretch*]{} parallel to the wind direction and perpendicular to it respectively. Denote its matrix by $S(a,b)$ which is given by $$S(a,b)=\left(\begin{array}{cc}1/a & 0\\
0 & 1/b\\ \end{array}\right).$$ Suppose $(x,y)$ is given in the original space with $(x^*,y^*)$ denoting the same point in the wind coordinates: $$(x^*,y^*)^t = R(\theta_0) (x,y)^t,
\label{eqn-trans}$$ where $t$ denotes the matrix transpose operation. We assume that the point $(x,y)$ is stretched along the axes of the wind coordinates: $$\left(\begin{array}{c}x^* \\ y^* \\ \end{array}\right) \mapsto \left(\begin{array}{c}x^*/a \\ y^*/b \\ \end{array}\right)=S(a,b)\left(\begin{array}{c}x^* \\ y^* \\ \end{array}\right)$$ We are interested to find the transformation formula in the original space. In order to do so, we find the coordinates of $(x^*,y^*)$ in the original space. To that end, we can apply the rotation $R(-\theta_0)=R(\theta_0)^t$ (where $t$ denotes matrix transpose). Hence by Equation \[eqn-trans\], the above mapping can be written in the original space as $$R(\theta_0)^t \left(\begin{array}{c}x \\ y \\ \end{array}\right) \mapsto S(a,b)R(\theta_0)^t \left(\begin{array}{c}x \\ y \\ \end{array}\right).$$ By multiplying $(R(\theta_0)^t)^{-1}=R(\theta_0)$ to the two sides, we can write the above mapping in the original space as $$\left(\begin{array}{c}x \\ y \\ \end{array}\right) \mapsto R(\theta_0)S(a,b)R(\theta_0)^t \left(\begin{array}{c}x \\ y \\ \end{array}\right).$$
We call $\gamma=a/b$ the [*relative stretch*]{} parameter. Then we can write $$A=(1/b)R(\theta_0)S(\gamma,1)R(\theta_0)^t,$$ we denote $R(\theta_0)S(\gamma,1)R(\theta_0)^t$ by $A[\gamma]$ which is a symmetric matrix.
The hope is that the covariance in the new space has a simple form so that after applying this transformation, we can model the covariance appropriately using a simple model. We assume that in the new space the covariance is isotropic, i.e. it is only a function of the Euclidean distance. The Euclidean distance between two points is given by $${\bf s}=\left(\begin{array}{c}x \\ y \\ \end{array}\right),\;\;{\bf s'}=\left(\begin{array}{c}x' \\ y' \\ \end{array}\right)$$ which in the original space is equal to $$d_E(\bf{s},\bf{s'})= {\bf h}^t {\bf h},$$ where ${\bf h}={\bf s} - {\bf s'}.$ However as indicated we would like to use the Euclidean distance in the new space as the distance used in isotropic covariance models: $$d({\bf s},{\bf s'})=d_E(A{\bf s},A{\bf s'}) = (A{\bf h})^t (A{\bf h})=\sqrt{{\bf h}^t A^tA {\bf h}}=(1/b^2)\sqrt{{\bf h}^t A[\gamma]^tA[\gamma]} {\bf h}.$$ Note that $$A[\gamma]^tA[\gamma] = R(\theta_0) S(\gamma,1) R(\theta_0)^tR(\theta_0) S(\gamma,1) R(\theta_0)^t=
A[\gamma^2]$$
As an example consider a gaussian covariance function: $$C({\bf h}) =\sigma^2 \exp(-{\bf h}^t{\bf h}/\phi),$$ the new covariance function is given by $$C(s,s') =\sigma^2 \exp(-({\bf h}^tA[\gamma^2]{\bf h})/(\phi b^2)),$$ which implies that the parameter $b$ is not identifiable and can be absorbed into the range parameter $\phi$ by the change of variables $(\phi b^2) \mapsto \phi $ to arrive at the anisotropy covariance function: $$C(s,s') =\sigma^2 \exp(-{\bf h}^tA[\gamma^2]{\bf h}/\phi).$$ Figure \[wind\_cov\_example.pdf\] depicts the application of this method to the exponential covariance function $C(s,s') = \exp(-({\bf h}^t{\bf h})^{1/2}).$
Conclusion
==========
We obtained a simple covariance function which achieves higher spatial correlation for pairs of points in the direction of wind and lower for pairs which are perpendicular and give the closed-form formula. It turns out that the covariance function is anisotropy and except for the wind angle depends on only one other parameter which we called the relative stretch parameter, $\gamma$. The relative stretch can be considered to be a function of the wind speed for modeling, for example by letting $\gamma = \exp(v \gamma')$. In that case $\gamma'=0$ correspond to no stretch case which is also the case when the wind speed is zero ($v=0$) as desired. $\gamma'>0$ corresponds to a stretch perpendicular to the wind vector and $\gamma'<0$ corresponds to a stretch parallel to the wind vector. The same formulation can be used for transforming a kernel for averaging a predictor such as a traffic variable which is a proxy for pollution source emission. In that context we can transform an isotropic kernel (with circle contours) which is utilized for averaging the source effect around a given spatial point $s$ to a kernel which is directed parallel to the wind (with ellipse contours) at that point. The resulting kernel in that case will have exactly the same form as we discussed here.
![The space deformation for $a=3, b=2/3, \theta=\pi/12$. The original space is considered to be the circle depicted in bold and the deformed space is given with in dashed. The bold perpendicular lines depict the original axes and the dashed perpendicular lines depict the new axes.[]{data-label="wind_cov_construct.pdf"}](wind_cov_construct.pdf){width="40.00000%"}
![The exponential covariance function transformation for a=3,b=1, $\theta=\pi/12$. The left panel is the original covariance function and the right panel is the covariance function of the deformed space.[]{data-label="wind_cov_example.pdf"}](wind_cov_example1.pdf "fig:"){width="40.00000%"}![The exponential covariance function transformation for a=3,b=1, $\theta=\pi/12$. The left panel is the original covariance function and the right panel is the covariance function of the deformed space.[]{data-label="wind_cov_example.pdf"}](wind_cov_example2.pdf "fig:"){width="40.00000%"}
[**Acknowledgements**]{}: The author gratefully acknowledges useful discussions with Drs Duncan Thomas, Kiros Berhane, and Meredith Franklin from University of Southern California. This work was partially supported by National Institute of Environmental Health Sciences (5P30ES007048, 5P01ES011627, 5P01ES009581); United States Environmental Protection Agency (R826708-01, RD831861-01); National Heart Lung and Blood Institute (5R01HL061768, 5R01HL076647); California Air Resources Board contract (94-331); and the Hastings Foundation.
[13]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
Sampson, P. D., and Guttorp, P. (1992) *Nonparametric estimation of nonstationary spatial covariance structure.* Journal of the American Statistical Association, 87:108–119
Banerjee, S., Gelfand, A. E., and Carlin B. P. (2003) *Hierarchical Modeling and Analysis for Spatial Data.* Chapman & Hall/CRC Monographs on Statistics & Applied
Gelfand, A. E., Diggle, P., Guttorp, P., and Fuentes, M. (2010) *Handbook of Spatial Statistics* Chapman & Hall/CRC Handbooks of Modern Statistical Methods
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abstract: 'We provide a rigorous mathematical foundation to the study of strongly rational, holomorphic vertex operator algebras $V$ of central charge $c = 8, 16$ and $24$ initiated by Schellekens. If $c = 8$ or $16$ we show that $V$ is isomorphic to a lattice theory corresponding to a rank $c$ even, self-dual lattice. If $c = 24$ we prove, among other things, that either $V$ is isomorphic to a lattice theory corresponding to a Niemeier lattice or the Leech lattice, or else the Lie algebra on the weight one subspace $V_1$ is semisimple (possibly $0$) of Lie rank less than $24$.'
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amssym.def amssym \#1[\_[\_[\#1]{}]{}]{} ł ø Ø
\[section\] \[th\][Proposition]{} \[th\][Corollary]{} \[th\][Lemma]{} \[th\][Remark]{} \[th\][Definition]{} \[th\][Conjecture]{} \[th\][Example]{}
\
Chongying Dong[^1] and Geoffrey Mason[^2]\
Department of Mathematics, University of California, Santa Cruz, CA 95064
Introduction
============
One of the highlights of discrete mathematics is the classification of the positive-definite, even, unimodular lattices $L$ of rank at most $24$, due originally to Minkowski, Witt and Niemeier (cf. \[CS\] for more information and an extensive list of references). One knows (loc. cit.) that such a lattice has rank divisible by $8$; that the $E_8$ root lattice is (up to isometry) the unique example of rank $8$; that the two lattices of type $E_8 + E_8$ and $\Gamma_{16}$ are the unique examples of rank $16$; and that there are $24$ inequivalent such lattices of rank $24$. In each case, the lattice may be characterized by the nature of the semi-simple root system naturally carried by the set of minimal vectors (i.e. those of squared length $2$).
The theory of vertex operator algebras is a newer subject which enjoys several parallels with lattice theory. Already in \[B\], Borcherds pointed out that one can naturally associate a vertex operator algebra $V_L$ to any positive-definite, even lattice $L$ (cf. \[FLM\] for a complete discussion). It is known that $V_L$ is rational and that it is holomorphic precisely when $L$ is self-dual (\[D\], \[DLM1\]) (we defer the formal introduction of technical definitions concerning vertex operator algebras until Section 2). Since the central charge $c$ of the vertex operator algebra $V_L$ is precisely the rank of $L$, the classification of holomorphic vertex operator algebras of central charge at most $24$ may be construed as a generalization of the corresponding problem for even, unimodular lattices. We will say that a holomorphic vertex operator algebra has [*small central charge*]{} in case it satisfies $c\leq 24.$
Schellekens was the first to consider the problem of classifying holomorphic vertex operator algebras $V$ of small central charge \[Sch\]. Based on extensive computation, Schellekens wrote down a list of $71$ integral $q$-expansions $q^{-1} + constant + 196884q +
...$ and, among other things, conjectured that the graded character of a $V$ satisfying $c=24$ is necessarily equal to one of these $71$ $q$-expansions. As is well-known, it is only the constant term that distinguishes the $q$-expansions from each other, the constant in question being the dimension of the Lie algebra naturally defined on the weight one subspace $V_1$ of $V$ (see below for more details). Schellekens in fact wrote down a list of $71$ Lie algebras (including levels) which are the candidates for $V_1$. (It turns out that if $V$ has central charge strictly less that $24$ then the graded character of $V$ is uniquely determined, so at least as far as the dimension of the weight one subspace is concerned, the case $c=24$ carries the most interest.)
The purpose of the present paper is to put the Schellekens program on a firm mathematical foundation within the general context of rational conformal field theory, and to make a start towards the classification (up to isomorphism) of the holomorphic vertex operator algebras of small central charge. In addition to unitarity, there are other (unstated) assumptions in \[Sch\]. This circumstance means that we cannot assume the results of (loc. cit.) and need to find new approaches. In some ways we will go much further than \[Sch\] in that we will be able to give an adequate characterization of the holomorphic lattice theories among all holomorphic theories of small central charge. On the other hand, although we are able to establish numerical restrictions on the nature of the Lie algebra on $V_1$ which show that there are only a finite number of possibilities, we are at present unable to show, assuming $c=24,$ that it is necessarily one of the $71$ on Schellekens’ list. The remaining obstacle is essentially that of establishing that the levels of the associated Kac-Moody Lie algebras are positive integers (which is immediate if unitarity is assumed).
We now state some of our main results, and for this purpose we will take $V$ to be a holomorphic vertex operator algebra of CFT type (see Section 2) which is $C_2$-cofinite. (In the language of \[DM\], $V$ is strongly rational and holomorphic.). By results of Zhu \[Z\], this implies that $c$ is a positive integer divisible by $8$.
\[t1\] Suppose that $c=8$. Then $V$ is isomorphic to the lattice theory $V_{E_8}$ associated to the $E_8$ root lattice.
\[t2\] Suppose that $c=16$. Then $V$ is isomorphic to a lattice theory $V_L$ where $L$ is one of the two unimodular rank $16$ lattices.
Thus for the cases $c=8$ and $16$, the classification of the holomorphic vertex operator algebras completely mirrors that of the corresponding lattices. These two theorems are commonly assumed in the physics literature, and are related to the uniqueness of the heterotic string (\[Sch\], \[GSW\]).
\[t3\] Suppose that $c=24$. Then the Lie algebra on $V_1$ is reductive, and exactly one of the following holds:
\(a) $V_1 = 0$.
\(b) $V_1$ is abelian of rank $24$. In this case $V$ is isomorphic to the lattice theory $V_{\Lambda}$ where $\Lambda$ is the Leech lattice.
\(c) $V_1$ is a semi-simple Lie algebra of rank $24$. In this case $V$ is isomorphic to the lattice theory $V_L$ where $L$ is the even, unimodular rank $24$ lattice whose root system is the same as the root system associated to $V_1$.
\(d) $V_1$ is a semi-simple Lie algebra of rank less than $24$.
We actually establish more than is stated here. It is a well-known conjecture \[FLM\] that the Moonshine module is characterized among all $c=24$ holomorphic theories by the condition $V_1 = 0$. Our methods are less effective when there is no Lie algebra, however we will show that in this case $V_2$ carries the structure of a simple, commutative algebra (of dimension 196,884). The commutativity and dimension formula are well-known; it is the simplicity that is novel here. (The inherent difficulty in dealing with $V_2$ is that it is not an associative algebra, indeed it is not even power associative, and there seem to be no useful identities which are satisfied.) In case (d) we show that the simple components ${\frak g}_i$ of $V_1$ have levels $k_i$ and dual Coxeter numbers $h_i^{\vee}$ such that the identity $$\label{1.1}
\frac{h_i^{\vee}}{k_i} = \frac{ \dim V_1 -24}{24}$$ holds for each $\g_i$. This implies that there are only finitely many choices for the family of pairs $(\g_i, k_i)$ determined by $V_1$. Note that the condition (\[1.1\]) was already identified by Schellekens \[Sch\]. We are in fact able to extract some further numerical restrictions on the levels $k_i$, but these fall well short of the expectation that they are all positive integers, and we forgo any discussion of this beyond (\[1.1\]). We also establish that if $V_1$ is semisimple in Theorem \[t3\], then the Virasoro element in $V$ coincides with the usual Virasoro element associated with an affine Lie algebra defined by the Sugawara construction. We expect the result to be useful in further analysis of the situation.
The main inspiration for the proof of our results originates from our recent paper \[DM\]. There, we introduced methods based on the theory of modular forms used in tandem with techniques from vertex operator algebra theory. In particular we obtained a simple numerical characterization of the lattice vertex operator algebras among all rational vertex operator algebras. The present paper is in many ways a continuation and elaboration of \[loc.cit.\]. When the central charge is small, the theory of modular forms gives very precise numerical information about the vertex operator algebra which allows us to show, under certain circumstances, that the numerical characterizations of \[DM\] are applicable. This leads to Theorems 1-3. Modular-invariance also underlies the other results that we obtain.
In addition to the Schellekens program that we have already discussed, another potential application of Theorem \[t3\] is to the FLM conjecture regarding the Moonshine module alluded to above. Namely, Theorem \[t3\] shows that the Leech lattice theory $V_{\Lambda}$ is the only $c=24$ holomorphic theory for which the Lie algebra $V_1$ is both non-zero and not semi-simple. On the other hand, the Moonshine module is closely related to $V_{\Lambda}$, being built from it by a $\Z_2$-orbifold construction \[FLM\].
The paper is organized as follows: we gather together some preliminaries in Section 2, and prove Theorems \[t1\] - \[t3\] together with the supplementary result in case(a) of Theorem 3 in Section 3. In section 4 we identify the Virasoro element with the Sugawara construction.
Preliminary results
===================
For general background on the theory of vertex operator algebras, we refer the reader to \[FLM\], \[FHL\]. As usual, for a state $v$ in a vertex operator algebra $V$, we denote the corresponding vertex operator by $$Y(v, z) = \sum_{n \in \Z}v_n z^{-n-1},$$ while the vertex operator corresponding to the conformal (Virasoro) vector is $$Y(\omega, z) = \sum_{n \in \Z} L(n)z^{-n -2}.$$
$V$ is called [*rational*]{} if all admissible $V$-modules are completely reducible (cf. \[DLM1\] for the definition of admissible module). It was shown in \[DLM2\] that this implies that $V$ has only finitely many inequivalent simple modules. A rational vertex operator algebra is called [*holomorphic*]{} if it has a unique simple module, namely the adjoint module $V$. $V$ is said to be $C_2$-[*cofinite*]{} in case the subspace spanned by elements $u_{-2}v$ for $u, v \in V$ is of finite codimension. Finally, we say that $V$ is of [*CFT-type*]{} in case the natural $\Z$-grading on $V$ takes the form $$\label{2.1}
V = V_0 \oplus V_1 \oplus \cdots\ \ {\rm with}\ \ V_0 = \C\1 .$$
Throughout the rest of this paper, we assume that $V$ is a $C_2$-cofinite, holomorphic vertex operator algebra of CFT-type of small central charge $c\leq 24$. In order to avoid the case of the trivial vertex operator algebra $V = \C$, we also assume that $V$ has dimension greater than one.
Next we discuss some consequences of these assumptions for the structure of $V$. Many of them are well-known. First, the weight $1$ subspace $V_1$ of $V$ carries a natural structure of Lie algebra given by $$\label{2.2}
[u,v] = u_0v$$ for $u,v \in V_1$. Because the adjoint module is the unique simple $V$-module, then the contragredient module $V'$ is necessarily isomorphic to $V$. This is equivalent to the existence of a non-degenerate, invariant bilinear pairing $$\< , \>: V \times V \to \C$$ which is necessarily symmetric. (For the theory of contragredient modules, cf. Section 3 of \[FHL\].) Because of (\[2.1\]), Li’s theory of invariant bilinear forms \[L1\] shows that we have $L(1)V_1 = 0$ and that $\< , \>$ is uniquely determined up to an overall scalar. It is convenient to fix the normalization so that $$\label{2.3}
\< \1,\1 \> = -1.$$ In particular, the restriction of $\< , \>$ to $V_1$ is given by $$\label{2.4}
\<u, v\>\1 = u_1v$$ for $u, v \in V_1.$
In the language of \[DM\], it follows from what we have said that $V$ is [*strongly rational,*]{} so that the results of (loc. cit.) apply. They tell us that the following hold:
\(I) $V_1$ is a reductive Lie algebra of Lie rank $l\leq c.$
\(II) $l = c$ if, and only if, $V$ is isomorphic to a lattice theory $V_L$ for some positive-definite, even, unimodular lattice $L$.
We also note that by results of Zhu \[Z\] (cf. \[DLM3\]), $c$ is necessarily a positive integer divisible by $8$. So in fact $c = 8, 16$, or $24.$
We refer the reader to \[Z\] and \[DM\] for an extended discussion of the role of modular-invariance in the theory of rational vertex operator algebras. We need to recall the genus one vertex operator algebra $(V,Y[\ ],\1, \omega-c/24)$ from \[Z\]. The new vertex operator associated to a homogeneous element $a$ is given by $$Y[a,z] = \sum_{n\in\Z}a[n]z^{-n-1} = Y(a,e^{z} -1)e^{z\wt{a}}$$ while the Virasoro element is $\tilde{\omega} = \omega-c/24$. Thus $$a[m] = \Res_z\left(Y(a,z)(\ln{(1+z)})^m(1+z)^{\wt{a}-1}\right)$$ and $$a[m] = \sum_{i=m}^\infty c(\wt{a},i,m)a(i)$$ for some scalars $c(\wt{a},i,m)$ such that $c(\wt{a},m,m)=1.$ In particular, $$a[0]=\sum_{i\geq 0}{\wt{a}-1 \choose i}a(i).$$ We also write $$L[z] = Y[\omega-c/24,z] = \sum_{n\in\Z} L[n]z^{-n-2}.$$ Then the $L[n]$ again generate a copy of the Virasoro algebra with the same central charge $c.$ Now $V$ is graded by the $L[0]$-eigenvalues, that is $$V=V_{[0]}\oplus V_{[1]}\oplus \cdots$$ where $V_{[n]}=\{v\in V|L[0]v=nv\}.$
We will need the following facts: Let $v \in V$ satisfy $L[0]v = kv.$ Then the graded trace $$\label{2.7}
Z(v, \tau) = \tr_V o(v) q^{L(0) - c/24} = q^{-c/24}\sum_{n\geq 0}
\tr_{V_n}o(v) q^n$$ is a modular form on $SL(2,\Z)$ (possibly with character), holomorphic in the complex upper half-plane $H$ and of weight $k$. Here, we have set $o(v)$ to be the [*zero mode*]{} of $v$, defined via $o(v) = v_{\wt v -
1}$ if $v$ is homogeneous, and extended by linearity to the whole of $V$. Moreover, $\tau$ will denote an element in $H$ and $q = e^{2 \pi
i \tau}$. In particular, if we take $v$ to be the vacuum element then (\[2.7\]) is just the graded trace $$\label{2.8}
\ch_qV=q^{-c/24} \sum_{n\geq 0} (\dim V_n) q^n$$ and is a modular function of weight zero on $SL(2,\Z)$. Because of the holomorphy of $\ch_q(V)$ in $H$ and our assumption that $c\leq 24$, one knows (\[K\], \[Sch\]) that (\[2.8\]) is uniquely determined up to an additive constant, and indeed is determined uniquely if $c\leq 16$. The upshot is this:
\[l2.1\] One of the following holds:
\(a) $c = 8$ and $\ch_q(V) = \Theta_{E_8}(q)/ \eta(q)^8 = q^{-1/3}(1 + 248q +
\cdots)$
\(b) $c = 16$ and $\ch_q(V) = (\Theta_{E_8}(q))^2/ \eta(q)^{16} =
q^{-2/3}(1 + 496q + \cdots)$
\(c) $c = 24$ and $\ch_q(V) = J(q) + const = q^{-1} + const + 196884q +\cdots.$
Here, we have introduced the theta function $\Theta_{E_8}(q)$ of the $E_8$ root lattice $$\Theta_{E_8}(q)=\sum_{\alpha\in E_8}q^{(\alpha,\alpha)/2}$$ (where $E_8$ denotes the root lattice of type $E_8$ normalized so that the squared length of a root is $2$) as well as the eta function $$\eta(q)=q^{1/24}\prod_{n\geq 1}(1-q^n);$$ $$J(q) = q^{-1} + 0 + 196884q +\cdots$$ is the absolute modular invariant normalized to have constant term zero (alias the graded character of the Moonshine Module \[FLM\]). We also need the ’unmodular’ Eisenstein series of weight two, namely $$E_2(q) = 1 - 24 \sum_{n\geq 1}\sigma_1(n) q^n$$ where $\sigma_1(n)$ is the sum of the divisors of $n.$
\[l2.2\] For states $u, v$ in $V_1$ we have $$Tr_V o(u)o(v) q^{L(0) - c/24} = \frac{ \< u, v\>}{24}
(\frac{ 24}{c}D_q(ch_qV) + E_2(q)ch_qV)$$ where $D_q = q\frac{ d}{ dq}$.
Proof: First recall the important identity, Proposition 4.3.5 in \[Z\]. If we pick a pair of states $u, v \in V_1$ this result yields the following: $$\label{2.9}
\tr_V o(u)o(v) q^{L(0) - c/24} = Z(u[-1]v, \tau) + 1/ 12
E_2(\tau) Z(u[1]v, \tau).$$ The term $Z(u[-1]v, \tau)$ is a modular form of weight 2, and we shall be able to write it down explicitly. Indeed, the l.h.s. of (\[2.9\]) has leading term $\kappa(u,v)q^{1-c/ 24}$ where $\kappa(u,v)$ denotes the usual Killing form on $V_1$, whereas the leading term of the second summand on the r.h.s. of (\[2.9\]) is equal to $\frac{-1}{12}\<u, v\>q^{-c/ 24}$. From this discussion we conclude that the leading term of $Z(u[-1]v, \tau)$ is equal to $\frac{1}{12}\<u,v\>q^{-c/24}$. But up to scalars, the unique form of weight $2$ on $SL(2,\Z)$ which is holomorphic in $H$ and has a pole of order $c/ 24$ at infinity is $D_q(ch_q(V))$. As a result, we see that the first term on the r.h.s. of (\[2.9\]) is equal to $\frac{2\<u, v\>}{c}
D_q(ch_q(V)).$ The lemma follows from this, (\[2.9\]), and (\[2.4\]).
Identifying coefficients of $q^{1 - c/24}$ in the formula of Lemma \[l2.2\] yields
\[c2.3\] $\kappa(u,v) = 2\<u , v\> (\frac{\dim V_1}{c} - 1)$.
Proof of the Theorems
=====================
We consider the three possibilities $c = 8, 16, 24$ in turn. In the first two cases the idea is to show that (II) [*always*]{} applies, while in the third case we study [*when*]{} it applies.
Case 1: $c = 8$. In this case dim $V_1 = 248$ by Lemma \[l2.1\] (a), so that $\kappa(u, v) = 60\<u, v\>$ by Corollary \[c2.3\]. Since $\< , \>$ is non-degenerate then so too is the Killing form. We conclude that in fact $V_1$ is semi-simple of dimension $248$, and by (I) the Lie rank is no greater than $8$. By the classification of semi-simple Lie algebras, we conclude that in fact $V_1$ is the Lie algebra of type $E_8$ and Lie rank $8$. Now (II) and the fact that there is a unique positive-definite, even, unimodular lattice of rank $8$, namely the $E_8$ root lattice, completes the proof of Theorem \[t1\].
Case 2: $c = 16$. This is very similar to case 1. Namely, we have dim $V_1 = 496$ by Lemma \[l2.1\] (b), whence $\kappa(u,v) = 60\<u, v\>$ once more. So again $V_1$ is semi-simple, and we proceed as in case 1, now using the fact that there are just the two positive-definite, even, unimodular lattices. Theorem \[t2\] follows.
Case 3: $c = 24$. Set $dim V_1 = d$. From Corollary 2.3 we see in this case that $$\label{3.1}
\kappa(u, v) = \<u, v\> (d - 24)/12.$$
If $d = 24$ then the Killing form is identically zero. Then $V_1$ is solvable by Cartan’s criterion and therefore abelian since $V_1$ is in any case reductive by (I). So if $d = 24$ then we have shown that $V_1$ has rank $24$ and is therefore a lattice theory $V_L$ for suitable $L$ by (II). The fact that $(V_L)_1$ is abelian tells us that $L$ has no roots i.e., no vectors of squared length two, and that $L$ is therefore the Leech lattice (cf. \[CS\]). This deals with case (b) of Theorem \[t3\].
Let us assume from now on that $d \ne 0,24$. Together with (\[3.1\]), this tells us that the Killing form on $V_1$ is again non-degenerate, so that $V_1$ is a semi-simple Lie algebra of Lie rank $l$ no greater than $24$ by (I). Moreover, if $l = 24$ then $V$ is a lattice theory by (II) once more. This confirms part (c) of Theorem \[t3\].
Next we consider the levels of the affine Lie algebras spanned by the vertex operators $Y(u, z)$, $u$ in $V_1$. For states $u, v \in V_1$ and integers $m, n$ we have $$\label{a1}
[u_m, v_n] = (u_0v)_{m + n} + m u_1v \delta_{m, - n},$$ whereas the usual relations for a Kac-Moody Lie algebra of level $k$ associated to a simple Lie algebra $\g$ take the form $$\label{a2}
[a_m, b_n] = [a, b]_{m + n} + k(a, b)m\delta_{m, - n}$$ where $(a, b)$ is the non-degenerate form on $\g$ normalized so that $(\alpha, \alpha) = 2$ for a long root $\alpha$.
Let $V_1$ be a direct sum $$\label{6.11}
V_1 = {\frak g}_{1, k_1} \oplus {\frak g}_{2, k_2} \oplus ... \oplus
{\frak g}_{n,k_n}$$ of simple Lie algebras ${\frak g}_i$ whose corresponding affine Lie algebra has level $k_i.$ By comparing (\[a1\]) and (\[a2\]), using Corollary \[c2.3\] we obtain for $u,v\in{\frak g}_{i}$ that $$\label{3.2}
\kappa_{{\frak g}_{i}}(u, v) =( d-24) k_i(u, v)/ 12$$ where $\kappa_{{\frak g}_{i}}$ denotes the restriction of the Killing form to ${\frak g}_{i}.$ Now $\kappa_{{\frak g}_{i}}(h_{\alpha}, h_{\alpha}) = 4 h_i^{\vee} $ for a long root $\alpha$, where $h_i^{\vee}$ is the dual Coxeter number of the root system associated to ${\frak g}_{i}$. Therefore, (\[3.2\]) tells us that for each simple component ${\frak g}_{i}$ of $V_1$, of level $k_i$ and dual Coxeter number $h_i^{\vee}$, the ratio $$\label{3.3}
h_i^{\vee}/ k_i =(d- 24)/ 24$$ is independent of ${\frak g}_{i}$.
\[p3.1\] Assume that case (a) of Theorem \[t3\] holds. Then $B =
V_2$ carries the structure of a (non-associative) simple, commutative algebra with respect to the product $a.b = a_1b$.
We take for granted the well-known facts that $B$ is indeed a non-associative, commutative algebra with respect to the indicated product, and that the pairing $\< , \>: B \times B \to \C$ defined by $$\label{3.4}
a_3b = \<a,b\>\1$$ endows $B$ with a non-degenerate, invariant trace form. That is, $\<
, \>$ is symmetric and satisfies $\<ab, c\> = \<a , bc\>$. Moreover $B$ has an identity element $1/2\omega$. Set $d = \dim B = 196884$.
Next we state two more results that we will need. Each can be established using modular-invariance arguments along the same lines as before. Alternatively, we may use results of Section 4 of \[M\]: $$\label{3.5}
\tr_B o(ab) =(d/3) \<a, b\>.$$ If $e^2 = e$ is in $B$ then $$\label{3.6}
\tr_B o(e)^2 = 4620 \<e, e\> + 20336 \<e,e\>^2.$$
Turning to the proof of the Proposition, we first show that $B$ is semisimple. Indeed, (\[3.5\]) guarantees that the form $\tr_B o(ab)$ is non-degenerate, and this is sufficient to establish that $B$ is semi-simple. To see this, recall a well-known result of Dieudonne (cf. \[S, Theorem 2.6\]) that an arbitrary algebra $B$ is semisimple if it has a non-degenerate trace form and contains no non-zero nilpotent ideals. We will show that indeed there are no non-zero nilpotent ideals in $B$. If not, we may choose a minimal non-zero nilpotent ideal $M$ in $B$. Note that $M^2 = 0$. Let $m$ be a non-zero element in $M$, and let $b$ in $B$ be arbitrary. Then $mb$ lies in $M$, so that $(mb)B\subset M$ and $(mb)M = 0$. This shows that each element $mb$ is nilpotent as a multiplication operator on $B$, and hence $\tr_B mb
= 0$ for all $b$. This contradicts the non-degeneracy of $\< , \>$.
From the last paragraph we know that $B$ can be written as an (orthogonal) direct sum of simple ideals $$B = B_1 + B_2 + \cdots + B_t.$$ We must show that $t = 1$. Write $1/2 \omega = e_1 + e_2 + ... + e_t$ where $e_i$ is the identity element of $B_i$. In particular, $e_i$ is a central idempotent of $B$. By (\[3.5\]) and (\[3.6\]) we obtain $$\label{3.7}
(d/3)\<e, e\> = 4620\<e, e\> +20336\<e, e\>^2$$ where $e$ is any of the idempotents $e_i$. Notice that (\[3.7\]) yields that $$\label{3.8}
\<e, e\>= 3.$$
It is well-known that (\[3.8\]) implies that the components of the vertex operator $Y(2e, z)$ generate a Virasoro algebra of central charge $24$, so that the total central charge must be $24t$. Hence, $t = 1$ as required.
Some Virasoro elements
======================
Throughout this section we assume that $V$ is a strongly rational, holomorphic vertex operator algebra of central charge $24$ as before, and we assume in addition that $V_1$ is a (non-zero) semisimple Lie algebra of Lie rank $l.$ Let $V_1$ have decomposition (\[6.11\]) into simple Lie algebras. Set $d_i = \dim {\frak g}_{i,k_i}.$ Let $( , )$ denote the normalized invariant bilinear form on ${\frak g}_{i,k_i}$ with the property that $(\alpha,\alpha) = 2$ for a long root $\alpha$ in ${\frak g}_{i,k_i}.$ It is known (cf. \[DL\], \[FZ\], \[K\], \[L2\]) that the element $$\label{4.1}
\omega_i=\frac{1}{2(k_i+h_i^{\vee})}\sum_{j=1}^{d_i}u^j_{-1}u^j_{-1}{\bf 1}$$ is a Virasoro element of central charge $c_i=\frac{k_i\dim {\frak
g}_i}{k_i+h_i^{\vee}},$ where $u^1, u^2, ... u^{d_i}$ is an orthonormal basis of ${\frak g}_i$ with respect to $( , )$.
There are three Virasoro elements in $V$ that are relevant to a further analysis of the situation. Namely, in addition to the original Virasoro element $\omega$ in $V$, we have $$\label{4.2}
\omega_{aff} = \sum_{i=1}^n\omega_i$$ and $$\label{4.3}
\omega_H = \frac{1}{2}\sum_{i=1}^l(h^i_{-1})^{2}{\bf 1}$$ where $h^{1}, ... , h^{l}$ is an orthonormal basis of a maximal abelian subalgebra $H$ of $V_1$ with respect to the inner product $\<,
\>.$ Note that as a consequence of equation (\[3.3\]), $\omega_{aff}$ has central charge $\sum c_i = 24$. We omit further discussion of $\omega_H$, but will prove:
\[p4.1\] $\omega_{aff} = \omega$.
Consequences of modular-invariance again underlie the proof of the Proposition, notably the absence of cusp-forms of small weight on $SL(2,\Z)$. We introduce some notation for Virasoro operators: in addition to the usual operators $L(n)$ associated to $\omega$ , we use $L^{aff}(n)$ for the corresponding operators associated to $\omega_{aff}.$ We also set $\omega' = \omega - \omega^{aff}$. We will soon see that $\omega'$ is itself a Virasoro element, and define its component operators to be $L'(n)$. We eventually want to prove of course that $\omega' = 0$. We proceed in a series of steps.
Step 1: $\omega'$ is a highest weight vector of weight 2 for the Virasoro algebra generated by ${L(n)}$.
The definition of $\omega'$ shows that it lies in $V_2$, so it suffices to show that $\omega'$ is annihilated by the operators $L(1)$ and $L(2)$. First calculate that if $u$ is an element of $V_1$ then $[L(1), u_{-1}] = u_0$. Then from the definitions it follows easily that $L(1)$ annihilates both $\omega_{aff}$ and $\omega_H$, whence it also annihilates $\omega'$. To establish that $L(2)$ annihilates $\omega'$ we must show that $$\label{4.4}
L(2)\omega_{aff} = L(2)\omega_H =12.$$ Now for $u$ in $V_1$ we have $[L(2), u_{-1}] = u_1$. Then $$L(2) u^{j}_{-1} u^{j}_{-1}{\1} = - \<u^{j}, u^{j}\> =k_i,$$ and (\[4.4\]) follows.
Step 2: $\omega'$ is a Virasoro vector of central charge $0$.
Use step 1, in particular that $L(1)$ annihilates $\omega_{aff}$, to see that $$[L(m), L^{aff}(n)] = (m-n) L^{aff}(m+n) -2(m^{3}
- m)\delta_{m, -n}.$$ The result follows easily from this.
Step 3: $Z_V(\omega', \tau) = 0$.
Since $\omega'$ is a highest weight vector of weight 2 for the Virasoro operators corresponding to $\omega$ then from our earlier discussion of (\[2.7\]) we see that $Z_V(\omega', \tau)$ is a modular form on $SL(2,Z)$ of weight 2 and is holomorphic in the upper half-plane. Moreover there is no pole in the q-expansion of $Z_V(\omega', \tau)$, so $Z_V(\omega', \tau)$ is in fact a holomorphic modular form of weight 2 on $SL(2,Z)$, hence must be zero.
Step 4: $\tr_VL'(0)^2q^{L(0)-1}=0.$
By Proposition 4.3.5 in \[Z\] we have $$\tr_VL'(0)^2q^{L(0)-1}=Z_V(L'[-2]\omega',\tau)-\sum_{k\geq
1}E_{2k}(\tau) Z_V(L'[2k-2]\omega',\tau)$$ where $Y[\omega',z]=\sum_{m\in\Z}L'[m]z^{-m-2}$ and the functions $E_{2k}(\tau)$ are Eisenstein series of weight $2k,$ normalized as in \[DLM3\]. $E_{2k}(\tau)$ is a holomorphic modular form on $SL(2,\Z)$ if $k>1$. Since $L'[2k-2]\omega'=0$ if $k\geq 2,$ $L'[0]\omega'=2\omega',$ and $L[0]L'[-2]\omega'=4L'[-2]\omega',$ we see that $$\tr_VL'(0)^2q^{L(0)-1}=Z_V(L'[-2]\omega',\tau)$$ is a modular form of weight 4 for $SL(2,\Z).$ Since $L'(0)V_1=0,$ it is in fact a cusp form, hence equal to zero.
Step 5. All the eigenvalues of $L^{aff}(0)$ on $V$ are real.
In order to see this recall the decomposition (\[6.11\]) and set $\g=V_1.$ Consider the affine Lie algebra $$\hat{\frak g}={\frak g}\otimes \C[t,t^{-1}]\oplus \C$$ with bracket $$[x\otimes t^p,y\otimes t^q]=[x,y]\otimes t^{p+q}+p\delta_{p+q,0}\<x,y\>$$ for $x,y\in {\frak g}$ and $p,q\in\Z.$ Since each $V_m$ is finite dimensional we see that $V$ has a composition series as a module for $\hat{\frak g}$ such that each factor is an irreducible highest weight $\hat{\frak g}$-module. So it is enough to show that $L^{aff}(0)$ has only real eigenvalues on any irreducible highest weight $\hat{\frak g}$-module. Note that such an irreducible highest weight $\hat{\frak g}$-module is a tensor product of irreducible highest weight $\hat{\frak g}_i$-modules $L(k_i,\Lambda_i)$ of level $k_i$ ($i=1,...,n$) for some dominant weight $\Lambda_i$ in the weight lattice of ${\frak g}_i$ as $V$ is a completely reducible ${\frak g}_i$-module. Here $L(k_i,\Lambda_i)$ is the unique irreducible quotient of the generalized Verma module $U(\hat{\frak g}_i)\otimes_{U({\frak g}_i\otimes \C[t]+\C)}L(\Lambda_i)$ where $L(\Lambda_i)$ is the highest weight module for ${\frak g}_i$ with highest weight $\Lambda_i$ and $x\otimes t^m$ acts as zero if $m>0$ and $x\otimes t^0$ acts as $x.$ So it is enough to show that $L_i(0)$ has only real eigenvalues on $L(k_i,\Lambda_i)$ where $Y(\omega_i,z)=\sum_{m\in\Z}L_i(m)z^{-m-2}.$
It is well-known that the eigenvalues of $L_i(0)$ on $L(k_i,\Lambda_i)$ are the numbers $\frac{(\Lambda_i+2\rho_i,\Lambda)}{2(k_i+h_i^{\vee})}+m$ (cf. \[DL\], \[K\]) for nonnegative integers $m$ where $\rho_i$ is the half-sum of the positive roots of ${\frak g}_i.$ Since $k_i$ is rational, it is clear that $\frac{(\Lambda_i+2\rho_i,\Lambda)}{2(k_i+h_i^{\vee})}+m$ is real, as required.
Step 6. $\omega'=0.$
By Step 5, all eigenvalues of $L'(0)$ are real on $V.$ This implies that $L'(0)^2$ has only nonnegative eigenvalues. By Step 4 we conclude that all the eigenvalues of $L'(0)$ are zero. Since $L'(0)\omega'=2\omega',$ we must have $\omega'=0,$ as desired.
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[^1]: Supported by NSF grant DMS-9987656 and faculty research funds granted by the University of California at Santa Cruz ([email protected]).
[^2]: Supported by NSF grant DMS-9700909 and faculty research funds granted by the University of California at Santa Cruz ([email protected]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We analyze the mixed frame equations of radiation hydrodynamics under the approximations of flux-limited diffusion and a thermal radiation field, and derive the minimal set of evolution equations that includes all terms that are of leading order in any regime of non-relativistic radiation hydrodynamics. Our equations are accurate to first order in $v/c$ in the static diffusion regime. In contrast, we show that previous lower order derivations of these equations omit leading terms in at least some regimes. In comparison to comoving frame formulations of radiation hydrodynamics, our equations have the advantage that they manifestly conserve total energy, making them very well-suited to numerical simulations, particularly with adaptive meshes. For systems in the static diffusion regime, our analysis also suggests an algorithm that is both simpler and faster than earlier comoving frame methods. We implement this algorithm in the Orion adaptive mesh refinement code, and show that it performs well in a range of test problems.'
author:
- 'Mark R. Krumholz[^1]'
- 'Richard I. Klein'
- 'Christopher F. McKee'
- John Bolstad
title: |
Equations and Algorithms for Mixed Frame\
Flux-Limited Diffusion Radiation Hydrodynamics
---
Introduction
============
Astrophysical systems described by radiation hydrodynamics span a tremendous range of scales and parameter regimes, from the interiors of stars [e.g. @kippenhahn94] to accretion disks around compact objects [e.g. @turner03] to dusty accretion flows around massive protostars [e.g. @krumholz05a; @krumholz07a] to galactic-scale flows onto AGN [e.g. @thompson05]. All of these systems have in common that matter and radiation are strongly interacting, and that the energy and momentum carried by the radiation field is significant in comparison to that carried by the gas. Thus an accurate treatment of the problem must include analysis of both the matter and the radiation, and of their interaction.
Numerical methods exist to simulate such systems in a variety of dimensionalities and levels of approximation. In three dimensions, treatments of the matter and radiation fields generally adopt the flux-limited diffusion approximation, first introduced by @alme73, for reasons of computational cost and simplicity [e.g. @hayes06]. Flux-limited diffusion is optimal for treating continuum transfer in a system such as an accretion disk, stellar atmosphere, or opaque interstellar gas cloud where the majority of the interesting behavior occurs in optically thick regions that are well described by pure radiation diffusion, but there is a surface of optical depth unity from which energy is radiated away. Applying pure diffusion to these problems would lead to unphysically fast radiation from this surface, so flux-limited diffusion provides a compromise that yields a computationally simple and accurate description of the interior, while also giving a reasonably accurate loss rate from the surface [@castor04].
However, the level of accuracy provided by this approximation has been unclear because the equations of radiation hydrodynamics for flux-limited diffusion have previously only been analyzed to zeroth order in $v/c$. In contrast, several authors have analyzed the radiation hydrodynamic equations in the general case to beyond first order in $v/c$ [e.g. @mihalas99; @castor04 and references therein]. In a zeroth order treatment, one neglects differences between quantities in the laboratory frame and the comoving frame. The problem with this approach is that in an optically thick fluid, the radiation flux only follows Fick’s law (${\mathbf{F}}\propto -\nabla E$) in the comoving frame, and in other frames there is an added advective flux of radiation enthalpy, as first demonstrated by @castor72. In certain regimes (i.e. the dynamic diffusion limit – see below) this advective flux can dominate the diffusive flux [@mihalas01; @castor04].
@pomraning83 does give a flux-limiter usable to first order in $v/c$, which is an approach to the problem of flux-limiting with relativistic corrections that is an alternative to the one we pursue in this paper. However, this approach does not correctly handle the dynamic diffusion limit, a case that as we show requires special attention because order $v^2/c^2$ terms can be important. Furthermore, @pomraning83 derives his flux-limiter directly from the transfer equation, so the computation provides little insight into the relative importance of radiation hydrodynamic terms, and the level of accuracy obtained by using the uncorrected flux-limiter, the most common procedure in astrophysical applications.
@mihalas82 were the first to derive the mixed frame equations of radiation hydrodynamics dynamics to order $v/c$ in frequency-integrated and frequency-dependent forms, and gave numerical algorithms for solving them. @lowrie99, @lowrie01, and @hubeny06 give alternate forms of these equations, as well as numerical algorithms for solving them. However, these treatments require that one solve the radiation momentum equation (and for the frequency-dependent equations calculate over many frequencies as well), rather than adopt the flux-limited diffusion approximation. While this preferable from a standpoint of accuracy, since it allows explicit conservation of both momentum and energy and captures the angular-dependence of the radiation field in a way that diffusion methods cannot, treating the radiation momentum equation is significantly more computationally costly than using flux-limited diffusion, making it difficult to use in three-dimensional calculations.
In this Paper we analyze the equations of radiation hydrodynamics under the approximations that the radiation field has a thermal spectrum and obeys the flux-limited diffusion approximation, and that scattering is negligible for the system. Our goal is to derive an accurate set of mixed frame equations, meaning that radiation quantities are written in the lab frame, but fluid quantities, in particular fluid opacities, are evaluated in the frame comoving with the fluid. This formulation is optimal for three-dimensional simulations, because writing radiation quantities in the lab frame lets us use an Eulerian grid on which the radiative transfer problem may be solved by any number of standard methods, while avoiding the need to model the direction- and velocity-dependence of the lab frame opacity and emissivity of a moving fluid.
In § \[equations\] we begin from the general lab frame equations of hydrodynamics to first order in $v/c$, apply the flux-limited diffusion approximation in the frame comoving with the gas where it is applicable, and transform the appropriate radiation quantities into the lab frame, thereby deriving the corresponding mixed frame equations suitable for implementation in numerical simulations. We retain enough terms to ensure that we achieve order unity accuracy in all regimes, and order $v/c$ accuracy for static diffusion problems. In § \[termanalysis\] we assess the significance of the higher order terms that appear in our equations, and consider where treatments omitting them are acceptable, and where they are likely to fail. We show that, in at least some regimes, the zeroth order treatments most often used are likely to produce results that are incorrect at order unity. We also compare our equations to the comoving frame equations commonly used in other codes. In § \[algorithm\] we take advantage of the ordering of terms we derive for the static diffusion regime to construct a radiation hydrodynamic simulation algorithm for static diffusion problems that is simpler and faster than those now in use, which we implement in the Orion adaptive mesh refinement code. In § \[tests\] we demonstrate it in a selection of test problems. Finally, we summarize our results in § \[summary\].
Derivation of the Equations {#equations}
===========================
In the discussion that follows, we adopt the convention of writing quantities measured in the frame comoving with a fluid with a subscript zero. Quantities in the lab frame are written without subscripts. We write scalars in italics (e.g. $a$), vectors in boldface (e.g. ${\mathbf{a}}$), and rank two tensors in calligraphy (e.g. ${\mathcal{A}}$). We indicate tensor contractions over a single index by dots (e.g. ${\mathbf{a}}\cdot{\mathbf{b}}=a^i b_i$), tensor contractions over two indices by colons (e.g.${\mathcal{A}}\colon{\mathcal{B}}=A^{ij} B_{ij}$), and tensor products of vectors without any operator symbol (e.g. $({\mathbf{a}}{\mathbf{b}})^{ij}
=a^i b^j$). Also note that we follow the standard convention in radiation hydrodynamics rather than the standard in astrophysics, in that when we refer to an opacity $\kappa$ we mean the total opacity, measured in units of inverse length, rather than the specific opacity, measured in units of length squared divided by mass. Since we are neglecting scattering, we may set the extinction $\chi=\kappa$.
Regimes of Radiation Hydrodynamics
----------------------------------
Before beginning our analysis, it is helpful to examine some characteristic dimensionless numbers for a radiation hydrodynamic system, since evaluating these quantities provides a useful guide to how we should analyze our equations. Let $\ell$ be the characteristic size of the system under consideration, $u$ be the characteristic velocity in this system, and ${\lambda_{\rm P}}\sim 1/\kappa$, be the photon mean free path. Following @mihalas99, we can define three distinct limiting cases by considering the dimensionless ratios $\tau\equiv
\ell/{\lambda_{\rm P}}$, which characterizes the optical depth of the system, and $\beta \equiv u/c$, which characterizes how relativistic it is. Since we focus on non-relativistic systems, we assume $\beta\ll 1$. We term the case $\tau \ll 1$, in which the radiation and gas are weakly coupled, the *streaming* limit. If $\tau \gg 1$ then radiation and gas are strongly coupled, and the system is in the diffusion limit. We can further subdivide the diffusion limit into the cases $\beta \gg \tau^{-1}$ and $\beta \ll \tau^{-1}$. The former is the *dynamic diffusion* limit, while the latter is the *static diffusion* limit. In summary, the limiting cases are $$\begin{aligned}
\tau\ll 1, & & \mbox{ (streaming limit)} \\
\tau\gg 1, & \;\beta \tau \ll 1, & \mbox{ (static diffusion limit)} \\
\tau\gg 1, & \;\beta \tau \gg 1, & \mbox{ (dynamic diffusion limit)}.\end{aligned}$$
Physically, the distinction between static and dynamic diffusion is that in dynamic diffusion radiation is principally transported by advection by gas, so that terms describing the work done by radiation on gas and the advection of radiation enthalpy dominate over terms describing either diffusion or emission and absorption. In the static diffusion limit the opposite holds. A paradigmatic example of a dynamic diffusion system is a stellar interior. The optical depth from the core to the surface of the Sun is $\tau \sim 10^{11}$, and typical convective and rotational velocities are $\gg 10^{-11} c = 0.3$ cm s$^{-1}$, so the Sun is strongly in the dynamic diffusion regime. In contrast, an example of a system in the static diffusion limit is a relatively cool, dusty, outer accretion disk around a forming massive protostar, as studied e.g. by @krumholz07a. The specific opacity of gas with the standard interstellar dust abundance to infrared photons is $\kappa/\rho
\sim 1$ cm$^2$ g$^{-1}$, and at distances of more than a few AU from the central star the density is generally $\rho {\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle <}
{\sim}\:$}}10^{-12}$ g cm$^{-3}$. For a disk of scale height $h\sim 10$ AU, the optical depth to escape is $$\begin{aligned}
\tau^{-1} & \approx &
6.7\times 10^{-3}
\left(\frac{\kappa/\rho}{\mbox{cm$^2$ g$^{-1}$}}\right)^{-1}
\nonumber \\
& & \qquad \left(\frac{\rho}{10^{-12}\mbox{ g cm$^{-3}$}}\right)^{-1}
\left(\frac{h}{10\mbox{ AU}}\right)^{-1}.\end{aligned}$$ The velocity is roughly the Keplerian speed, so $$\beta \approx 1.4 \times 10^{-4}
\left(\frac{M_*}{10\;{M_{\odot}}}\right)^{1/2}
\left(\frac{r}{10\mbox{ AU}}\right)^{-1/2},$$ where $M_*$ is the mass of the star and $r$ is the distance from it. Thus, this system is in a static diffusion regime by roughly two orders of magnitude.
In the analysis that follows, our goal will be to obtain expressions that are accurate for the leading terms in all regimes. This is somewhat tricky, particularly for diffusion problems, because we are attempting to expand our equations simultaneously in the two small parameters $\beta$ and $1/\tau$. The most common approach in radiation hydrodynamics is to expand expressions in powers of $\beta$ alone, and only analyze the equations in terms of $\tau$ after dropping terms of high order in $\beta$. However, this approach can produce significant errors, because terms in the radiation hydrodynamic equations proportional to the opacity are multiplied by a quantity of order $\tau$. Thus, in our derivation we will repeatedly encounter expressions proportional to $\beta^2\tau$, and in a problem that is either in the dynamic diffusion limit or close to it ($\beta\tau
{\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle >}
{\sim}\:$}}1$), it is inconsistent to drop these terms while retaining ones that are of order $\beta$. We therefore retain all terms up to order $\beta^2$ in our derivation unless we explicitly check that they are not multiplied by terms of order $\tau$, and can therefore be dropped safely.
The Equations of radiation hydrodynamics {#equationderivation}
----------------------------------------
We now start our derivation, beginning from the lab frame equations of radiation hydrodynamics [@mihalas82; @mihalas99; @mihalas01] $$\begin{aligned}
\label{massconservation}
\frac{\partial \rho}{\partial t} + \nabla\cdot (\rho {\mathbf{v}}) & = & 0 \\
\label{mom1}
\frac{\partial}{\partial t} (\rho {\mathbf{v}}) + \nabla \cdot (\rho {\mathbf{v}}{\mathbf{v}})
& = & -\nabla P + {\mathbf{G}}\\
\label{en1}
\frac{\partial}{\partial t} (\rho e) + \nabla \cdot \left[\left(\rho e +
P\right) {\mathbf{v}}\right] & = & c G^0 \\
\label{raden1}
\frac{\partial E}{\partial t} + \nabla \cdot {\mathbf{F}}&=& - c G^0 \\
\label{radmom1}
\frac{1}{c^2} \frac{\partial {\mathbf{F}}}{\partial t} + \nabla\cdot{\mathcal{P}}& =
& -{\mathbf{G}}\end{aligned}$$ where $\rho$, ${\mathbf{v}}$, $e$, and $P$ are the density, velocity, specific energy (thermal plus kinetic), and thermal pressure of the gas, $E$, ${\mathbf{F}}$, and ${\mathcal{P}}$ are the radiation energy density, flux, and pressure tensor, $$\begin{aligned}
c E & = & \int_0^{\infty} d\nu \int d\Omega \, I({\mathbf{n}}, \nu) \\
{\mathbf{F}}& = & \int_0^{\infty} d\nu \int d\Omega \, {\mathbf{n}}I({\mathbf{n}}, \nu) \\
c {\mathcal{P}}& = & \int_0^{\infty} d\nu \int d\Omega \, {\mathbf{n}}{\mathbf{n}}I({\mathbf{n}},
\nu),\end{aligned}$$ $(G^0,{\mathbf{G}})$ is the radiation four-force density $$\begin{aligned}
cG^0 & = & \int_0^{\infty} d\nu \int d\Omega\,[\kappa({\mathbf{n}},\nu)
I({\mathbf{n}}, \nu) - \eta({\mathbf{n}},\nu)], \\
c{\mathbf{G}}& = & \int_0^{\infty} d\nu \int d\Omega\,
[\kappa({\mathbf{n}},\nu) I({\mathbf{n}}, \nu) - \eta({\mathbf{n}},\nu)] {\mathbf{n}},\end{aligned}$$ and $I({\mathbf{n}},\nu)$ is the intensity of the radiation field at frequency $\nu$ traveling in direction ${\mathbf{n}}$. Here $\kappa({\mathbf{n}},\nu)$ and $\eta({\mathbf{n}},\nu)$ are the direction- and frequency-dependent radiation absorption and emission coefficients in the lab frame. Intuitively, we can understand $cG^0$ as the rate of energy absorption from the radiation field minus the rate of energy emission for the fluid, and ${\mathbf{G}}$ as the rate of momentum absorption from the radiation field minus the rate of momentum emission. Equations (\[massconservation\]) – (\[en1\]) are accurate to first order in $v/c$, while equations (\[raden1\]) – (\[radmom1\]) are exact. Note that no terms involving opacity or optical depth appear explicitly in any of these equations, so the fact that they are accurate to first order in $\beta$ means that they include all the leading order terms.
In order to derive the mixed-frame equations, we must now evaluate the radiation four-force $(G^0,{\mathbf{G}})$ in terms of lab frame radiation quantities and comoving frame emission and absorption coefficients. @mihalas01 show that, if the flux spectrum of the radiation is direction-independent, the radiation four-force on a thermally-emitting material to all orders in $v/c$ is given in terms of moments of the radiation field by $$\begin{aligned}
G^0 & = & \gamma [\gamma^2 {\kappa_{\rm 0E}}+ (1-\gamma^2) {\kappa_{\rm 0F}}] E
- \gamma{\kappa_{\rm 0P}}a_R T_0^4
\nonumber \\
& & {} - \gamma ({\mathbf{v}}\cdot{\mathbf{F}}/c^2)
[{\kappa_{\rm 0F}}- 2\gamma^2 ({\kappa_{\rm 0F}}-{\kappa_{\rm 0E}})]
\nonumber \\
& & {}
- \gamma^3 ({\kappa_{\rm 0F}}-{\kappa_{\rm 0E}}) ({\mathbf{v}}{\mathbf{v}})\colon{\mathcal{P}}/c^2, \\
{\mathbf{G}}& = & \gamma{\kappa_{\rm 0F}}({\mathbf{F}}/c)
- \gamma {\kappa_{\rm 0P}}a_R T_0^4 ({\mathbf{v}}/c)
\nonumber \\
& & {} -
[\gamma^3 ({\kappa_{\rm 0F}}- {\kappa_{\rm 0E}}) ({\mathbf{v}}/c) E + \gamma{\kappa_{\rm 0F}}({\mathbf{v}}/c)\cdot {\mathcal{P}}]
\nonumber \\
& & {} + \gamma^3 ({\kappa_{\rm 0F}}-{\kappa_{\rm 0E}}) [ 2 {\mathbf{v}}\cdot{\mathbf{F}}/c^3 -
({\mathbf{v}}{\mathbf{v}})\colon{\mathcal{P}}/c^3] {\mathbf{v}},\end{aligned}$$ where $\gamma = 1/\sqrt{1-v^2/c^2}$ is the Lorentz factor and $T_0$ is the gas temperature. The three opacities that appear are the Planck-, energy-, and flux-mean opacities, which are defined by $$\begin{aligned}
{\kappa_{\rm 0P}}& = &
\frac{\int_0^{\infty} d\nu_0 \, \kappa_0(\nu_0)
B(\nu_0, T_0)}{B(T_0)} \\
{\kappa_{\rm 0E}}& = &
\frac{\int_0^{\infty} d\nu_0 \, \kappa_0(\nu_0)
E_0(\nu_0)}{E_0} \\
\label{fluxmeandef}
{\kappa_{\rm 0F}}& = &
\frac{\int_0^{\infty} d\nu_0 \, \kappa_0(\nu_0)
{\mathbf{F}}_0(\nu_0)}{{\mathbf{F}}_0},\end{aligned}$$ where $E_0(\nu_0)$ and ${\mathbf{F}}_0(\nu_0)$ are the comoving frame radiation energy and flux per unit frequency, $E_0$ and ${\mathbf{F}}_0$ are the corresponding frequency-integrated energy and flux, and $B(\nu,T)= (2 h \nu^3/c^2) / (e^{h\nu/k_B T} - 1)$ and $B(T) = c a_R T^4/(4\pi)$ are the frequency-dependent and frequency-integrated Planck functions.
Note that we have implicitly assumed that the opacity and emissivity are directionally-independent in the fluid rest frame, which is the case for any conventional material. We have also assumed that the flux spectrum is independent of direction, allowing us to replace the flux-mean opacity vector with a scalar. This may not be the case for an optically thin system, or one in which line transport is important, but since we are limiting our application to systems to which we can reasonably apply the diffusion approximation, this is not a major limitation.
To simplify $(G^0, {\mathbf{G}})$, first we assume that the radiation has a blackbody spectrum, so that $E_0(\nu_0)\propto B(\nu_0, T_0)$. In this case, clearly $${\kappa_{\rm 0E}}= {\kappa_{\rm 0P}}.$$ Second, we adopt the flux-limited diffusion approximation (see below), so in optically thick parts of the flow ${\mathbf{F}}_0(\nu_0) \propto
-\nabla E_0(\nu_0)/\kappa_0(\nu_0)$ (Fick’s Law). This implies that ${\mathbf{F}}_0(\nu_0) \propto -[\partial B(\nu_0, T_0)/\partial T_0] (\nabla
T_0) / \kappa_0(\nu_0)$, and substituting this into (\[fluxmeandef\]) shows that the flux-mean opacity ${\kappa_{\rm 0F}}$ is equal to the Rosseland-mean opacity, defined by $${\kappa_{\rm 0R}}^{-1} = \frac{\int_0^{\infty} d\nu_0 \,
\kappa_0(\nu_0)^{-1}
\frac{\partial B(\nu_0,T_0)}{\partial T_0}}
{\int_0^{\infty} d\nu_0 \,
\frac{\partial B(\nu_0,T_0)}{\partial T_0}}.$$ In optically thin parts of the flow, $|{\mathbf{F}}_0(\nu_0)|\rightarrow c
E_0(\nu_0)$, so in principle we should have ${\kappa_{\rm 0F}}={\kappa_{\rm 0E}}$. However, interpolating between these cases is complex, and the flux-limited diffusion approximation is of limited accuracy for optically thin flows in complex geometries. Moreover, our approximation that the radiation spectrum is that of a blackbody at the local radiation temperature is itself problematic in the optically thin limit, so setting ${\kappa_{\rm 0F}}={\kappa_{\rm 0P}}$ would not necessarily be more accurate than using ${\kappa_{\rm 0R}}$. We therefore choose to optimize our accuracy in the optically thick part of the flow and set $${\kappa_{\rm 0F}}={\kappa_{\rm 0R}}.$$ With these two approximations, the only two opacities remaining in our equations are ${\kappa_{\rm 0R}}$ and ${\kappa_{\rm 0P}}$, both of which are independent of the spectrum of the radiation field and the direction of radiation propagation, and which may therefore be tabulated as a function of temperature for a given material once and for all.
Next, we expand $(G^0,{\mathbf{G}})$ in powers of $v/c$, retaining terms to order $v^2/c^2$. In performing this expansion, we note that $|{\mathbf{F}}|\le cE$, and $\mbox{Tr}({\mathcal{P}})=E$. The resulting expression for the radiation four-force is $$\begin{aligned}
G^0 & = & {\kappa_{\rm 0P}}\left(E - \frac{4\pi B}{c}\right)
+ \left({\kappa_{\rm 0R}}-2{\kappa_{\rm 0P}}\right) \frac{{\mathbf{v}}\cdot{\mathbf{F}}}{c^2}
\nonumber \\
& & {}
+ \frac{1}{2} \left(\frac{v}{c}\right)^2 \left[2 ({\kappa_{\rm 0P}}-{\kappa_{\rm 0R}}) E + {\kappa_{\rm 0P}}\left(E-\frac{4\pi B}{c}\right)\right]
\nonumber \\
& & {}
+ ({\kappa_{\rm 0P}}-{\kappa_{\rm 0R}}) \frac{{\mathbf{v}}{\mathbf{v}}}{c^2}\colon{\mathcal{P}}+ O\left(\frac{v^3}{c^3}\right)
\label{G01}
\\
{\mathbf{G}}& = & {\kappa_{\rm 0R}}\frac{{\mathbf{F}}}{c}
+ {\kappa_{\rm 0P}}\left(\frac{{\mathbf{v}}}{c}\right) \left(E - \frac{4\pi
B}{c}\right)
\nonumber \\
& & {}
- {\kappa_{\rm 0R}}\left[\frac{{\mathbf{v}}}{c}E + \frac{{\mathbf{v}}}{c}\cdot {\mathcal{P}}\right]
+ \frac{1}{2}\left(\frac{v}{c}\right)^2 {\kappa_{\rm 0R}}\frac{{\mathbf{F}}}{c}
\nonumber \\
& & {}
+ 2 ({\kappa_{\rm 0R}}-{\kappa_{\rm 0P}}) \frac{({\mathbf{v}}\cdot{\mathbf{F}}){\mathbf{v}}}{c^3} +
O\left(\frac{v^3}{c^3}\right)
\label{Gvec1}\end{aligned}$$
It is helpful at this point, before we making any further approximations, to examine the scalings of these terms with the help of our dimensionless parameters $\beta$ and $\tau$. In the streaming limit, radiation travels freely at $c$ and emission and absorption of radiation by matter need not balance, so $|{\mathbf{F}}|\sim c
E$ and $4\pi B/c-E \sim E$. For static diffusion, @mihalas99 show that $|{\mathbf{F}}|\sim c E/\tau$ and $4\pi B/c - E \sim E/\tau^2$. For dynamic diffusion, radiation travels primarily by advection, so $|{\mathbf{F}}| \sim v E$. We show in Appendix \[dyndiffusionscaling\] that for dynamic diffusion $4\pi B/c - E \sim \beta^2 E$. Note that the scaling $4\pi B/c-E\sim (\beta/\tau) E$ given in @mihalas99 appears to be incorrect, as we show in the Appendix. Using these values, we obtain the scalings shown in Table \[Gscalings\] for the terms in (\[G01\]) and (\[Gvec1\]).
[ccccc]{} $G^0$ & ${\kappa_{\rm 0P}}(E-4\pi B/c)$ & ${\mbox{\boldmath$\tau$}}$ & $\mathbf{1/{\mbox{\boldmath$\tau$}}}$ & $\mathbf{{\mbox{\boldmath$\beta$}}^2{\mbox{\boldmath$\tau$}}}$\
$G^0$ & $({\kappa_{\rm 0R}}-2{\kappa_{\rm 0P}}) ({\mathbf{v}}\cdot{\mathbf{F}}/c^2)$ & $\beta \tau$ & $\beta$ & $\mathbf{{\mbox{\boldmath$\beta$}}^2 {\mbox{\boldmath$\tau$}}}$\
$G^0$ & $(v/c)^2({\kappa_{\rm 0P}}-{\kappa_{\rm 0R}}) E$ & $\beta^2 \tau$ & $\beta^2 \tau$ & $\mathbf{{\mbox{\boldmath$\beta$}}^2 {\mbox{\boldmath$\tau$}}}$\
$G^0$ & $(1/2)(v/c)^2 {\kappa_{\rm 0P}}(4\pi B/c-E)$ & $\beta^2 \tau$ & $\beta^2/\tau$ & $\beta^4 \tau$\
$G^0$ & $({\kappa_{\rm 0R}}-{\kappa_{\rm 0P}}) ({\mathbf{v}}{\mathbf{v}}/c^2)\colon{\mathcal{P}}$ & $\beta^2 \tau$ & $\beta^2 \tau$ & $\mathbf{{\mbox{\boldmath$\beta$}}^2 {\mbox{\boldmath$\tau$}}}$\
${\mathbf{G}}$ & ${\kappa_{\rm 0R}}{\mathbf{F}}/c$ & $\mathbf{{\mbox{\boldmath$\tau$}}}$ & $\mathbf{1}$ & ${\mbox{\boldmath$\beta$}}{\mbox{\boldmath$\tau$}}$\
${\mathbf{G}}$ & ${\kappa_{\rm 0P}}({\mathbf{v}}/c)(4\pi B/c-E)$ & $\beta \tau$ & $\beta/\tau$ & $\beta^3 \tau$\
${\mathbf{G}}$ & ${\kappa_{\rm 0R}}[({\mathbf{v}}/c)E + ({\mathbf{v}}/c)\cdot{\mathcal{P}}]$ & $\beta \tau$ & $\beta \tau$ & ${\mbox{\boldmath$\beta$}}{\mbox{\boldmath$\tau$}}$\
${\mathbf{G}}$ & $(1/2)(v/c)^2 {\kappa_{\rm 0R}}{\mathbf{F}}/c$ & $\beta^2 \tau$ & $\beta^2$ & $\beta^3 \tau$\
${\mathbf{G}}$ & $2 ({\kappa_{\rm 0R}}-{\kappa_{\rm 0P}}) ({\mathbf{v}}\cdot{\mathbf{F}}){\mathbf{v}}/c^3$ & $\beta^2 \tau$ & $\beta^2$ & $\beta^3 \tau$\
The Table shows that, despite the fact that we have kept all terms that are formally order $\beta^2$ or more, in fact we only have leading-order accuracy in the dynamic diffusion limit, because in this limit the order unity and order $\beta$ terms in $G^0$ vanish to order $\beta^2$. To obtain the next-order terms, we would have had to write $G^0$ to order $\beta^3$. A corollary of this is that treatments of the dynamic diffusion limit that do not retain order $\beta^2$ terms are likely to produce equations that are incorrect at order unity, since they will have dropped terms that are of the same order as the ones that have been retained.
At this point we could begin dropping terms that are insignificant at the order to which we are working, but it is cumbersome to construct a table analogous to Table \[Gscalings\] at every step of our derivation. It is more convenient to continue our analysis retaining all the terms in (\[G01\]) and (\[Gvec1\]), and to drop terms only periodically.
Before moving on, there is a subtlety in (\[G01\]) and (\[Gvec1\]) that is worth commenting on. Consider a gray fluid, one in which ${\kappa_{\rm 0R}}={\kappa_{\rm 0P}}={\kappa_0}$. In $cG^0$, the term that describes the work done by radiation, $-{\kappa_0}{\mathbf{v}}\cdot {\mathbf{F}}/c$, has the opposite sign from what one might naively expect. Using $cG^0$ in the gas energy equation (\[en1\]) in this case implies that the gas energy increases when ${\mathbf{v}}$ and ${\mathbf{F}}$ are anti-aligned, i.e. when gas moves into an oncoming photon flux. We can understand the origin of this somewhat counter-intuitive behavior by considering the example of a fluid in thermal equilibrium with a radiation field in its rest frame (i.e.$4\pi B = c E_0$). In the comoving frame, the radiation four-force behaves as one intuitively expects: at leading order the rate at which the radiation field transfers momentum density to the gas is ${\mathbf{G}}_0 = {\kappa_0}{\mathbf{F}}_0/c$, and the rate at which the gas energy density changes as a result is $c G^0_0 = {\kappa_0}{\mathbf{v}}\cdot
{\mathbf{F}}_0/c$ [@mihalas01 their equations 53a and 53b]. Thus, gas loses energy when it moves opposite the direction of the flux, and hence opposite the force.
However, now consider the fluid as seen by an observer in a frame boosted by velocity $-{\mathbf{v}}$ relative to the fluid. The observer sees the radiation energy density as $E$, which differs from $E_0$ by $2 {\mathbf{v}}\cdot {\mathbf{F}}/c^2$ (see equation \[etransform\]), and this difference is the reason that the work term in $G^0$ is $-{\kappa_0}{\mathbf{v}}\cdot{\mathbf{F}}/c^2$. Physically, this happens because an observer who sees the fluid moving at velocity ${\mathbf{v}}$ also sees the radiation and gas as being out of thermal equilibrium ($4\pi B \neq c E$), since $E$ and $E_0$ are different. This disequilibrium leads the radiation and gas to exchange energy at a rate that is opposite in direction and twice as large as the radiation work, ${\kappa_0}{\mathbf{v}}\cdot{\mathbf{F}}/c$. This is why the “work" term has the opposite sign than the one we might expect. Thus, for the rest of this paper, while for convenience we continue to refer to $({\kappa_{\rm 0R}}-2{\kappa_{\rm 0P}}) {\mathbf{v}}\cdot {\mathbf{F}}/c$ and the terms to which it gives rise as “work" terms, it is important to keep in mind that in reality this term contains contributions from two different effects of comparable magnitude, the “Newtonian work" ${\kappa_{\rm 0R}}{\mathbf{v}}\cdot{\mathbf{F}}/c$ and the post-Newtonian term $-2{\kappa_{\rm 0P}}{\mathbf{v}}\cdot{\mathbf{F}}/c$ describing the imbalance between emission and absorption that an observer sees solely because the fluid is moving.
With this point understood, we now adopt the flux-limited diffusion approximation [@alme73], under which we drop the radiation momentum equation (\[radmom1\]) and set the radiation flux in the comoving frame to $$\label{fldflux}
{\mathbf{F}}_0 = -\frac{c\lambda}{{\kappa_{\rm 0R}}} \nabla E_0,$$ where $\lambda$ is a dimensionless number called the flux-limiter. Many functional forms for $\lambda$ are possible. For the code implementation we describe later, we adopt the @levermore81 flux-limiter, given by $$\begin{aligned}
\lambda & = & \frac{1}{R} \left(\coth R - \frac{1}{R}\right) \\
R & = & \frac{|\nabla E_0|}{{\kappa_{\rm 0R}}E_0}.\end{aligned}$$ However, our derivation is independent of this choice. Regardless of their exact functional form, all flux limiters have the property that in an optically thick medium $\lambda\rightarrow 1/3$, thereby giving ${\mathbf{F}}_0\rightarrow -[c/(3{\kappa_{\rm 0R}})]\nabla E_0$, the correct value for diffusion. In an optically thin medium, $\lambda\rightarrow ({\kappa_{\rm 0R}}E_0/|\nabla E_0|) {\mathbf{n}}_0$, where ${\mathbf{n}}_0$ is the unit vector antiparallel to $\nabla E_0$, so the flux approaches ${\mathbf{F}}_0\rightarrow c E_0 {\mathbf{n}}_0$, and the propagation speed of radiation is correctly limited to $c$.
For the @levermore81 flux-limiter we adopt, the corresponding approximate value for the radiation pressure tensor is [@levermore84] $$\label{fldpressure}
{\mathcal{P}}_0 = \frac{E_0}{2} \left[(1-R_2){\mathcal{I}}+ (3 R_2 - 1)
{\mathbf{n}}_0 {\mathbf{n}}_0\right],$$ where ${\mathcal{I}}$ is the identity tensor of rank 2 and $$R_2 = \lambda + \lambda^2 R^2.$$ Physically, this approximation interpolates between the behavior in very optically thick regions, where $R_2\rightarrow 1/3 +
O(1/\tau^2)$, the radiation pressure is isotropic, and off-diagonal components vanish, and optically thin regions, where $R_2 \rightarrow
1$ and the radiation pressure tensor is zero orthogonal to ${\mathbf{n}}_0$ and $E_0$ parallel to it.
Note that for pure diffusion @mihalas99 and @castor04 show that the pressure tensor reduces to $(E_0/3) {\mathcal{I}}$ plus off-diagonal elements of order $\beta/\tau$ or $\beta^2$. Our approximation does not quite reproduce this, since in the diffusion limit it gives ${\mathcal{P}}_0 = (E_0/3) {\mathcal{I}}$ plus off-diagonal elements of order $\tau^{-2}$. We might therefore worry that, in the static diffusion regime where $\beta \ll \tau^{-1}$, we will have an incorrect term. However, examination of our final equations below shows that all terms arising from off-diagonal elements of ${\mathcal{P}}_0$ are smaller than order $\beta$ in the static diffusion limit, so adopting the @levermore84 approximation for the pressure tensor does not introduce any incorrect terms at order $\beta$ in the final equations.
To use the approximations (\[fldflux\]) and (\[fldpressure\]) to evaluate the radiation four-force, we must Lorentz transform them to express the radiation quantities in the lab frame. The Lorentz transforms for the energy, flux, and pressure to second order in $v/c$ are [@mihalas99] $$\begin{aligned}
E & = & E_0 + 2\frac{{\mathbf{v}}\cdot{\mathbf{F}}_0}{c^2} + \frac{1}{c^2}\left[v^2 E_0 + ({\mathbf{v}}{\mathbf{v}})\colon{\mathcal{P}}_0\right]
\label{etransform}
\\
{\mathbf{F}}& = & {\mathbf{F}}_0 + {\mathbf{v}}E_0 + {\mathbf{v}}\cdot{\mathcal{P}}_0 + \frac{1}{2c^2} \left[v^2 {\mathbf{F}}_0+3 {\mathbf{v}}({\mathbf{v}}\cdot{\mathbf{F}}_0)\right]
\label{ftransform}
\\
{\mathcal{P}}& = & {\mathcal{P}}_0 + \frac{{\mathbf{v}}{\mathbf{F}}_0 + {\mathbf{F}}_0{\mathbf{v}}}{c^2} +
\frac{1}{c^2}\left[{\mathbf{v}}{\mathbf{v}}E_0 + {\mathbf{v}}({\mathbf{v}}\cdot{\mathcal{P}}_0)\right].
\label{ptransform}\end{aligned}$$ Note that in the expression for ${\mathcal{P}}$ we have simplified the final term using the fact that ${\mathcal{P}}_0$ is a symmetric tensor.
Using the same scaling arguments we used to construct Table \[Gscalings\], we see that ${\mathcal{P}}$ and ${\mathcal{P}}_0$ differ at order $\beta$ in the streaming limit, at order $\beta/\tau$ for static diffusion, and at order $\beta^2$ for dynamic diffusion. Since this is below our accuracy goal, we need not distinguish ${\mathcal{P}}$ and ${\mathcal{P}}_0$. The same is true of $E$ and $E_0$. However, ${\mathbf{F}}$ is different. In the comoving frame in an optically thick system, one is in the static diffusion regime, so ${\mathbf{F}}_0 \sim c E_0/\tau$. Since ${\mathbf{v}}E_0$ and ${\mathbf{v}}\cdot{\mathcal{P}}_0$ are of order $\beta c E_0$, and in dynamic diffusion $\beta \gg 1/\tau$, this means that ${\mathbf{v}}E_0$ and ${\mathbf{v}}\cdot{\mathcal{P}}_0$ are the dominant components of ${\mathbf{F}}$ in dynamic diffusion, and must therefore be retained. Thus, $$\label{labflux}
{\mathbf{F}}= -\frac{c\lambda}{{\kappa_{\rm 0R}}} \nabla E + {\mathbf{v}}E +
{\mathbf{v}}\cdot{\mathcal{P}},
\label{fluxtransform}$$ which is simply the rest frame flux plus terms describing the advection of radiation enthalpy.
Substituting (\[fldpressure\]) with ${\mathcal{P}}={\mathcal{P}}_0$ and (\[fluxtransform\]) into the four-force density (\[G01\]) and (\[Gvec1\]), and continuing to retain terms to order $v^2/c^2$, gives $$\begin{aligned}
G^0 & = & {\kappa_{\rm 0P}}\left(E-\frac{4\pi B}{c}\right)
+ \left(\frac{\lambda}{c}\right)\left(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right)
{\mathbf{v}}\cdot \nabla E
\nonumber \\
& & {}
- \frac{{\kappa_{\rm 0P}}}{c^2} E \left[\frac{3-R_2}{2} v^2 + \frac{3
R_2-1}{2} ({\mathbf{v}}\cdot{\mathbf{n}})^2\right]
\nonumber \\
& & {}
+ \frac{1}{2}\left(\frac{v}{c}\right)^2 {\kappa_{\rm 0P}}\left(E-\frac{4\pi
B}{c}\right)
\label{gztrans}
\\
{\mathbf{G}}& = & - \lambda \nabla E + {\kappa_{\rm 0P}}\frac{{\mathbf{v}}}{c}
\left(E - \frac{4\pi B}{c}\right)
\nonumber \\
& & {}
-\frac{1}{2}\left(\frac{v}{c}\right)^2 \lambda \nabla E
\nonumber \\
& & {}
+ 2\lambda\left(\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right) \frac{({\mathbf{v}}\cdot\nabla
E){\mathbf{v}}}{c^2}.
\label{gtrans}\end{aligned}$$ Here ${\mathbf{n}}$ is the unit vector antiparallel to $\nabla E$. We again remind the reader that, although these equations contain terms of order $\beta^2$, they are not truly accurate to order $\beta^2$ because we did not retain all the $\beta^2$ when applying the Lorentz transform to the flux and pressure. However, these equations include all the terms that appear at the order of accuracy to which we are working, and by retaining terms of order $\beta^2$ we guarantee that these terms will be preserved.
Inserting $(G^0,{\mathbf{G}})$ and the lab frame flux (\[labflux\]) into the gas momentum and energy equations (\[mom1\]) and (\[en1\]), and the radiation energy equation (\[raden1\]), and again retaining terms to order $v^2/c^2$ gives $$\begin{aligned}
\frac{\partial}{\partial t}(\rho {\mathbf{v}}) & = &
- \nabla\cdot(\rho{\mathbf{v}}{\mathbf{v}}) -\nabla P
- \lambda \nabla E
\nonumber \\
& & {}
- {\kappa_{\rm 0P}}\frac{{\mathbf{v}}}{c^2}
\left(4\pi B - c E\right)
-\frac{1}{2}\left(\frac{v}{c}\right)^2 \lambda \nabla E
\nonumber \\
& & {}
+ 2\lambda\left(\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right) \frac{({\mathbf{v}}\cdot\nabla E){\mathbf{v}}}{c^2}.
\label{mom2}\\
\frac{\partial}{\partial t}(\rho e) & = &
- \nabla\cdot[(\rho e + P){\mathbf{v}}] - {\kappa_{\rm 0P}}(4\pi B - c E)
\nonumber \\
& & {}
+ \lambda\left(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right) {\mathbf{v}}\cdot\nabla E
\nonumber \\
& & {}
- \frac{{\kappa_{\rm 0P}}}{c} E \left[\frac{3 - R_2}{2} v^2 +
\frac{3 R_2 - 1}{2} ({\mathbf{v}}\cdot{\mathbf{n}})^2
\right]
\nonumber \\
& & {}
- \frac{1}{2}\left(\frac{v}{c}\right)^2 {\kappa_{\rm 0P}}\left(4 \pi B - c E\right)
\label{en2}
\\
\frac{\partial}{\partial t}E
& = &
\nabla\cdot\left(\frac{c\lambda}{{\kappa_{\rm 0R}}} \nabla E\right) + {\kappa_{\rm 0P}}(4\pi B - c E)
\nonumber \\
& & {}
-\lambda \left(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right) {\mathbf{v}}\cdot \nabla E
\nonumber \\
& & {}
+\frac{{\kappa_{\rm 0P}}}{c} E \left[
\frac{3-R_2}{2} v^2 + \frac{3R_2-1}{2} ({\mathbf{v}}\cdot{\mathbf{n}})^2
\right]
\nonumber \\
& & {}
- \nabla \cdot \left[
\frac{3-R_2}{2} {\mathbf{v}}E + \frac{3 R_2-1}{2} {\mathbf{v}}\cdot({\mathbf{n}}{\mathbf{n}})
E\right]
\nonumber \\
& & {}
+ \frac{1}{2}\left(\frac{v}{c}\right)^2 {\kappa_{\rm 0P}}\left(4 \pi B - c E\right).
\label{raden2}\end{aligned}$$ At this point we construct Table \[eqscalings\] showing the scalings of the radiation terms to see which must be retained and which are superfluous. In constructing the table, we take spatial derivatives to be of characteristic scaling $1/\ell$, i.e. we assume that radiation quantities vary on a size scale of the system, rather than over a size scale of the photon mean free path. In the streaming limit, $\lambda\sim \tau$ and $R_2\sim 1+O(\tau)$. In the diffusion limit $\lambda\sim 1/3$ and $R_2\sim 1/3+O(\tau^{-2})$.
[ccccc]{} M & $\lambda\nabla E$ & ${\mbox{\boldmath$\tau$}}$ & $\mathbf{1}$ & $\mathbf{1}$\
M & ${\kappa_{\rm 0P}}({\mathbf{v}}/c^2) (4\pi B - c E)$ & $\beta\tau$ & $\beta/\tau$ & $\beta^3\tau$\
M & $(1/2)(v/c)^2 \lambda \nabla E$ & $\beta^2 \tau$ & $\beta^2$ & $\beta^2$\
M & $2\lambda({\kappa_{\rm 0P}}/{\kappa_{\rm 0R}}-1) ({\mathbf{v}}\cdot\nabla E){\mathbf{v}}/c^2$ & $\beta^2\tau$ & $\beta^2$ & $\beta^2$\
G and R & ${\kappa_{\rm 0P}}(4\pi B - c E)$ & ${\mbox{\boldmath$\tau$}}$ & $\mathbf{1/{\mbox{\boldmath$\tau$}}}$ & $\mathbf{{\mbox{\boldmath$\beta$}}^2 {\mbox{\boldmath$\tau$}}}$\
G and R & $\lambda(2{\kappa_{\rm 0P}}/{\kappa_{\rm 0R}}-1) {\mathbf{v}}\cdot\nabla E$ & $\beta \tau$ & $\beta$ & ${\mbox{\boldmath$\beta$}}$\
G and R & ${\kappa_{\rm 0P}}(v^2/c) [(3-R_2)/2]E$ & $\beta^2 \tau$ & $\beta^2 \tau$ & $\mathbf{{\mbox{\boldmath$\beta$}}^2 {\mbox{\boldmath$\tau$}}}$\
G and R & ${\kappa_{\rm 0P}}[({\mathbf{v}}\cdot{\mathbf{n}})^2/c] [(3R_2 - 1)/2]E$ & $\beta^2 \tau$ & $\beta^2/\tau$ & $\beta^2/\tau$\
G and R & $(1/2)(v/c)^2 {\kappa_{\rm 0P}}(c E - 4\pi B)$ & $\beta^2\tau$ & $\beta^2/\tau$ & $\beta^4\tau$\
R & $\nabla \cdot[(c\lambda/{\kappa_{\rm 0R}})\nabla E]$ & $\mathbf{1}$ & $\mathbf{1/{\mbox{\boldmath$\tau$}}}$ & $1/\tau$\
R & $\nabla \cdot\{[(3-R_2)/2] {\mathbf{v}}E\}$ & $\beta$ & $\beta$ & ${\mbox{\boldmath$\beta$}}$\
R & $\nabla \cdot\{[(3 R_2-1)/2] {\mathbf{v}}\cdot({\mathbf{n}}{\mathbf{n}}) E\}$ & $\beta$ & $\beta/\tau^2$ & $\beta/\tau^2$\
Using Table \[eqscalings\] to drop all terms that are not significant at leading order in any regime, we arrive at our final equations: $$\begin{aligned}
\frac{\partial}{\partial t}(\rho {\mathbf{v}}) & = &
- \nabla\cdot(\rho{\mathbf{v}}{\mathbf{v}}) -\nabla P
- \lambda \nabla E
\label{momentumconservation}\\
\frac{\partial}{\partial t}(\rho e) & = &
- \nabla\cdot[(\rho e + P){\mathbf{v}}] - {\kappa_{\rm 0P}}(4\pi B - c E)
\nonumber \\
& & {}
+ \lambda\left(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right) {\mathbf{v}}\cdot\nabla E
\nonumber \\
& & {}
- \frac{3-R_2}{2} {\kappa_{\rm 0P}}\frac{v^2}{c} E
\label{gasenergy}
\\
\frac{\partial}{\partial t}E
& = &
\nabla\cdot\left(\frac{c\lambda}{{\kappa_{\rm 0R}}} \nabla E\right) + {\kappa_{\rm 0P}}(4\pi B - c E)
\nonumber \\
& & {}
-\lambda \left(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right) {\mathbf{v}}\cdot \nabla E
\nonumber \\
& & {}
+ \frac{3-R_2}{2} {\kappa_{\rm 0P}}\frac{v^2}{c} E
\nonumber \\
& & {}
- \nabla\cdot\left(\frac{3-R_2}{2}{\mathbf{v}}E\right).
\label{radenergy}\end{aligned}$$ These represent the equations of momentum conservation for the gas, energy conservation for the gas, and energy conservation for the radiation field, which, together with the equation of mass conservation (\[massconservation\]), fully describe the system under the approximations we have adopted. They are accurate and consistent to leading order in the streaming and dynamic diffusion limits. They are accurate to first order in $\beta$ in the static diffusion limit, since we have had to retain all order $\beta$ terms in this limit because they are of leading order in dynamic diffusion problems. Also note that if in a given problem one never encounters the dynamic diffusion regime, it is possible to drop more terms, as we discuss in § \[algorithm\].
The equations are easy to understand intuitively. The term $-\lambda\nabla E$ in the momentum equation (\[momentumconservation\]) simply represents the radiation force ${\kappa_{\rm 0R}}{\mathbf{F}}/c$, neglecting distinctions between the comoving and laboratory frames which are smaller than leading order in this equation. Similarly, the terms $\pm{\kappa_{\rm 0P}}(4\pi B - cE)$ and $\pm\lambda(2{\kappa_{\rm 0P}}/{\kappa_{\rm 0R}}-1){\mathbf{v}}\cdot\nabla E$ in the two energy equations (\[gasenergy\]) and (\[radenergy\]) represent radiation absorbed minus radiation emitted by the gas, and the work done by the radiation field as it diffuses through the gas. The factor $(2{\kappa_{\rm 0P}}/{\kappa_{\rm 0R}}-1)$ arises because the term contains contributions both from the Newtonian work and from a relativistically-induced mismatch between emission and absorption. The term proportional to ${\kappa_{\rm 0P}}E/c$ represents another relativistic correction to the work, this one arising from boosting of the flux between the lab and comoving frames. In the radiation energy equation (\[radenergy\]), the first term on the left hand side is the divergence of the radiation flux, i.e. the rate at which radiation diffuses, and the last term on the right hand side represents advection of the radiation enthalpy $E+{\mathcal{P}}$ by the gas.
It is also worth noting that equations (\[en2\]) and (\[raden2\]) are manifestly energy-conserving, since every term in one equation either has an obvious counterpart in the other with opposite sign, or is clearly an advection. In contrast, the momentum equation (\[momentumconservation\]) is not manifestly momentum-conserving, since there is a force term $-\lambda\nabla E$ with no equal and opposite counterpart. This non-conservation of momentum is an inevitable side-effect of using the flux-limited diffusion approximation, since this approximation amounts to allowing the radiation field to transfer momentum to the gas without explicitly tracking the momentum of the radiation field and the corresponding transfer from gas to radiation.
The Importance of Higher Order Terms {#termanalysis}
====================================
Our dynamical equations result from retaining at least some terms that are formally of order $\beta^2$. Even though our analysis shows that these terms can be the leading ones present, due to cancellations of lower order terms, one might legitimately ask whether they are ever physically significant. In § \[ordcomparison\] we address this question by comparing our equations to those that result from lower order treatments. In § \[framecomparison\], we also compare our equations with those generally used in comoving frame formulations of radiation hydrodynamics.
To make our work in this section more transparent, and since we are more interested in physical intuition than rigorous derivation here, we specialize to the diffusion regime in gray materials. Thus, we set $\lambda = R_2 = 1/3$ and ${\kappa_{\rm 0P}}={\kappa_{\rm 0R}}={\kappa_0}$. A more general analysis produces the same conclusions, but is more mathematically cumbersome. We also focus on the radiation energy equation, since all the terms that appear in the gas energy equation also appear in it, and because there are no higher order terms present in the momentum equation. Under these assumptions, our radiation energy equation (\[radenergy\]) becomes $$\begin{aligned}
\frac{\partial}{\partial t}E
& = &
\nabla\cdot\left(\frac{c}{3 {\kappa_0}} \nabla E\right) + {\kappa_0}(4\pi B - c E)
\nonumber \\
& & {}
- \frac{4}{3}\nabla \cdot({\mathbf{v}}E)
- \frac{1}{3} {\mathbf{v}}\cdot \nabla E
+ \frac{4}{3}{\kappa_0}\frac{v^2}{c} E.
\label{difffull}\end{aligned}$$
Comparison to Lower Order Equations {#ordcomparison}
-----------------------------------
A common approach in radiation-hydrodynamic problems is to expand the equations in $\beta$, rather than in both $\beta$ and $\tau$ as we have done, and drop at least some terms that are of order $\beta^2$ in every regime [e.g @mihalas99]. To determine how equations derived in this manner compare to our higher order treatment, we compare our simplified energy equation (\[difffull\]) to the corresponding equation one would obtain by following this procedure with (\[radenergy\]). This resulting energy equation is $$\begin{aligned}
\frac{\partial}{\partial t} E & = & \nabla\cdot\left(\frac{c}{3{\kappa_0}}
\nabla E \right) + {\kappa_0}(4 \pi B - c E)
\nonumber \\
& &
{} - \frac{4}{3}\nabla\cdot({\mathbf{v}}E) - \frac{1}{3} {\mathbf{v}}\cdot\nabla E.
\label{difffirstorder}\end{aligned}$$ *It is important to caution at this point that, as we show below, equation (\[difffirstorder\]) is not accurate to leading order in at least some cases, and should not be used for computations unless one carefully checks that the missing terms never become important in the regime covered by the computation.*
Compared to the energy equation (\[difffull\]) that we obtain by retaining all leading order terms in $\beta$ and $\tau$, (\[difffirstorder\]) is missing the term $(4/3) {\kappa_0}v^2 E/c$. If we think of the flux as having two parts, a “diffusion" part proportional to $\nabla E$ that comes from radiation diffusion in the comoving frame, and a “relativistic" part proportional to ${\mathbf{v}}E + {\mathbf{v}}\cdot{\mathcal{P}}$ that comes from the Lorentz transformation between lab and comoving frames, then it is natural to describe the ${\mathbf{v}}\cdot\nabla E$ term as the “diffusion work” arising from the combination of the diffusion flux and the post-Newtonian emission-absorption mismatch (as discussed in § \[equationderivation\]), and the ${\kappa_0}v^2 E/c$ as the “relativistic work” arising from the relativistic flux. The presence or absence of this relativistic work term is the difference between our leading order-accurate equation and the equation one would derive by dropping $\beta^2$ terms. Analyzing when, if ever, this term is physically important lets us identify in which situations a lower order treatment may be inadequate.
If we use Table \[eqscalings\] to compare the relativistic work term to the emission/absorption term, we find that $({\kappa_0}v^2 E/c) / [{\kappa_0}(4\pi B -
c E)]$ is of order $\beta^2 \tau^2$ for static diffusion, and of order unity for dynamic diffusion. Thus, the term is never important in a static diffusion problem, but is always important for a non-uniform, non-equilibrium dynamic diffusion problem system. We add the caveats about non-uniformity and time-dependence because in a system where there is no radiation-gas energy exchange, the relativistic work term will be small due to a cancellation. The example in Appendix \[dyndiffusionscaling\] shows that in an equilibrium, uniform medium, the terms ${\kappa_0}(4\pi B-c E)$ and $(4/3)
{\kappa_0}v^2 E/c$ cancel exactly at orders up to $\beta^2$. We expect any system where variations occur on a scale for which $\beta\tau\gg 1$ to resemble such a uniform, equilibrium medium, and thus we do not expect the term $(4/3) {\kappa_0}v^2 E/c$ to be important in such a system.
That said, there is still clearly a problem with omitting the relativistic work term in a system where $\beta\tau \sim 1$. In this case, Table \[eqscalings\] implies that *every* term on the right hand side is roughly equally important regardless of whether we use the static of dynamic diffusion scalings. To illustrate this point, consider a radiation-dominated shock. The width of such a shock is set by the balance between radiation diffusing upstream from the hot post-shock region into the cold pre-shock region, and advection of the radiation back downstream by the pre-shock gas. This condition requires that $\beta\tau\sim 1$ across the shock [@mihalas99], so its width $w
\sim {\lambda_{\rm P}}/\beta$. Since $E$ changes by of order unity across this distance, its spatial derivative is of order $\nabla E\sim
E/w\sim (\beta/{\lambda_{\rm P}}) E$. Applying this to (\[difffull\]), we find that each term on the right hand side is of order $\beta^2 (c/{\lambda_{\rm P}})
E$. Since the terms like $-(4/3) \nabla \cdot ({\mathbf{v}}E)$ describing advection and $\nabla\cdot[c/(3{\kappa_0})\nabla E]$ describing diffusion are obviously important in the structure of the shock, causing order unity changes in $E$, and the relativistic work term is comparable, it follows that the relativistic work term is equally important. One can obtain the correct structure within a radiation-dominated shock only by retaining the relativistic work term.
An interesting point to note here is that omitting the relativistic work term will not produce errors upstream or downstream of a shock, because $\beta\tau \gg 1$ in these regions. Furthermore, the jump conditions across a shock should be correct. The omitted term affects radiation-gas energy exchange, not total energy conservation, and all that is required to get the correct jump conditions are conservation of mass and energy, plus correct computation of the upstream and downstream radiation pressures. The lower order treatment will therefore only make errors within the shock. Whether this is physically important, or it is sufficient to get the jump conditions correct, depends on whether one is concerned with structures on scales for which $\beta\tau\sim 1$. An astrophysical example of a system where one does care about structures on this scale is a radiation-dominated accretion disk subject to photon bubble instability [@turner03]. Such disks are in the dynamic diffusion regime over the entire disk, but photon bubbles form on small scales within them, and individual bubbles may have $\beta\tau\sim 1$ across them.
Comparison to Comoving Frame Formulations {#framecomparison}
-----------------------------------------
Many popular numerical treatments of radiation hydrodynamics [e.g. @turner01; @whitehouse04; @hayes06] use a comoving formulation of the equations rather than our mixed frame formulation. It is therefore useful to compare our equations to the standard comoving frame equations. In the comoving formulation, the evolution equation for the radiation field is usually the first law of thermodynamics for the comoving radiation field [@mihalas82], $$\label{firstlaw}
\rho \frac{D}{Dt}\left(\frac{E_0}{\rho}\right) +
{\mathcal{P}}_0\colon(\nabla{\mathbf{v}}) = {\kappa_0}(4 \pi B - cE_0) - \nabla \cdot {\mathbf{F}}_0.$$ This equation is accurate to first order in $\beta$ in the sense that it contains all the correct leading order terms and all terms that are smaller than them by order $\beta$ or less.
To compare this to our mixed frame radiation energy equation (\[radenergy\]), we replace the comoving frame energy $E_0$ in (\[firstlaw\]) with the lab frame energy $E$ using the Lorentz transformation (\[etransform\]) and retain all terms that are of leading order in any regime. In practice, this means that we set $E_0
= E$ inside the time derivative, since the difference between $E$ and $E_0$ is at most $\beta/\tau$ or $\beta^2$ for static or dynamic diffusion. However, when replacing $E_0$ with $E$ in the heating/cooling term $4\pi B - c E_0$, we must retain all the terms in (\[etransform\]) because the leading term $4\pi B - c E$ is itself only of order $\tau^{-2}$ or $\beta^2$ relative to $E$, so the difference between $E$ and $E_0$ can be of leading order.[^2] This gives a transformed equation $$\begin{aligned}
\lefteqn{\rho \frac{D}{Dt}\left(\frac{E}{\rho}\right) +
{\mathcal{P}}_0\colon(\nabla{\mathbf{v}}) = {\kappa_0}(4 \pi B - cE) - \nabla \cdot
{\mathbf{F}}_0
} \qquad\qquad
\nonumber \\
& & {}
+ 2 {\kappa_0}\frac{{\mathbf{v}}\cdot{\mathbf{F}}_0}{c} + \frac{{\kappa_0}}{c} \left[ v^2 E +
({\mathbf{v}}{\mathbf{v}})\colon{\mathcal{P}}_0\right].
\label{firstlaw1}\end{aligned}$$
If we now adopt the diffusion approximation ${\mathbf{F}}_0 = -c/(3{\kappa_0})
\nabla E_0$ and ${\mathcal{P}}_0=(1/3) E_0 {\mathcal{I}}$, use the Lorentz transformation to replace $E_0$ with $E$ throughout, and again only retain terms that are of leading in order in some regime, then it is easy to verify that (\[firstlaw1\]) reduces to (\[difffull\]). Thus, our evolution equation is equivalent to the comoving frame first law of thermodynamics for the radiation field, *provided that one retains all the leading order terms with respect to $\beta$ and $\tau$, including some that are of order $\beta^2$, when evaluating the Lorentz transformation*.
While the equations are equivalent, the mixed frame formulation has two important advantages over the comoving frame formulation when it comes to practical computation. First, we are able to write the equations in a manner that allows a numerical solution algorithm to conserve total energy to machine accuracy. We present such an algorithm in § \[algorithm\]. In contrast, it is not possible to write a conservative update algorithm using the comoving frame equations. The reason for this is that a conserved total energy only exists in an inertial frame, and for a fluid whose velocity is not a constant in space and time, the comoving frame is not inertial. The lack of a conserved energy is a serious drawback to comoving frame formulations.
A second advantage of the mixed-frame formulation is that it is far more suited to implementation in codes with dynamically modified grid structures such as adaptive mesh refinement methods. Since the radiation energy is a conserved quantity, it is obvious how to refine or coarsen it in a conservative manner. On the other hand, there is no obviously correct method for refining or coarsening the comoving frame energy density, because it will not even be defined in the same reference frames before and after the refinement procedure.
An Optimized Algorithm for Static Diffusion Radiation Hydrodynamics {#algorithm}
===================================================================
Operator Splitting
------------------
Our analysis shows that for static diffusion, the terms involving diffusion and emission minus absorption of radiation always dominate over those involving radiation work and advection. In addition, some terms are always smaller than order $\beta$. This suggests an opportunity for a significant algorithmic improvement over earlier approaches while still retaining order $\beta$ accuracy in the solution. In a simulation, one must update terms for the radiation field implicitly, because otherwise stability requirements limit the update time step to values comparable to the light-crossing time of a cell. Standard approaches [e.g. @turner01; @whitehouse04; @whitehouse05; @hayes06] therefore update all terms involving radiation implicitly except the advection term and the radiation force term in the gas momentum equation.
However, implicit updates are computationally expensive, so the simpler the terms to be updated implicitly can be made, the simpler the algorithm will be to code and the faster it will run. Since the work and advection terms are non-dominant, we can produce a perfectly stable algorithm without treating them implicitly. Even if this treatment introduces numerically unstable modes in the work or advection terms, they will not grow because the radiation diffusion and emission/absorption terms, which are far larger, will smooth them away each time step.
For the case of static diffusion, we therefore adopt the order $v/c$ equations (\[massconservation\]) and (\[momentumconservation\]) for mass and momentum conservation. For our energy equations, we adopt (\[gasenergy\]) and (\[radenergy\]), but drop terms that are smaller than order $\beta$ for static diffusion. This gives $$\begin{aligned}
\frac{\partial}{\partial t}(\rho e) & = &
-\nabla[(\rho e + P){\mathbf{v}}] - {\kappa_{\rm 0P}}(4\pi B - c E)
\nonumber \\
& & {} + \lambda \left(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right)
{\mathbf{v}}\cdot\nabla E
\label{gasenstatdiff}
\\
\frac{\partial}{\partial t}E & = &
\nabla\cdot\left(\frac{c\lambda}{{\kappa_{\rm 0R}}} \nabla E\right)
+ {\kappa_{\rm 0P}}(4\pi B - c E)
\nonumber \\
& & {}
-\lambda \left(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1\right) {\mathbf{v}}\cdot \nabla E
\nonumber \\
& & {}
- \nabla \cdot \left(\frac{3-R_2}{2}{\mathbf{v}}E\right)
\label{radenstatdiff}\end{aligned}$$ To solve these, we operator split the diffusion and emission/absorption terms, which we treat implicitly, from the work and advection terms, which we treat explicitly. To do this, we write our gas/radiation state as $${\mathbf{q}}=
\left(
\begin{array}{c}
\rho \\
\rho {\mathbf{v}}\\
\rho e \\
E
\end{array}
\right),$$ and our evolution equations as $$\frac{\partial {\mathbf{q}}}{\partial t}
= {\mathbf{f}}_{\rm e-nr} + {\mathbf{f}}_{\rm e-rad} + {\mathbf{f}}_{\rm i-rad},$$ where we have broken our right hand side up into non-radiative terms to be handled explicitly, $${\mathbf{f}}_{\rm e-nr} =
\left(
\begin{array}{c}
-\nabla\cdot(\rho{\mathbf{v}}) \\
-\nabla\cdot(\rho{\mathbf{v}}{\mathbf{v}})-\nabla P \\
-\nabla\cdot[(\rho e+P){\mathbf{v}}] \\
0
\end{array}
\right),$$ radiative terms to be handled explicitly, $${\mathbf{f}}_{\rm e-rad} =
\left(
\begin{array}{c}
0 \\
-\lambda\nabla E \\
\lambda (2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1) {\mathbf{v}}\cdot\nabla E \\
-\lambda(2\frac{{\kappa_{\rm 0P}}}{{\kappa_{\rm 0R}}}-1){\mathbf{v}}\cdot\nabla E
- \nabla\cdot\left(\frac{3-R_2}{2}{\mathbf{v}}E\right)
\end{array}
\right),$$ and radiative terms that must be handled implicitly, $${\mathbf{f}}_{\rm i-rad} =
\left(
\begin{array}{c}
0 \\
\mathbf{0} \\
-{\kappa_{\rm 0P}}(4\pi B - cE) \\
\nabla\cdot\left(\frac{c\lambda}{{\kappa_{\rm 0R}}} \nabla E\right)
+ {\kappa_{\rm 0P}}(4\pi B - cE)
\end{array}
\right).$$
Update Scheme
-------------
For each update cycle, we start with the state ${\mathbf{q}}^n$ at the old time. We first perform an implicit update to the radiation and gas energy densities using ${\mathbf{f}}_{\rm i-rad}$. Any number of methods are possible for this. For our implementation of this algorithm in the Orion adaptive mesh refinement (AMR) code, we use the method of @howell03, which we will not discuss in detail here. To summarize, the algorithm involves writing the equations using second order accurate spatial discretization and a time discretization that limits to backwards Euler for large values of $\partial E/\partial
t$ (to guarantee stability) and to Crank-Nicolson when $\partial
E/\partial t$ is small (to achieve second order time accuracy). This yields a matrix equation for the radiation and gas energy densities at the new time, which may be solved on both individual grids and over a hierarchy of nested grids (as is necessary for AMR) using standard multigrid techniques. The output of this procedure is an intermediate state ${\mathbf{q}}^{n,*}$ which has been updated for ${\mathbf{f}}_{\rm i-rad}$.
Once the implicit update is done, we compute the ordinary hydrodynamic update. As with the implicit update, this may be done using the hydrodynamics method of one’s choice. For our implementation, we use the Godunov method described by @truelove98, @klein99, and @fisher02. This update gives us ${\mathbf{q}}^{n,\dagger}$, the state updated for ${\mathbf{f}}_{\rm i-rad}$ and ${\mathbf{f}}_{\rm e-nr}$. The only modification we make to the standard update algorithm is to include a radiation pressure term in the effective sound speed used to compute the Courant condition. Thus, we take $$c_{\rm eff} = \sqrt{\frac{\gamma P + (4/9) E (1 - e^{-{\kappa_{\rm 0R}}\Delta
x})}{\rho}}$$ and set the time step to $$\Delta t = C \frac{\Delta x}{\max(|{\mathbf{v}}|+c_{\rm eff})},$$ where $\gamma$ is the ratio of specific heats for the gas, $C$ is the Courant factor (usually 0.5), and the maximum is evaluated over all cells. For AMR, this condition is applied independently on each level $l$, and the time step is set using the values of $\Delta t^l$ in the standard AMR manner [e.g. @klein99]. The factor $(1 -
e^{-{\kappa_{\rm 0R}}\Delta x})$ gives us a means of interpolating between optically thick cells, where radiation pressure contributes to the restoring force and thus increases the effective signal speed, and optically thin cells, where radiation does not provide any pressure.
Finally, we compute the force and advection terms in ${\mathbf{f}}_{\rm e-rad}$. In our implementation we compute all of these at cell centers using second order centered differences. For $\nabla
E$ this is $$(\nabla E)^{n,*}_{i,j,k} =
\left(
\begin{array}{c}
\frac{E^{n,*}_{i+1,j,k}-E^{n,*}_{i-1,j,k}}{2\Delta x}, \\
\frac{E^{n,*}_{i,j+1,k}-E^{n,*}_{i,j-1,k}}{2\Delta y}, \\
\frac{E^{n,*}_{i,j,k+1}-E^{n,*}_{i,j,k-1}}{2\Delta z} \\
\end{array}
\right).$$ Other derivatives are computed in an analogous manner. We then find the new state by $${\mathbf{q}}^{n+1} = {\mathbf{q}}^{n,\dagger} + {\mathbf{f}}_{\rm e-rad} \Delta t.$$
This update is manifestly only first order-accurate in time for the explicit radiation terms, but there is no point in using a more complex update because our operator splitting of some of the radiation terms means that we are performing our explicit update using a time-advanced radiation field, rather than the field at a half time step. (@truelove98 show that one can avoid this problem for gravitational body forces because the potential is linear in the density, so it is possible to derive the half-time step potential from the whole time step states. No such fortuitous coincidence occurs for the radiation field.) This necessarily limits us to first order accuracy in time for the terms we treat explicitly. However, since these terms are always small compared to the dominant radiation terms, the overall scheme should still be closer to second order than first order in accuracy.
Advantages and Limitations of the Method
----------------------------------------
Our algorithm has two significant advantages in comparison to other approaches, in particular those based on comoving frame formulations of the equations [e.g. @turner01; @whitehouse05; @hayes06]. In any of these approaches, since the radiation work terms are included in the implicit update, one must solve an implicit quartic equation arising from the combination of the terms ${\kappa_{\rm 0P}}(4\pi B - c E)$ and ${\mathcal{P}}\colon\nabla {\mathbf{v}}$. This may be done either at the same time one is iterating to update the flux divergence term $\nabla\cdot{\mathbf{F}}$ [@whitehouse05], or in a separate iteration to be done once the iterative solve for the flux divergence update is complete [@turner01; @hayes06]. In contrast, since our iterative update involves only ${\kappa_{\rm 0P}}(4\pi B - c E)$ and $\nabla\cdot{\mathbf{F}}$, using the @howell03 algorithm we may linearize the equations and never need to solve a quartic, leading to a simpler update algorithm and a faster iteration step. Moreover, by using the @howell03 time-centering, we obtain second order accuracy in time whenever $E$ is changing slowly, as opposed to the backwards Euler differencing of @turner01, @whitehouse05, and @hayes06, which is always first order-accurate in time. Thus, our algorithm provides a faster and simpler approach than the standard one.
A second advantage of our update scheme is that it retains the total energy-conserving character of the underlying equations. In each of the update steps involving radiation, for ${\mathbf{f}}_{\rm e-rad}$ and ${\mathbf{f}}_{\rm i}$, the non-advective update terms in the radiation and gas energy equations are equal and opposite. Thus, it is trivial to write the update scheme so that it conserves total energy to machine precision. This property is particularly important for turbulent flows with large radiation energy gradients, such as those that occur in massive star formation [e.g. @krumholz07a], because numerical non-conservation is likely to be exacerbated by the presence of these features. In contrast, in comoving frame formalisms such as those of @turner01, @whitehouse04, and @hayes06 the exchange terms in their gas and radiation energy equations are not symmetric. As a result, their update schemes do not conserve total energy exactly. The underlying physical reason for this asymmetry is that total energy is conserved only in inertial frames such as the lab frame; it is not conserved in the non-inertial comoving frame. For this reason, there is no easy way to write a conservative update scheme from a comoving formulation.
Our algorithm also has two significant limitations, one obvious and one subtle. The obvious limit is that our algorithm is only applicable for static diffusion problems. For dynamic diffusion problems, e.g.stellar interiors or radiation-dominated shocks, our scheme is unstable unless an appropriately small timestep is used. Whether this instability is due to the explicit advection term, the explicit work term, or both is not clear. Since codes such as ZEUS [@hayes06] treat the advection explicitly without instability, however, it seems likely that the work term is the culprit. Regardless of the cause, even if we were to use a time step small enough to guarantee stability, since the work and advection terms can be comparable to or larger than the diffusion and heating/cooling terms for dynamic diffusion, an algorithm that treats all the terms implicitly or all explicitly, rather than our mix, is likely to be more accurate.
The subtle limitation is in our treatment of the hydrodynamics. We perform the hydrodynamic update using a Riemann solver unmodified for the presence of radiation force, work, and heating and cooling terms. These terms should change the characteristic velocities of the wave families in ways that depend on the radiation hydrodynamic regime of the system. For example, in optically thick systems we should have a radiation-acoustic mode rather than a simple sound wave, and in optically thin systems where the radiation time scale is short compared to the mechanical time scale, a gas may act as if it were isothermal even if it has $\gamma\ne 1$. In some cases, failure to modify the Riemann solver appropriately for these effects may produce substantial errors, including a reduction in the order of accuracy of the method from second to first [@pember93; @lowrie01; @miniati06]. The severity of these effects for a given problem depends the degree of stiffness of the radiation source terms. It should also be noted that the other radiation diffusion methods most commonly used for three-dimensional problems also suffer from this defect, so this is not a comparative disadvantage of our method relative to others.
Tests of the Static Diffusion Algorithm {#tests}
=======================================
Here we describe five tests of our static diffusion algorithm, done using our implementation of the algorithm in the Orion AMR code, various aspects of which are described in detail by @puckett92 [multifluid hydrodynamics], @truelove98 [hydrodynamics and gravity], @klein99 [hydrodynamics and gravity], @fisher02 [gravity], @howell03 [radiation transport], @krumholz04 [sink particles], and @crockett05 [magnetohydrodynamics]. For all of these tests we use a single fluid with no magnetic fields and no self-gravity.
Non-Equilibrium Marshak Wave
----------------------------
As an initial check of the gas-radiation energy exchange in our code in a case when radiation pressure is not significant and the gas is at rest, we simulate the non-equilibrium Marshak wave problem. In this problem, a zero-temperature, motionless, gaseous medium occupying all space at $z>0$ is subject to a constant radiation flux $F_{\rm inc}
\hat{\mathbf{z}}$ incident on its surface at $z=0$. The gas is held stationary, appropriate for early times before hydrodynamic motions become significant. The medium is gray, with opacity ${\kappa_{\rm 0R}}={\kappa_{\rm 0P}}=\kappa$, and the constant-volume specific heat capacity of the gas is taken to have the same $T^3$ dependence as that of the radiation, i.e. $c_v =
[\partial (e-v^2/2) / \partial T_g]_v = \alpha T_g^3$, where $T_g$ is the gas temperature. The gas is not assumed to be in thermal equilibrium with the radiation field, so the gas and radiation temperatures may be different.
@su96 give a semi-analytic solution to the time-dependent behavior of the radiation energy density $E(z,t)$ and gas temperature $T_g(z,t)$ for this problem. They introduce the dimensionless position and time variables $x \equiv \sqrt{3}\kappa z$ and $\tau \equiv (4 a_R
c \kappa/\alpha) t,$ and the “retardation” parameter $\epsilon\equiv 4
a_R/\alpha$, and show that the dimensionless radiation energy density $$\begin{aligned}
u(x,\tau) & \equiv & \left(\frac{c}{4}\right)
\left[\frac{E(z,t)}{F_{\rm inc}}\right] \\
& = & 1 - \frac{2\sqrt{3}}{\pi}
\int_0^{1} d\eta\; e^{-\tau \eta^2}
\left\{\frac{\sin[x\gamma_1(\eta)+\theta_1(\eta)]}
{\eta\sqrt{3+4\gamma_1^2(\eta)}}\right\}
\nonumber
\\ & & {}
- \frac{\sqrt{3}}{\pi} e^{-\tau}
\int_0^1 d\eta\; \left(e^{-\tau/(\epsilon \eta)} \right.
\nonumber
\\ & & \quad {}
\left.\left\{\frac{\sin[x\gamma_2(\eta)+\theta_2(\eta)]}
{\eta(1+\epsilon\eta)\sqrt{3+4\gamma_2^2(\eta)}}\right\}\right),
\label{uintegral}\end{aligned}$$ where $$\begin{aligned}
\gamma_1(\eta) & = & \eta \sqrt{\epsilon + \frac{1}{1-\eta^2}}, \\
\gamma_2(\eta) & = &
\sqrt{(1-\eta)\left(\epsilon+\frac{1}{\eta}\right)},\end{aligned}$$ and $$\theta_n(\eta) =
\cos^{-1} \sqrt{\frac{3}{3+4\gamma_n^2(\eta)}}.$$ The dimensionless gas energy density is $$\begin{aligned}
v(x,\tau) & \equiv & \left(\frac{c}{4}\right) \left[\frac{a_R
T_g^4(z,t)}{F_{\rm inc}}\right] \\
& = &
u(x,\tau) - \frac{2\sqrt{3}}{\pi}
\int_0^1 d\eta \;
\left(
e^{-\tau(1-\eta^2)}
\right.
\nonumber
\\ & & \quad
\left.
\left\{
\frac{
\sin[x\gamma_3(\eta)+\theta_3(\eta)]
}{
\sqrt{
4-\eta^2+4\epsilon\eta^2(1-\eta^2)}
}
\right\}
\right)
\nonumber
\\ & & {}
+ \frac{\sqrt{3}}{\pi} e^{-\tau}
\int_0^1 d\eta \;
\left(
e^{-\tau/(\epsilon \eta)}
\right.
\nonumber
\\ & & \quad
\left.
\left\{
\frac{
\sin[x \gamma_2(\eta)+\theta_2(\eta)]
}{
\eta\sqrt{3+4\gamma_2^2(\eta)}
}
\right\}
\right),
\label{vintegral}\end{aligned}$$ where $$\gamma_3(\eta) =
\sqrt{(1-\eta^2)\left(\epsilon+\frac{1}{\eta^2}\right)}.$$ Numerical evaluation of the integrals (\[uintegral\]) and (\[vintegral\]) for $u$ and $v$ is not trivial because the integrands perform an infinite number of oscillations about zero as $\eta\rightarrow 0$. Correct computation of the result when $\tau$ is small and $x$ is large requires careful numerical analysis to ensure that the positive and negative contributions cancel properly (J. Bolstad, 2007, in preparation).
We compare the properly computed semi-analytic results for $u$ and $v$ to a calculation performed with Orion using $\kappa=1$ cm$^{-1}$ and $\alpha=32 a_r/c$ (so $\epsilon=0.5$). The computational domain goes from $0$ to $15$ cm (and thus to an optical depth $\kappa z=15$), and is resolved by 100 equally-sized cells. For this test, since we are comparing to a pure diffusion result, we set the flux limiter $\lambda=1/3$ everywhere.
Figures \[marshakrad\] and \[marshakgas\] compare the semi-analytic dimensionless radiation and gas energy densities with the values computed by Orion. At $\tau=0.001$ the agreement is fairly poor due to low numerical resolution, since the wave only reaches an optical depth of $\kappa z\sim 0.2$ and $\kappa z = 0.15$ is the size of an individual computational cell. However, at later times when the wave is resolved by a reasonable number of cells, the agreement between the code result and the semi-analytic solution is excellent.
Radiating Blast Wave
--------------------
We next compare to a test problem in which the gas is not at rest: a Sedov-type blast wave with radiation diffusion. @reinicke91 gave the first similarity solution to the problem of a point explosion with heat conduction, and following @shestakov99 and @shestakov01, we can adapt this solution to the case of a point explosion with radiation diffusion. This tests our code’s ability to follow coupled radiation-hydrodynamics in cases where radiation pressure is small.
We first summarize the semi-analytic solution. Consider an $n=3$ dimensional space filled with an adiabatic gas with equation of state $P=(\gamma-1)\rho e \equiv \Gamma \rho T$, where $\Gamma$ is the gas constant. The Planck mean opacity ${\kappa_{\rm 0P}}$ of the gas is very high, so the gas and radiation temperatures are always equal. The Rosseland mean opacity has a powerlaw form ${\kappa_{\rm 0R}}=\kappa_{0R,0} \rho^m T^{-n}$, and we assume that it is always high enough to place us in the diffusion regime, so $\lambda = 1/3$. Note that the choice of $-n=-3$ as the exponent of the opacity powerlaw is a necessary condition for applying the @reinicke91 conduction solution to our radiation diffusion problem. Moreover, the similarity solution does not include radiation energy density or pressure, so we consider only temperatures for which the gas energy density and pressure greatly exceed the radiation energy density and pressure, i.e. $\rho e \gg a_R
T^4$.
Under the assumptions described above, we may re-write the gas and radiation energy equations (\[gasenergy\]) and (\[radenergy\]) as a single conduction-type equation for the temperature, $$\rho c_v \frac{\partial}{\partial t}T = \nabla(\chi_0 \rho^a T^b \nabla
T),$$ where $c_v=\partial e/\partial T = \Gamma/(\gamma-1)$ is the constant-volume specific heat of the gas, $\chi_0=4 c a_R/(3
\kappa_{0R,0})$, $a=-m$, and $b=n+3$. This equation has the same form as the conduction equation considered by @reinicke91.
Consider now a point explosion at the origin of a spherically symmetric region with an initial powerlaw density distribution $\rho(r,t=0) = g_0 r^{-k_\rho}$. Initially the gas temperature $T$ and pressure $P$ are negligible. The explosion occurs at the origin at time zero, so the initial gas energy density is $(\rho e)(r,t=0)=E_0
\delta({\mathbf{r}})$. @reinicke91 show that if the initial density profile has a powerlaw index $$\label{krhoeq}
k_{\rho} = \frac{(2b-1) n + 2}{2b - 2a + 1},$$ then one may obtain a similarity solution via the change of variables $$\begin{aligned}
\xi & = & \frac{r}{\zeta t^\alpha} \\
G(\xi) & = & \frac{\rho(r,t)}{g_0 r^{-k_{\rho}}} \\
U(\xi) & = & v(r,t)\frac{t}{\alpha r} \\
\Theta(\xi) & = & T(r,t) \Gamma \left(\frac{\alpha r}{t}\right)^2.\end{aligned}$$ Here, $\xi$, $G(\xi)$, $U(\xi)$, and $\Theta(\xi)$ are the dimensionless distance, density, velocity, and temperature, $$\alpha = \frac{2b - 2a + 1}{2b - (n+2)a + n},$$ and $\zeta$ is a constant with units of $[\mbox{length}][\mbox{time}]^{-\alpha}$ whose value is determined by a procedure we discuss below.
With this similarity transformation, the equations of motion and heat conduction reduce to $$\begin{aligned}
U' - (1-U)(\ln G)' + (n-k_{\rho}) U & = & 0 \\
(1-U) U' + U(\alpha^{-1}-U) & = &
\nonumber \\
\Theta[\ln (\xi^{2-k_{\rho}} G\Theta)]',\end{aligned}$$ and $$\begin{aligned}
\lefteqn{
2[U' + nU - \mu (\alpha^{-1}-1)]
=
\mu (1-U) [\ln(\xi^2\Theta)]'
}
\nonumber \\
& &
{}
+ \beta_0 \Theta^b G^{a-1} \xi^{(2b-1)/\alpha}
\cdot
\left((\ln \Theta)'' +
[\ln (\xi^2\Theta )]'
\right.
\\
& &
\left.
{}\cdot
\left\{n-2+a[\ln(\xi^{-k_{\rho}} G)]'
+
(b+1)
[\ln (\xi^2 \Theta)]'\right\}\right),\end{aligned}$$ where $()'\equiv d()/d\ln \xi$, $\mu=2/(\gamma-1)$, and $$\beta_0 = \frac{2\chi_0 (\alpha \zeta^{1/\alpha})^{2b-1}}{\Gamma^{b+1}
g_0^{1-\alpha}} \mbox{sgn}(t).$$ This constitutes a fourth-order system of non-linear ordinary differential equations. All physical solutions to these equations pass through two discontinuities, a heat front and a shock front, with the heat front at larger radius. However, the jump conditions for these discontinuities are easy to determine, and one can integrate between them. For a given $\beta_0$, the solution depends only on the dimensionless parameter $$\Omega = \frac{2\chi_0}{\Gamma^{b+1} g_0^{1-a}}
\left(\frac{E_0}{g_0}\right)^{b-1/2},$$ which measures the strength of the explosion. Large values of $\Omega$ constitute “strong” explosions, and the ratio of heat front radius to shock front radius is a monotonically increasing function of $\Omega$. It is important at this point to add a cautionary note: in deriving the similarity solution, we assumed that radiation energy density is negligible in comparison to gas energy density. This cannot strictly be true at early times, since at $t=0$ the temperature diverges at the origin, and the radiation energy density varies as $T$ to a higher power than the gas energy density. However, the true behavior should approach the similarity solution at later times.
While we have reduced the gas dynamical equations to a system of ordinary differential equations that is trivial to integrate, solving the full problem is complex because the equations still depend on the unknown parameter $\beta_0$, which in turn depends on $\zeta$. To solve the problem, we must determine $\beta_0$ from the given initial conditions. @reinicke91 describe the iteration procedure required to do this in detail, and we only summarize it here. To find a solution, one first chooses a value $\xi_h>1$ for the dimensionless radius of the heat front, applies the boundary conditions at the front, and guesses a corresponding value of $\beta_0$. For each $\xi_h$ there exists a unique $\beta_0$ for which it is possible to integrate the equations back from $\xi=\xi_h$ to the location of the shock front at $\xi=\xi_s$, apply the shock jump conditions, and continue integrating back to the origin at $\xi=0$ without having the solution become double-valued and thus unphysical. One iterates to identify the allowed value of $\beta_0$ for the chosen $\xi_h$, and this gives the unique density, velocity, and temperature profiles allowed for that $\xi_h$. However, the solution one finds in this way may not correspond to the desired value of $\Omega$. @reinicke91 show that $$\Omega = \beta_0 \left[2\pi \int_0^{\xi_h} \xi^{n-k_{\rho}+1} G (U^2 +
\mu \Theta)\, d\xi\right]^{b-1/2}.$$ Thus, each choice of $\xi_h$ corresponds to a particular value of $\Omega$, and one must iterate a second time to find the value of $\xi_h$ that gives the value of $\Omega$ determined from the input physical parameters of the problem. Alternately, instead of specifying a desired value of $\Omega$, one may specify a ratio $R=\xi_h/\xi_s$, which also determines a unique value for $\xi_h$.
For our comparison between the semi-analytic solution and Orion, we adopt the parameters $\gamma=7/5$, $c_v=1/(\gamma-1)$, $a=-2$, $b=6$, $g_0=\chi_0=1$, and $E_0 = 135$, which yields a strength $\Omega = 1.042\times 10^{12}$ and a ratio $R=2.16$. In the simulation, we turn off terms in the code involving radiation pressure and forces, and we set $\lambda=1/3$ exactly. We use one-dimensional spherical polar coordinates rather than Cartesian coordinates; the solution procedures for this are identical to the ones outlined in § \[algorithm\], with the exception that the gradient and divergence operators have their spherical rather than Cartesian forms, and the cell-centered finite differences are modified appropriately. Our computational domain goes covers $0 \leq r \leq 1.05$, resolved by 256, 512, or 1024 cells, and has reflecting inner and outer boundary conditions. To initialize the problem we set initial density to the powerlaw profile $\rho=r^{-k_{\rho}}$ (with $k_{\rho}$ set from equation \[krhoeq\]), the initial velocity to zero, and the initial energy density to a small value, except in the cell adjacent to the origin, where its value is $\rho e = 135/(\gamma-1)$.
Figures \[blastwaveden\], \[blastwavevel\], and \[blastwavetemp\] compare the semi-analytic density, velocity, and temperature profiles to the values we obtain from Orion after running to a time $t=0.06$. As the plots show, the Orion results agree very well with the semi-analytic solution, and the agreement improves with increasing resolution. In the lowest resolution run, there is a small oscillation in the density and velocity about a third of the way to the shock, which is likely due to the initial blast energy being deposited in a finite-volume region rather than as a true $\delta$ function. However, this vanishes at higher resolutions. Overall, the largest errors are in the temperature in the shocked gas.
As a metric of convergence, we plot the error of our simulation relative to the analytic solution as a function of resolution in Figure \[blastwaveerr\]. We do this for the quantities $r_h$ and $r_s$, the positions of the shock and heat fronts, and their ratio $R$. For this purpose, we define the location of the heat and shock fronts for the simulations as the positions of the cell edges where $dT/dr$ and $d\rho/dr$ are most negative. As the plot shows, at the highest resolution the errors in all three quantities are ${\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle <}
{\sim}\:$}}3\%$, and the calculation appears to be converging. The order of convergence is roughly 0.6 in all three quantities. It is worth noting that computing the locations of the heat and shock fronts is a particularly strong code test, because obtaining the correct propagation velocities for the two fronts requires that the code conserve total energy very well. Non-conservative codes have significant difficulties with this test [@timmes06].
Radiation Pressure Tube {#radtube}
-----------------------
Our third test is to simulate a tube filled with radiation and gas. The gas within the tube is optically thick, so the diffusion approximation applies. The two ends of the tube are held at fixed radiation and gas temperature, and radiation diffuses through the gas from one end of the tube to the other. The radiation flowing through the tube exerts a force on the gas, and the gas density profile is such that, with radiation pressure, the gas is in pressure balance and should be stationary. For computational simplicity, we set the Rosseland- and Planck-mean opacities per unit mass of the gas to a constant value $\kappa$. A simulation of this system tests our code’s ability to compute accurately the radiation pressure force in the very optically thick limit.
We first derive a semi-analytic solution for the configuration of the tube satisfying our desired conditions. Since the gas is very optically thick and we are starting the system in equilibrium, we set $T_{\rm rad} = T_{\rm gas} \equiv T$. The fluid is initially at rest. The condition of pressure balance amounts to setting $\partial (\rho
{\mathbf{v}})/\partial t + \nabla\cdot(\rho{\mathbf{v}}{\mathbf{v}})=0$ in equation (\[momentumconservation\]), so that the radiation pressure force balances the gas pressure gradient. Thus, we have $$\begin{aligned}
\frac{dP}{dx} + \lambda \frac{dE}{dx} & = & 0 \\
\left(\frac{k_B}{\mu} \rho + \frac{4}{3} a_R T^3\right) \frac{dT}{dx} +
\frac{k_B}{\mu} T \frac{d\rho}{dx} & = & 0.
\label{radtube1}\end{aligned}$$ In the second step we have set $E=a_R T^4$ and $P=\rho k_B T/\mu$, where $\mu$ is the mean particle mass, and we have set $\lambda=1/3$ as is appropriate for the optically thick limit. The radiation energy equation (\[radenstatdiff\]) for our configuration is simply $$\begin{aligned}
\frac{d}{dx} \left( \frac{c \lambda}{\kappa \rho} \frac{dE}{dx}\right)
& = & 0 \\
\frac{d^2 T}{dx^2} + 3\frac{1}{T} \left(\frac{dT}{dx}\right)^2 -
\frac{1}{\rho} \left(\frac{d\rho}{dx}\right)
\left(\frac{dT}{dx}\right) & = & 0.
\label{radtube2}\end{aligned}$$
Equations (\[radtube1\]) and (\[radtube2\]) are a pair of coupled non-linear ordinary differential equations for $T$ and $\rho$. The combined degree of the system is three, so we need three initial conditions to solve them. Thus, let the tube run from $x=x_0$ to $x=x_1$, with temperature, density, and density gradient $T_0$, $\rho_0$, and $(d\rho/dx)_0$ at $x_0$. For a given choice of initial conditions, it is trivial to solve (\[radtube1\]) and (\[radtube2\]) numerically to find the density and temperature profile. We wish to investigate both the radiation pressure and gas pressure dominated regimes, so we choose parameters to ensure that our problem covers both. The choice $x_0=0$, $x_1=128$ cm, $\rho_0=1$ g cm$^{-3}$, $(d\rho/dx)_0 = 5\times 10^{-3}$ g cm$^{-4}$, and $T_0=2.75 \times 10^7$ K satisfies this requirement if we adopt $\mu=2.33 \,m_{\rm P}=3.9\times
10^{-24}$ g and $\kappa=100$ cm$^2$ g$^{-1}$. Figure \[radtubesol\] shows the density, temperature, and pressure as a function of position for these parameters.
We solve the equations to obtain the density and temperature as a function of position, and then set these values as initial conditions in a simulation. The simulation has 128 cells along the length of the tube on the coarsest level. We impose Dirichlet boundary conditions on the radiation field, with the radiation temperature at each end of the tube set equal to its value as determined from the analytic solution. We use symmetry boundary conditions on the hydrodynamics, so that gas can neither enter nor leave the computational domain. To ensure that our algorithm does not encounter problems at the boundaries between AMR levels, we refine the central $1/4$ of the problem domain to double the resolution of the base grid. We evolve the system for 10 sound crossing times and measure the amount by which the density and temperature change relative to the exact solution. We plot the relative error, defined as (numerical solution $-$ analytic solution) / (analytic solution), in the density, gas temperature, and radiation temperature in Figure \[radtubeerr\]. As the plot shows, our numerical solution agrees with the analytic result to better than $0.5\%$ throughout the computational domain. The density error is smallest in the higher resolution central region, as expected. There is a very small increase in error at level boundaries, but it is still at the less than $0.5\%$ level.
Radiation-Inhibited Bondi Accretion
-----------------------------------
The previous test focuses on radiation pressure forces in the optically thick limit. To test the optically thin limit, we simulate accretion onto a radiating point particle. We consider a point mass $M$ radiating with a constant luminosity $L$ accreting from a background medium. The medium consists of gas which has zero velocity and density $\rho_{\infty}$ far from the particle. We take the gas to be isothermal with constant temperature $T$, and enforce that it is not heated or cooled radiatively by setting its Planck opacity ${\kappa_{\rm 0P}}=0$. We set the Rosseland opacity of the gas to a constant non-zero value ${\kappa_{\rm 0R}}$, and choose $\rho_{\infty}$ such that the computational domain is optically thin. In this case, the radiation free-streams away from the point mass, and the radiation energy density and radiative force per unit mass on the gas are $$\begin{aligned}
\label{eanalyt}
E &= & \frac{L}{4\pi r^2 c} \\
{\mathbf{f}}_{\rm r} & = & \frac{{\kappa_{\rm 0R}}L}{4\pi r^2 c}
\left(\frac{{\mathbf{r}}}{r}\right),\end{aligned}$$ where ${\mathbf{r}}$ is the radial vector from the particle and $r$ is its magnitude. The gravitational force per unit mass is ${\mathbf{f}}_{\rm g}=- (G
M/r^2) ({\mathbf{r}}/r)$, so the net force per unit mass is $${\mathbf{f}}= {\mathbf{f}}_{\rm r} + {\mathbf{f}}_{\rm g} = -(1 - f_{\rm Edd}) \frac{G M}{r^2}
\left(\frac{{\mathbf{r}}}{r}\right),$$ where $$f_{\rm Edd} = \frac{{\kappa_{\rm 0R}}L}{4\pi G M c}$$ is the fraction of the Eddington luminosity with which the point mass is radiating.
Since the addition of radiation does not alter the $1/r^2$ dependence of the specific force, the solution is simply the standard @bondi52 solution, but for an effective mass of $(1-f_{\rm
Edd}) M$. The accretion rate is the Bondi rate $$\dot{M}_B = 4\pi \xi r_B^2 c_s \rho_{\infty},$$ where $$r_B = (1-f_{\rm Edd}) \frac{G M}{c_s^2}$$ is the Bondi radius for the effective mass, $c_s$ is the gas sound speed at infinity, and $\xi$ is a numerical factor of order unity that depends on the gas equation of state. For an isothermal gas, $\xi=e^{3/2}/4$, and the radial profiles of the non-dimensional density $\alpha\equiv \rho/\rho_{\infty}$ and velocity $u\equiv v/c_s$ are given by the solutions to the non-linear algebraic equations [@shu92] $$\begin{aligned}
x^2 \alpha u & = & \xi \\
\frac{u^2}{2} + \ln \alpha - \frac{1}{x} & = & 0,\end{aligned}$$ where $x\equiv r/r_B$ is the dimensionless radius.
To set up this test, we make use of the Lagrangian sink particle algorithm of @krumholz04, coupled with the “star particle” algorithm of @krumholz07a which allows the sink particle to act as a source of radiation. We refer readers to those papers for details on the sink and star particle algorithms. We simulate a computational domain $5\times 10^{13}$ cm on a side, resolved by $256^3$ cells, with a particle of mass $M=10$ ${M_{\odot}}$ and luminosity $L=1.6\times 10^{5}$ ${L_{\odot}}$ at its center. We adopt fluid properties $\rho_{\infty} =
10^{-18}$ g cm$^{-3}$, ${\kappa_{\rm 0R}}=0.4$ cm$^2$ g$^{-1}$, and $c_s=1.3\times
10^{7}$ cm s$^{-1}$, corresponding to a gas of pure, ionized hydrogen with a temperature of $10^6$ K. With these values, $f_{\rm Edd} = 0.5$, $r_B = 4.0\times 10^{12}$ cm, and $\dot{M}_B = 2.9 \times 10^{17}$ g s$^{-1}$. We use inflow boundary conditions on the gas and Dirichlet boundary conditions on the radiation field, with the radiation energy density on the boundary set to the value given by equation (\[eanalyt\]).
Figure \[bondisol\] compares the steady-state density $\alpha$ and velocity $u$ computed by Orion to the analytic solution. The agreement is excellent, with differences between the analytic and numerical solutions of $\sim 1\%$ everywhere except very near the accretion radius at $x=0.25$. The maximum error is $\sim 10\%$ at the surface of the accretion region; this is comparable to the error in density for non-radiative Bondi accretion with similar resolution in @krumholz04. In comparison, the solution is nowhere near the solution that would be obtained without radiation. After running for $5$ Bondi times ($=r_B/c_s$), the average accretion rate is $2.4
\times 10^{17}$ g s$^{-1}$. While this differs from the analytic solution by $19\%$, the error is also not tremendously different from that obtained by @krumholz04 when the Bondi radius was resolved by 4 accretion radii, and is nowhere near the value of $1.2\times 10^{18}$ g s$^{-1}$ which would occur without radiation.
We should at this point mention one limitation of our algorithm, as applied on an adaptive grid, that this test reveals. The $1/r^2$ gradient in the radiation energy density is very steep, and we compute the radiation force by computing gradients in $E$. We found that, in an AMR calculation, differencing this steep gradient across level boundaries introduced significant artifacts in the radiation pressure force. With such a steep gradient, we were only able to compute the radiation pressure force accurately on fixed grids, not adaptive grids. This is not a significant limitation for most applications though, since for any appreciable optical depth the gradient will be much shallower than $1/r^2$. As the radiation pressure tube test in § \[radtube\] demonstrates, in an optically thick problem the errors that arise from differencing across level boundaries are less than 1%.
Advecting Radiation Pulse
-------------------------
The previous two tests check our ability to compute the radiation pressure force accurately in the optically thick and optically thin limits. However, they do not strongly test radiation advection by gas. To check this, we simulate a diffusing, advecting radiation pulse. The initial condition is a uniform background of gas and radiation far from the pulse. Centered on $x=0$ there is an increase in the radiation energy density and a corresponding decrease in the gas density, so that the initial condition is everywhere in pressure balance. As radiation diffuses out of the pulse, pressure support is lost and the gas moves into the lower density region. We cannot solve this problem analytically, but we can still perform a very useful test of the methodology by comparing a case in which the gas is initially at rest with respect to the computational grid with a case in which the gas is moving at a constant velocity with respect to the grid. The results should be identical when shifted to lie on top of one another, but the work and advection terms will be different in the stationary case than in the advected case. Checking that the results do not change when we advect the problem enables us to determine if our code is correctly handling the advection of radiation by the gas.
For our simulations, we use equal initial gas and radiation temperatures, with temperature and density profiles $$\begin{aligned}
T & = & T_0 + (T_1-T_0) \exp\left(-\frac{x^2}{2 w^2}\right) \\
\rho & = & \rho_0 \frac{T_0}{T}+ \frac{a_R \mu}{3 k_B} \left(\frac{T_0^4}{T} -
T^3\right),\end{aligned}$$ with $T_0 = 10^7$ K, $T_1=2\times 10^7$ K, $\rho_0 = 1.2$ g cm$^{-3}$, $w = 24$ cm, $\mu=2.33 \,m_{\rm P}=3.9\times 10^{-24}$ g, and $\kappa=100$ cm$^2$ g$^{-1}$. The density, temperature, and pressure profiles are shown in Figure \[radadvectsetup\]. In the bottom panel, the solid line is the total pressure, the dashed line is the gas pressure, and the dot-dashed line is the radiation pressure. As the figure indicates, the system is initially in pressure balance.
We compare two runs, one where the velocity is zero everywhere and another with a uniform initial velocity $v=10^6$ cm s$^{-1}$ in the $x$ direction. In both runs the simulation domain extends from $-512$ to $512$ cm, resolved by 512 cells with no adaptivity. We use periodic boundary conditions on the gas and the radiation, and run for $4.8\times
10^{-5}$ s, long enough for the pulse to have been advected over its own initial width twice.
To check our results, we shift the advected run by 48 cm in the $-x$ direction, so that it should lie on top of the unadvected run. Figure \[radadvectfinal\] shows the configuration of the advected and unadvected runs at this point. We then plot the relative difference between the advected and unadvected runs, defined as $(\mbox{unadvected} - \mbox{advected})
/ \mbox{unadvected}$, in Figure \[radadvecterr\]. We do not differentiate between the gas and radiation temperatures, because they are identical at the $10^{-3}$ level. We do not plot the error in velocity because the velocities in the unadvected run are close to zero over most of the computational domain. As the plot shows, the difference between the advected and unadvected runs is less than $2\%$ everywhere in the simulation.
Summary
=======
We derive the correct equations for mixed frame flux-limited diffusion radiation hydrodynamics. The error in our equations if of order $v^2/c^2$ in the static diffusion limit, and of order $v/c$ in the dynamic diffusion and streaming limits. We give the equations in a form that is well-suited to implementation in numerical simulations, because they make it trivial to maintain exact conservation of total energy. Our analysis reveals that lower order formulations of the equations, which neglect differences between the laboratory and comoving frames, are incorrect at order unity for systems in the dynamic diffusion limit. It remains to be seen how serious this defect is in practice, but analytic arguments suggest that at a minimum one ought to be very careful in applying zeroth order codes to problems where there are interesting or important structures on scales for which $\beta\tau \sim 1$. We give the equations that are correct to leading order for dynamic diffusion, which do not suffer from this problem.
Our analysis also reveals that, for static diffusion problems, one can obtain a significant algorithmic simplification and speedup compared to algorithms based on comoving frame formulations of the equations by treating non-dominant radiation terms explicitly rather than implicitly. This advance is possible even though the underlying equations of our method conserve total energy to machine precision while comoving frame formulations of the equations do not. This property is particularly important for flows that are turbulent or otherwise involve large gradients in gas or radiation properties, since these are the problems most likely to suffer from numerical non-conservation. We demonstrate an implementation of this method in the Orion adaptive mesh refinement code, and show that it provides excellent agreement with analytic and semi-analytic solutions in a series of test problems covering a wide range of radiation-hydrodynamic regimes.
We thank J. I. Castor and J. M. Stone for helpful comments on the manuscript, and T. A. Thompson for helpful discussions. We also thank the referee for comments that improved the paper. Support for this work was provided by NASA through Hubble Fellowship grant \#HSF-HF-01186 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555 (MRK); NASA ATP grants NAG 5-12042 and NNG06GH96G (RIK and CFM); the US Department of Energy at the Lawrence Livermore National Laboratory under contract W-7405-Eng-48 (RIK and JB); and the NSF through grants AST-0098365 and AST-0606831 (CFM). This research was also supported by grants of high performance computing resources from the Arctic Region Supercomputing Center; the NSF San Diego Supercomputer Center through NPACI program grant UCB267; the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098, through ERCAP grant 80325; and the US Department of Energy at the Lawrence Livermore National Laboratory under contract W-7405-Eng-48.
Scalings in the Dynamic Diffusion Limit {#dyndiffusionscaling}
=======================================
Here we show that the emission minus absorption term $4\pi B/c - E$ is of order $\beta^2 E$ in the dynamic diffusion limit. @mihalas99 argue that in this limit $4\pi B/c - E$ is of order $(\beta/\tau)E$. However, this conclusion is based on their analysis of the second order equilibrium diffusion approximation [@mihalas99 pg. 461-466], in which they retain terms of order $\beta/\tau$ while dropping those of order $\beta^2$. While this is correct for static diffusion, in the dynamic diffusion limit $\beta^2
\gg \beta/\tau$, so the approach in @mihalas99 is not consistent, and is insensitive to terms of order $\beta^2$.
We will not give a general proof that $4\pi B/c - E\sim \beta^2 E$ for dynamic diffusion, but we can establish it by a simple thought experiment. Consider a system that is infinitely far into the dynamic diffusion limit, in the sense that $\tau=\infty$: an infinite uniform medium that is at rest and in perfect thermal equilibrium between the radiation field and the gas. In the rest frame of the medium, these assumption require $E_0=4\pi B/c$, ${\mathbf{F}}_0 = 0$, and ${\mathcal{P}}_0 = (E_0/3){\mathcal{I}}$. Now consider an observer moving at velocity ${\mathbf{v}}$ relative to the medium. In the observer’s frame, $4\pi B/c$ is the same because the gas temperature $T_0$ is a world-scalar, and the Lorentz transform to all orders for the energy gives $$\begin{aligned}
E & = & \gamma^2 \left(E_0 + 2\frac{{\mathbf{v}}\cdot{\mathbf{F}}_0}{c^2} +
\frac{({\mathbf{v}}{\mathbf{v}})}{c^2}\colon{\mathcal{P}}_0\right) \\
& = & \gamma^2 \left[1+\frac{1}{3}\left(\frac{v^2}{c^2}\right)\right]
\left(\frac{4 \pi B}{c}\right) \\
& = & \left(\frac{4 \pi B}{c}\right) \left[1 +
\frac{4}{3}\left(\frac{v^2}{c^2}\right) +
O\left(\frac{v^4}{c^4}\right)\right].\end{aligned}$$ Thus, for this case it is clear that $4\pi B/c - E\sim \beta^2 E$ to leading order.
Note that using the correct scaling is necessary to obtain sensible behavior from the equations in the dynamic diffusion limit. If one assumes that $4\pi B/c - E \sim (\beta/\tau) E$, then in the gas and radiation energy equations (\[gasenergy\]) and (\[radenergy\]) in the dynamic diffusion limit, the term ${\kappa_{\rm 0P}}(v^2/c)[(3-R_2)/2] E$ is of higher order than any other term except perhaps the time derivative. Since this term is non-zero for any system with non-zero velocity, opacity, and radiation energy density, this means that there would be no way for the time derivative term to ever vanish. Thus, a system in the dynamic diffusion limit could never be in equilibrium unless its velocity or radiation energy were zero everywhere. Clearly this cannot be correct, since it predicts that our static, infinite, uniform medium cannot be in equilibrium when seen by an observer moving by at velocity ${\mathbf{v}}$, even though it is manifestly in equilibrium in its own rest frame. On the other hand, if we take $4\pi B/c - E = (4/3) (v^2/c^2) E$, as computed from the Lorentz transform, it is trivial to verify that equations (\[gasenergy\]) and (\[radenergy\]) correctly give $\partial(\rho e)/\partial t = \partial E/\partial t = 0$, and $(G^0,
{\mathbf{G}}) = (0, \mathbf{0})$ as well. The observer sees a flux that does work on the gas, but this is precisely canceled by a mismatch between emission and absorption of radiation by the gas, leading to zero net energy transfer.
[^1]: Hubble Fellow
[^2]: Note that our need to retain difference between $E$ and $E_0$ here is different from the situation when we first applied the Lorentz transformation to derive (\[gztrans\]) and (\[gtrans\]). In that case we did not need to retain the distinction between $E$ and $E_0$, because in deriving (\[gztrans\]) and (\[gtrans\]) there were no terms involving $E_0$ explicitly. Instead, $E_0$ appeared only implicitly, as part of the flux ${\mathbf{F}}$, and non-leading order corrections to ${\mathbf{F}}$ are not of leading order in any regime. In contrast, $E_0$ does appear explicitly in (\[firstlaw\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
A Čech closure space $(X,u)$ is a set $X$ with a (Čech) closure operator $u$ which need not be idempotent. Many properties which hold in topological spaces hold in Čech closure spaces as well.
The notions of proper (splitting) and admissible (jointly continuous) topologies are introduced on the sets of continuous functions between Čech closure spaces. It is shown that some well-known results of Arens and Dugundji [@AD] and Iliadis and Papadopoulos [@IP] are true in this setting.
We emphasize that Theorems 1–10 encompass the results of A. di Concilio [@C] and Georgiou and Papadopoulos [@GP1; @GP2] for the spaces of continuous-like functions as $\theta$-continuous, strongly and weakly $\theta$-continuous, weakly and super-continuous.
address: |
Faculty of Mathematics, University of Belgrade\
Beograd, Yugoslavia
author:
- Mila Mršević
title: Proper and admissible topologies in the setting of closure spaces
---
(474,66)(0,0) (0,66)(1,0)[40]{}[(0,-1)[24]{}]{} (43,65)(1,-1)[24]{}[(0,-1)[40]{}]{} (1,39)(1,-1)[40]{}[(1,0)[24]{}]{} (70,2)(1,1)[24]{}[(0,1)[40]{}]{} (72,0)(1,1)[24]{}[(1,0)[40]{}]{} (97,66)(1,0)[40]{}[(0,-1)[40]{}]{} (143,66)[(0,0)\[tl\][Proceedings of the Ninth Prague Topological Symposium]{}]{} (143,50)[(0,0)\[tl\][Contributed papers from the symposium held in]{}]{} (143,34)[(0,0)\[tl\][Prague, Czech Republic, August 19–25, 2001]{}]{}
[^1] [^2]
Čech closure spaces
===================
An operator $u : \mathcal{P}(X) \rightarrow \mathcal{P}(X)$ defined on the power set $\mathcal{P}(X)$ of a set $X$ satisfying the axioms:
- $u(\emptyset) = \emptyset$,
- $A \subset u(A)$ for every $A \subset X$,
- $u(A \cup B) = u(A) \cup u(B)$ for all $A,B \subset X$,
is called a [*Čech closure operator*]{} and the pair $(X,u)$ is a [*Čech closure space*]{}. For short, the space will be noted by $X$ as well, and called a [*closure space*]{}.
A subset $A$ is [*closed*]{} in the closure space $(X,u)$ if $u(A)=A$ holds. It is [*open*]{} if its complement is closed. The empty set and the whole space are both open and closed.
The ${\rm int}_u:\mathcal{P}(X) \rightarrow \mathcal{P}(X)$ is defined by means of the closure operator in the usual way: ${\rm int}_u ={\rm c}\circ u\circ {\rm c}$, where ${\rm c}:\mathcal{P}(X) \rightarrow \mathcal{P}(X)$ is the complement operator. A subset $U$ is a [*neighbourhood*]{} of a point $x$ (subset $A$) in $X$ if $x\in {\rm int}_uU (A\subset {\rm int}_uU )$ holds. We denote by $\mathcal{N}(x)$ the collection of all neighbourhoods (the [*neighbourhood system)*]{} at the point $x$. By (C3), the intersection of two (and thus finitely many) neighbourhoods at $x$ is a neighbourhood at $x$ again. The condition (C1) is equivalent to ${\rm int}_uX = X$, that is to $X\in\mathcal{N}(x)$ for every $x\in X$, and ${\rm int}_u A \subset A$ for every $A \subset X$ is equivalent to (C2).
In a closure space $(X,u)$ a family $\mathcal{U}(x) \subset \mathcal{N}(x)$ is a neighbourhood (local) base at a point $x$ if the following axioms are satisfied:
- $\mathcal{U}(x) \ne \emptyset$ for every $x \in X$,
- $x \in U$ for every $U \in \mathcal{U}(x)$,
- $U_{1}, U_{2} \in \mathcal{U}(x) \Rightarrow
(\exists U \in \mathcal{U}(x)) U \subset U_{1} \cap U_{2}$.
A family $\mathcal{U}(x) \subset \mathcal{N}(x)$ is a neighbourhood (local) subbase at a point $x$ if the conditions (Nb1) and (Nb2) are fulfilled.
If a collection $\{\mathcal{U}(x) \mid x \in X\}$ of filters on $X$ satisfies the conditions (Nb1)–(Nb3), then there is exactly one closure operator $u$ for $X$ such that $\mathcal{U}(x)$ is a neighbourhood base at $x$ for each $x\in X$. The operator $u$ is defined by: $$u(A) =
\{ x\in X \mid U\in \mathcal{U}(x) \Rightarrow U \cap A \ne \emptyset \}.$$
Let $(X,u_{1})$ and $(X,u_{2})$ be closure spaces. The closure $u_{1}$ is [*coarser*]{} than the closure $u_{2}$, or $u_{2}$ is [*finer*]{} than $u_{1}$, denoted by $u_{1} \leq u_{2}$, if $u_{1}{(A) \supset u}_{2}(A)$ for every $A \subset X$. So defined relation $\leq$ is a partial order on the set of all closure spaces.
Let $\{ u_{\alpha }\}$ be a collection of closure operators on a set $X$. The infimum (meet) and supremum (join) operators for $\{ u_{\alpha }\}$ are the operators $u_{0} = {\land u}_{\alpha }$ and $u = {\lor u}_{\alpha }$ respectively, defined by: $\mathcal{U}_{0}(x) = \bigcap_{\alpha} \mathcal{N}_{\alpha}(x)$ is a neighbourhood base (system) and $\mathcal{U}(x) = \bigcup_{\alpha} \mathcal{N}_{\alpha}(x)$ is a neighbourhood subbase at $x\in X$, for $u_{0}$ and $u$ respectively.
Many topological notions can be defined in the class of closure spaces by means of neighbourhoods.
Let ${\rm M} $ be a directed set and $(x_{\mu})_{\mu \in {\rm M}}$ a net in $(X,u)$. The net $(x_{\mu})$ converges to a point $x \in X$ if for every neighbourhood $U$ of $x$ there is a $\mu \in {\rm M}$ such that for every $\mu' \in {\rm M}$, $\mu' \geq \mu \Rightarrow x_{\mu '} \in U$. Similarly, $x$ is [*an accumulation point*]{} of the net $(x_{\mu})$ if for every neighbourhood $U$ of $x$ and every $\mu \in {\rm M}$ there is a $\mu' \in {\rm M}$ such that $\mu' \geq \mu$ and $x_{\mu'} \in U$. For every point $x$ the neighbourhood system $\mathcal{N}(x)$ is a filter on $X$ such that $x\in \bigcap \mathcal{N}(x)$. Moreover it is a set directed by the inverse inclusion $\supset$ and every net $(x_{U})_{U\in \mathcal{N}(x)}$ with $x_{U} \in U$, converges to $x$.
Let $(X,u)$ and $(Y,v)$ be two closure spaces. A function $f:(X,u)\rightarrow (Y,v)$ is [*continuous at*]{} $x \in X$ if “close points are mapped into close ones”, that is if the following holds $$A \subset X \land x \in u(A) \Rightarrow f(x) \in v(f(A)).$$ This condition is equivalent to:
- the inverse image of every neighbourhood of $f(x)$ is a neighbourhood of $x$;
- for every net $(x_{\mu})$ that converges to $x$, the net $(f(x_{\mu}))$ converges to $f(x)$;
- if $x$ is an accumulation point of a net $(x_{\mu}), f(x)$ is an accumulation point of the net $(f(x_{\mu}))$.
A function $f:(X,u) \rightarrow (Y,v)$ is [*continuous*]{} if it is continuous at every point of $X$. This condition is equivalent to:
- $f(u(A)) \subset v(f(A))$ for every $A \subset X$;
- $u(f^{-1}(B)) \subset f^{-1}(v(B))$ for every $B \subset Y$.
The [*product of a family $\{ (X_{\alpha}, u_{\alpha}) \mid \alpha \in {\rm A} \}$ of closure spaces*]{}, denoted by $\Pi (X_{\alpha} ,u_{\alpha})$, is the set $X = \Pi_{\alpha \in {\rm A}}X_{\alpha}$ endowed with the closure operator $u$ defined by means of neighbourhoods: for every $x \in X$ the family $$\mathcal{U}(x) =
\{ \pi_{\alpha}^{-1} (V) \mid
\alpha \in {\rm A}, V \in \mathcal{N}_{\alpha}(x_{\alpha }) \}$$ is a neighbourhood subbase at $x$ in $(X,u)$. Here $\pi_{\alpha }$ are the projections, while $\mathcal{N}_{\alpha}(x_{\alpha })$ is the neighbourhood system at $x_{\alpha} = \pi_{\alpha}(x)$ in $X_{\alpha}$. There exists exactly one closure operator $u$ such that $\mathcal{U}(x)$ is a local subbase at $x$ in $(X,u)$ for every $x \in X$. Canonical neighbourhoods of $x$ are of the form $\bigcap_{i=1}^{k} \pi_{\alpha _{i}}^{-1}(V_{i})$.
The projections are continuous as well as the restrictions of continuous functions. The composition of two continuous mappings is continuous and a mapping $f:(X,u) \rightarrow \Pi (Y_{\alpha}, v_{\alpha})$ is continuous at $x \in X$ if and only if each composition $\pi_{\alpha}\circ f$ is continuous at $x$. Also the product $f: \Pi (X_{\alpha} ,u_{\alpha}) \rightarrow \Pi (Y_{\alpha}, v_{\alpha})$ of a family $\{ f_{\alpha }\}$ of continuous mappings is continuous.
A well-known example of a Čech closure operator which is not a Kuratowski closure operator in general, is the so called $\theta$-closure. It was defined by Veličko [@V] in the following way: Let $(X,\mathcal{T})$ be a topological space and let $A \subset X$. A point $x \in X$ is in the $\theta$-[*closure*]{} of $A$, denoted by ${\rm cl}_{\theta}A$ (or $\mathcal{T}{\rm cl}_{\theta}A$), if each closed neighbourhood of $x$ intersects $A$. Neighbourhood bases in $(X,{\rm cl}_{\theta})$ consist of closed neighbourhoods (or closures of open neighbourhoods) in $(X,\mathcal{T})$ at every point $x$.
Let $(X,\mathcal{T})$ be the product space of a family $\{(X_{\alpha},\mathcal{T}_{\alpha})\}$ of topological spaces. The $\theta$-closure space of $(X,\mathcal{T})$ is the product of the $\theta$-closure spaces of $(X_{\alpha},\mathcal{T}_{\alpha})$, i.e. $(X,\mathcal{T}{\rm cl}_{\theta}) =
\Pi (X_{\alpha},\mathcal{T}_{\alpha}{\rm cl}_{\theta})$.
A function $f:(X,\mathcal{T}) \rightarrow (Y,\mathcal{V})$ is $\theta$-[*continuous at*]{} $x \in X$ if for every neighbourhood $V$ of $f(x)$ there is a neighbourhood $U$ of $x$ such that $f(\overline{U}) \subset \overline{V}$. A function $f:(X,\mathcal{T}) \rightarrow (Y,\mathcal{V})$ is $\theta$-[*continuous*]{} if it is $\theta$-continuous at each of its points. Every continuous function is $\theta$-continuous, but the converse does not hold in general.
$\theta$-continuity is not a continuity concept in the class of topological spaces, but it is in the class of Čech closure spaces. Namely,
A function $f:(X,\mathcal{T}) \rightarrow (Y,\mathcal{V})$ is $\theta$-continuous if and only if the function $f:(X,\mathcal{T}{\rm cl}_{\theta}) \rightarrow
(Y,\mathcal{V}{\rm cl}_{\theta})$ is a continuous mapping of (Čech) closure spaces.
Hence the following characterizations of $\theta$-continuity:
A function $f:(X,\mathcal{T})\rightarrow (Y,\mathcal{V})$ is $\theta$-continuous if and only if:
1. $f(\mathcal{T}{\rm cl}_{\theta}A) \subset
\mathcal{V}{\rm cl}_{\theta}f(A)$ for every $A\subset X$;
2. $\mathcal{T}{\rm cl}_{\theta} f^{-1}(B) \subset
f^{-1}(\mathcal{V}{\rm cl}_{\theta}B)$ for every $B\subset Y$.
The next statement follows from the definitions and the properties of $\theta$-closure.
Let $X, Y, Z$ be topological spaces. A function $g:Z\times X\rightarrow Y$ is $\theta$-continuous if and only if the function $g:(Z,{\rm cl}_{\theta}) \times (X,{\rm cl}_{\theta}) \rightarrow
(Y,{\rm cl}_{\theta})$ is continuous.
A closure space $(X,u)$ is:
- [*regular*]{} if for each point $x$ and each subset $A$ such that $x \notin u(A)$, there exist neighbourhoods $U$ of $x$ and $V$ of $A$ such that $U \cap V = \emptyset$;
- [*compact*]{} if each net in $(X,u)$ has an accumulation point.
A closure space $(X,u)$ is regular if and only if for each point $x$ and each neighbourhood $U$ of $x$, there is a neighbourhood $U_{1}$ of $x$ such that $u(U_{1})\subset U$.
Compactness can be characterized by means of covers. [@vC 41 A.9. Theorem] An [*interior cover*]{} of $(X,u)$ is a cover $\{ G_{\alpha }\}$ such that the collection $\{{\rm int}_{u}G_{\alpha }\}$ covers $X$. The space is compact if and only if every interior cover has a finite subcover.
We give the following
A collection $\{ G_{\alpha }\}$ is an [*interior cover*]{} of a set $A$ in $(X,u)$ if the collection $\{ {\rm int}_{u}G_{\alpha }\}$ covers $A$. A subset $A$ is [*compact*]{} if every interior cover of $A$ has a finite subcover.
All notions not explained here can be found in [@vC].
Proper and admissible topologies in the setting of closure spaces
=================================================================
Let $X,Y$ and $Z$ be three nonempty sets. For every function $g:Z\times X \rightarrow Y$ there is a function ${\rm E}(g)$ or $g^{\ast}$ from $Z$ to $Y^X$, the set of all functions from $X$ to $Y$, defined by $(g^{\ast}(z))(x)=g(z,x)$. The mapping ${\rm E}:Y^{Z\times X} \rightarrow (Y^{X})^{Z}$ is called the [*exponential function*]{}. By $\varepsilon $ we denote the [*evaluation mapping*]{} from ${Y}^{X}\times X$ to $Y$ defined by $\varepsilon(f,x) = f(x)$.
If $X,Y$ and $Z$ are topological or closure spaces, in particular sets of continuous functions can be considered. Now on $Y^X$ will mean the set of [*all continuous functions*]{} from $X$ to $Y$. The set $Y^X$ can be endowed with different topologies. The question is: Find the topologies on the set of functions such that
1. ${\rm E}(g) = g^{\ast} \in (Y^{X})^{Z}$ for every $g \in Y^{Z\times X}$, that is, for every continuous $g:Z\times X \rightarrow Y$ the function $g^{\ast}$ is continuous; and conversely,
2. $g \in Y^{Z\times X}$ for every $g^{\ast} \in (Y^{X})^{Z}$, that is, for every continuous $g^{\ast}:Z \rightarrow Y^{X}$ the function $g$ is continuous.
Following the definitions and notations used by Arens and Dugundji [@AD], Kuratowski [@K] and Iliadis and Papadopoulos [@IP] for the sets of continuous functions defined in the setting of topological spaces, we give the following definitions.
Let $(X,u)$ be a closure space and $(A_{\lambda })_{\lambda \in \Lambda}$ be a net in $\mathcal{P}(X)$. The [*upper limit*]{} of the net $(A_{\lambda})$, denoted by $\overline{\lim\limits_{\Lambda }}A_{\lambda}$, is the set of all points $x \in X$ such that for every $\lambda_{0}\in \Lambda$ and every neighbourhood $U$ of $x$ in $X$, there is a $\lambda \in \Lambda$ such that $\lambda{\geq \lambda}_{0}$ and $A_{\lambda} \cap U \ne \emptyset$. (See, for example, [@AD] and [@IP].)
Let $(X,u)$ and $(Y,v)$ be closure spaces and $Y^X$ be the collection of all continuous functions $f:(X,u) \rightarrow (Y,v)$. A closure operator $\sigma$ on $Y^X$ is called [*proper (splitting)*]{} if for any closure space $(Z,w)$
- $g:(Z,w)\times (X,u) \rightarrow (Y,v)$ is continuous $\Rightarrow
E(g) = g^{\ast}:(Z,w) \rightarrow (Y^{X},\sigma)$ is continuous;
$\sigma$ is called [*admissible (jointly continuous)*]{} if for every space $(Z,w)$
- $g^{\ast}:(Z,w) \rightarrow (Y^{X},\sigma)$ is continuous $\Rightarrow
g:(Z,w)\times (X,u) \rightarrow (Y,v)$ is continuous.
Let $f, f_{\lambda} \in Y^{X}$, $\lambda \in \Lambda$, where $\Lambda$ is a directed set. The net $(f_{\lambda})$ [*converges continuously*]{} to $f$, denoted by $f_{\lambda}\buildrel cc\over {\longrightarrow f}$, if
- the net $f_{\lambda }(x_{\mu })$, $(\lambda,\mu) \in \Lambda \times {\rm M}$, converges to $f(x)$ in $(Y,v)$ whenever the net $(x_{\mu})$ converges to $x$ in $(X,u)$.
The convergence of a net $(f_{\lambda})$ to $f$ in the space $(Y^{X},\sigma)$ will be denoted by $f_{\lambda }\buildrel \sigma\over {\longrightarrow f}$.
The next results follow from definitions and the proofs are analogous to the corresponding for the topological case. (See [@AD] and [@IP]).
A closure operator $\sigma$ on $Y^{X}$ is admissible if and only if the evaluation mapping $\varepsilon:(Y^{X},\sigma)\times (X,u) \rightarrow (Y,v)$ is continuous.
Let $(f_{\lambda})_{\lambda\in \Lambda}$ be a net in $Y^{X}$. The net $(f_{\lambda})_{\lambda \in \Lambda}$ converges continuously to $f \in Y^{X}$ if and only if for every $x\in X$ and every neighbourhood $V$ of $f(x)$ there is a neighbourhood $U$ of $x$ and a $\lambda _{0} \in \Lambda $ such that $f_{\lambda }(U) \subset V$ for all $\lambda{\geq \lambda}_{0}$.
*Let $(f_{\lambda })_{\lambda \in \Lambda }$ be a net in $Y^{X}$. The net $(f_{\lambda})_{\lambda \in \Lambda }$ converges continuously to $f\in Y^{X}$ if and only if the following holds:*
- $\overline{\lim\limits_{\Lambda }} f_{\lambda}^{-1}(B) \subset f^{-1}(v(B))$ for every subset $B$ in $Y$.
Let $f_{\lambda }\buildrel cc\over {\longrightarrow f}$, $B \subset Y$ and $x \in \overline{\lim\limits_{\Lambda }} f_{\lambda}^{-1}(B)$. For every neighbourhood $V$ of $f(x)$, by Theorem 2, there is a neighbourhood $U$ of $x$ and a $\lambda_{0}\in \Lambda$ such that $f_{\lambda}(U)\subset V$ for all $\lambda{\geq \lambda}_{0}$. By the assumption, there is a $\lambda' \geq \lambda_{0}$ such that $f_{\lambda'}^{-1}(B) \cap U \ne \emptyset$. It follows that $f_{\lambda'}(U) \cap B \ne \emptyset$. Thus $$(\forall V \in \mathcal{N}(f(x)))\ V \cap B \ne \emptyset$$ implies $f(x) \in v(B)$, hence $x \in f^{-1}(v(B))$.
For the converse, let $(f_{\lambda})_{\lambda \in \Lambda }$ be a net in $Y^{X}$ such that the condition (4) holds. For every $x \in X$ and every neighbourhood $V$ of $f(x)$, $$f(x) \in {\rm int}_vV = {\rm c}(v({\rm c}(V)))
\mbox{ implies }
f(x) \notin v(V^{\rm c}).$$ Thus $x \notin f^{-1}(v(V^{\rm c}))$ implies $x \notin \overline{\lim\limits_{\Lambda}} f_{\lambda}^{-1}(V^{\rm c})$. Hence $$\begin{array}{lll}
(\exists U \in \mathcal{N}(x))
(\exists\lambda _{0} \in \Lambda)
(\forall \lambda \in \Lambda)\
\lambda \geq \lambda_{0}&
\Rightarrow&
f_{\lambda}^{-1}(V^{\rm c}) \cap U = \emptyset\\
&
\Rightarrow&
V^{\rm c} \cap f_{\lambda }(U) = \emptyset.
\end{array}$$ Thus $$(\exists U \in \mathcal{N}(x))
(\exists \lambda_{0}\in \Lambda)
(\forall \lambda \in \Lambda) \
\lambda \geq \lambda _{0} \Rightarrow f_{\lambda}(U)\subset V.$$ By Theorem 2, $f_{\lambda}\buildrel cc\over {\longrightarrow f}$.
In Theorem 3 the condition “for every $B\subset Y$” can be replaced by: for every $B=V^{\rm c}$, where $V$ is a neighbourhood basic element.
Let $(f_{\lambda })_{\lambda \in \Lambda }$ be a net in $Y^{X}$ and $Y$ be a topological space. The net $(f_{\lambda})_{\lambda \in \Lambda} $ converges continuously to $f\in Y^{X}$ if and only if the following holds:
- $\overline{\lim\limits_{\Lambda}} f_{\lambda}^{-1}(B) \subset f^{-1}(B)$ for every closed subset $B$ in $Y$.
Let a net $(f_{\lambda})$ converges continuously to $f\in Y^{X}$ and $B$ be a closed set in $Y$. By (4), $$\overline{\lim \limits_{\Lambda}} f_{\lambda}^{-1}(B) \subset
f^{-1}(\overline B) =
f^{-1}(B).$$ Conversely, suppose that (4\*) holds and let $B$ be a subset in $Y$. By (4\*), for the closed subset $\overline B$, $\overline{\lim\limits_{\Lambda}} f_{\lambda}^{-1}(\overline B) \subset
f^{-1}(\overline B)$, and by isotony of the upper limit, $\overline{\lim\limits_{\Lambda}} f_{\lambda}^{-1}(B) \subset
\overline {\lim\limits_{\Lambda}} f_{\lambda}^{-1}(\overline B)$. Hence the statement.
Let $\sigma $ and $\sigma'$ be two closure operators on $Y^{X}$.
1. If $\sigma'$ is finer than $\sigma$ and $\sigma'$ is proper, then $\sigma$ is proper.
2. If $\sigma \leq \sigma'$ and $\sigma$ is admissible, then $\sigma'$ is admissible.
3. If $\sigma$ is proper and $\sigma'$ is admissible, then $\sigma \leq \sigma'$.
4. If there is a closure operator on $Y^{X}$ which is both proper and admissible, it is unique.
(3). If $\sigma'$ is admissible, the evaluation mapping $$\varepsilon:(Y^{X},\sigma')\times (X,u) \rightarrow (Y,v)$$ is continuous by Theorem 1. Since $\sigma$ is proper, for $Z = ({Y}^{X},\sigma')$, the identity $$\varepsilon^{\ast}=1_{Y^{X}}:(Y^{X},\sigma') \rightarrow
(Y^{X},\sigma)$$ is continuous, hence $\sigma'$ is finer than $\sigma$.
Let $\{\sigma_{\alpha}\}$ be a collection of closure operators on $Y^{X}$.
1. If $\sigma_{\alpha }$ is proper for every $\alpha$, then the infimum and supremum, $\land \sigma_{\alpha}$ and $\lor \sigma_{\alpha}$, are proper.
2. If $\sigma_{\alpha}$ is admissible for every $\alpha$, then the supremum, $\lor \sigma_{\alpha}$ is admissible as well.
For the nontrivial part of (1), let $\sigma_{\alpha }$ be proper for every $\alpha$. That is, for any space $(Z,w)$, continuity of $g:(Z,w)\times (X,u) \rightarrow (Y,v)$ implies $g^{\ast}:(Z,w) \rightarrow (Y^{X},\sigma_{\alpha})$ is continuous. In order to prove continuity of $g^{\ast}:(Z,w) \rightarrow (Y^{X}, \lor\sigma_{\alpha})$, for any $z\in Z$ and every neighbourhood $G$ of $g^{\ast}(z)$, there are finitely many $G_{i} \in \mathcal{N}_{\alpha_{i}}(g^{\ast}(z))$ such that $\bigcap G_{i}\subset G$. By continuity of $g^{\ast}:(Z,w) \rightarrow (Y^{X},\sigma_{\alpha})$, for every $i$ there is a $W_{i}\in \mathcal{N}(z)$ such that $g^{\ast}(W_{i})\subset G_{i}$. Then $$W = \bigcap W_{i} \in \mathcal{N}(z)$$ and $$g^{\ast}(W) \subset \bigcap g^{\ast}(W_{i}) \subset \bigcap G_{i}
\subset G$$ holds. Thus $g^{\ast}:(Z,w) \rightarrow (Y^{X}, \lor\sigma_{\alpha})$ is continuous at $z$.
A closure operator $\sigma$ on $Y^{X}$ is proper (splitting) if and only if continuous convergence of a net implies its convergence in $(Y^{X}, \sigma)$, and it is admissible (jointly continuous) if and only if the reverse holds, that is, convergence of a net in $(Y^{X},\sigma)$ implies its continuous convergence.
A closure operator $\sigma $ on $Y^{X}$ is:
1. proper if and only if continuity of a mapping $g:Z\times (X,u) \rightarrow (Y,v)$ implies continuity of the mapping $g^{\ast}:Z \rightarrow (Y^{X},\sigma)$ for every topological space $Z$ being either a ${\rm T}_1$-space having at most one non-isolated point or the Sierpinski space;
2. admissible if and only if the reverse holds, that is, continuity of a mapping $g^{\ast}:Z \rightarrow (Y^{X},\sigma)$ implies continuity of the mapping $g:Z\times (X,u) \rightarrow (Y,v)$ for every topological space $Z$ being either a ${\rm T}_1$-space having at most one non-isolated point or the Sierpinski space.
In Theorem 6 the conditions on the space $Z$ can be replaced by: $Z$ is a topological space having at most one non-isolated point.
In the sequel the finest proper topology on $Y^{X}$, which exists by Corollary 2, is characterized by means of convergence classes and upper limits, analogously to the topological situation. (Cf. [@IP].)
Let $\mathcal{C}(\sigma)$ be the [*convergence class*]{} of the closure space $(Y^{X},\sigma)$, that is $$\mathcal{C}(\sigma) =
\{ ((f_{\lambda})_{\lambda \in \Lambda}, f) \mid
f_{\lambda}, f \in Y^{X} \mbox{ and }
f_{\lambda}\buildrel \sigma\over {\longrightarrow f}
\}.$$ It can be easily seen that $\mathcal{C}(\sigma)$ satisfies the following axioms: (cf. [@vC 35 A.2. Theorem] and [@Ke 2.9 Theorem])
(i) (CONSTANTS)
: If $(f_{\lambda })_{\lambda \in \Lambda}$ is a net such that $f_{\lambda} = f$ for every $\lambda \in \Lambda$, then $(f_{\lambda})$ converges to $f$, that is, $((f_{\lambda})_{\lambda \in \Lambda}, f) \in \mathcal{C}(\sigma)$;
(ii) (SUBNETS)
: If a net $(f_{\lambda })$ converges to $f$, so does each subnet of $(f_{\lambda})$, i.e. if $((f_{\lambda})_{\lambda \in \Lambda}, f) \in \mathcal{C}(\sigma)$, then $((g_{\mu})_{\mu \in {\rm M}}, f) \in \mathcal{C}(\sigma)$ for every subnet $(g_{\mu})$ of $(f_{\lambda})$;
(iii) (DIVERGENCE)
: If a net $(f_{\lambda })$ does not converge to $f$, then there is a subnet $(g_{\mu})$ of $(f_{\lambda })$ no subnet of which converges to $f$, i.e. $((f_{\lambda })_{\lambda \in \Lambda},f) \notin \mathcal{C}(\sigma)$, then there is a subnet $(g_{\mu})$ of $(f_{\lambda})$ such that $((h_{\nu})_{\nu \in {\rm N}},f) \notin \mathcal{C}(\sigma)$ for every $(h_{\nu})$ subnet of $(g_{\mu})$.
(iii). Let $(f_{\lambda})$ be a net in $Y^{X}$, $f\in Y^{X}$ such that $((f_{\lambda})_{\lambda \in \Lambda},f)\notin \mathcal{C}(\sigma)$. It means that $$(\exists G_0 \in \mathcal{N}(f))
(\forall \lambda \in \Lambda)
(\exists \lambda' \in \Lambda) \lambda'
\geq
\lambda \land f_{\lambda'} \notin G_{0}.$$ Thus there is a cofinal subset ${\rm M} \subset \Lambda $ such that $f_{\mu }\notin G_{0}$ for every $\mu \in {\rm M} $. $(f_{\mu })_{\mu \in {\rm M} }$ is a subnet of $(f_{\lambda })$, no subnet of which converges to $f$.
The space $(Y^{X}, \sigma)$ is topological if and only if its convergence class satisfies the axiom of (ITERATED LIMITS). (Cf. [@Ke 2.9 Theorem] and [@vC 15 B.13. and 35 A.3. Theorems].)
Denote by $\mathcal{C}^{\ast}$ the class of all pairs $((f_{\lambda})_{\lambda \in \Lambda}, f)$ such that $(f_{\lambda})$ is a net in $Y^{X}$ which converges continuously to $f\in Y^{X}$, i.e. $$\mathcal{C}^{\ast} =
\{ ((f_{\lambda})_{\lambda \in \Lambda}, f) \mid
f_{\lambda}, f \in Y^{X} \mbox{ and }
f_{\lambda}\buildrel cc\over {\longrightarrow f}
\}.$$ By Theorem 5, $\sigma$ is proper if and only if $\mathcal{C}^{\ast} \subset \mathcal{C}(\sigma)$ and it is admissible if and only if the reverse inclusion holds: $\mathcal{C}(\sigma) \subset \mathcal{C}^{\ast}$.
The class $\mathcal{C}^{\ast}$ satisfies the axioms (CONSTANTS), (SUBNETS) and (DIVERGENCE).
For (CONSTANTS) and (SUBNETS) is clear.
(DIVERGENCE). Let $(f_{\lambda})$ be a net in $Y^{X}$, $f\in Y^{X}$ and let $$((f_{\lambda})_{\lambda \in \Lambda}, f) \notin \mathcal{C}^{\ast}.$$ By Theorem 3 there is a subset $B\subset Y$ such that $\overline{\lim\limits_{\Lambda}} f_{\lambda}^{-1}(B)$ is not contained in $f^{-1}(v(B))$. Let $$x \in \overline{\lim\limits_{\Lambda}} f_{\lambda}^{-1}(B) \setminus
f^{-1}(v(B)).$$ Let $\mathcal{N}(x)$ be the set of all neighbourhoods of $x$ directed by inverse inclusion and let ${\rm M} = \Lambda \times \mathcal{N}(x)$. If $\mu = (\lambda ,U) \in \Lambda \times \mathcal{N}(x)$, let $\varphi:{\rm M} \rightarrow \Lambda$ be defined by $\varphi(\mu) \in \Lambda$ such that $\varphi(\mu) = \varphi(\lambda, U) \geq \lambda$ and $f_{\varphi(\mu)}^{-1}(B) \cap U \ne \emptyset$. The net $(g_{\mu})_{\mu \in {\rm M}}$, where $g_{\mu} = f_{\varphi(\mu)}$, is a subnet of $(f_{\lambda})$.
Let $(h_{\nu})$ be a subnet of $(g_{\mu})$ and $\psi :{\rm N} \rightarrow {\rm M} $ be the corresponding map. In order to prove that $((h_{\nu})_{\nu \in {\rm N}}, f) \notin \mathcal{C}^{\ast}$, let $\nu_0 \in {\rm N} $ and $U \in \mathcal{N}(x)$. If $\psi(\nu_0) = (\lambda_0,U_0) \in {\rm M}$, set $\hat U = U_{0} \cap U \in \mathcal{N}(x)$ and $\mu_0 = (\lambda_0,\hat U)$. There is ${\nu_1\in {\rm N}}$ such that $\nu_{1}\geq \nu_0$ and $\nu \geq \nu_{1} \Rightarrow \psi (\nu) \geq \mu_0$. For any $\nu \geq \nu _{1}$ and $\psi (\nu) = (\lambda, \tilde U)$ we have $$h_{\nu }^{-1}(B) \cap U =
f_{\varphi(\psi(\nu))}^{-1}(B) \cap U \supset
f_{\varphi(\psi(\nu))}^{-1}(B) \cap \hat U \supset
f_{\varphi(\psi (\nu))}^{-1}(B) \cap \tilde U \ne
\emptyset.$$ It means that $x \in \overline{\lim\limits_{{\rm N}}} h_{\nu}^{-1}(B)$ and hence $\overline{\lim\limits_{{\rm N}}} h_{\nu }^{-1}(B)$ is not contained in $f^{-1}(v(B))$. Thus the axiom (DIVERGENCE) is satisfied.
$\mathcal{C}^{\ast }$ is the convergence class of the finest proper topology on $Y^{X}$ if and only if $\mathcal{C}^{\ast}$ satisfies the axiom (ITERATED LIMITS).
A subset $G$ in $Y^{X}$ is open in the finest proper topology if and only if for every $f\in G$ and for every net $(f_{\lambda})_{\lambda \in \Lambda}$ in $Y^{X}$ such that [(4)]{} holds, there exists a $\lambda _{0} \in \Lambda $ such that $f_{\lambda}\in G$ for every $\lambda {\geq \lambda}_{0}$.
($\Leftarrow $:) Let $\tau$ be the collection of subsets in $Y^{X}$ with the given property.
$\tau$ is a topology on $Y^{X}$: for $G_{1}, G_{2} \in \tau$, $f \in G_{1} \cap G_{2}$ and a net $(f_{\lambda})_{\lambda \in \Lambda}$ satisfying (4), there are $\lambda_{1}, \lambda_2 \in \Lambda$ such that $f_{\lambda} \in G_i$ for all $\lambda \geq \lambda_{i}$, $i=1,2$. Then $f_{\lambda} \in G_{1} \cap G_{2}$ for all $\lambda \geq \lambda_{0} = \max \limits \{ \lambda_{1}, \lambda_{2} \}$. It follows that $G_{1} \cap G_{2} \in \tau$. Similarly, if $\{ G_{\alpha} \} \subset \tau$ and $G = \bigcup_{\alpha} \{G_{\alpha}\}$, let $f\in G$ and $(f_{\lambda})_{\lambda \in \Lambda}$ be a net which satisfies (4). There is an $\alpha _{0}$ such that $f\in G_{\alpha _{0}}$, and since (4) holds, there exists a ${\lambda_0 \in \Lambda}$ such that $f_{\lambda}\in G_{\alpha_{0}}\subset G$ for every $\lambda \geq \lambda_{0}$. Thus $G \in \tau$.
$\tau$ is proper: let $(f_{\lambda })_{\lambda \in \Lambda}$ be a net such that $f_{\lambda}\buildrel cc\over {\longrightarrow f}$. By Theorem 3, $\overline{\lim\limits _{\Lambda }} f_{\lambda}^{-1}(B) \subset
f^{-1}(v(B))$ for every subset $B$ in $Y$. Let $f\in G\in \tau$. By the assumption, there exists a ${\lambda _0 \in \Lambda}$ such that $f_{\lambda} \in G$ for every $\lambda \geq \lambda_{0}$, that is $f_{\lambda}\buildrel \tau\over {\longrightarrow f}$. By Theorem 5, $\tau$ is proper.
$\tau$ is the finest proper topology: let $\sigma$ be a proper topology on $Y^{X}$ and $H \in \sigma$. Let $f\in H$ and a net $(f_{\lambda })_{\lambda \in \Lambda}$ satisfy (4). By Theorem 3, $f_{\lambda}\buildrel cc\over {\longrightarrow f}$. Since $\sigma$ is proper, $f_{\lambda }\buildrel \sigma\over {\rightarrow f}$. By definition of convergence, there exists a ${\lambda_0 \in \Lambda}$ such that $f_{\lambda} \in H$ for every $\lambda \geq \lambda_{0}$. By definition of $\tau$, $H \in \tau$. Thus $\sigma \subset \tau$.
($\Rightarrow $:) Let a subset $G$ in $Y^{X}$ be open in the finest proper topology $\tau$, let $f\in G$ and $(f_{\lambda })$ be a net satisfying (4). By Theorem 3, $f_{\lambda }\buildrel cc\over {\longrightarrow f}$. Since $\tau$ is proper, $f_{\lambda} \buildrel \tau\over {\longrightarrow f}$. By definition of convergence, there exists a ${\lambda_0 \in \Lambda}$ such that $f_{\lambda }\in G$ for every $\lambda \geq \lambda_{0}$.
In order to give nontrivial examples of admissible and proper topologies and to get results analogous to Theorems 4.1 and 4.21 in [@AD], we consider the following sets. Let $$\mathcal{V} = \{ V\subset Y | {\rm int}_{v}V \ne \emptyset \}.$$ For $A \subset X$ and $V \in \mathcal{V}$, set $$(A,V) = \{ f\in Y^{X} | f(A)\subset V \}.$$
Let $\mathcal{A}$ be a family of subsets of $X$. The collection $$\{ (A,V) | A \in \mathcal{A}, V \in \mathcal{V}, V = {\rm int}_{v}V \},$$ is a subbase for a topology on $Y^{X}$, which will be called the $\mathcal{A}$-[*topology*]{}.
Let $\mathcal{C}$ be an interior cover of $X$. The collection $$\{ (u(K),V) | V \in \mathcal{V} \mbox{ and } K \subset X
\mbox{ is such that } u(K) \subset C
\mbox{ for some } C \in \mathcal{C} \},$$ is a subbase for a topology on $Y^{X}$, which will be called the $\mathcal{C}$-[*topology*]{}.
Let $(X,u)$ be a regular closure space and $(Y,v)$ be arbitrary. For every interior cover $\mathcal{C}$ of $X$, the $\mathcal{C}$-topology is admissible.
By Theorem 1, it is enough to prove that the evaluation mapping is continuous. Let $f \in Y^{X}$, $x \in X$ and $V \in \mathcal{N}(f(x))$, $f(x) = \varepsilon(f,x)$. By continuity of $f$, the set $U = f^{-1}(V) \in \mathcal{N}(x)$. Choose a $C \in \mathcal{C}$ so that $x \in {\rm int}_{u}C$. Then $U \cap C \in \mathcal{N}(x)$ and by regularity of $X$, there is a $U_{1} \in \mathcal{N}(x)$ such that $$x \in {\rm int}_{u}U_{1} \subset
U_{1} \subset
u(U_{1}) \subset U \cap C.$$ For the subbasic element $(u(U_1),V)$ in the $\mathcal{C}$-topology, $\varepsilon((u(U_1),V),U_1) \subset V$ since for every $f_{1} \in (u(U_{1}),V)$ and each $x \in U_1$, $\varepsilon(f_1,x_1) = f_1(x_1) \in V$.
Let $(X,u)$ and $(Y,v)$ be closure spaces and $\mathcal{A}$ be a collection of compact subsets in $(X,u)$. The $\mathcal{A}$-topology is always proper.
Let $g:(Z,w)\times (X,u) \rightarrow (Y,v)$ be a continuous function. In order to prove continuity of the mapping $g^{\ast}:(Z,w) \rightarrow (Y^{X},\sigma)$, where $Y^{X}$ is endowed with the $\mathcal{A}$-topology, let $z\in Z$ and $f=g^{\ast}(z)$. For a subbasic element $(K,V)$ containing $f$, where $K$ is a compact set in $X$, $$f(K) = g(\{z\} \times K) \subset {\rm int}_{v}V = V.$$ By continuity of $g$, $$(\forall x \in K) \
g(z,x) \in {\rm int}_{v}V
\Leftrightarrow
V \in \mathcal{N}(g(z,x))$$ implies $$(\forall x\in K)
(\exists W_{x}\in \mathcal{N}(z))
(\exists U_{x}\in \mathcal{N}(x)) \ g(W_{x}\times U_{x}) \subset V.$$
$(\forall x \in K) U_x \in \mathcal{N}_{x}$ implies $\{ U_{x} \mid x \in K\}$ is an interior cover of the compact set $K$, so there is a finite subcover $\{U_{x_{i}} \mid i=1,\cdots k\}$. Set $W = \bigcap_{i=1}^{k}W_{x_{i}}$. Then $W \in \mathcal{N}(z)$. It follows that $$g(W \times K) \subset
g(\bigcup_{i=1}^{k}{(W}_{x_{i}}\times U_{x_{i}})) \subset
V.$$ Thus $$(\forall z' \in W) \
g(z',K) \subset V \Rightarrow g^{\ast}(W) \subset (K,V).$$
Some special cases including $\theta$-closure
=============================================
Let $(X,u)$ be a closure space and $(Y,\mathcal{V})$ be a topological space. A function $f:(X,u) \rightarrow (Y,\mathcal{V})$ is continuous if and only if the function $f:(X,\hat u) \rightarrow (Y, \mathcal{V})$ between topological spaces is continuous, where $\hat u$ is the topological modification of the closure operator $u$. In that case the problem is reduced to the topological case since $Y^{X} = C((X,\hat u),Y)$ and the topological modification of the product of closure spaces is the product of topological modifications.
It was already remarked that $\theta$-continuous functions are continuous functions of the corresponding Čech closure spaces. Compact sets in $(X,{\rm cl}_{\theta })$ are (quasi-)H-closed (g-H-closed) in $(X,\mathcal{U})$. Thus Theorems 3.1–3.6 and 4.2 in [@C] are special cases of Theorems 2, 1, 4, 5 and 10 respectively.
If $(Y,\mathcal{V})$ is regular, the $\theta$-topology $\mathcal{V}_{\theta} = \mathcal{V}$. Then for a topological space $(X,\mathcal{U})$, a function $f:(X,\mathcal{U}) \rightarrow (Y,\mathcal{V})$ is $\theta$-continuous if and only if $f:(X,{\rm cl}_{\theta}) \rightarrow (Y,\mathcal{V})$ is continuous, which is equivalent to $f:(X,\mathcal{U}_{\theta}) \rightarrow (Y,\mathcal{V})$ be continuous [@vC Thm 16.B.4]. Note that the topological modification of $\mathcal{U}{\rm cl}_{\theta }$ is $\mathcal{U}_{\theta}$, the topology of $\theta$-open sets in $(X,\mathcal{U})$.
A large number of continuous-like mappings between topological spaces is known in the literature. Recently, Georgiou and Papadopoulos [@GP1; @GP2] considered some of them and investigated splitting and jointly continuous topologies on the sets of these functions. Let us remark that all these examples and the main results are special cases of our subjects of investigations. For, let $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ be topological spaces. It follows from the definitions that a function $f:(X,\mathcal{U})\rightarrow (Y,\mathcal{V})$ is:
- [*strongly $\theta $-continuous*]{} (cf. [@GP1]) at a point $x$ (on the set $X$) if and only if $f:(X,\mathcal{U}{\rm cl}_{\theta}) \rightarrow (Y,\mathcal{V})$ is continuous at $x$ (on the set $X$);
- [*super-continuous*]{} (cf. [@GP2]) at $x$ (on the set $X$) if and only if $f:(X,\mathcal{U}_s) \rightarrow (Y,\mathcal{V})$ is continuous at $x$ (on the set $X$), where $\mathcal{U}_s$ is the semi-regularization topology of $\mathcal{U}$ (see [@GMRV] for example);
- [*weakly continuous*]{} (cf. [@GP2]) at $x$ (on the set $X$) if and only if $f:(X,\mathcal{U}) \rightarrow (Y,\mathcal{V}{\rm cl}_{\theta})$ is continuous at $x$ (on the set $X$);
- [*weakly $\theta $-continuous*]{} (cf. [@GP2]) at a point $x$ (on $X$) if and only if $f:(X,\mathcal{U}_s) \rightarrow (Y,\mathcal{V}{\rm cl}_{\theta})$ is continuous at $x$ (on $X$).
Also $\theta$-[*convergence*]{} of a net $(x_{\mu})$ in $(X,\mathcal{U})$ means convergence of $(x_{\mu})$ in the corresponding closure space $(X,\mathcal{U}{\rm cl}_{\theta})$, while [*weak $\theta$-convergence*]{} of a net $(x_{\mu})$ in $(X,\mathcal{U})$ is convergence of $(x_{\mu})$ in $(X,\mathcal{U}_s)$.
Similarly, [*$\theta $-continuous convergence*]{} (respectively: [*strongly $\theta$-continuous convergence, weakly $\theta$-continuous convergence, weakly continuous convergence*]{} and [*super continuous convergence*]{}) of a net $(f_{\lambda })$ in $Y^X$ is continuous convergence of $(f_{\lambda })$ for the corresponding closure spaces. Thus we are concerned with a change of topology, better to say: change of closure operator, technique. So the main results in [@GP1] and [@GP2] are special cases of the above Theorems 1–10.
[10]{}
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Eduard [Č]{}ech, *Topological spaces*, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1966. [MR ]{}[35 \#2254]{}
Anna Di Concilio, *On $\theta$-continuous convergence in function spaces*, Rend. Mat. (7) **4** (1984), no. 1, 85–94 (1985). [MR ]{}[87d:54024]{}
D. B. Gauld, M. Mr[š]{}evi[ć]{}, I. L. Reilly, and M. K. Vamanamurthy, *Continuity properties of functions*, Topology, theory and applications (Eger, 1983), North-Holland, Amsterdam, 1985, pp. 311–322. [MR ]{}[863 913]{}
D. N. Georgiou and B. K. Papadopoulos, *Strongly $\vartheta$-continuous functions and topologies on function spaces*, Papers in honour of Bernhard Banaschewski (Cape Town, 1996), Kluwer Acad. Publ., Dordrecht, 2000, pp. 433–444. [MR ]{}[2002b:54012]{}
[to3em]{}, *Weakly continuous, weakly $\vartheta$-continuous, super-continuous and topologies on function spaces*, Sci. Math. Jpn. **53** (2001), no. 2, 233–246. [MR ]{}[1 828 262]{}
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N. V. Veli[č]{}ko, *${H}$-closed topological spaces*, Mat. Sb. (N.S.) **70 (112)** (1966), 98–112, English translation in [*American Mathematical Society Translations. Series 2, Vol. 78*]{}, Amer. Math. Soc., Providence, R.I., 1968; MR [**39**]{} \#1253; ISBN 0-8218-1778-7. [MR ]{}[33 \#6576]{}
[^1]: This article will be expanded and submitted for publication elsewhere.
[^2]: Mila Mršević, [*Proper and admissible topologies in the setting of closure spaces*]{}, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 205–216, Topology Atlas, Toronto, 2002
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies vorticity equations and that it produces a family of conserved quantities.'
address: |
D.V. Voiculescu\
Department of Mathematics\
University of California at Berkeley\
Berkeley, CA 94720-3840
author:
- 'Dan-Virgil Voiculescu'
title: 'A Hydrodynamic Exercise in Free Probability: Setting Up Free Euler Equations'
---
[^1]
Introduction {#sec1}
============
In ${\mathbb R}^n$ equipped with the Gaussian measure, the coordinate functions can be viewed as i.i.d. Gaussian random variables and the divergence-free vector fields correspond to infinitesimal increments of the random variables which preserve the joint distribution. In free probability the analogue of the Gaussian random variables are the field operators on the full Fock space of ${\mathbb C}^n$ with respect to vacuum expectations and the von Neumann algebra they generate is isomorphic to the von Neumann algebra of a free group as shown in [@14]. We studied in [@18] the infinitesimal increments for the field operators, which preserve the non-commutative joint distribution and we found that the polynomial increments among these provide a Lie algebra which is dense in the space of all such increments. Thus we have a good description of a dense part of the analogue in free probability of the divergence-free vector fields.
Since divergence-free vector fields are the basic ingredient for the hydrodynamic Euler equations we became curious about pursuing the analogy further and finding the free Euler equations. Our approach is in two steps, one formal and one analytic. The formal step consists in getting the equations on the dense Lie algebra following the general recipe in ([@1],[@2]). Of course the framework of the dense Lie algebra is much too restrictive for solutions. Fortunately, in the analytic step we are able to greatly relax the smoothness requirements on the divergence-free vector fields. Indeed, our results in [@18] imply that the Leray projection commutes with the number operator, which in particular makes the use of free hypercontractivity [@3] possible. The equations we find can be stated either in projection form (i.e., using a Leray projection) or with a pressure term. The pressure term is a cyclic gradient, roughly, and using the exact sequence for cyclic gradients we found in [@17], one of the maps provides a replacement for the curl. We define in this way a cyclic vorticity, which satisfies a vorticity equation and we obtain conserved quantities which are the moments of the cyclic vorticity. In a sense this is similar to the situation on even-dimensional manifolds; however, the cyclic vorticity is not a non-commutative tensor-field, but a non-commutative scalar.
The paper has eight sections, including this introduction.
Section \[sec2\] contains preliminaries of a general algebraic nature from [@18].
Section \[sec3\] deals with preliminaries concerning semicircular systems, the free probability analogue of i.i.d. Gaussian random variables [@14], [@15], [@16] the main aim being the results in [@18] about the Lie algebra of infinitesimal automorphisms of a semicircular system. We also recall some free hypercontractivity facts from [@3].
Section \[sec4\] is an application of Arnold’s procedure ([@1], [@2]) for setting up Euler equations on a Lie algebra to the Lie algebra of infinitesimal automorphisms of a semicircular system [@18]. This is a formal derivation of the Euler equations, the context of non-commutative polynomials being too restrictive to expect interesting solutions.
Section \[sec5\] provides analytic facts which will be used to make sense of more general solutions of the free Euler equations. In particular, we deal with an algebra ${\mathscr B}_{\infty,1}$ of smooth elements in the style of [@9], [@12], [@19], which is a kind of Sobolev or Besov space. We also keep track of the special properties of the analogue of the Leray projection arising from its commutation with the number operator, or equivalently with the free Ornstein–Uhlenbeck semigroup.
In Section \[sec6\] we show that in view of the analytic facts about ${\mathscr B}_{\infty,1}$ in the preceding section, we can make sense of the free Euler equations when non-commutative polynomials are replaced by ${\mathscr B}_{\infty,1}$, roughly.
In Section \[sec7\] the cyclic vorticity is introduced, based on the idea that the role of the curl should be played by a map appearing in the exact sequence for cyclic gradients [@17]. We show that for a certain subalgebra ${\mathscr C}_{\infty,1}$ instead of ${\mathscr B}_{\infty,1}$ the non-commutative moments of the cyclic vorticity are conserved quantities.
Section \[sec8\] is devoted to concluding remarks, in particular about corresponding Navier–Stokes equations and about a possible substitute for boundary conditions.
Algebraic Preliminaries ([@18]) {#sec2}
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Let $s_1,\dots,s_n$ be an $n$-tuple of self-adjoint elements in a von Neumann algebra $M$, which are algebraically free, that is there is no non-trivial algebraic relation among them. In this section we collect purely algebraic preliminaries, while later on, under additional assumptions on $s_1,\dots,s_n$ we will go beyond algebra.
The ${\mathbb C}$-subalgebra of $M$ generated by $1,s_1,\dots,s_n$ will be denoted by ${\mathbb C}{\langle}s_1,\dots,s_n{\rangle}$ or ${\mathbb C}_{{\langle}n{\rangle}}$ and consists of the non-commutative polynomials in the indeterminates $s_1,\dots,s_n$.
By ${\partial}_j: {\mathbb C}_{{\langle}n{\rangle}} \to {\mathbb C}_{{\langle}n{\rangle}} \otimes {\mathbb C}_{{\langle}n{\rangle}}$, $1 \le j \le n$, we will denote the partial free difference quotient derivations, that is the derivations so that ${\partial}_jX_k = {\delta}_{jk}1 \otimes 1$, $1 \le j,k \le n$. We will also use the partial cyclic derivatives ${\delta}_j: {\mathbb C}_{{\langle}n{\rangle}} \to {\mathbb C}_{{\langle}n{\rangle}}$ defined by ${\delta}_j = \mu\ \circ \sim \circ\ {\partial}_j$ where $\sim$ is the flip $\sim(a \otimes b) = b \otimes a$ and $\mu: {\mathbb C}_{{\langle}n{\rangle}} \otimes {\mathbb C}_{{\langle}n{\rangle}} \to {\mathbb C}_{{\langle}n{\rangle}}$ is the multiplication map $\mu(a \otimes b) = ab$, a linear map of ${\mathbb C}_{{\langle}n{\rangle}}$ bimodules. On monomials we have $$\begin{aligned}
{\partial}_j(s_{i_1}\dots s_{i_m}) &= \sum_{\{k \mid i_k=j\}} s_{i_1} \dots s_{i_{k-1}} \otimes s_{i_{k+1}} \dots s_{i_m} \\
\delta_j(s_{i_1}\dots s_{i_m}) &= \sum_{\{k \mid i_k = j\}} s_{i_{k+1}} \dots s_{i_m}s_{i_1} \dots s_{i_{k-1}}.
\end{aligned}$$ We will also consider the cyclic gradient ${\delta}: {\mathbb C}_{{\langle}n{\rangle}} \to ({\mathbb C}_{{\langle}n{\rangle}})^n = {\mathbb C}_{{\langle}n{\rangle}} \oplus \dots \oplus {\mathbb C}_{{\langle}n{\rangle}}$, ${\delta}P = {\delta}_1P \oplus \dots \oplus {\delta}_nP $ and the free difference quotients gradient ${\partial}: {\mathbb C}_{{\langle}n{\rangle}} \to ({\mathbb C}_{{\langle}n{\rangle}} \otimes {\mathbb C}_{{\langle}n{\rangle}})^n$, ${\partial}P = {\partial}_1P \oplus \dots \oplus {\partial}_nP $.
If ${\mathscr E}$ is a ${\mathbb C}_{{\langle}n{\rangle}}$-bimodule and $b \in {\mathscr E}$ we consider the ${\mathbb C}_{{\langle}n{\rangle}}$-bimodules map $m_b: {\mathbb C}_{{\langle}n{\rangle}} \otimes {\mathbb C}_{{\langle}n{\rangle}} \to {\mathscr E}$ so that $m_b(P \otimes Q) = PbQ$. If $b_j \in {\mathscr E}$, $1 \le j \le n$ the map $$D_{(b_1,\dots,b_n)}: {\mathbb C}_{{\langle}n{\rangle}} \to {\mathscr E}$$ defined by $\sum_{1 \le j \le n} m_{b_j} \circ {\partial}_j$, is a derivation of ${\mathbb C}_{{\langle}n{\rangle}}$ into ${\mathscr E}$.
Since $s_1,\dots,s_n$ are algebraically free, there are evaluation homomorphisms $\varepsilon_{(a_1,\dots,a_n)}: {\mathbb C}_{{\langle}n{\rangle}} \to {\mathscr A}$ where $a_1,\dots,a_n$ are elements of the unital ${\mathbb C}$-algebra ${\mathscr A}$, which map $s_j$ to $a_j$, $1 \le j \le n$, that is in essence $P (s_1,\dots,s_n)$ is mapped to $P (a_1,\dots,a_n)$.
If $b_1,\dots,b_n \in M$, then since $M$ is a ${\mathbb C}_{{\langle}n{\rangle}}$-bimodule, we have $$\frac {d}{d\varepsilon} P(s_1+\varepsilon b_1,\dots,s_n + \varepsilon b_n)|_{\varepsilon = 0} = D_{(b_1,\dots,b_n)}P$$ where $P \in {\mathbb C}_{{\langle}n{\rangle}}$ and $\varepsilon \in {\mathbb C}$. If, moreover, we also have a trace $\tau: M \to {\mathbb C}$ then we have $$\frac {d}{d\varepsilon} \tau(P(s_1+\varepsilon b_1,\dots,s_n+\varepsilon b_n)|_{\varepsilon = 0} = \sum_{1 \le j \le n} \tau(b_j{\delta}_jP).$$ This means that if we endow $M^n$ with the scalar product $${\langle}(a_j)_{1 \le j \le n},(b_j)_{1 \le j \le n}{\rangle}= \sum_{1 \le j \le n} \tau(a_jb_j)$$ then the cyclic gradient ${\delta}P$ is the gradient at $(s_1,\dots,s_n)$ of the map $$M^n \ni (a_1,\dots,a_n) \to \tau(P(a_1,\dots,a_n)) \in {\mathbb C}.$$
If ${\mathscr E}$ is a ${\mathbb C}_{{\langle}n{\rangle}}$-bimodule we shall use the notation ${\mbox{Vect}}\ {\mathscr E}$ for ${\mathscr E}^u$. In particular ${\mathscr E}$ can be ${\mathbb C}_{{\langle}n{\rangle}}$ or $M$.
The space ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}}$ is a Lie algebra under the bracket $$\{P,Q\} = (D_PQ_j - D_QP_j)_{1 \le j \le n}$$ where $P = (P_j)_{1 \le j \le n}$, $Q = (Q_j)_{1 \le j \le n}$, which is the analogue of the Poisson bracket. For the analogue of hydrodynamic equations, we shall use like in Example $5.2$ on page 20 of [@2], the commutator $[P,Q] = -\{P,Q\}$.
If $\tau$ is a trace on $M$, we define $${\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}} = \{P \in {\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}} \mid \sum_{1 \le j \le n} \tau(P_j({\delta}_jR)) = 0,\ \forall\ R \in {\mathbb C}_{{\langle}x{\rangle}}\}.$$ Then ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}}$ is a Lie subalgebra of ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}}$. It is a non-commutative analogue of a Lie algebra of divergence-free vector fields (in an algebraic context where the vector fields can be required to be polynomial functions).
Since $s_1,\dots,s_n$ are self-adjoint, ${\mathbb C}_{{\langle}n{\rangle}}$ is a $*$-algebra. If we evaluate $P \in {\mathbb C}_{{\langle}n{\rangle}}$ at $a_1,\dots,a_n \in M$ we will have $(P(a_1,\dots,a_n))^* = P^*(a_1^*,\dots,a_n^*)$. Note also that ${\delta}_jP^* = ({\delta}_jP)^*$ and ${\partial}_jP^* = \sim({\partial}_jP)^*$ where on ${\mathbb C}_{{\langle}n{\rangle}} \otimes {\mathbb C}_{{\langle}n{\rangle}}$ we use the involution $(\xi \otimes \eta)^* = \xi^* \otimes \eta^*$.
The involution on ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}}$ is defined componentwise and we have $D_{P^*}Q_j^* = (D_PQ_j)^*$. Then $P \rightsquigarrow P^*$ is a conjugate-linear automorphism of the Lie algebra ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}}$. In particular, the selfadjoint part $${\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}}^{sa} = \{P \in {\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}} \mid P = P^*\}$$ is a real Lie subalgebra of ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n{\rangle}}$. Similarly, since ${\delta}R^* = ({\delta}R)^*$ we have $${\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}}^{sa} = \{P \in {\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}} \mid P = P^*\}$$ is a real Lie algebra and $${\mbox{Vect}}\ {\mathbb C}_{{\langle}n,\tau{\rangle}}^{sa} + i {\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}}^{sa} = {\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}}^{sa}.$$
Remark also, that if $a = (a_1,\dots,a_n) \in {\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}}$ and $b \in {\mathbb C}_{{\langle}n{\rangle}}$, then $$\tau(D_ab) = \sum_{1 \le j \le n} \tau(a_j{\delta}_jb) = 0.$$ In particular, if $c \in {\mathbb C}_{{\langle}n{\rangle}}$, then $\tau(D_a(bc)) = 0$ gives the “integration by parts” formula $$\tau((D_ab)c) = -\tau(b(D_ac)).$$
Semicircular Preliminaries {#sec3}
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From now on in this paper we shall assume $(M,\tau)$ is the von Neumann algebra $W^*(s_1,\dots,s_n)$ generated by a semicircular system $s_1,\dots,s_n$ and $\tau$ is the unique normal trace state. So, $s_1,\dots,s_n$ have $(0,1)$ semicircle distributions and are freely independent in $(M,\tau)$, which is actually isomorphic to the $II_1$ factor generated by the regular representation of the free group $F_n$.
Let ${\mathbb C}^n$ have the Hilbert space structure with orthonormal basis $e_1,\dots,e_n$ and let $${\mathscr T}({\mathbb C}^n) = \bigoplus_{k \ge 0} ({\mathbb C}^n)^{\otimes k}$$ be the full Fock space where $({\mathbb C}^n)^{\otimes 0} = {\mathbb C}1$ where 1 is the vacuum vector and let $l_j$, $r_j$, $1 \le j \le n$ be left and right creation operators $$l_j\xi = e_j \otimes \xi,\ r_j\xi = \xi \otimes e_j.$$
Then $L^2(M,\tau)$ can be identified with ${\mathscr T}({\mathbb C}^n)$ so that $s_j = l_j + l_j^*$, $1 \le j \le n$ and $$e_{i_1}^{\otimes k_1} \otimes \dots \otimes e_{i_p}^{\otimes k_p} = P_{k_1}(s_{i_1})\dots P_{k_p}(s_{i_p})1$$ where $i_j \ne i_{j+1}$ $(1 \le j \le p - 1)$, $k_j > 0$ $(1 \le j \le p)$ and $P_k$, $k \ge 0$ are the Chebyshev polynomials, that is the orthogonal polynomials on $[-2,2]$ w.r.t. the semicircle measure (an instance of the more general Gegenbauer polynomials). Note also that the involution $\xi \to \xi^*$ on $L^2(M,\tau)$ corresponds to the antiunitary operator $${\mathscr J}(ce_{k_1} \otimes \dots \otimes e_{k_m}) = {\bar c}e_{k_m} \otimes \dots \otimes e_{k_1}.$$
The results in ([@18], $7.1$–$7.5$) show that ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}}$ and ${\delta}{\mathbb C}_{{\langle}n{\rangle}}$ have good properties as subspaces of $(L^2(M,\tau))^n \simeq ({\mathscr T}({\mathbb C}^n))^n$, keeping in mind that ${\delta}s_{i_1}\dots s_{i_p} = {\delta}s_{i_p}s_{i_1}\dots s_{i_{p-1}}$. We have ${\mbox{Vect}}\ {\mathbb C}_{{\langle}n\mid \tau{\rangle}} = \sum_{k \ge 0} {\mathscr X}_k$, ${\delta}{\mathbb C}_{{\langle}n{\rangle}} = \sum_{k \ge 0} {\mathscr Y}_k$ where ${\mathscr X}_k,{\mathscr Y}_k \subset (({\mathbb C}^n)^{\otimes k})^n = {\mathscr X}_k + {\mathscr Y}_k$ and $${\mathscr X}_k = \{((l^*_j - r^*_j)\xi)_{1 \le j \le n} \mid \xi \in ({\mathbb C}^n)^{\otimes k+1}\}.$$ Moreover ${\mathscr J}{\mathscr X}_k = {\mathscr X}_k$, ${\mathscr J}{\mathscr Y}_k = {\mathscr Y}_k$ and ${\mathscr X}_k,{\mathscr Y}_k$ are orthogonal both w.r.t. the symmetric scalar product $\sum_{1 \le j \le n} \tau(a_jb_j)$ as well as w.r.t. the sesquilinear one $\sum_{1 \le j \le n} \tau(a_
jb_j^*)$.
The Hilbert spaces ${\mathscr T}({\mathbb C}^n)$ and the left creation operators are part of a functor, the free analog of the Gaussian functor [@14]. We will need here only the free analogue of the Ornstein–Uhlenbeck semigroup $(P_t)_{t \ge 0}$ in order to use the free hypercontractivity results of [@3]. On ${\mathscr T}({\mathbb C}^n)$ the action of $P_t$ is given by $P_t\xi = e^{-kt}\xi$ if $\xi \in ({\mathbb C}^n)^{\otimes k}$. The operators $P_t$ are self-adjoint contractions and extend to all $L^p(M,\tau)$, $1 \le p \le \infty$, where they act as contractions. Moreover, on $L^{\infty}(M,\tau) \simeq M$, $P_t$ is a unit-preserving completely positive contraction.
Note also that $(P_t)^n {\mathscr X}_k = {\mathscr X}_k$, $(P_t)^n{\mathscr Y}_k = {\mathscr Y}_k$, where ${\mathbb C}_{{\langle}n\mid \tau{\rangle}} = \sum_{k \ge 0} {\mathscr X}_k$, ${\delta}{\mathbb C}_{{\langle}n{\rangle}} = \sum_{k \ge 0} {\mathscr Y}_k$, ${\mathscr X}_k,{\mathscr Y}_k \subset (({\mathbb C}^n)^{\otimes k})^n$. In particular, if $\Pi$ is an analogue of the Leray projection, that is the orthogonal projection of $(L^2(M,\tau))^n$ onto $\overline{{\mathbb C}_{{\langle}n\mid \tau{\rangle}}}$ we have $\Pi {\mathscr J}= {\mathscr J}\Pi$, $\Pi(P_t)^n = (P_t)^n\Pi$, $\Pi{\mathbb C}_{{\langle}n{\rangle}} = {\mathbb C}_{{\langle}n\mid \tau{\rangle}}$ and $\Pi{\mathbb C}^{sa}_{{\langle}n{\rangle}} = {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$.
Formal derivation of the Euler equations on the Lie algebra ${\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$ {#sec4}
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The recipe in I §4 and §5 of [@2] for Euler equations on a real Lie algebra ${\mathbf g}$ endowed with a scalar product ${\langle}\cdot,\cdot{\rangle}$ will be applied to ${\mathbf g}= {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$ and the scalar product $${\langle}(a_j)_{1 \le j \le n},(b_j)_{1 \le j \le n}{\rangle}= \sum_{1 \le j \le n} \tau(a_jb_j).$$ The key computation we will need to perform will provide the bilinear map $B: {\mathbf g}\times {\mathbf g}\to {\mathbf g}$ so that $${\langle}[a,b],c{\rangle}= {\langle}B(c,a),b{\rangle}.$$ The scalar product being non-degenerate, $B$ is unique; however, the Lie algebra ${\mathbf g}$ is not finite-dimensional and the existence of $B$ will be a consequence of the good properties of $\Pi$ the analogue of the Leray projection. The identification of $B$ is the subject of the next lemma.
[**Lemma 1.**]{} [ *Let $a,b,c \in {\mathbb C}^{sa}_{{\langle}n,\tau{\rangle}}$, $a = (a_k)_{1 \le k \le n}$, $b = (b_k)_{1 \le k \le n}$, $c = (c_k)_{1 \le k \le n}$. Then if $$B(c,a) = \Pi(D_ac_k + \sum_{1 \le j \le n} m_{c_j}(\sim {\partial}_ka_j))_{1 \le k \le n}$$ we have $$-{\langle}[a,b],c{\rangle}= {\langle}B(c,a),b{\rangle}.$$ Moreover, we have $$B(a,a) = \Pi(D_a a_k)_{1 \le k \le n}.$$* ]{}
[*Proof.*]{} We have $$\begin{aligned}
{\langle}[a,b],c{\rangle}&= \sum_{1 \le j \le n} \tau((D_ab_j)c_j) - \sum_{1 \le j \le n} \tau((D_ba_j)c_j) \\
&= \sum_{1 \le j \le n} \tau(-b_j(D_ac_j)) - \sum_{1 \le j \le n} \tau((D_ba_j)c_j)
\end{aligned}$$ where we used “integration by parts” since $a \in {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$. Note further, that $$\tau((D_ba_j)c_j) = \sum_{1 \le k \le n} \tau((m_{b_k}(\partial_ka_j))c_j)$$ and note also that if $\xi,\eta \in {\mathbb C}_{{\langle}n{\rangle}}$ then $$\begin{aligned}
\tau((m_{b_k}(\xi \otimes \eta))c_j) &= \tau(\xi b_k\eta c_j) \\
&= \tau((\eta c_j\xi)b_k) = \tau((m_{c_j}(\sim(\xi \otimes \eta)))b_k)
\end{aligned}$$ which gives that $$\tau((m_{b_k}(\partial_ka_j)c_j) = \tau((m_{c_j}(\sim \partial_ka_j))b_k)$$ and hence $$\tau((D_ba_j)c_j) = \sum_{1 \le k \le n} \tau((m_{c_j}(\sim \partial_ka_j))b_k).$$ Putting all this together gives $$-{\langle}[a,b],c{\rangle}= \sum_{1 \le k \le n} \tau((-D_ac_k)b_k) - \sum_{1 \le k \le n} \tau\left(\left(\sum_{1 \le j \le n}(m_{c_j}(\sim\partial_ka_j)\right)b_k\right)$$ where we changed indexing from $j$ to $k$ in the first sum and switched summations in the rest. This gives the formula for $B(c,a)$, the projection $\Pi$ is applied in order that $B(c,a) \in {\mathbf g}$ and we get this, since $$\Pi\ {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n{\rangle}} = {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}.$$
To get the last assertion about $B(a,a)$ we must show that $$\sum_{1 \le j \le n} \tau((D_ba_j)a_j) = 0$$ if $b \in {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$. Since $\tau$ is a trace we have $$\tau((D_ba_j)a_j) = \frac {1}{2} \tau(D_ba^2_j) = 0$$ which gives the desired result.
The hydrodynamic Euler equations for an element $v(t) \in {\mathbf g}$ evolving in time is ([@2] top of page 20) $${\dot v} = -B(v,v).$$ In view of the Lemma we proved, the Euler equations for $$v(t) = (v_k(t))_{1 \le k \le n} \in {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$$ will be: $$(\dot{v}_k)_{1\le k \le n} + \Pi(D_vv_k)_{1 \le k \le n} = 0.$$ In our case $a - \Pi a \in \delta{\mathbb C}_{{\langle}n{\rangle}}$ so we can also figure a form of equations involving a “pressure” $p(t) \in {\mathbb C}^{sa}_{{\langle}n{\rangle}}$ and taking the form $$\dot{v}_k + D_vv_k + \delta_kp = 0\qquad 1 \le k \le n.$$ Thus comparing with the classical equations the gradient of the pressure becomes in the free probability setting a cyclic gradient of the pressure. To continue the comparison, we did not write a continuity equation since this corresponds to the requirement $v(t) \in {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$, that is $v = \Pi v$.
As a final comment, the computations we did in this section did not use the assumption that $s_1,\dots,s_n$ is a semicircular $n$-tuple; however, without this assumption we have no control over ${\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n\mid \tau{\rangle}}$, which may be zero and we also have no control about the projection $\Pi$.
Analytic preparations {#sec5}
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We collect here a few analytic facts which we will use in the next section to define solutions of the free Euler equations which may not be polynomial.
Using [@3] an element $\xi \in L^1(M,\tau)$ can be described as a formal series $\sum_{k \ge 0} \xi_k$ where $\sum_{k \ge 0} e^{-kt} \xi_k \in L^2(M,\tau)$ (or equivalently $\sum_{k \ge 0} e^{-2kt}|\xi_k|_2^2 < \infty$) for all $t > 0$ and so that $$\sup_{t > 0} \left| \sum_{k \ge 0} e^{-kt}\xi_k\right|_1 < \infty.$$ Moreover $|\xi|_1$ is precisely the above $\sup_{t > 0}$ and $\left|\xi - \sum_{k \ge 0} e^{-kt}\xi_k\right|_1 \to 0$ as $t \downarrow 0$. Note also that actually $\sum_{k \ge 0} e^{-kt}\xi_k \in M = L^{\infty}(M,\tau)$ and equals $P_t\xi$ by [@3].
We may then use this to work with vector fields with $L^p$ components $1 \le p \le \infty$, that is ${\mbox{Vect}}\ L^p(M,\tau)$, $1 \le p \le \infty$ the elements of which are $n$-tuples $\xi = \left( \sum_{k \ge 0} \xi_{k,j}\right)_{1 \le j \le n}$ where $\sum_{k \ge 0} \xi_{k,j} \in L^p(M,\tau)$, $1 \le j \le n$, where $L^p$ is viewed as a subspace of $L^1$. We shall denote by $Q_k$ the projection $Q_k\xi = (\xi_{k,j})_{1 \le j \le n}$ in ${\mbox{Vect}}\ L^1(M,\tau)$. We have $Q_k(P_t)^n\xi = (P_t)^n Q_k\xi = e^{-kt}Q_k\xi$ and also $Q_k\xi = e^{kt}Q_k(P_t)^n\xi$.
The $L^1$-“divergence-free” vector fields will be denoted by ${\mbox{Vect}}(L^1(M,\tau)\mid \tau)$ and can be defined in several equivalent ways. One definition is by requiring that $\xi = (\xi_j)_{1 \le j \le n}$, $\xi_j \in L^1(M,\tau)$, $1 \le j \le n$ satisfy $\sum_{1 \le j \le n} \tau(\xi_j(\delta_jR)) = 0$ for all $R \in {\mathbb C}_{{\langle}n{\rangle}}$. Since ${\delta}{\mathbb C}_{{\langle}n{\rangle}} = \sum_{k \ge 0} {\mathscr Y}_k$, ${\mathbb C}_{{\langle}n\mid \tau{\rangle}} = \sum_{k \ge 0} {\mathscr X}_k$ where ${\mathscr X}_k,{\mathscr Y}_k \subset (({\mathbb C}^n)^{\otimes k})^n$ and $({\mathbb C}^n)^{\otimes k} = {\mathscr X}_k \oplus {\mathscr Y}_k$ it is easily seen that ${\mbox{Vect}}\ (L^1(M,\tau)\mid \tau)$ consists of the $\xi \in {\mbox{Vect}}(L^1(M,\tau))$ so that $Q_k\xi \in {\mathscr X}_k$. Denoting by $\Pi_k$ the projection of ${\mbox{Vect}}\ L^1(M,\tau)$ onto ${\mathscr X}_k$, this second definition of ${\mbox{Vect}}(L^1(M,\tau)\mid \tau)$ is that $\xi \in {\mbox{Vect}}\ L^1(M,\tau)$ satisfies $\Pi_k\xi = Q_k\xi$ for all $k \ge 0$. From here it is also easily seen that ${\mbox{Vect}}(L^1(M,\tau) \mid \tau)$ can be defined as the closure of ${\mathbb C}_{{\langle}n\mid \tau{\rangle}}$ in ${\mbox{Vect}}(L^1(M,\tau))$. Note also that in view of the equivalence of these 3 definitions we also have that $\xi \in {\mbox{Vect}}(L^1(M,\tau))$ is in ${\mbox{Vect}}(L^2(M,\tau) \mid \tau)$ iff $P_t\xi \in {\mbox{Vect}}(L^1(M,\tau \mid \tau)$ for some $t > 0$. By ${\mbox{Vect}}(L_{sa}^1(M,\tau))$ and ${\mbox{Vect}}(L_{sa}^1(M,\tau)\mid \tau)$ we shall denote the corresponding subspaces of self-adjoint vector fields, that is $\xi = (\xi_j)_{1 \le j \le n}$ with $\xi_j^* = \xi_j$, $1 \le j \le n$.
We pass now to introducing a certain subalgebra of $M$ with a suitable smoothness requirement (see [@9], [@12], [@19] for related constructions). We define $${\mathscr B}_{\infty,1} = \{a \in M \mid [a,r_j-r^*_j] \in {\mathscr C}_1\}$$ where $a \in M$ is identified with the left multiplication operator $L_a$ on $L^2(M,\tau) \simeq {\mathscr T}({\mathbb C}^n)$ and ${\mathscr C}_1$ (or ${\mathscr C}_1(L^2(M,\tau))$) denotes the trace-class operators on $L^2(M,\tau)$. We shall denote by $|\cdot|_p$ the $p$-norm on $L^p(M,\tau)$, $1 \le p \le \infty$ and by $\|\cdot\|_p$ the $p$-norm on the Schatten–von Neumann classes ${\mathscr C}_p$. Then $$\||a\|| = \|a\| + \max_{1 \le j \le n} \|[a,r_j-r^*_j]\|_1$$ is a Banach algebra norm on ${\mathscr B}_{\infty,1}$.
We will need some basic facts arising from $L^1(M,\tau)$ being the predual of $M$. There is a unique contractive linear map $$\Phi: {\mathscr C}_1(L^2(M,\tau)) \to L^1(M,\tau)$$ so that $$\mbox{Tr}(Xa) = \tau(\Phi(X)a)$$ for all $a \in M$ and $X \in {\mathscr C}_1$. If $\xi,\eta \in L^2(M,\tau)$ and $E_{\xi,\eta}$ denotes the rank one operator ${\langle}\cdot,{\mathscr J}\eta{\rangle}\xi$, then $\Phi(E_{\xi,\eta}) = \xi\eta$. More generally every $X \in {\mathscr C}_1$ can be written in the form $\sum_k E_{\xi_k,\eta_k}$ with $\sum_k |\xi_k|_2|\eta_k|_2 < \infty$ and then $\Phi(X) = \sum_k \xi_k\eta_k$. Using the $L_a,R_a$ notation for left and respectively right multiplication operators by $a$ on $L^2(M,\tau)$ we have $\Phi(R_aE_{\xi,\eta}) = \xi a\eta$ and $[R_a,X] \in \ker \Phi$ for all $a \in M$, $X \in {\mathscr C}_1$. It is also easily seen that $\Phi(XL_a) = \Phi(X)a$ an $\Phi(L_aX) = a\Phi(X)$ if $a \in M$, $X \in {\mathscr C}$, and $\Phi(X^*) = \Phi(X)^*$.
If $a \in {\mathscr B}_{\infty,1}$, then $[a,r_j-r^*_j] \in {\mathscr C}_1$, $1 \le j \le n$ and if $b_j \in M$, $1 \le j \le n$ we have that $2^{-1} \sum_{1 \le j \le n} R_{b_j}[a,r_j-r_j^*] \in {\mathscr C}_1$ and $$\Phi\left( 2^{-1} \sum_{1\le j \le n} R_{b_j}[a,r_j-r_j^*]\right)$$ defines a map from ${\mathscr B}_{\infty,1}$ to $L^1(M,\tau)$. This map extends the derivation $D_b$ where $b = (b_j)_{1 \le j \le n}$ from ${\mathbb C}_{{\langle}n{\rangle}}$ to ${\mathscr B}_{\infty,1}$ and takes values in $L^1(M,\tau)$. Indeed, we have $[s_j,r_k-r_{k^*}] = [l_j + l_j^*,r_k-r_k^*] = 2E_{1,1} \delta_{jk}$. A straightforward computation shows that on monomials $a = s_{i_1}\dots s_{i_p}$ we have that $D_b$ and $\Phi(2^{-1}\sum_j R_{b_j}[a,r_j-r_j^*])$ are equal. We shall denote this extension of $D_b$ to a continuous linear map ${\mathscr B}_{\infty,1} \to L^1(M,\tau)$ by ${\mathscr D}_b$.
Consider also $\Psi: {\mathscr C}_1 \to L^1(M,\tau)$ the map $\Psi(X) = \Phi({\mathscr J}X^*{\mathscr J})$. Since ${\mathscr J}(E_{\xi,\eta})^*{\mathscr J}= E_{\eta,\xi}$ we will have $\Psi(E_{\xi,\eta}) = \eta\xi$. Let ${\tilde \delta}: {\mathscr B}_{\infty,1} \to L^1(M,\tau)$ be defined by ${\tilde \delta}_j(a) = \Psi(2^{-1}[a,r_j-r_j^*])$, $1 \le j \le n$. If $a \in {\mathbb C}_{{\langle}n{\rangle}}$ we have ${\tilde \delta}_j(a) = \delta_j(a)$, so that ${\tilde \delta}_j$ is an extension of $\delta_j$ to ${\mathscr B}_{\infty,1}$.
An easy computation also proves the following.
[**Lemma.**]{} [ *${\mathscr D}_b: {\mathscr B}_{\infty,1} \to L^1(M,\tau)$ is a derivation and then ${\mathscr D}_{b^*}(a^*) = ({\mathscr D}_b(a))^*$, in particular if $b=b^*$ then ${\mathscr D}_b(a^*)=({\mathscr D}_b(a))^*$. Moreover we have $$|{\mathscr D}_ba|_1 \le 2^{-1} \sum_{1 \le j \le n} \|b_j\|\|[a,r_j-r_j^*]\|_1.$$ Similarly we have $$|{\tilde \delta}_j(a)|_1 \le 2^{-1}\|[a,r_j-r_j^*]\|_1.$$* ]{}
The free Euler equations in ${\mathscr B}_{\infty,1}$ {#sec6}
=====================================================
In this section we show that the free Euler equations we found in the formal setting of section \[sec4\], with some adjustments, make sense also for ${\mbox{Vect}}({\mathscr B}_{\infty,1}^{sa} \mid \tau)$, that is for divergence-free vector fields with components in ${\mathscr B}_{\infty,1}^{sa}$.
We recall the equations were $$({\dot v}_k)_{1 \le k \le n} + \Pi(D_vv_k)_{1 \le k \le n} = 0$$ and the requirement $v(t) \in {\mbox{Vect}}\ {\mathbb C}^{sa}_{{\langle}n|\tau{\rangle}}$ was playing the role of the continuity equation. We shall assume that each $v_k(t) \in {\mathscr B}^{sa}_{\infty,1}$, $1 \le k \le n$ where $t \in [0,T)$ some $T > 0$ is differentiable as a function of $t$ with values in $L^1(M,\tau)$ (which is a weaker requirement than as a function with values in the Banach space ${\mathscr B}_{\infty,1}$). We shall also assume that $$v(t) = (v_k(t))_{1 \le k \le n} \in {\mbox{Vect}}(L^1(M,\tau)\mid \tau)$$ for all $t \in [0,T]$.
With the above requirements we get clearly that $({\dot v}_k(t))_{1 \le k \le n} \in {\mbox{Vect}}(L_{sa}^1(M,\tau)\mid \tau)$.
Since $(v_k(t))_{1 \le k \le n} \in {\mathscr B}_{\infty,1}$ we have that ${\mathscr D}_{v(t)} v_k(t) \in L^1(M,\tau)$ by the results of section \[sec5\]. This suggests that we replace $D_v(t)$ which we had defined only to act on ${\mathbb C}_{{\langle}n{\rangle}}$, by ${\mathscr D}_{v(t)}$ which acts on $B_{\infty,1}$.
The resulting ${\mathscr D}_{v(t)}v(t)$ being in ${\mbox{Vect}}\ L^1(M,\tau)$ we face the problem that the free Leray projection is defined in ${\mbox{Vect}}\ L^2(M,\tau)$. Since the Leray projection commutes with the number operator we can use instead of $\Pi$ the projections $\Pi_m$, $m \ge 0$ and we get a sequence of equations $$-Q_m{\dot v}(t) = \Pi_m({\mathscr D}_{v(t)}v_k(t))_{1 \le k \le n},\ m > 0.$$ An essentially equivalent way is to use the Ornstein–Uhlenbeck semigroup with some $\epsilon > 0$. The equations become $$-P_{\epsilon}{\dot v}(t) = \Pi P_{\epsilon}({\mathscr D}_{v(t)}v_k(t))_{1 \le k \le n}.$$
Other possibilities one may consider would be to use a suitably defined cyclic gradient of pressure term. The equations give $$-{\dot v}(t) = ({\mathscr D}_{v(k)}v_k(t))_{1 \le k \le n} + q(t)$$ where $$q(t) \in {\mbox{Vect}}\ L^1_{sa}(M,\tau)$$ is continuous as a function of $t$ and should satisfy $\Pi_kq(t) = 0$ for all $k \ge 0$. This implies each $Q_kq(t) \in \delta{\mathbb C}_{{\langle}n{\rangle}}$. In view of the exact sequence for cyclic gradients [@17], this gives $$\sum_{1 \le j \le n} [s_j,Q_kq(t)] = 0.$$ It is easy then to infer from here that $$\sum_{1 \le j \le n} [s_j,q(t)] = 0.$$
Cyclic vorticity and conserved quantities {#sec7}
=========================================
In the classical setting applying the curl to the Euler equations makes the gradient of the pressure disappear and leads to the vorticity equations and the expectation values of powers of the vorticity, which is a differential form, are roughly the source of conserved quantities. In the free probability setting, the pressure term is a cyclic gradient and instead of the curl we have a map in the exact sequence we gave in [@17] which will make the pressure term disappear. The new quantity we get is in $M$ and expectation values of its powers, or differentiable functions of it, provide conserved quantities, a situation somewhat reminiscent of the classical Euler equations in even dimension. Since the analogy works reasonably well, we will call the quantity we get, the cyclic vorticity.
If $v \in {\mbox{Vect}}(L_{sa}^1(M,\tau))$ we define $$\Omega = i \sum_{1 \le j \le m} [s_j,v_j]$$ to be its cyclic vorticity. This is just $i\theta(v)$ with $\theta$ the extension to $L^1$ of the map in the exact sequence for cyclic gradients [@17] and the coefficient $i$ has been added to make sure that $$\Omega \in {\mbox{Vect}}(L^1_{sa}(M,\tau)).$$
Assume now $v(t) \in {\mbox{Vect}}({\mathscr B}_{\infty,1}^{sa}) \cap {\mbox{Vect}}(L_{sa}^1(M,\tau)\mid \tau)$ is differentiable as a ${\mbox{Vect}}(L^1(M,\tau))$-valued function of $t \in [0,T)$ and satisfies the free Euler equations in the form with pressure-term, discussed at the end of section \[sec6\], that is $$-{\dot v}(t) = ({\mathscr D}_{v(t)}v_k(t))_{1 \le k \le n} + q(t)$$ where $q(t) \in {\mbox{Vect}}(L_{sa}^1(M,\tau))$ is continuous as a function of $t$.
Since $\sum_{1 \le j \le n}[s_j,q_j(t)] = 0$ as discussed in section \[sec6\], we have $$\begin{aligned}
i{\dot \Omega}(t) &= -\sum_{1 \le j \le n} [s_j,{\dot v}_j(t)] \\
&= \sum_{1 \le j \le n}([s_j,{\mathscr D}_{v(t)}v_j(t)] + [s_j,q_j(t)]) \\
&= \sum_{1 \le j \le n}({\mathscr D}_{v(t)}[s_j,v_j(t)] - [{\mathscr D}_{v(t)}s_j,v_j(t)]) \\
&= \sum_{1 \le j \le n}({\mathscr D}_{v(t)}[s_j,v_j(t)] - [v_j(t),v_j(t)]) \\
&= i^{-1}{\mathscr D}_{v(t)}\Omega(t).
\end{aligned}$$ This gives the cyclic vorticity differential equation $$-{\dot \Omega}(t) = {\mathscr D}_{v(t)}\Omega(t).$$ Note also that $\Omega(t) \in {\mbox{Vect}}({\mathscr B}_{\infty,1}^{sa})$.
The usual vorticity is divergence-free and this provides a further equation. To see what the analogue of this is in our setting, we need to return to the exact sequence for cyclic gradients [@17] and the range of the map $\theta$. The range of $\theta$ is contained in $\mbox{Ker } C$, where $C$ is the map $Cs_{i_1}\dots s_{i_p} = \sum_{1 \le j \le p} s_{i_j}\dots s_{i_p}s_{i_1}\dots s_{i_{j-1}}$ also in $\mbox{Ker } \tau$. On the other hand $Ca = \sum_{1 \le j \le n} s_j\delta_ja$. Using the discussion at the end of section \[sec5\] and the Lemma, the map $C$ has an extension ${\tilde C}a = \sum_{1 \le j \le n} s_j{\tilde \delta}_ja$. Introducing the subalgebra ${\mathscr C}_{\infty,1}$, the closure of ${\mathbb C}_{{\langle}n{\rangle}}$ in ${\mathscr B}_{\infty,1}$, we find that if $v(t) \in {\mathscr C}_{\infty,1}$, then we have $${\tilde C}\Omega(t) = 0 \mbox{ and } \tau(\Omega(t)) = 0.$$
As a consequence of this differential equation, if $m \in {\mathbb N}$ we have $$-\frac {\partial}{\partial t} \tau(\Omega^m(t)) = \tau({\mathscr D}_{v(t)}\Omega^m(t)).$$ If $\Omega^m(t)$ would be in ${\mathbb C}_{{\langle}m{\rangle}}$, the fact that $v(t) \in {\mbox{Vect}}((L^1(M,\tau)\mid \tau)$ would imply $\tau({\mathscr D}_{v(t)}\Omega^m(t)) = 0$ and hence $\tau(\Omega^m(t))$ would be a conserved quantity. Building on this observation, we will use ${\mathscr C}_{\infty,1}$ the subalgebra of ${\mathscr B}_{\infty,1}$ which is the closure of ${\mathbb C}_{{\langle}n{\rangle}}$ with respect to the norm $\||\cdot\||$. Using the lemma at the end of section \[sec5\], we have
[**Lemma.**]{} [ *If $v \in$ [Vect]{}$(M,\tau)$ and $a \in {\mathscr C}_{\infty,1}$, then we have $$\tau({\mathscr D}_va) = 0.$$* ]{}
We then conclude that
[**Theorem.**]{} [ *If $v(t) \in$ [Vect]{}$({\mathscr C}^{sa}_{\infty,1}) \cap$ [Vect]{}$(L_{sa}^1(M,\tau) \mid \tau)$ is differentiable as a [Vect]{}$(L^1(M,\tau))$-valued function of $t \in [0,T)$ and satisfies the free Euler equations in the form with pressure-term, then $\tau(\Omega^m(t))$ is a constant where $\Omega(t)$ is the cyclic vorticity and $m > 0$.* ]{}
Clearly the theorem can be stated more generally, that $\tau(\varphi(\Omega(t)))$ is constant, where $\varphi$ is a bounded Borel function, since we know that $\Omega(t)$ is bounded and its distribution is constant.
Concluding remarks {#sec8}
==================
We collect here various remarks related to the free Euler equations.
What is the analogue of boundary conditions for our free Euler equations? A possible answer may be: the requirement that the non-commutative vector field $v(t)$ be the generator of a one-parameter group of automorphisms.
We didn’t go beyond Euler equations in this paper, but clearly free Navier–Stokes equations can be imagined where the viscosity term would involve the number operator. In particular, the commutation of the Leray projection with the number operator can still be expected to help.
We should also mention that replacing Gaussian random variables by non-commutative generalizations, was considered in other questions concerning fluids, quite early, in the pioneering paper [@5] which led to later non-commutative probability work [@6], [@7].
We should mention that transportation for classical PDE [@8], [@13] being relevant to Euler equations and having now free transportation developments [@4], [@10], [@11], in particular the last two paper achieving very strong results, there seem to be reasons for optimism about going deeper into the analytic problems of the free Euler equations.
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[^1]: Research supported in part by NSF Grant DMS-1665534.
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"pile_set_name": "ArXiv"
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---
author:
- 'Katsuaki <span style="font-variant:small-caps;">Kodama</span>$^1$, Naoki <span style="font-variant:small-caps;">Igawa</span>$^1$, Shin-ichi <span style="font-variant:small-caps;">Shamoto</span>$^1$, Kazutaka <span style="font-variant:small-caps;">Ikeda</span>$^2$, Hidetoshi <span style="font-variant:small-caps;">Oshita</span>$^2$, Naokatsu <span style="font-variant:small-caps;">Kaneko</span>$^2$, Toshiya <span style="font-variant:small-caps;">Otomo</span>$^2$, and Kentaro <span style="font-variant:small-caps;">Suzuya</span>$^3$'
title: 'Local Lattice Distortion Caused by Short Range Charge Ordering in LiMn$_2$O$_4$ '
---
Introduction
============
Charge ordering observed in strongly correlated electron system is one of typical examples of self-organization phenomena caused by many-body effect. It can be regarded as a crystallization of valence electrons. Because the charge ordering is accompanied with a periodic lattice distortion corresponding with the periodicity of the arrangement of the localized electrons, it can be detected as an appearance of a superlattice reflection in diffraction data.[@schiffer; @ohwada; @chen] However, in several materials which have the same band-filling as those of the charge ordered mateirals and exhibit metal-insulator transitions, the superlattice reflection and/or the structural phase transtion can not be observed in their insulating phases. It is considered that in such materials, although the electrons are localized by their Coulomb repulsion similar to charge ordered state, the arrangement of the localized electrons does not have a long range ordering, causing the lattice distortion without long range periodicity. It can be regarded as a glass state of valence electrons. In comparison with the charge ordered state, detailed study on such glass-like state of the charge has not been performed because only the local probe measurements such as NMR and M$\ddot{\textrm{o}}$ssbauer spectroscopy can be applied.[@hanasaki; @kuzushita] In this paper, we focus on LiMn$_2$O$_4$ as a candidate of such materials in which the valence electron is freezed like a glass.
LiMn$_2$O$_4$ has been studied as a candidate of cathod materials in secondary lithium ion batteries, [@thackeray1; @thackeray2; @bruce; @goodenough; @guyomard] and as a frustrated magnetic materials.[@mukai; @kamazawa] This compound has a cubic spinel structure with space group of $Fd\bar{3}m$ at room temperature. Mn atom is surrounded by six O atoms and they form MnO$_6$ octahedron. All Li, Mn and O atoms are crystallorgraghically equivalent, respectively. With decreasing temperature, the compound exhibits a structural phase transition at around 260K. [@RC; @tomeno; @shimakawa; @wills; @oikawa; @piszora] The low temperature phase has an orthorhombic structure with space group of $Fddd$ and “$3a \times 3a \times a$” super cell relative to the cubic phase, where $a$ is lattice parameter of the cubic phase.[@RC; @piszora] Five kinds of inequivalent Mn sites are included in the unit cell. From the bond valence sum calculation, valences of three Mn and two Mn sites are estimated to be about +3 and +4, and then the orthorhombic phase is in the charge ordered state.[@RC] However, even in the cubic phase, the temperature dependence of the electrical resistivity is not metallic although the metallic conductivity can be expected from the simple band picture since the valence band which consists of 3$d$ orbits is partically filled because of the averaged valence of +3.5 of Mn ions.[@shimakawa; @sugiyama; @molenda1; @molenda2] These results suggest that the valence electrons are localized at Mn sites like a glass due to their Coulomb repulsion, and the arrangement of Mn$^{3+}$ and Mn$^{4+}$ ions does not have a long range ordering in the cubic phase.
The lattice distortion without a long range periodicity (local lattice distortion) is an evidence of the glass-like freezing of the valence electrons. In order to investigate the local structure around Mn ion in the cubic phase, extended x-ray-absorption fine-structure (EXAFS) measurements have been performed.[@exafs1; @exafs2] However, the results of EXAFS measurements do not give a clear answer on the existence of the local lattice distortion ; all Mn-O distances of MnO$_6$ octahedra have same bond length in ref. 23 suggesting no local lattice distortion, whereas two kinds of Mn-O distance are reported in ref. 22 suggesting the local lattice distortion induced by the existence of Mn$^{3+}$ and Mn$^{4+}$. The EXAFS measurement can probe only short atomic distance, for example, first or second nearest neighboring atomic distances. It can also probe the local structure only around the selected atoms. Since it is difficult to detect the local lattice distortion only from the Mn-O distances because of the small difference between Mn$^{3+}$-O and Mn$^{4+}$-O distances (about 0.1 $\mathrm{\AA}$), the other atomic distances, for example, O-O distance should also be observed. Then in order to investigate the local lattice distortion in this compound, atomic pair distribution function (PDF) which can detect all atomic pair correlation should be used.
In this paper, we report the results of the PDF analysis on neutron powder diffraction data on LiMn$_2$O$_4$ at room temperature where the average crystal structure is the cubic with the space group of $Fd\bar{3}m$. The obtained atomic pair distribution function is fitted for the orthorhombic structure with space group of $Fddd$ much better than the cubic structure, indicating the existence of the local lattice distortion. The locally distorted structure has MnO$_6$ octahedra with long and short Mn-O distances which almost correspond with the distances of Mn$^{3+}$-O and Mn$^{4+}$-O, respectively, suggesting that the valence electrons are localized at Mn sites with short range periodicity like a glass.
Experiments
===========
Powder sample of $^7$LiMn$_2$O$_4$ for present neutron measurements was prepared by following method. Here, we prepared $^7$Li-enriched sample in order to avoid the neutron absorption by natural abundance of $^6$Li. Starting materials are powders of $^7$LiOH$\cdot$H$_2$O and (CH3COO)$_2$Mn$\cdot$H$_2$O with stoichiometric composition and they were mixed in 2-propanol. The mixture was dried at 400 for 1 hour to decarbonate. After the calcinations, the powder was ground and then heated at 850 for 10 hours. The sample of about 1.8 g was used in neutron diffraction measurements.
The powder neutron diffraction data for the Rietveld and PDF analyses were collected by using the neutron total scattering spectrometer NOVA installed in the Japan Proton Accelerator Research Complex (J-PARC). The powder sample of about 1.7 g was set in vanadium-nickle alloy holder with a diameter of 0.6 cm. The data were collected at room temperature for about 8 hours.
Results and Discussions
=======================
Analysis on averaged crystal structure by using Rietveld analysis
-----------------------------------------------------------------
Neutron powder diffraction pattern of $^7$LiMn$_2$O$_4$ obtained at 90 bank of NOVA is shown in Fig. 1 by crosses. In the plotted pattern, the contaminations of background intensities from the sample cell and the spectrometer are subtracted and the neutron absorption effect is calibrated.
![ (Color online) Observed (crosses) and calculated (solid line) neutron powder diffraction pattern of $^7$LiMn$_2$O$_4$. Observed pattern is collected by using NOVA at room temperature. Vertical bars show the calculated position of Bragg reflections. The solid line at the bottom of the figure is the difference between observed and calculated intensities. Inset : foot of the main peak with $d \sim 2.4~\mathrm{\AA}$ are extended. []{data-label="fig.1"}](kodamafig1.eps){width="8.5cm"}
Structural analysis on the neutron powder diffraction pattern is performed by using the program Z-Rietveld (ver.0.9.37.4).[@zcode1; @zcode2] The reported space group of $Fd\bar{3}m$ is used. In the analysis, the occupation factors of Li and O atoms are also refined. The obtained structural parameters of LiMn$_2$O$_4$ are shown in Table I.
Atom Site Occ. $x$ $y$ $z$ $B$ ($\mathrm{\AA)}^{2}$)
------ ------- ---------- ----------- -------- -------- ---------------------------
Li 8$a$ 0.984(1) 1/8 1/8 1/8 0.77(1)
Mn 16$d$ 1 1/2 1/2 1/2 0.61(1)
O 32$e$ 0.985(1) 0.2634(1) 0.2634 0.2634 0.99(1)
: Atomic positions of LiMn$_2$O$_4$ determined by Rietveld analysis of neutron powder diffraction data at room temperature. Space group of $Fd\bar{3}m$ (origin choice 2) is used in the analysis. Obtained lattice parameter is $a$=8.24333(2) Å. The $R$-factors, $R_\textrm{wp}$, $R_\textrm{p}$, $R_\textrm{e}$, $R_\textrm{B}$, and $R_\textrm{F}$ are 9.16 $\%$, 7.53 $\%$, 0.62 $\%$, 4.27 $\%$, 11.56 $\%$, respectively. []{data-label="t1"}
The errors of the parameters shown in the table are mathematical standard deviations obtained by the analysis. The diffraction pattern calculated by using refined parameters is shown in Fig. 1 by solid line. The calculated line reproduces the observed pattern. All Mn-O bonds are equivalent and the Mn-O distance is 1.957(1) Å, which is intermediate between what is expected for Mn$^{3+}$-O and Mn$^{4+}$-O distances. The deficiencies of Li and O sites are about 1.5 $\%$. If the analyis is performed for the occupation factors of Li and O fixed at 1.0, the fitting does not almost change. (For, example, $R_\textrm{wp}$ becomes 9.23 $\%$). Then the deficiencies of Li and O sites are almost negligible. Here, we can also neglect the possibility that excess Li atom occupies so-called $B$-site which is occupied by Mn atom because the present sample exhibits a structural phase transition from cubic to orthorhombic phase between 270 K and 240 K [@igawa] whereas the samples of Li$_{1+x}$Mn$_{2-x}$O$_4$ with $0 < x \lesssim 0.15$ retain the cubic symmetry in the whole temperature region.[@yamada; @kamazawa] Although the analysis on the structure model with space group of $Fddd$ has also been carried out, meaningful improvement of fitting was not achieved. These results indicate that the present sample is almost stoichiometric LiMn$_2$O$_4$ and the averaged structure is the cubic with the space group of $Fd\bar{3}m$ at room temperature, as reported preceding studies.
In the inset figure, the foot of the main peak at $d\sim 2.4~\mathrm{\AA}$ is extended. Broad shoulder structures are observed at the foot of the main peak as shown by asterisks. The $d$ values of the broad shoulders at the larger and smaller $d$ sides almost correspond to the values of 2 10 0 and 10 2 0, and 4 8 2 and 8 4 2 reflections in the $Fddd$ orthorhombic structure, respectively. These diffuse scatterings suggest the $Fddd$ orthorhombic lattice distortion with a short range correlation.
Analyses on local structure by using PDF analysis
-------------------------------------------------
Figures 2(a) and 2(b) show the structure function $S(Q)$ and the atomic pair distribution function $G(r)$ of $^7$LiMn$_2$O$_4$ at room temperature. The data are obtained from the neutron scattering intensity which is collected at back-scattering bank. $G(r)$ can be obtained by follwing Fourier transformation. $$G(r)=\frac{2}{\pi} \int Q[S(Q)-1]sin(Qr)dQ.$$ In the present analysis, $S(Q)$ in the range of $1.21 \le Q \le 50$ $\mathrm{\AA}^{-1}$ is transformed into $G(r)$ by using the program installed at NOVA.[@otomo]
![(a) The structure function $S(Q)$ obtained from the neutron diffraction data at room temperature. (b) The atomic pair distribution function (PDF) $G(r)$ obtained from $S(Q)$.[]{data-label="fig.2"}](kodamafig2.eps){width="7.5"}
In Fig. 3(a), the fitting result by using the cubic structure with the space group of $Fd\bar{3}m$ which can reproduce the diffraction pattern in the previous subsection, is shown by solid line. In the analysis, occupation factors of all atoms are fixed at 1.0 because the deficiencies of Li and O atoms are only about 1.5 $\%$. The structural refinements on obtained $G(r)$ are performed by using the program PDFFIT.[@proffen2] The data in the region of $1.4 \le r \le 10~ \mathrm{\AA}$ are used for the fitting.
![(Color online) Atomic pair distribution function (PDF) of $^7$LiMn$_2$O$_4$ obtained at room temperature (open circle). Solid lines are the fittings for cubic (a) and orthorhombic (b) structures with space groups of $Fd\bar{3}m$ and $Fddd$, respectively. Lines at lower position show the differences between observed data and fitting results. Weighted $R$-factors obtained by fitting the data in the region of $1.4 \le r \le 10~ \mathrm{\AA}$ are 15.4 $\%$ and 6.94 $\%$ for cubic and orthorhombic structures, respcetively.[]{data-label="fig.3"}](kodamafig3.eps){width="8"}
The fitting line roughly reproduces the observed data. However, the shapes of the first negative peak at about 1.9 $\mathrm{\AA}$ and positive peaks around 2.8 $\mathrm{\AA}$ can not be reproduced by the line. The first negative peak almost consists of Mn-O atomic correlation in MnO$_6$ octahedron. Although Li-O correlation also contributes to the negative peak, the intensity is about 1/8 of the intensity of Mn-O correlation. The second positive peaks are superposition of Mn-Mn correlation between neighboring MnO$_6$ octahedra and O-O correlations which correspond to the correlation in the octahedron and the correlation between apical O atoms of the neighboring octahedra. The intensity of Mn-Mn correlation peak is about 1/4 of the intensity of the O-O peaks. In the cubic phase, the negative peak should be sharp and symmetric because the all Mn-O bonds are equivalent. However, the negative peak has a shoulder structure at larger $r$ side, indicating the existence of the inequivalent Mn-O bonds. Although, to fit the such asymmetric shape of the negative peak, the calculated negative peak becomes broad by adopting the large thermal factors of $B_\mathrm{Li}$=1.12, $B_\mathrm{Mn}$=0.947 and $B_\mathrm{O}$=1.11 $\mathrm{\AA}^2$ which are obtained by PDF analysis, the observed peak shape can not be reproduced by calculated $G(r)$. The Mn-Mn and O-O correlation peaks should be roughly symmetric three peak structure in the cubic phase because Mn-Mn distance is 2.92 $\mathrm{\AA}$ and O-O distances are 2.63, 2.92 and 3.21 $\mathrm{\AA}$ (the inensity ratio of O-O peaks is about 1:2:1). Because the large thermal factors broaden the calculated positive peaks of the Mn-Mn and O-O correlations, the calculated shape of the superposition of these positive peaks seems to be broad single peak around 2.9 $\mathrm{\AA}$ and can not reproduce the observed complicated structure. The weighted $R$ factor, $R_\textrm{wp}$, obtained by fitting the data in the region of $1.4 \le r \le 10~ \mathrm{\AA}$ is 15.4 $\%$, indicating that the fitting is not satisfactory.
Then we use the orthorhombic structure with space group of $Fddd$ corresponding with the crystal structure in the charge ordered phase because the diffuse scattering which may be due to the $Fddd$ orthorhombic structure with short range correlation is observed in the diffraction pattern, as mention in the previous subsection. The fitting result by using the orthorhombic structure is shown in Fig. 3(b) by solid line. In this structure, five and nine inequivalent Mn and O atoms are contained in the unit cell, respectively. Because the atomic distances of 27 kinds of Mn-O bonds in MnO$_6$ octahedra distribute from about 1.82 to 2.23 $\mathrm{\AA}$, shoulder structure of the first negative peak can be reproduced. At the same time, the calculated Mn-Mn and O-O correlation peaks can also reproduce the complicated peak structure around 2.8 $\mathrm{\AA}$. As a result, the fitting is improved and the $R_\textrm{wp}$ value is 6.94 $\%$ much smaller than the $R_\textrm{wp}$ obtained from the cubic structure. In order to check the possibility of other local lattice distortions, fittings by using structure models with the maximal subgroups of $Fd\bar{3}m$ are performed. Figures 4(a), 4(b) and 4(c) show the fitting results by using the structure models with space groups of $F4\bar{3}m$, $I4_1/amd$ and $R\bar{3}m$, respectively.
![(Color online)Atomic pair distribution function (PDF) of $^7$LiMn$_2$O$_4$ obtained at room temperature (open circle). Solid lines are the fittings for structures with space groups $F4\bar{3}m$ (a), $I4_1/amd$ (b) and $R\bar{3}m$ (c), respcetively. Lines at lower position show the differences between observed data and fitting results. Weighted $R$-factors obtained by fitting the data in the region of $1.4 \le r \le 10~ \mathrm{\AA}$ are 14.2 $\%$, 15.3 $\%$ and 15.2 $\%$ for $F4\bar{3}m$, $I4_1/amd$ and $R\bar{3}m$, respcetively.[]{data-label="fig.3"}](kodamafig4.eps){width="7"}
Here, space groups $F4_132$ and $Fd\bar{3}$ are not used for the fitting because the atomic sites of Li, Mn and O atoms in these space groups correspond with the sites in $Fd\bar{3}m$. The meaningful improvements of fitting are not achieved and $R$-factors do not becomes smaller significantly for the above three structural models despite the lower symmetries. The complicated shapes of the first negative and second positive peaks are not reproduced in these models although the structure models with $F4\bar{3}m$, $I4_1/amd$ and $R\bar{3}m$ contain two, two and three kinds of Mn-O bonds and four, five and seven kinds of O-O bonds, respectively. These results show that the cubic phase of LiMn$_2$O$_4$ has an orthorhombic local lattice distortion which corresponds with the structure in the charge ordered phase. The broad diffuse scattering observed in the diffraction pattern which is mentioned in the previous section, is consistent with the orthorhombic local lattice distortion detected by the present PDF analysis and indicates the short range ordering of this local lattice distortion.
In Table II, the structural parameters of the locally distorted structure of LiMn$_2$O$_4$ which are determined by the PDF analysis using the orthorhombic structure are shown.
Atom Site $x$ $y$ $z$ $B$ (Å$^{2}$)
------- ------- ----------- ----------- ----------- ---------------
Li(1) 8$a$ 1/8 1/8 1/8 0.29(1)
Li(2) 16$f$ 1/8 0.7898(2) 1/8 0.29
Li(3) 16$e$ 0.7950(2) 1/8 1/8 0.29
Li(4) 32$h$ 0.2965(1) 0.3016(1) 0.1162(1) 0.29
Mn(1) 16$d$ 1/4 1/4 1/2 0.33(1)
Mn(2) 32$h$ 0.0833(1) 0.0836(1) 0.5083(1) 0.33
Mn(3) 32$h$ 0.0879(1) 0.3298(1) 0.2512(1) 0.33
Mn(4) 32$h$ 0.2517(1) 0.1674(1) 0.2505(1) 0.33
Mn(5) 32$h$ 0.1644(1) 0.2458(1) 0.2529(2) 0.33
O(1) 32$h$ 0.1744(1) 0.1685(1) 0.2585(2) 0.56(1)
O(2) 32$h$ 0.0782(1) 0.0049(1) 0.4818(1) 0.56
O(3) 32$h$ 0.0785(1) 0.3312(1) 0.4756(1) 0.56
O(4) 32$h$ 0.2526(1) 0.1721(1) 0.4719(1) 0.56
O(5) 32$h$ 0.0035(1) 0.0082(1) 0.2487(2) 0.56
O(6) 32$h$ 0.2530(1) 0.0897(1) 0.2370(1) 0.56
O(7) 32$h$ 0.1627(1) 0.3238(1) 0.2370(2) 0.56
O(8) 32$h$ 0.0908(1) 0.2467(1) 0.2279(1) 0.56
O(9) 32$h$ 0.0843(1) 0.1610(1) 0.5150(1) 0.56
: Atomic positions of LiMn$_2$O$_4$ determined by PDF analysis on neutron powder diffraction data at room temperature. Space group of $Fddd$ (origin choice 2) is used in the analyis. Occupation factors of all atoms are fixed at 1.0. Obtained lattice parameter is $a$=24.6388(3) Å, $a$=24.7978(3) Å, and $c$=8.21781(8) Å. The $R$-factor, $R_\textrm{wp}$, is 6.94 $\%$. []{data-label="t2"}
Here, the thermal factors of each atom are common in order to reduce the refined parameters. In Table III, the atomic distances between Mn and O atoms averaged in MnO$_6$ octahedra determined from the parameters in Table II are shown.
--------- ------------------ ----------------------------- ------------------ -----------------------------
$d~\mathrm{\AA}$ $\Delta$ ($\times 10^{-4}$) $d~\mathrm{\AA}$ $\Delta$ ($\times 10^{-4}$)
Mn(1)-O 1.986(2) 10.6 2.003(2) 20.6
Mn(2)-O 2.000(3) 20.7 1.995(4) 19.4
Mn(3)-O 2.013(2) 45.2 2.021(5) 36.6
Mn(4)-O 1.895(3) 5.9 1.903(4) 4.6
Mn(5)-O 1.920(3) 10.0 1.916(4) 6.1
--------- ------------------ ----------------------------- ------------------ -----------------------------
: Averaged atomic distances between Mn and O atoms in MnO$_6$ octahedra and distortion parameters of MnO$_6$ octahedra, $\Delta$ obtained from the structural parameters in Table II. Distrotion parameter of MnO$_6$ octahedron with averaged atomic distance $d$ is defined as $\Delta = 1/6 \Sigma_{n=1} ^6 [(d_n-d)/d]^2$. Values of $d$ and $\Delta$ determined by Rietveld analysis at 230 K [@RC] are also shown in the right side of the table. []{data-label="t3"}
The errors of the parameters shown in the tables are mathematical standard deviations obtained by the analysis. For comparison, the Mn-O distances in the orthorhombic phase determined by Rietveld analysis[@RC] are also shown in the right side of the table. The Mn(1)-O, Mn(2)-O and Mn(3)-O distances are about 2.00 $\mathrm{\AA}$, whereas the Mn(4)-O and Mn(5)-O distances are about 1.90 $\mathrm{\AA}$. The former and latter values are apparantly larger and smaller than the value of Mn-O distance in the averaged structure, 1.957(1) $\mathrm{\AA}$, respectively. Moreover, these values are very similar to the values obtained by Rietveld analysis on the charge ordered phase.[@RC] From these results, the valences of Mn(1), Mn(2) and Mn(3) ions are about +3 whereas the valences of Mn(4) and Mn(5) ions are about +4. Even in the cubic phase, valence electrons are localized at Mn sites similar to the case of the charge ordered phase with the orthorhombic structure.
Here, we emphasize that as mentioned in previous subsection, the averaged (periodic) structure at room temperature is the cubic structure with single Mn site. Because the arrangement of valence electrons localized at Mn sites has only short range correlation, the lattice distortion caused by the localized electrons is not observed in the averaged structure determined by conventional structural analysis. It can be regarded as a glass state of valence electrons whereas a charge ordered state can be regareded as a crystal state of the electrons. In the cases of amorphous alloys and metallic glasses, although a unique atomic arrangement with a short range ordering, for example, atomic distances and coordination numbers of the nearest neighbor atoms, is observed, the atomic arrangement does not have a long range periodicity.[@inoue] Similar situation may be achieved in the arrangement of the localized electrons in the cubic phase of LiMn$_2$O$_4$. Although the local arrangement of the electrons is unique and is consistent with the charge ordered state, the arrangement does not have a long range periodicity. The long range ordering of the arrangement of the electrons seems to be constricted by the geometrical frustration due to the atomic arrangement of Mn. In this state, the non-metallic electrical conductivity is compatible with the existence of only single Mn site with the valence of +3.5 in the averaged structure. Such glass-like state of valence electrons may be possible in other compounds with mixed valence states and non-metallic electrical conductivities.
In the orthorhombic phase, the distortion parameters, $\Delta$, of the octahedra including Mn(1), Mn(2) and Mn(3) whose valences are +3, are larger than the $\Delta$ of the octahedra including Mn(4) and Mn(5) with the valences of +4, as shown in the right side of Table III. Here, $\Delta$ is defined as $\Delta = 1/6 \Sigma_{n=1} ^6 [(d_n-d)/d]^2$.[@RC] This result indicates that the charge ordering transition at about 260 K is accompanied with an orbital ordering at Mn$^{3+}$ sites due to Yahn-Teller effect. Distortion parameters of MnO$_6$ octahedra at room temperature obtained by the present PDF analysis are also shown in Table III of the left side. The relationship between the Mn valences and distortion parameters seems to be qualitatively consistent with the relationship in the chrage ordered phase; the distortion parameters of MnO$_6$ octahedra including Mn$^{3+}$ ions tend to be larger than those of MnO$_6$ octahedra including Mn$^{4+}$. It suggest the possibility that the short range orbital ordering of Mn$^{3+}$ ion also exist in the cubic phase of LiMn$_2$O$_4$. Such short range orbital ordering is also observed in LaMnO$_3$ at higher temperature than the orbital ordering temperature.[@qiu] However, as shown in the table, the distortion parameter of the octahedron of Mn(1) whose valence is +3, almost corresponds with the parameter of the octahedron of Mn(5) with the valence of +4, indicating that the relationship between the Mn valences and distortion parameters is incomplete at room temperature. The possibility of the short range orbital ordering of Mn$^{3+}$ ion should be revealed by, for example, the measurement just sbove the structural transition temperature.
The structural phase transition from the cubic to orthorhombic structures at around 260 K accompanied with the charge ordering and orbital ordering at Mn$^{3+}$ sites is first order. In first order phase transtion, the short range correlation of the low temperature phase is generally absent above the transition temperature. However, as mentioned above, the short range cluster with the local lattice distortion corresponding with the low temperature phase is also observed in LaMnO$_3$.[@qiu] In LaMnO$_3$, the size of the cluster with the lattice distortion gradually develops with decreasing temperature, and discontinuosly vanishes accompanied with the first order transition. In the present compound, the temperature dependence of the correlation length (periodicity) near the structural phase transition temperature should also be estimated by PDF analysis to confirm the first order transition.
Summary
=======
We have performed the neutron powder diffraction measurement on $^7$LiMn$_2$O$_4$ at room temperature. Although the averaged structure determined by the Rietveld analysis is the cubic spinel, the local structure determined by PDF analysis is the orthorhombic corresponding with the charge ordered phase. The Mn-O distances of the locally distorted orthorhombic structure are almost consistent with the distances of Mn$^{3+}$-O and Mn$^{4+}$-O, indicating that the Mn$^{3+}$ and Mn$^{4+}$ sites are arranged with short range periodicity. In the cubic phase of LiMn$_2$O$_4$, the valence electrons are localized like a glass at Mn sites, resulting in the non-metallic electrical conductivity.
Acknowledgment {#acknowledgment .unnumbered}
==============
The neutron scattering experiment was approved by the Neutron Scattering Program Advisory Committee of IMSS, KEK (Proposal No. 2009S06). This work was supported by a Grant-in-Aid for Scientific Research (C) (24510129) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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{
"pile_set_name": "ArXiv"
}
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author:
- 'S. Sasaki$^{1,*}$'
- 'K. Hashimoto$^{1,*,{\dag}}$'
- 'R. Kobayashi$^1$'
- 'K. Itoh$^1$'
- 'S. Iguchi$^1$'
- 'Y. Nishio$^2$'
- 'Y. Ikemoto$^3$'
- 'T. Moriwaki$^3$'
- 'N. Yoneyama$^4$'
- 'M. Watanabe$^5$'
- 'A. Ueda$^6$'
- 'H. Mori$^6$'
- 'K. Kobayashi$^7$'
- 'R. Kumai$^7$'
- 'Y. Murakami$^7$'
- 'J. M$\ddot{\rm{u}}$ller$^8$'
- 'T. Sasaki$^1$'
title: 'Crystallization and vitrification of electrons in a glass-forming charge liquid'
---
[ **Charge ordering (CO) is a phenomenon in which electrons in solids crystallize into a periodic pattern of charge-rich and charge-poor sites owing to strong electron correlations. This usually results in long-range order. In geometrically frustrated systems, however, a glassy electronic state without long-range CO has been observed. We found that a charge-ordered organic material with an isosceles triangular lattice shows charge dynamics associated with crystallization and vitrification of electrons, which can be understood in the context of an energy landscape arising from the degeneracy of various CO patterns. The dynamics suggest that the same nucleation and growth processes that characterize conventional glass-forming liquids guide the crystallization of electrons. These similarities may provide insight into our understanding of the liquid-glass transition.** ]{}
{width="0.8\linewidth"}
The physics of glassy materials represents a fascinating problem in solid-state theory ([*[1]{}*]{}). Although progress has been made over the past several decades toward clarifying the dynamical aspects of the glass transition, the processes by which liquids acquire the glassy state upon cooling are not fully understood ([*[2]{}*]{}). The most fundamental nonequilibrium dynamic phenomena associated with the liquid-glass transition process are crystallization and vitrification. These phenomena are competing and mutually exclusive but are closely related to each other ([*[3, 4]{}*]{}). In glass-forming liquids, crystallization below the melting point $T_m$ can be avoided when the system is cooled quickly enough, leading to a supercooled liquid state accompanied by an increase in viscosity (Fig.1A) ([*[2, 5]{}*]{}). Upon further cooling, owing to the viscous retardation of crystallization, the supercooled liquid state freezes into a glassy state; that is, vitrification occurs at the glass transition temperature $T_g$. Thus, the relationship between crystallization and vitrification is key to the understanding of the liquid–glass transition. Here, we demonstrate that this general picture can be extended to crystallization and vitrification of strongly correlated electrons realized in a geometrically frustrated charge-ordered organic system, $\theta_m$-(BEDT-TTF)$_2$TlZn(SCN)$_4$ ([*[6]{}*]{}), where BEDT-TTF denotes bis(ethylenedithio)tetrathiafulvalene. Surprising similarities between our system and conventional glass formers are observed in the crystallization and vitrification processes, which highlight the universal nature of the liquid-glass transition.
The quasi-two-dimensional (quasi-2D) organic materials $\theta$-(BEDT-TTF)$_2$$X$ consist of alternating stack of conducting BEDT-TTF and insulating anion $X$ layers; the BEDT-TTF molecules form a triangular lattice ([*[6–8]{}*]{}). The charge transfer between these two layers leads to a 2D quarter-filled hole band system (that is, one hole per two BEDT-TTF molecules), in which the intersite Coulomb repulsions give rise to an instability towards charge ordering (CO) ([*[9]{}*]{}). Indeed, $\theta$-(BEDT-TTF)$_2$RbZn(SCN)$_4$ (henceforth $\theta$-RbZn), for example, undergoes a CO transition at 190 K ([*[6, 7, 10, 11]{}*]{}), where the charge carries are localized periodically with a horizontal stripe pattern (see the phase diagram in fig.S1C). Such a periodic CO state can be regarded as a “charge-crystal" state ([*[11]{}*]{}). In contrast, above the CO transition temperature, the charge of +0.5 per one BEDT-TTF molecule is distributed uniformly in space; therefore, such a delocalized state can be referred to as a “charge-liquid" state.
In $\theta$-RbZn, when the sample is cooled faster than a critical cooling rate ($\sim$5 K/min), charge crystallization is kinetically avoided, leading to a “charge-glass" state where the charge is randomly quenched. For comparison, in $\theta$-(BEDT-TTF)$_2$CsZn(SCN)$_4$ ($\theta$-CsZn), which has a more isotropic triangular lattice than $\theta$-RbZn, the critical cooling rate becomes much slower (fig.S1, C and D). As a result, the charge-liquid state inevitably results in a charge-glass state even upon very slow cooling ($<$ 0.1 K/min) ([*[12]{}*]{}). However, the mechanism of formation of the glassy electronic state—which has been discussed experimentally ([*[11–14]{}*]{}) and theoretically ([*[15]{}*]{}) in terms of the geometrically frustrated triangular lattice—still remains rather elusive.
The system $\theta$-(BEDT-TTF)$_2$TlZn(SCN)$_4$, which exists in two different crystal forms with orthorhombic and monoclinic symmetries (fig.S1, A and B) ([*[6, 16]{}*]{}), may play a key role in the understanding of the charge-glass state. Orthorhombic $\theta_o$-(BEDT-TTF)$_2$TlZn(SCN)$_4$ ($\theta_o$-TlZn), which has the same structural symmetry as $\theta$-RbZn and $\theta$-CsZn, exhibits a CO transition with a horizontal charge modulation at 240 K ([*[16]{}*]{}). Because the anisotropy of the triangular lattice is large, the critical cooling rate for glass formation remains quite high ([*[13, 14]{}*]{}). In contrast, the monoclinic $\theta_m$-(BEDT-TTF)$_2$TlZn(SCN)$_4$ ($\theta_m$-TlZn) shows a CO transition with a diagonal charge modulation at $T_m = 170$ K (Fig.1, B to D) ([*[16]{}*]{}). The different CO patterns of the two systems can be related to a difference in the strength of electron-lattice coupling, as pointed out by theoretical studies ([*[17]{}*]{}). Because the lattice distortion of $\theta_m$-TlZn at the CO transition is much smaller than that of $\theta_o$-TlZn, and because most of the entropy change of $\theta_m$-TlZn is of electronic origin (fig.S2), $\theta_m$-TlZn more likely approximates a system where the observed effects are purely electronic in nature. This is consistent with the extended Hubbard model (EHM) in the absence of electron-phonon coupling, in which the diagonal CO pattern rather than the horizontal one is expected ([*[17]{}*]{}).
The temperature-dependent resistivity $\rho(T)$ of $\theta_m$-TlZn shows a strong dependence on sweeping rate below $T_m$ (Fig.1B), although the triangular lattice for $\theta_m$-TlZn is more anisotropic than that for $\theta_o$-TlZn. In $\theta_m$-TlZn, the long-range CO transition can be avoided by rapid cooling ($\geq$ 50 K/min), and charge vitrification occurs through a supercooled charge-liquid state (Fig.1, B and E). In the cooling/heating cycle of the $\rho$-$T$ profile, a clear hysteresis loop associated with the glass transition is observed at 145 to 165 K (Fig.1B, inset), quite similar to what is observed in $\theta$-CsZn ([*[12]{}*]{}). We tentatively define the glass transition temperature $T_g$ as the temperature above which the resistivity starts to branch off.
{width="0.8\linewidth"}
To clarify the origin of charge-glass formation in $\theta_m$-TlZn, we performed resistance noise measurements, which are a powerful probe to detect the slow dynamics associated with electronic glassiness ([*[11, 12, 18]{}*]{}). Figure2A shows a typical normalized noise power spectral density of the resistance fluctuations $S_R/R^2$. The baseline of $S_R/R^2$ fits well to generic $1/f$ with a slightly deviating frequency exponent, yielding $1/f^{\alpha}$ with $\alpha \sim $ 0.8 to 0.9. For clarity, $f^{\alpha} \times S_R/R^2$ is plotted in Fig.2B, which clearly shows that the resistance fluctuations exhibit a broad peak structure. The data are well fitted to the distributed Lorentzian model ([*[11, 12]{}*]{}) with a characteristic center frequency $f_c$ ($=\sqrt{f_{c1} f_{c2}}$, where $f_{c1}$ and $f_{c2}$ are the high- and low-cutoff frequencies, respectively), from which we derived the temperature evolution of the relaxation time $\tau_c = 1/(2\pi f_c)$. The peak structure in $f^{\alpha} \times S_R/R^2$ becomes broader and more asymmetric with decreasing temperature (Fig.2, B and C), showing that the dynamics become more heterogeneous. In addition, $\tau_c$ slows drastically over several orders of magnitude (Fig.2D); extrapolating our data under the assumption that $\tau_c$ obeys an Arrhenius law \[as observed for $\theta$-CsZn ([*[12]{}*]{}) and as expected from recent Monte Carlo simulations ([*[15]{}*]{})\], we find that $\tau_c$ may be as high as $10^2$ s around $T_g$. These results indicate the emergence of slow dynamics accompanied by increasing dynamic heterogeneity upon approaching the charge glass transition. We note that the charge vitrification in the present case is distinctly different from the drastic slowing down of charge carrier dynamics and onset of non-Gaussian fluctuations observed in noise measurements as a precursor of metal-insulator transitions (MITs) ([*[18]{}*]{}). Electronic glassiness in MIT systems seemingly only becomes stabilized by disorder in the presence of strong electronic correlations ([*[18, 19]{}*]{}) and is not observed for clean samples.
Clarifying the relationship between the heterogeneous slow dynamics and local charge configurations may be key to the understanding of charge-glass formation ([*[3, 20]{}*]{}). To this end, we investigated the imbalance of charge distribution—that is, charge disproportionation—at a microscopic level by means of a charge-sensitive vibrational mode $\nu_{27}$ of the BEDT-TTF molecule. The $\nu_{27}$ mode is known as a local probe of the molecular charge $\rho_c$ ([*[21]{}*]{}) and splits into two modes, $\nu_{\rm{27I}}$ and $\nu_{\rm{27N}}$, in the presence of charge disproportionation between the A and B sites in the unit cell (Figs.1C and 2E), where the subscripts $\rm{I}$ and $\rm{N}$ denote the hole-rich and hole-poor sites, respectively. Figure2F shows the temperature dependence of the polarized optical conductivity spectra. A clear peak around 1420 cm$^{-1}$ is assigned to the $\nu_{\rm{27I}}$ mode ($\rho_c \sim 0.85$) ([*[16]{}*]{}). There is a slight difference in the peak frequency of 1.5 cm$^{-1}$ between the slow and rapid cooling processes, which corresponds to 0.01 in charge distribution on the BEDT-TTF molecule.
A sizable intensity of $\nu_{\rm{27I}}$ was observed above $T_m$ (Fig.2G), indicating the presence of charge disproportionation above $T_m$. Because the A and B sites are crystallographically equivalent above $T_m$ owing to the screw axis along the $b$ axis ([*[16]{}*]{}), the time-averaged charge distribution above $T_m$ is +0.5 per one BEDT-TTF molecule (Fig.1C). Therefore, the splitting of $\nu_{27}$ above $T_m$ implies that charge disproportionation above $T_m$ is not static but dynamically fluctuates on a time scale slower than that of the molecular vibrational $\nu_{27}$ motion. Because the intensity of $\nu_{\rm{27I}}$ is considered to reflect the volume of dynamically fluctuating charge clusters switching between the locally ordered and charge-liquid states, its increase with decreasing temperature suggests that the heterogeneous slow dynamics observed in the noise measurements are caused by the dynamically fluctuating charge clusters.
![[**Semimacroscopic degeneracy of striped CO patterns and energy landscape.**]{} ([**A**]{}) Schematics of various striped CO patterns. The magenta circles represent the charge-rich sites. $V_1$ and $V_2$ ($V_1>V_2$) are the nearest-neighbor Coulomb interactions. Because all these states are degenerate in the classical limit of the $t$-$V$ model, the classical ground state can be described by the superposition of these states, which has a degeneracy of $2^{L_c}$, where $L_c$ is the system length in the $c$ direction. ([**B**]{}) Illustration of an energy landscape with multiple local minima separated by barriers having an energy scale of the hopping integral $t_{\rm{hopping}}$ and/or the long-range Coulomb interaction $V$, which are on the order of $\sim$ 1 eV. ](Fig3.eps){width="0.95\linewidth"}
{width="0.9\linewidth"}
To obtain insights into the structural origin of the heterogeneous slow dynamics, we performed x-ray diffuse scattering measurements. Oscillation photographs measured at various temperatures are shown in Fig.2, H to K. At room temperature, only Bragg reflections are observed (Fig.2K), whereas in the charge-crystal state, clear satellite peaks appear at $\bm q_0 = (1/2\,1/2)$, compatible with the diagonal CO pattern (Fig.2H), which is attributed to the periodicity of charge-rich and charge-poor sites accompanied by a periodical change of the C=C double bond length of the BEDT-TTF molecules. Theoretical calculations for the $\theta$-type materials based on the EHM ([*[22]{}*]{}) have suggested that the diagonal CO pattern is most stable when $V_1 > V_2$ (where $V_1$ and $V_2$ are the nearest-neighbor Coulomb interactions), which is consistent with the observations in the charge-crystal state of $\theta_m$-TlZn. In contrast, in the charge-glass state, diffuse lines at $\bm q_d = (1/2\, l)$ are observed (Fig.2I). The diffuse lines can be ascribed to geometric frustration.
The classical ground states of the $t$-$V$ model, which is the spinless version of the EHM, on an isosceles triangular lattice are known to be disordered owing to geometric frustration when $V_1 \geq V_2$ ([*[23–25]{}*]{}). For $V_1 = V_2$, the ground state is “macroscopic" disordered with a degeneracy of $\sim 2^{N-1}$ (where $N$ is the number of lattice sites). On the other hand, for $V_1>V_2$, $V_1$ preferentially determines the two-fold periodic striped CO pattern along the $b$ direction, but the geometric frustration along the diagonal directions arising from the isosceles triangular lattice gives rise to a “semimacroscopic" degeneracy of $2^{L_c}$, where $L_c$ is the system length in the $c$ direction (Fig.3A). However, introducing a small quantum hopping term or a long-range Coulomb potential lifts the degeneracy, which drives the system to the diagonal CO pattern ([*[15, 22]{}*]{}), although the degeneracy is presented in a wide temperature range above the ordering temperature ([*[15]{}*]{}). This situation may induce an energy landscape with multiple local minima, as illustrated in Fig.3B—that is, a metastable state with an amorphous stripe-glass structure as proposed in ([*[15]{}*]{})—which in turn causes the heterogeneous slow dynamics in $\theta_m$-TlZn. Indeed, frustration is a key concept for understanding glass transitions in a variety of systems ([*[3]{}*]{}). For example, crystallization in metallic glasses is prevented if locally favored structures such as icosahedral order do not match the symmetry of the system ([*[3, 26]{}*]{}). Likewise, in $\theta_m$-TlZn, locally favored short-range electronic ordering with $\bm q_d = (1/2\, l)$ induced by geometric frustration may hinder long-range CO with $\bm q_0 = (1/2\,1/2)$, thereby causing the slow dynamics. Our results provide an experimental demonstration of recent theoretical considerations that frustration, in combination with strong quantum effects, plays an important role in the realization of quantum charge-glass states in clean systems, essentially free from impurities and defects ([*[27–30]{}*]{}).
We next examine the charge crystallization process in detail to clarify the relationship between crystallization and vitrification of electrons in $\theta_m$-TlZn. Figure4A displays the time evolution of the resistivity during the charge crystallization process from the supercooled charge-liquid or charge-glass state ([*[25]{}*]{}). The magnitude of the resistivity, which is a measure of the crystallization progress, increases with time and then saturates. The relaxation time becomes faster with decreasing temperature, and then slower below 157 K, which is referred to as the “nose temperature"; this characteristic temperature dependence of the relaxation time can be explained by the theory of nucleation and growth at a first-order liquid-crystal phase transition ([*[25]{}*]{}).
To quantitatively evaluate the CO volume fraction from the resistivity, we used the effective medium percolation theory ([*[31]{}*]{}). This theory describes a percolating current passing through an inhomogeneous mixture of conducting and insulating media ([*[25]{}*]{}). Through a generalized effective medium equation ([*[32]{}*]{}), we derived the time evolution of the CO volume fraction, $\phi(t)$, from the time-dependent resistivity data at various temperatures (Fig.4, B and C). In the high-temperature region, $\phi(t)$ can be fitted over the whole time range by the Johnson-Mehl-Avrami-Kolmogorov (JMAK) formula describing a conventional nucleation and growth process, where $\phi(t) = 1-\exp{(-kt^n)}$ (here, $k$ and $n$ are the JMAK parameters) ([*[25, 33]{}*]{}) (Fig.4E). By contrast, near the nose temperature, $\phi(t)$ can be fitted to the JMAK formula only in the early stage of crystal growth (Fig.4E). For later times, a crossover to the Ostwald ripening that describes a rearrangement of crystal grain boundaries, where $\phi(t) = 1-(1+k^{'}t)^{-1/3}$ (here, $k^{'}$ is a constant) ([*[34]{}*]{}), explains the observed more moderate time evolution of $\phi(t)$. Such a process is often observed in the final stage of crystal growth ([*[25]{}*]{}). Interestingly, below 145 K, $\phi(t)$ again exhibits a steep increase over the whole time range, which can be fitted to the JMAK formula (Fig.4F); the temperature of 145 K is close to $T_g$. Crystallization below $T_g$ is studied in many fields of materials science, and an enhancement of the crystallization rate at $T_g$ has been reported ([*[35, 36]{}*]{}). The origin has been discussed, for example, in terms of a crystal–glass interface ([*[35, 36]{}*]{}). In this scenario, the volume contraction upon crystallization below $T_g$ provides free volume for atoms or molecules surrounding the crystal, which leads to a mobility increase at the crystal–glass interface, resulting in an enhancement of the crystal growth rate at $T_g$. We speculate that the same surface dynamics occurs at the charge crystal–glass interface in $\theta_m$-TlZn.
Figure4G displays the contour map of the CO fraction plotted in the time-temperature plane \[a so-called time-temperature-transformation (TTT) diagram\]. The obtained TTT diagram clearly reflects the two characteristic features discussed above: the nose structure around 160 K, and the enhancement of crystal growth close to $T_g$. These observations suggest that the crystallization process of electrons in solids can be described by the nucleation and growth process of a liquid—as observed in conventional glass-forming liquids such as structural and metallic glasses ([*[35–38]{}*]{})—and that charge crystallization and vitrification are closely related.
Our study reveals that when electrons in a strongly correlated system are put on a geometrically frustrated lattice, their dynamics are similar to those known from structural relaxation in conventional glass-forming liquids, although the additional role of the lattice degrees of freedom for charge-glass formation on a geometrically frustrated system requires further investigation. The convenient time and temperature scales of the present material and the possibility of inferring volume fractions from easily accessible charge transport will enable investigations of aging, memory effects, cooperativity, and the presence or absence of an underlying true phase transition from a different perspective.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank K. Yoshimi, T. Kato, M. Naka, H. Shiba, C. Hotta, and K. Yamamoto for fruitful discussions, and J. Kudo and M. Kurosu for technical assistance. Synchrotron radiation measurements were performed at SPring-8 with the approvals of the Japan Synchrotron Radiation Research Institute (2014B1340, 2014B1752, 2015A1777, 2015B1752, 2015B1756, 2016A0073, and 2016B0073). X-ray diffraction study was performed under the approval of the Photon Factory Program Advisory Committee (proposal 2014S2-001). Supported by the Deutsche Forschungsgemeinschaft within the Transregional Collaborative Research Center SFB/TR49 (J.M.); Grants-in-Aid for Scientific Research (grants 24340074, 25287080, 26610096, 26102014, 15H00984, 15H00988, 15K13511, 15K17688, 16H04010, 16K05430, 16K05744, 17H05138, and 17H05143) from MEXT and JSPS; a Grant-in-Aid for Scientific Research on Innovative Areas “p-Figuration" (grant 26102001); and the Canon Foundation.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by Thomassen [@thomassen] and a linear-time fixed-parameter algorithm has been obtained by Mohar [@mohar2]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of an $O(1)$-approximation is not ruled out, the currently best-known upper bound is a trivial $O(n/g)$-approximation that follows from bounds on the Euler characteristic.
In this paper, we give the first non-trivial approximation algorithm for this problem. Specifically, we present a polynomial-time algorithm which given a graph $G$ of Euler genus $g$ outputs an embedding of $G$ into a surface of Euler genus $g^{O(1)}$. Combined with the above $O(n/g)$-approximation, our result also implies a $O(n^{1-\alpha})$-approximation, for some universal constant $\alpha>0$. Our approximation algorithm also has implications for the design of algorithms on graphs of small genus. Several of these algorithms require that an embedding of the graph into a surface of small genus is given as part of the input. Our result implies that many of these algorithms can be implemented even when the embedding of the input graph is unknown.
author:
- 'Ken-ichi Kawarabayashi[^1]'
- 'Anastasios Sidiropoulos[^2]'
bibliography:
- 'bibfile.bib'
title: |
Beyond the Euler characteristic:\
Approximating the genus of general graphs
---
Introduction
============
A *drawing* of a graph $G$ into a surface ${\cal S}$ is a mapping ${\varphi}$ that sends every vertex $v\in V(G)$ into a point ${\varphi}(v)\in {\cal
S}$ and every edge into a simple curve connecting its endpoints, so that the images of different edges are allowed to intersect only at their endpoints. The [[*Euler genus*]{}]{} of a surface ${\cal S}$, denoted by ${\mathsf{eg}}({\cal S})$, is defined to be $2-\chi({\cal S})$, where $\chi({\cal S})$ is the Euler characteristic of ${\cal S}$. This parameter coincides with the usual notion of genus, except that it is twice as large if the surface is orientable. For a graph $G$, the Euler genus of $G$, denoted by ${\mathsf{eg}}(G)$, is defined to be the minimum Euler genus of a surface ${\cal S}$, such that $G$ can be embedded into ${\cal S}$. In this paper we consider the following basic problem: Given a graph $G$, approximate ${\mathsf{eg}}(G)$. This is a fundamental problem in graph theory and its exact version is one of the basic problems listed by Garey and Johnson [@garey2002computers]. Part of the original motivation for the study of the genus of graphs goes back to the Heawood problem which concerns the maximum chromatic number of graphs embeddable in a fixed surface. The solution of the Heawood problem turned out to be equivalent to determining the genus of complete graphs (cf. [@ringel]). The practical interest for planar embeddings, and more generally, embeddings into low-genus surfaces arises, for instance, in problems concerning VLSI. Moreover, “nearly planar” networks can be used to model a plethora of natural objects and phenomena. Algorithmic interest comes from the fact that graphs of bounded genus naturally generalize the family of planar graphs and share many important properties with them. Moreover, graphs of small genus play a central role in the seminal work of Robertson and Seymour on graph minors and the proof of Wagner’s conjecture.
Apart from bounds on the genus of specific families of graphs, there are no general results available. This can be explained by the result of Thomassen [@thomassen] who showed that computing the genus of a given graph exactly is NP-hard. Nevertheless, closely related problems have been extensively studied by many researchers. For example, a seminal result of Hopcroft and Tarjan [@lin1] gives a linear time algorithm for testing planarity of graphs, and for computing a planar embedding if one exists. Extending this planarity result, many researchers have focused on the case when the Euler genus $g$ is a fixed constant. Filotti, Miller, and Reif [@FMR] were the first to give a polynomial time algorithm for this problem. In their solution, the degree of the polynomial bound on the time complexity depends on $g$. Djidjev and Reif [@dr] improved the result of [@FMR] by presenting a polynomial time algorithm for each fixed orientable surface, where the degree of the polynomial is fixed. In addition, linear time algorithms have been devised for embedding graphs into the projective plane [@projective] and the torus [@torus]. Mohar [@mohar1; @mohar2] finally gave a linear time algorithm for embedding a graph into an arbitrary fixed surface. This is one of the deepest results in this area, generalizing linear time algorithms for planarity [@lin2; @lin4; @lin1; @lin3]. A relatively simple linear-time algorithm was given by Kawarabayashi, Mohar, and Reed [@DBLP:conf/focs/KawarabayashiMR08]. From the work of Robertson and Seymour [@RobertsonS90b], the family of graphs of genus at most $g$ is characterized as the class of graphs that exclude as a minor all graphs from a finite family. However, this family of excluded minors is not known explicitly even for small values of $g$ and can generally be very large (it contains two graphs for $g=0$ [@kuratowski1930probleme; @wagner1937eigenschaft], and 35 graphs for $g=1$ [@DBLP:journals/jgt/Archdeacon81; @DBLP:journals/jct/GloverHW79]). The dependence of the running time of all of the above mentioned exact algorithms is at least exponential in $g$.
#### Our results.
We consider the problem of approximating ${\mathsf{eg}}(G)$, when ${\mathsf{eg}}(G)$ is not fixed. Perhaps surprisingly, despite its central importance, essentially nothing is known for this problem on general graphs. Let us first briefly describe what is currently known. Euler’s characteristic implies that any $n$-vertex graph of Euler genus $g$ has at most $O(n + g)$ edges. Since any graph can be drawn into a surface that has one handle for every edge, this immediately implies a O$(n/g)$-approximation, which is a $\Theta(n)$-approximation in the worst case. In other words, even though we currently cannot exclude the existence of an $O(1)$-approximation, the state of the art only gives a trivial $O(n)$-approximation.
We give the first non-trivial approximation algorithm for ${\mathsf{eg}}(G)$ on general graphs. Our result can be summarized as follows.
\[thm:main1\] There exists a polynomial-time algorithm which given a graph $G$ and an integer $g$, either correctly decides that ${\mathsf{eg}}(G)>g$, or outputs an embedding of $G$ into a surface of Euler genus $O(g^{256} \log^{189}n)$.
Combined with the above trivial $O(n)$-approximation, our result implies the first non-trivial approximation algorithm for approximating the Euler genus of general graphs.
There is a polynomial-time $O(n^{1-\alpha})$-approximation algorithm for Euler genus, for some universal constant $\alpha>0$.
Kawarabayashi, Mohar and Reed [@DBLP:conf/focs/KawarabayashiMR08] have obtained an exact algorithm for computing ${\mathsf{eg}}(G)$ with running time $2^{O({\mathsf{eg}}(G))} n$. This implies a polynomial-time algorithm when ${\mathsf{eg}}(G) = O(\log n)$. Combining this result with Theorem \[thm:main1\], we immediately obtain the following Corollary.
\[cor:main1\] There exists a polynomial-time algorithm which given a graph $G$ and an integer $g$, either correctly decides that ${\mathsf{eg}}(G)>g$, or outputs an embedding of $G$ into a surface of Euler genus $g^{O(1)}$.
#### Previous work on approximating the genus of graphs.
For special classes of graphs, some $o(n)$-approximation guarantees are known. Chekuri and Sidiropoulos [@DBLP:conf/focs/ChekuriS13] have recently obtained a polynomial-time algorithm which given a graph $G$ of maximum degree $\Delta$ computes an embedding of $G$ into a surface of Euler genus at most $\Delta^{O(1)} ({\mathsf{eg}}(G))^{O(1)} \log^{O(1)} n$. We remark that there are graphs of maximum degree three that have Euler genus $\Omega(n)$. Therefore, this result does not imply anything better that a $\Theta(n)$-approximation for the Euler genus of general graphs. Mohar has obtained a $O(1)$-approximation for graphs $G$ that contain a vertex $a$ such that $G-a$ is planar and 3-connected (note that this is a special class of 1-apex graphs). Finally, Makarychev, Nayyeri, and Sidiropoulos [@MNS12] obtained an algorithm that given a Hamiltonian graph $G$ along with a Hamiltonian path $P$, computes an embedding of $G$ into a surface of Euler genus $g^{O(1)} \log^{O(1)} n$ where $g$ is the orientable genus of $G$. We remark that the Hamiltonicity assumption is a major restriction of this algorithm. On the lower-bound side, Mohar [@Mo-apex] showed that computing ${\mathsf{eg}}(G)$ remains NP-hard even when the input is a 1-apex graph. We emphasize that essentially no inapproximability result is known for ${\mathsf{eg}}(G)$, even on graphs of bounded degree.
#### Further algorithmic implications.
Our result has a general consequence for the design of algorithms on graphs of small genus. Most of the known algorithms for problems on such graphs require that an embedding of the graph is given as part of the input. Our result implies that many of these algorithms can be implemented even when the embedding is unknown. Prior to our work, such a general reduction was known only for the case of graphs of bounded degree (see [@DBLP:conf/focs/ChekuriS13]). As an illustrative example, consider the Asymmetric TSP. For this problem, Erickson and Sidiropoulos [@DBLP:conf/compgeom/EricksonS14] recently gave a $O(\log g/\log\log g)$-approximation algorithm for graphs *embedded* into a surface of Euler genus $g$. Our result implies a polynomial-time algorithm with the same asymptotic approximation guarantee for the case of graphs that are *embeddable* into a surface of Euler genus $g$ (that is, without having an embedding as part of the input). We refer the reader to [@DBLP:conf/focs/ChekuriS13] for a more detailed discussion of such implications.
Overview of the algorithm
-------------------------
We now give a high-level description of our approach. Our algorithm builds on the recent approximation algorithm for the bounded degree case, due to Chekuri and Sidiropoulos [@DBLP:conf/focs/ChekuriS13].
#### Tools from the bounded-degree case.
The approach from [@DBLP:conf/focs/ChekuriS13] is based on ideas from the fixed-parameter case and the theory of graph minors [@DBLP:conf/focs/KawarabayashiMR08; @RS5; @DBLP:journals/jct/RobertsonST94]. However, we remark that the implementation of certain steps from the exact case is quite challenging due to the fact that the parameters are not fixed in the approximate setting.
The algorithm of [@DBLP:conf/focs/ChekuriS13] proceeds as follows. First, while the input graph has sufficiently large treewidth, it finds a subgraph that can be removed without significantly affecting the solution. This is done by computing a *flat* grid minor. Here, a planar subgraph $\Gamma$ of some graph $G$ is called *flat* (w.r.t. $G$) if there exists a planar drawing ${\varphi}$ of $\Gamma$, such that for all edges $\{u,v\} \in E(\Gamma)$, with $u\in V(\Gamma)$, and $v\in V(G) \setminus V(\Gamma)$, $u$ is on the outer face of ${\varphi}$. Eventually, they arrive at a graph of small treewidth. For such a graph they can compute a small set of vertices $X$ whose removal leaves a planar graph. Since in their case the degree is bounded, they can add $X$ back to the planar graph by introducing at most a constant number of handles for every vertex in $X$. In summary, the approach of Chekuri and Sidiropoulos [@DBLP:conf/focs/ChekuriS13] reduces the problem of computing the genus of a graph to the following two sub-problems:
[**Sub-problem 1: Computing flat grid minors.**]{} Suppose that we are given a graph $G$ of genus $g$ and large treewidth, say $t>g^{c}$, for some sufficiently large constant $c$. We wish to find a *flat* subgraph that contains a $(c'g \times c'g)$-grid minor, for some sufficiently large constant $c'$.
[**Sub-problem 2: Embedding $k$-apex graphs.**]{} Given a graph $G$ and some $X\subseteq V(G)$, such that $H=G\setminus X$ is planar, we wish to compute an embedding of $G$ into a surface of genus $g^{O(1)} \cdot |X|^{O(1)}$. In general, there might be edges between the vertices in $X$. We may remove all such edges, and add them to the final embedding by increasing the resulting genus by at most an additive factor $O(|X|^2)$, which does not affect our asymptotic bounds. We may therefore assume in the rest of the this high-level overview that $X$ is an independent set.
Chekuri and Sidiropoulos [@DBLP:conf/focs/ChekuriS13] obtain algorithms for both of these sub-problems. Indeed, the second problem is trivial for them. Unfortunately, since we are dealing with graphs of unbounded degree, their algorithms are not applicable in our case. We next describe our algorithms for these sub-problems on general graphs.
#### Computing flat grid minors.
Our algorithm for Sub-problem 1 follows an approach similar to the one used for the bounded-degree case in [@DBLP:conf/focs/ChekuriS13]. We start by removing a small number of vertices that make the graph planar. Since the original graph has large treewidth, it must also have a large grid minor. The removal of a small number of vertices can only destroy a small part of this grid minor. The main difficulty is to prove that some part of this remaining grid minor must be flat in the original graph. We establish this property by arguing that if no such flat grid minor exists, then the graph must contain a $K_{3,b\cdot {\mathsf{eg}}(G)}$ minor, for some sufficiently large constant $b$, which contradicts the fact that the Euler genus of $G$ is ${\mathsf{eg}}(G)$.
#### Embedding $k$-apex graphs.
We now discuss our algorithm for Sub-problem 2. Recall that a graph $G$ is called $k$-apex if there exists some $X\subseteq V(G)$, with $|X|\leq k$, such that $H=G\setminus X$ is planar. Our algorithm for approximating the Euler genus of $k$-apex graphs is the main technical contribution of this work. Indeed, prior to our work, a similar algorithm was only known for special cases of $1$-apex graphs, and even the case of 2-apex graphs was completely open. The problem is that each apex in $X$ may have many (e.g. $\Omega(n)$) neighbors in $G\setminus X$. This makes a major difference between our proof and the bounded-degree case in [@DBLP:conf/focs/ChekuriS13]. This is because in the latter case, there is only a small number of edges between $X$ and $G-X$, so it is possible to add a handle for each edge. On the other hand, in our case, we cannot do this simply because we may have to add linearly many handles for each vertex in $X$. Most of the technical effort in this paper goes into bounding the number of handles added in this step. Our algorithm for $k$-apex graphs proceeds in several steps. At each step we simplify the graph via a sequence of operations. Roughly speaking, every simplification operation either reduces the number of apices, or it simplifies the structure of the planar piece $H$. Let us now describe the key ingredients of our approach in more detail.
**1. Simplification via vertex splitting.** We introduce an operation called *vertex splitting*. This allows us to “split” a vertex of the planar piece into two vertices, as depicted in Figure \[fig:splitting\_embedding\]. The benefit of this operation is that given an embedding of the new graph, we can efficiently compute an embedding of the original graph, without significantly increasing the Euler genus of the underlying surface (see Figure \[fig:splitting\_embedding\]).
**2. Reduction to the 2-apex case.** A key step in our algorithm is to reduce the problem of embedding $k$-apex graphs to the problem of embedding $2$-apex graphs. This is done in several steps, by performing appropriate sequences of splitting operations. We first compute a sequence of splitting operations such that in every resulting planar piece, every connected component is either incident to at most two apices, or every 1-separator is incident to at most one apex. In the former case, we have obtained a 2-apex instance, which we show how to handle below. In the latter case, we compute another sequence of splitting operations such that in the resulting graph, every component is either “nearly locally 2-apex” or has a 2-connected planar piece. We shall deal with each one of these cases separately.
**3. Embedding $2$-apex graphs and their generalizations.** Our algorithm for 2-apex graphs starts by decomposing the input graph into simpler pieces using a sequence of splitting operations. In the resulting graph every piece is either 2-connected planar, or it has a 2-connected planar piece. Therefore, the case of 2-apex graphs is reduced to the case of $2$-apex graphs with a 2-connected planar piece, which we addressed below. In reality, our algorithm has to embed graphs that can be more complicated than 2-apex graphs. More specifically, we need to design an algorithm for embedding graphs that have at most one maximal 2-connected component that is $k$-apex, and a small number of remaining components (not necessarily 2-connected) that are 2-apex. We call these graphs *nearly locally 2-apex*. Our algorithm for embedding these graphs uses similar ideas to the 2-apex case, but needs to perform a significantly more complicated decomposition step.
**4. Embedding $k$-apex graphs with a 2-connected planar piece.** Next, we obtain an algorithm for embedding $k$-apex graphs where the planar piece $H$ is 2-connected. This is done by splitting any 2-connected $k$-apex graph of small genus into a small number of simpler structures, that we call *centipedes* and *butterflies*. This is done by splitting any 2-connected $k$-apex graph of small genus into a small number of simpler structures, that we call *centipedes* and *butterflies* (see Figure \[fig:butterfly\]). These are special subgraphs with at most four apices. Ultimately, the problem of embedding these graphs can be reduced to the problem of embedding 1-apex graphs, which we address below.
**6. Embedding 1-apex graphs.** Finally, after the above sequence of reductions, we arrive at a small number of instances that are 1-apex. Unfortunately, even the case of approximating the Euler genus of 1-apex graphs was open prior to our work. In fact, the only previous result was a $O(1)$-approximation for the orientable genus of 1-apex graphs where the planar piece is 3-connected, due to Mohar [@DBLP:journals/jct/Mohar01]. We remark that the 3-connectedness assumption simplifies the problem significantly. We overcome this limitation by generalizing Mohar’s argument using the theory of SPQR decompositions (for the definition of an SPQR decomposition and further exposition we refer the reader to [@DBLP:conf/focs/BattistaT89]). Finally, since we are dealing with the Euler genus instead of orientable genus, we also have to extend Mohar’s argument to the non-orientable case.
**7. Further complications: Extremities.** In the above description of the key steps of our approach we have omitted certain complications that arise when splitting the input graph into simpler subgraphs. More specifically, performing a splitting operation can occasionally create a certain number of components that are simple to embed. We call such components *extremities* (see Figure \[fig:extremity\]). This does not significantly affect our approach at the high level, but makes the statements of our intermediate reduction steps somewhat more technical.
Organization
------------
The rest of the paper is organized as follows. Section \[sec:prelim\] gives some basic definitions and facts. Section \[sec:algo\] states formally the reduction of the problem to Sub-problem 1 (computing a flat grid minor) and Sub-problem 2 (embedding $k$-apex graphs), and presents our main algorithm. Section \[sec:splitting\] presents the vertex splitting operation and proves some of its basic properties. It also formalizes the issues concerning extremities that arise when performing vertex splitting operations. Section \[sec:k-apex\] presents our algorithm for embedding $k$-apex graphs (i.e. Sub-problem 1). This algorithm uses several other algorithms as sub-routines. Due to lack of space, these algorithms are deferred to the full version of the paper, which is attached at the end of this extended abstract. All the proofs can be found in the full version. This algorithm uses several other algorithms as sub-routines, which are discussed in subsequent Sections. Section \[sec:2-connected\] presents our algorithm for embedding $k$-apex graphs with a 2-connected planar piece. Section \[sec:1-separators\] introduces certain intermediate results that allow us to generalize the algorithm for a 2-connected planar piece; in particular, we present tools that allow us to simplify a graph by splitting along certain kinds of 1-separators. These tools will later be used when dealing with nearly locally 2-apex graphs. Section \[sec:2-apex\] presents our algorithm for 2-apex graphs. Section \[sec:2-apex\_generalizations\] combines the results from Sections \[sec:1-separators\] and \[sec:2-apex\] to obtain an algorithm for embedding generalizations of 2-apex graphs, namely, nearly locally 2-apex graphs. Section \[sec:1-apex\] presents our algorithm for embedding 1-apex graphs, generalizing Mohar’s theorem on face covers [@Mo-apex]. Section \[sec:flat\_grids\] presents our algorithm for computing a flat grid minor (i.e. Sub-problem 1). Finally, in Section \[sec:CS\_summary\], for the sake of completeness, we give a high-level overview of the reduction from the general problem to Sub-problems 1 and 2, from the work of Chekuri and Sidiropoulos [@DBLP:conf/focs/ChekuriS13].
Preliminaries {#sec:prelim}
=============
Before proceeding, we review some basic definitions and facts used throughout this paper. We use $n$ to denote the number of vertices. For basic graph theoretic definitions we refer the reader to the book by Diestel [@diestel], and for an in-depth treatment of topological graph theory, to the monograph by Mohar and Thomassen [@MT]. We will only consider [[*$2$-cell embeddings*]{}]{} of graphs into surfaces; that is, we always assume that every face is homeomorphic to a disk. Such embeddings can be represented combinatorially by means of a [[*local rotation*]{}]{} and [[*signature*]{}]{} (see [@MT] for details). The local rotation and signature define a [[*rotation system*]{}]{}.
Tree-Decomposition and Treewidth
--------------------------------
A [[*tree decomposition*]{}]{} of a graph $G$ is a pair $(T,R)$, where $T$ is a tree and $R$ is a family $\{R_t : t \in V(T)\}$ of vertex sets $R_t\subseteq V(G)$, such that the following two properties hold:
1. $\bigcup_{t \in V(T)} R_t = V(G)$, and every edge of $G$ has both ends in some $R_t$.
2. If $t,t',t''\in V(T)$ and $t'$ lies on the path in $T$ between $t$ and $t''$, then $R_t \cap R_{t''} \subseteq R_{t'}$.
The *width* of a tree decomposition $(T,R)$ is $\max\{|R_t|:
t\in V(T)\}-1$, and the [[*treewidth*]{}]{} of $G$ is defined as the minimum width taken over all tree decompositions of $G$. The *adhesion* of our decomposition $(T, R)$ for $tt' \in T$ is $R_t\cap R_{t'}$. One of the most important results about graphs whose treewidth is large is the existence of a large grid minor or, equivalently, a large wall. Let us recall that an [[*$r$-wall*]{}]{} is a graph which is isomorphic to a subdivision of the graph $W_r$ with vertex set $V(W_r) = \{ (i,j) : 1\le i \le r,\ 1\le j \le r \}$ in which two vertices $(i,j)$ and $(i',j')$ are adjacent if and only if one of the following possibilities holds:
- $i' = i$ and $j' \in \{j-1,j+1\}$.
- $j' = j$ and $i' = i + (-1)^{i+j}$.
We can also define an $(a \times b)$-wall in a natural way, so that an $r$-wall is the same as an $(r\times r)$-wall. It is easy to see that if $G$ has an $(a \times b)$-wall, then it has an $(\lfloor\frac{1}{2}a\rfloor \times b)$-grid minor, and conversely, if $G$ has an $(a \times b)$-grid minor, then it has an $(a \times
b)$-wall. Let us recall that the $(a \times b)$-grid is the Cartesian product of paths $P_a\times P_b$.
The main result in [@RS5] says the following (see also [@ChekuriChuzhoy; @rein; @yusuke; @reed1; @DBLP:journals/jct/RobertsonST94]).
\[gridgeneral\] For every positive integer $r$, there exists a constant $f(r)$ such that if a graph $G$ is of treewidth at least $f(r)$, then $G$ contains an $r$-wall.
A *biconnected component tree decomposition* of a given graph $G$ consists of a tree-decomposition $({\cal T},R)$ such that for every $\{t,t'\} \in E({\cal T})$, $R_t \cap R'_t$ consists of a single vertex and for every $t \in {\cal T}$, $R_t$ consists of a 2-connected graph (i.e., a block). ${\cal T}$ is called a *biconnected component tree*.
For a vertex $v$ in a graph $G$ we write $N_G(v) = \{u\in V(G) : \{u,v\} \in E(G)\}$. For some $X\subseteq V(G)$ we denote by $G[X]$ the subgraph of $G$ induced on $X$. For $X, Y \in V(G)$ with $X \cap Y =\emptyset$, $E(X,Y)$ denotes the set of edges with one endpoint in $X$ and the other endpoint in $Y$.
Let $G$ be a graph, $v\in V(G)$, and let ${\cal C}$ be the set of connected components of $G\setminus v$. We say that the graph $G' = \bigsqcup_{C\in {\cal C}} G[C\cup v]$ is obtained by *cutting* $G$ along $v$, where $\sqcup$ denotes disjoint union. For a set $X=\{v_1,\ldots,v_k\}\subseteq V(G)$ we say that a graph $G'$ is obtained by cutting $G$ along $X$ if there exists a sequence of graph $G_0,\ldots,G_k$ with $G_0=G$, $G_k=G'$, and such that for each $i\in \{1,\ldots,k\}$ the graph $G_i$ is obtained by cutting $G_{i-1}$ along $v_i$. Let $C$ be a maximal 2-connected component of $G$.
We recall the following result on the genus of the complete bipartite graph [@harary1994graph].
\[lem:K33\] For any $n,m\geq 1$, ${\mathsf{eg}}(K_{m,n}) = \left\lceil(m-2)(n-2)/4\right\rceil$.
We recall the following result due to Richter [@DBLP:journals/jct/Richter87] (see also [@DBLP:journals/combinatorica/DeckerGH85], [@stahl1980permutation]).
\[lem:2sum\] Let $G_1,\ldots,G_k$ be non-planar graphs, and for each $i\in \{1,\ldots,k\}$ let $\{s_i,t_i\}\in E(G_i)$. Let $G$ be the graph obtained by identifying every $s_i$ into a vertex $s$, every $t_i$ into a vertex $t$, and every edge $\{s_i,t_i\}$ into an edge $\{s,t\}$. Then, ${\mathsf{eg}}(G) = \Omega\left( \sum_{i=1}^k {\mathsf{eg}}(G_i) \right)$.
We shall also use the following elementary fact.
\[lem:contract\_U\] Let $G$ be a graph, and let $U\subseteq V(G)$. Let $G'$ be the graph obtained by contracting $U$ into a single vertex. Then, ${\mathsf{eg}}(G') \leq {\mathsf{eg}}(G)+|U|-1$ and ${\mathsf{genus}}(G')\leq {\mathsf{genus}}(G)+|U|-1$.
We will give only the proof for Euler genus, since the argument for orientable genus is identical. Let $J$ be the graph obtained from $G$ by removing all edges with both endpoints in $U$. We have ${\mathsf{eg}}(J) \leq {\mathsf{eg}}(G)$. Let also $J'$ be the graph obtained from $J$ by adding the edges of a tree spanning $U$. We have ${\mathsf{eg}}(J') \leq {\mathsf{eg}}(J) + |U|-1$. Since $G'$ is a minor of $J'$, we have ${\mathsf{eg}}(G')\leq {\mathsf{eg}}(J') \leq {\mathsf{eg}}(G)+|U|-1$, as required.
The algorithm {#sec:algo}
=============
In this section we present our algorithm for approximating the Euler genus of a graph. We begin by stating formally the reduction from the general problem to Sub-problems 1 and 2, due to [@DBLP:conf/focs/ChekuriS13]. For a graph $G$ and a minor $\Gamma$ of $G$ we say that a mapping $\mu:V(\Gamma)\to 2^{V(G)}$ is a *minor mapping* for $\Gamma$ if for each $u\in V(\Gamma)$ the graph $G[\mu(u)]$ is connected, and by contracting $G[\mu(u)]$ into a single vertex for each $u\in V(\Gamma)$, we obtain $\Gamma$. The reduction can now be stated as follows.
\[lem:CS\_summary\] Suppose that the following conditions hold:
[(1)]{} There exists a polynomial-time algorithm which given a $n$-vertex graph $G$ of treewidth $t$ and an integer $g\geq 1$, either correctly decides that ${\mathsf{eg}}(G)>g$, or outputs a flat subgraph $G'\subset G$, such that $X$ contains a $\left(\Omega(r)\times \Omega(r)\right)$-grid minor $M$, for some $r=r(n,g,t)$. Moreover, in the latter case, the algorithm also outputs a minor mapping for $M$.
[(2)]{} Given an $n$-vertex graph $G$, an integer $g$, and some $X\subset V(G)$ such that $G\setminus X$ is planar, either correctly decides that ${\mathsf{eg}}(G) > g$, or it outputs a drawing of $G$ into a surface of Euler genus at most $\gamma$, for some $\gamma=\gamma(n, g, |X|)$.
Then there exists a polynomial-time algorithm which given a $n$-vertex graph $G$ and an integer $g\geq 1$, either correctly decides that ${\mathsf{eg}}(G)>g$ or outputs an embedding of $G$ into a surface of Euler genus at most $\gamma(n, g, k) + k$, for some $k=O(t' g \log^{3/2}n)$, where $t'$ is some integer satisfying $r(n,g,t')=O(g)$.
For the sake of completeness, we include an overview of the proof of Lemma \[lem:CS\_summary\] in Section \[sec:CS\_summary\].
The next lemma states our result for computing a flat grid minor.
\[lem:flat\_grid\] There exists a polynomial-time algorithm which given a graph $G$ of treewidth $t$, and an integer $g\geq 1$, either correctly decides that ${\mathsf{eg}}(G)>g$, or it outputs a flat subgraph $G'\subset G$, such that $G'$ contains a $\left(\Omega(r)
\times \Omega(r)\right)$-grid minor $M$, for some $r=\Omega\left(\frac{t^{1/2}}{g^{4} \log^{15/4}
n}\right)$. In the latter case, the algorithm also outputs a minor mapping for $M$.
The next theorem gives our approximation algorithm for embedding $k$-apex graphs. This is the main technical result of our paper. The proof is discussed in subsequent sections.
\[lem:vertex\_insertion\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Then there exists a polynomial-time algorithm which given $G$, $g$, and $X$, outputs an embedding of $G$ into a surface of Euler genus $O(g^{25} \cdot |X|^{21})$.
Given the above results, we are now ready to prove our main Theorem.
The algorithm given by Lemma \[lem:flat\_grid\] satisfies condition (1) of Lemma \[lem:CS\_summary\] with $r(n,g,t) = \Omega\left(\frac{t^{1/2}}{g^{4} \log^{15/4}
n}\right)$. The algorithm given by Lemma \[lem:vertex\_insertion\] satisfies condition (2) of Lemma \[lem:CS\_summary\] with $\gamma(n,g,|X|) = O(g^{25} \cdot |X|^{21})$. Let $t'$ be some integer satisfying $r(n,g,t')=O(g)$. We have $t'= O(g^{10} \log^{15/2} n)$. It now follows by Lemma \[lem:CS\_summary\] that there exists a polynomial-time algorithm that either correctly decides that ${\mathsf{eg}}(G)>g$ or outputs an embedding of $G$ into a surface of Euler genus at most $g'=\gamma(n, g, k) + k$, for some $k=O(t' g \log^{3/2}n) = O(g^{11} \log^{9}n)$. Therefore, $g'=O(g^{25} \cdot k^{21}) = O(g^{25} \cdot (g^{11} \log^{9}n)^{21}) = O(g^{256} \log^{189}n)$, completing the proof.
Vertex splitting {#sec:splitting}
================
We now formalize the notion of *vertex splitting* that was discussed in the Introduction.
Let $H$ be a planar graph and let ${\varphi}$ be a drawing of $H$ into the plane. Let $v\in V(H)$. Let $E^*$ be the set of edges incident to $v$. Consider the circular ordering $\tau$ of $E^*$ induced by ${\varphi}$. Partition $E^*$ into $E^*=E_1 \cup E_2$ where for each $i\in \{1,2\}$ the edges in $E_i$ form a contiguous subsequence in $\tau$. Let $H'$ be the graph obtained from $H$ by removing $v$ and introducing two new vertices $v_1,v_2$. For each $e=\{v,w\}\in E_i$ we add the edge $\{v_i,w\}$ in $H'$. This operation is called a *$(H,{\varphi})$-splitting*. We also say that the splitting operation is *performed on $v$ with partition* $\{E_1,E_2\}$. If $H$ is a subgraph of some graph $G$, then when performing a splitting operation on $v\in V(H)$ we also remove all edges in $E(G)$ between $v$ and $V(G)\setminus V(H)$ (see Figure \[fig:splitting\_embedding\] for an example).
Let $H$ be planar graph and let ${\varphi}$ be a drawing of $H$ into the plane. A sequence $\sigma=p_1,\ldots,p_t$ where each $p_i$ is a $(H,{\varphi})$-splitting is called a *$(H,{\varphi})$-splitting sequence*. The result of performing $\sigma$ on $H$ is a graph $H'$ defined as follows. For every $v\in V(H)$ let $P_v$ be the set of all splittings in $\sigma$ that are performed on $v$. Let $E_v$ be the set of edges incident to $v$ in $H$. For each $p\in P_v$ let $\{E_{v,1}^p, E_{v,2}^p\}$ be the partition of $E_v$ induced by the splitting $p$. Let $\{E_{v,1},\ldots,E_{v,r_v}\}$ be the common refinement of all these partitions of $E_v$. Note that $r_v\leq 2 |P_v|$. We remove $v$ and we add vertices $v_1,\ldots,v_{r_v}$ in $H'$. For every $e=\{v,u\} \in E_{v,i}$ we add the edge $\{v_i,u\}$ in $H'$. Repeating this process for all $v\in V(H)$ concludes the definition of $H'$.
Let $H$ be a planar graph and let ${\varphi}$ be a planar drawing of $H$. Let $\sigma$ be a $(H,{\varphi})$-splitting sequence. Let $H'$ be the graph obtained by performing $\sigma$ on $H$. Let ${\cal C}$ be the set of connected components of $H$ that are also connected components of $H'$. In other words, ${\cal C}$ contains all connected components of $H$ on which $\sigma$ does not perform any splittings. Let also ${\cal C}'$ be the remaining connected components of $H$. For any $C\in {\cal C}'$ let ${\cal F}_C$ be the collection of connected components in $H'$ that are obtained from $C$ after performing $\sigma$. Let ${\cal D} = \bigcup_{C\in {\cal C}'} {\cal F}_C$. Then we refer to the elements in ${\cal C}\cup {\cal D}$ as the *fragments* of $H'$ (w.r.t. $\sigma$). Note that if $\sigma$ has length $k$, then there are at most $k+2$ fragments of $H'$.
Let $U$ be a set and let $\{A,B\}$, $\{A',B'\}$ be partitions of $U$. We say that the partitions are *crossing* if their common refinement consists of four non-empty sets. Otherwise we say that they are *non-crossing*.
Let $H$ be a planar graph and let ${\varphi}$ be a drawing of $H$ into the plane. A $(H,{\varphi})$-splitting sequence $\sigma$ is called *monotone* if the following condition is satisfied. Let $v \in V(H)$ be an arbitrary vertex. Let $E_v$ be the set of edges incident to $v$ in $H$. Then, for any pair $p,q\in \sigma$ of splittings performed on $v$ the partitions of $E_v$ induced by $p$ and $q$ are non-crossing.
Having defined monotonicity, we need the following result.
\[lem:monotone\_splitting\] Let $H$ be a planar graph and let ${\varphi}$ be a drawing of $H$ into the plane. Let $\sigma$ be a $(H,{\varphi})$-splitting sequence of length $m$. Let $H'$ be the graph obtained by performing the $(H,{\varphi})$-splitting sequence $\sigma$. Then, there exists a monotone $(H,{\varphi})$-splitting sequence $\sigma'$ of length at most $m$, such that the graph resulting by performing $\sigma'$ on $H$ is $H'$.
Let $v\in V(H)$, and let $E_v\subset E(H)$ be the set of edges in $H$ that are incident to $v$. Let $\xi$ be the circular ordering of $E_v$ around $v$ induced ${\varphi}$. Suppose that at most $t$ splittings in $\sigma$ are performed in $v$. Then, it follows by induction on $t$ that $v$ corresponds to some $S_v\subseteq V(H')$, with $|S_v|\leq t$. Moreover, for each $v'\in S_v$, the edges incident to $v'$ in $H'$ correspond to a subset $E_{v,v'}\subseteq E_v$, where $E_{v,v'}$ forms a contiguous segment in $\xi$. We may therefore add for each $v'\in S_v$ a splitting in $\sigma'$ with partition $\{E_{v,v'}, E_v\setminus E_{v,v'}\}$. It is immediate that all these splittings are monotone. Repeating the same process for all $v\in V(H)$, we obtain the desired splitting sequence $\sigma$. It is also immediate that for every $v\in V(H)$, the number of splittings in $\sigma'$ that are performed on $v$ is at most the number of such splittings in $\sigma$. Therefore, the length of $\sigma'$ is at most $k$. This concludes the proof.
The following lemma shows that performing a small number of splitting operations does not increase the genus significantly.
\[lem:genus\_splittings\] Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Let ${\varphi}$ be a planar drawing of $H$. Let $G'$ be the graph obtained from $G$ after performing a $(H,{\varphi})$-splitting sequence of length $k$ on $H$. Then ${\mathsf{genus}}(G') \leq {\mathsf{genus}}(G) + O(k \cdot |X|)$, and ${\mathsf{eg}}(G') \leq {\mathsf{eg}}(G) + O(g\cdot |X|)$. Moreover, given a drawing of $G'$ into a surface of orientable (resp. non-orientable) genus $\gamma$, we can compute a drawing of $G$ into a surface of orientable (resp. non-orientable) genus $\gamma + O(k\cdot |X|)$ in polynomial time.
By Lemma \[lem:monotone\_splitting\] we may assume that $\sigma$ is monotone. Let $H'$ be the graph obtained by performing the splitting sequence $\sigma$ on $H$. Each $v\in V(H)$ corresponds to some $S_v\subseteq V(H')$. For every $v\in V(G)$, let $E_v$ be the set of edges in $H$ that are incident to $v$. Let $\xi$ be the circular ordering of $E_v$ around $v$ induced by ${\varphi}$. Since $\sigma$ is monotone, it follows that each $s\in S_v$ is incident in $H'$ to some set of edges $E_{v,s} \subseteq E_v$, that appear in a contiguous subsegment of the circular ordering $\xi$. By monotonicity, and by induction on the length of the splitting sequence $\sigma$, it follows that when we perform the $i$-splitting in $\sigma$, some vertex $v'_i$ in the current graph that corresponds to some $v_i$ is split into two vertices $v'_{i,1}$ and $v'_{i,2}$. We set $v'_{i,1}$ to be the parent of $v'_{i,2}$. By monotonicity of $\sigma$, it follows that this parent relationship defines a tree in $H'$. Starting from $H'$, we add all the edges of all these trees and for all vertices $v\in V(H)$. Let $H''$ be the resulting graph obtained from $H'$. Let also $G''$ be the corresponding graph obtained from $G'$. It is immediate that $H''$ is obtained by adding at most $k$ edges in $H'$. Therefore, ${\mathsf{eg}}(G'') \leq {\mathsf{eg}}(G') + k$. Let $G'''$ be the graph obtained by contracting each one of these trees into a single vertex. Finally, for every spitting operation performed on some $v\in V(H)$, we might remove at most $|X|$ edges between $v$ and $X$. Therefore, adding at most $k\cdot |X|$ edges to $G'''$, we obtain the graph $G$. Therefore, ${\mathsf{eg}}(G) \leq {\mathsf{eg}}(G''') + k\cdot |X| \leq {\mathsf{eg}}(G'') + k\cdot |X| \leq {\mathsf{eg}}(G') + (k+1)\cdot |X|$. Finally, let $\psi$ be a drawing of $G$ into a surface ${\cal S}$ of Euler genus $\gamma$. By adding at most $O(k\cdot |X|)$ handles in ${\sigma'}$, one for every new edge in $G'''$, we can extend $\psi$ into a drawing $\psi''$ of $G''$ into a surface of Euler genus $\gamma+O(k\cdot |X|)$, concluding the proof.
The following lemma shows how to compute an embedding of a graph given an embedding of the graph obtained after performing a sequence of splitting operations.
\[lem:glueing\_fragmented\] Let $G$ be a graph and let $X\subseteq V(G)$ be an independent set such that $H=G\setminus X$ is planar (and possibly disconnected). Let ${\varphi}$ be a planar drawing of $H$. Let $\sigma$ be a $(H,{\varphi})$-splitting sequence of length $k$. Let $H'$ be the graph obtained by performing $\sigma$ on $H$. Let ${\cal F}$ be the set of fragments of $H'$. Suppose that for any $C\in {\cal F}$ there exits an embedding $\psi_C$ of $G'[V(C)\cup X]$ into a surface ${\cal S}_C$ of Euler genus $\gamma_C$. Then, there exists an embedding $\psi$ of $G$ into a surface of Euler genus at most $O(k\cdot |X|) + \sum_{C\in {\cal F}} \gamma_C$. Moreover, there exists a polynomial-time algorithm which given $G$, $X$, ${\varphi}$, $\sigma$, and $\{\psi_C\}_{C\in {\cal F}}$ outputs $\psi$ (see Figure \[fig:splitting\_embedding\]).
Let $G'$ be the graph obtained from $G$ after applying $\sigma$ on $H$. For any $C,C'\in {\cal F}$, $N_{G'}(C)\cap N_{G'}(C')\subseteq X$. Therefore, we can extend $\psi_C$ to $C'$ by adding at most $|X|$ cylinders, each connecting a puncture in the surface ${\cal S}_C$ (that accommodates $C$) to a puncture in the surface ${\cal S}_{C'}$ (that accommodates $C'$). The resulting surface has Euler genus at most ${\mathsf{eg}}({\cal S}_C) + {\mathsf{eg}}({\cal S}_{C'}) + |X|$. Using the same procedure we can inductively extend this embedding to all the fragments of $H'$. For every fragment $C''$ we increase the genus of the embedding by at most ${\mathsf{eg}}({\cal S}_{C''}) + |X|$. Since $\sigma$ has length $k$, the number of fragments of $H'$ is at most $k+2$. Thus we obtain an embedding $\psi'$ of $G'$ into a surface of Euler genus at most $\gamma'=\sum_{C\in {\cal F}} ({\mathsf{eg}}({\cal S}_C) + |X|) = O(k\cdot |X|) + \sum_{C\in {\cal F}} \gamma_C$.
Finally, using Lemma \[lem:genus\_splittings\] we can compute an embedding of $G$ into a surface of Euler genus at most $\gamma' + O(k\cdot |X|) = O(k\cdot |X|) + \sum_{C\in {\cal F}} \gamma_C$, concluding the proof.
Extremities
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We now introduce some machinery that allows us to handle some issues that arise from splitting operations. More specifically, when performing a spitting operation, we might create a certain number of pieces that are easy to embed, called *extremities*. We formalize this notion next.
Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Let ${\cal C}$ be a collection of maximal 2-connected components of $H$ such that $C=\bigcup_{A\in {\cal C}} A$ is connected. Suppose further that there exist some 1-separator $v$ of $H$ such that all edges between $C$ and $H\setminus C$ are incident to $v$. Moreover, suppose that there exists $x\in X$ such that $N_G[V(C) \setminus \{v\}] \cap X \subseteq \{x\}$. Finally, assume that that $G[C\cup \{x\}]$ admits a planar drawing such that $x$ and $v$ are in the same face. Then we say that $C$ is an *extremity* (w.r.t. $X$). Figure \[fig:extremity\] depicts an example of an extremity. We refer to $v$ as the *portal* of $C$. The *extremity number* of $G$ is defined to be the minimum integer $M$ such that any family of pairwise edge-disjoint maximal extremities of $G$ has size at most $M$.
The next lemma allows us to compute embeddings of graphs, while ignoring their extremities. In other words, if we obtain an embedding of a resulting graph (after contracting extremities), then we can extend to the embedding of the original graph in the same surface.
\[lem:contracting\_extremities\] Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Let $C_1,\ldots,C_t$ be a collection of pairwise edge-disjoint extremities. Let $G'$ be the graph obtained by contracting each $C_i$ into a single vertex $v_i$ and removing parallel edges (see Figure \[fig:extremity\]). Then there exists a polynomial-time algorithm which given an embedding of $G'$ into a surface of Euler genus $\gamma$ outputs an embedding of $G$ into a surface of Euler genus $\gamma$.
Suppose that ${\varphi}'$ is an embedding of $G'$ into some surface ${\cal S}'$. Consider some extremity $C_i$. If $C_i=H$, then the assertion is immediate, so suppose that $C_i\neq H$. Let $v$ be the portal of $C_i$. We have either $N(C_i\setminus \{v\})\cap X=\emptyset$ or $\{u\}=N(C_i\setminus \{v\})\cap X$. In the later case, there exists at most one edge $e$ in $G'$ between $u$ and $v$ that corresponds to the edges between $u$ and $V(C_v)\setminus \{v\}$ in $G$. Let ${\cal D}$ be a disk in ${\cal S}'$ that intersects ${\varphi}'(G')$ only on ${\varphi}'(v)$. Moreover, if $e$ is defined, then we can ensure that ${\cal D}$ lies inside a face that contains $e$. By choosing the disks for distinct extremities to be sufficiently small, we can further ensure that they are pairwise disjoint. By the definition of an extremity it follows that there exists a planar drawing $\psi_{C_i}$ of $G[C_i\cup \{u\}]$ in which $u$ and $v$ are in the same face $F$. We extend ${\varphi}'$ to $H[C_i]$ by mapping $H[C_i]$ into ${\cal D}$ according to $\psi_{C_i}$, so that the face bounded by $F$ contains the boundary of ${\cal D}$. We can also extend the embedding to the edges in $G$ between $u$ and $C_i$ by mapping them along curves that are contained in a sufficiently small neighborhood of ${\varphi}'(e)\cup {\cal D}$. Repeating the same process for all extremities $C_i$ results in the desired embedding ${\varphi}$.
Finally, we argue that splitting operations cannot create a significant number of extremities.
\[lem:creating\_extremities\_splitting\] Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Suppose that the extremity number of $G$ if $M$. Let ${\varphi}$ be a planar drawing of ${\varphi}$. Let $H'$ be the graph obtained by performing a splitting operation on some 1-separator of $H$, and let $G'$ be the corresponding graph obtained from $G$. Then the extremity number of $G'$ is at most $M+2\cdot |X|$.
Suppose we perform a splitting operation on some 1-separator $v$ of $H$. Then $v$ corresponds to exactly copies $v_1$ and $v_2$ in $H'$. Any maximal extremity in $G'$ that is not an extremity in $G$ must contain either $v_1$ or $v_2$. By the definition of an extremity, it follows any two new maximal and edge-disjoint extremities in $H'$ that contain $v_1$ must be incident to distinct vertices in $X$ (otherwise their union must also be an extremity, contradicting their maximality). Therefore, there can be at most $|X|$ distinct new maximal and pairwise edge-disjoint extremities that contain each $v_i$, $i\in \{1,2\}$. Thus, there can be at most $M+2\cdot |X|$ extremities in $G'$, concluding the proof.
\[lem:creating\_extremities\_splitting\_sequence\] Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Suppose that the extremity number of $G$ is $M$. Let ${\varphi}$ be a planar drawing of ${\varphi}$. Let $H'$ be the graph obtained by performing a splitting sequence $\sigma$ of length $\ell$ on $H$, where each splitting operation in $\sigma$ is performed on some 1-separator of $H$, and let $G'$ be the corresponding graph obtained from $G$. Then the extremity number of $G'$ is at most $M+2\ell\cdot |X|$.
If follows immediately from Lemma \[lem:creating\_extremities\_splitting\] and induction on $\ell$.
Embedding $k$-apex graphs {#sec:k-apex}
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In this section we present our algorithm for embedding $k$-apex graphs. This uses several other algorithms as sub-routines, that we discuss in the full version. This uses several other algorithms as sub-routines, that we discuss in subsequent sections. We first state a preliminary result that allows us to assume that every vertex of the planar piece is incident to at most two apices.
\[lem:all3\] Let $G$ be a graph of genus $g$ and let $X\subset V(G)$ such that $H=G\setminus X$ is planar. Let $x_1,x_2,x_3\in X$ be distinct vertices. Let $U = V(H) \cap N_G(x_1) \cap N_G(x_2) \cap N_G(x_3)$. Then $|U|=O(g)$.
The following lemma allows us to reduce the $k$-apex case to two sub-cases: (1) 2-apex graphs, and (2) $k$-apex graphs where every 1-separator of the planar piece is incident to at most one apex.
\[lem:2apices\_or\_simple1separators\] Let $G$ be a graph of Euler genus $g$ and let $X\subset V(G)$ be an independent set such that $H=G\setminus X$ is planar. Let ${\varphi}$ be a planar drawing of $H$. Suppose that every vertex $v\in V(H)$ is incident to at most two vertices in $X$, that is $|N(v) \cap X|\leq 2$. Then there exists a $(H,{\varphi})$-splitting sequence $\sigma$ of length $O(g\cdot |X|^3)$ such that if we let $H'$ be the graph obtained by performing $\sigma$ on $H$, then for every connected component $C$ of $H'$ at least one of the following conditions is satisfied:
[(1)]{} $C$ is incident to at most two vertices in $X$, that is $|N_G(C)\cap X| \leq 2$.
[(2)]{} Every $1$-separator $v$ of $C$ is incident to at most one vertex in $X$, that is $|N(v)\cap X| \leq 1$.
\(1) $C$ is incident to at most two vertices in $X$, that is $|N_G(C)\cap X| \leq 2$.
\(2) Every $1$-separator $v$ of $C$ is incident to at most one vertex in $X$, that is $|N(v)\cap X| \leq 1$.
The next two lemmas summarize our embedding algorithms for the above two cases.
\[lem:embedding\_2-apex\] Let $G$ be a planar graph and let $X\subseteq V(G)$, with $|X|=2$, such that $H=G\setminus X$ is planar. Then there exists a polynomial-time algorithm which given $G$ and $X$, outputs a drawing of $G$ into a surface of Euler genus $O(({\mathsf{eg}}(G))^{15})$.
\[lem:embedding\_1-apex\_1-separators\] Let $G$ be a graph of genus $g$ and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Suppose that every 1-separator $v$ of $H$ is incident to at most one vertex in $X$, that is $|N_G(v)\cap X|\leq 1$. Suppose further that the extremity number of $G$ is $M$. Then there exists a polynomial-time algorithm which given $G$, $g$, $M$, and $X$ outputs an embedding of $G$ into a surface of Euler genus $O(g^{24} \cdot |X|^{18} + g^{22} \cdot |X|^{14} \cdot M)$.
Given all the above results, we are ready to present our algorithm for embedding $k$-apex graphs.
The algorithm consists of the following steps.
**Step 1: Deleting edges between vertices in $X$.** There can be at most $O(|X|^2)$ edges between the vertices in $X$. We remove all such vertices, and we extend the drawing at the end to include these edges by adding at most $O(|X|^2)$ additional handles. We can therefore assume for the remainder that there are no edges between the vertices in $|X|$.
**Step 2: Contracting extremities.** We compute a maximal collection ${\cal E}$ of pairwise edge-disjoint maximal extremities in $G$. For every extremity $C\in {\cal E}$ let $v_C$ be its portal. There exits $x_C\in X$ such that $N_G(V(C)\setminus \{v_C\}) \cap X \subseteq \{x_C\}$. We contract $C$ into $v_C$, thus replacing $G[C\cup \{x_C\}]$ by a single edge $e_C$ between $x_C$ and $v_C$. Repeating for all $C\in {\cal E}$, we obtain a graph $G'$ with extremity number 0. We will show next how to compute an embedding for $G'$ into a surface of Euler genus $\gamma$. Given this embedding we can extend it to $G$ using Lemma \[lem:contracting\_extremities\], obtaining an embedding of $G$ into a surface of Euler genus $\gamma$. We may therefore assume for the remainder of the algorithm that the extremity number of $G$ is 0.
**Step 3: Bounding the number of apices that are incident to any vertex in $H$.** By Lemma \[lem:all3\] for any three distinct vertices $x_1,x_2,x_3\in X$ there can be at most $O(g)$ vertices in $V(H)$ that are incident to all of $x_1,x_2,x_3$. Therefore, there can be at most $O(g\cdot |X|^3)$ vertices in $V(H)$ that are incident to at least three vertices in $X$. For every such a vertex $v$ we remove all except for two edges between $v$ and $X$. The total number of edges we remove is $O(g\cdot |X|^4)$. Since we leave at least two edges between $v$ and $X$, it follows that the extremity number of $G$ remains 0. We next show how to compute an embedding for the resulting graph. We will extend the drawing to these edges at the end of the algorithm, by adding at most $O(g\cdot |X|^4)$ additional handles. We may therefore assume for the remainder that every $v\in V(H)$ is incident to at most two vertices in $X$.
**Step 4: Splitting into pieces that are either 2-apex or contain only simple 1-separators.** Fix any planar drawing ${\varphi}$ of $H$. By Lemma \[lem:2apices\_or\_simple1separators\] we can find a $(H,{\varphi})$-splitting sequence $\sigma$ of length $k=O(g\cdot |X|^3)$, such that for every fragment $C$ of the resulting graph $H'$, either $C$ is incident to at most two vertices in $X$, or every 1-separator in $C$ is connected to at most one apex. By Lemma \[lem:creating\_extremities\_splitting\_sequence\], the extremity number of the resulting graph is at most $M=O(g\cdot |X|^4)$.
**Step 5: Embedding the fragments.** Let ${\cal F}$ be the set of fragments of $H'$. We have $|{\cal F}|=O(k)=O(g\cdot |X|^3)$. Consider some $C\in {\cal F}$. If $C$ is incident to at most two vertices in $X$, then by Lemma \[lem:embedding\_2-apex\] we can compute an embedding of $G[V(C)\cup X]$ into a surface of Euler genus $O(g^{15})$. Otherwise, every 1-separator in $C$ is incident to at most one vertex in $X$. Thus, by Lemma \[lem:embedding\_1-apex\_1-separators\] we can compute an embedding of $G[V(C)\cup X]$ into a surface of Euler genus $O(g^{24}\cdot |X|^{18} + g^{22} \cdot |X|^{14} \cdot M) = O(g^{24}\cdot |X|^{18})$. Thus in either case, for each $C\in {\cal F}$ we can compute an embedding ${\varphi}_C$ of $G[V(C)\cup X]$ into a surface of Euler genus $O(g^{24}\cdot |X|^{18})$.
**Step 6: Combining the embeddings of all fragments into a single embedding.** By Lemma \[lem:glueing\_fragmented\] we can combine all the embeddings $\{{\varphi}_C\}_{C\in {\cal F}}$ to obtain an embedding ${\varphi}$ of $G$ into a surface of Euler genus $O(k\cdot |X|) + \sum_{C\in {\cal F}} {\mathsf{eg}}({\varphi}_C) = O(g\cdot |X|^4) + |{\cal F}| \cdot O(g^{24} \cdot |X|^{18}) = O(g^{25} \cdot |X|^{21})$. This concludes the proof.
Embedding graphs with a 2-connected planar piece {#sec:2-connected}
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In this section we present the algorithm for embedding $k$-apex graphs with a 2-connected planar piece. Our approach is to decompose the graph into a small number of graphs with a constant number of apices. The key ingredient is the notion of *coupled* edge sets, which we now introduce.
Let $G$ be a graph and let $X\subseteq V(G)$ and $H=G\setminus X$. Let $x_1,x_2\in X$ be distinct vertices. Let $P$ be a path in $H$. We say that $E'$ is *$(P, \{x_1,x_2\})$-coupled* (or just $P$-coupled when $x_1$ and $x_2$ are clear from the context) if the following conditions are satisfied:
[(1)]{} $E' \subseteq E(V(P), \{x_1,x_2\})$.
[(2)]{} For every internal vertex $v$ of $P$ $E(\{v\},\{x_1,x_2\}) \subseteq E'$.
Figure \[fig:coupled\] depicts an example of a coupled set of edges.
Intuitively, we will give a decomposition of the edges between the planar piece and the apex set into a small number of coupled sets. In order to do that, we need the notion of *interleaving number*. This allows us to argue that when the decomposition fails, we can find a large $K_{3,r}$ minor, which contradicts the fact that the Euler genus is small.
Let $G$ be a graph and let $X\subseteq V(G)$ be such that $H=G\setminus X$ is planar. Let $x_1,x_2,x_3\in X$ be distinct vertices. Let $P$ be a path in $H$. Let $E' = E(V(P), \{x_1,x_2,x_3\})$. The $(x_1,x_2,x_3)$-interleaving number of $P$ is defined to be the minimum integer $k$ such that the following holds: There exists a collection of edge-disjoint subpaths $P_1,\ldots,P_k$ of $P$ and a decomposition $E'=E_1\cup\ldots\cup E_k$ such that for each $i\in \{1,\ldots,k\}$, $E_i$ is $P_i$-coupled (in particular, it follows that there exists $y_i \in \{x_1, x_2, x_3\}$ such that no edge in $E_i$ is incident to $y_i$). See Figure \[fig:interleaving\] for an example.
Having defined the above concepts, we now prove the following two lemmas that are used in the subsequent decomposition.
\[lem:interleaving\_number\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ be such that $H=G\setminus X$ is planar. Then, for any triple $x_1,x_2,x_3\in X$ of distinct vertices and for any path $P$ in $H$ the $\{x_1,x_2,x_3\}$-interleaving number of $P$ is at most $O(g)$.
Suppose that the $\{x_1,x_2,x_3\}$-interleaving number of $P$ is $k$. By the definition of interleaving number there exists a collection of edge-disjoint subpaths $P_1,\ldots,P_k$ of $P$ and a decomposition $E'=E_1\cup\ldots\cup E_k$ such that for each $i\in \{1,\ldots,k\}$ $E_i$ is $P_i$-coupled, and moreover, for any $k'<k$ there exists no such collections. This means that for any $i\in \{1,\ldots,k-1\}$ $E_i \cup E_{i+1}$ contains edges incident to all three vertices in $\{x_1,x_2,x_3\}$.
By the choose of $k$, there can be at most two paths in $P_1,\ldots,P_k$ that share the same vertex $v\in V(P)$. It follows that for any $i\in \{1,\ldots,\lfloor 3/4 \rfloor\}$, if we let $Q_i=P_{3i+1}\circ P_{3i+2}$, then $Q_i$ is incident to all vertices in $\{x_1,x_2,x_3\}$, and for any $i\neq j\in \{1,\ldots,\lfloor k/3 \rfloor\}$ $V(Q_i)\cap V(Q_j) = \emptyset$. We can now construct a $K_{3,\lfloor k/3 \rfloor}$ minor in $G$ with the left side $\{x_1,x_2,x_3\}$, and for any $i\in \{1,\ldots,\lfloor k/3 \rfloor\}$ the right side contains a vertex obtained by contracting $Q_i$. It follows by Lemma \[lem:K33\] that $k=O(g)$, as required.
We also need the following decomposition.
\[lem:interleaving\_number\_X\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ be such that $H=G\setminus X$ is planar. Let $P$ be a path in $H$. Let $E'=E(V(P), X)$. There exists a multi-set[^3] of edge-disjoint subpaths $P_1,\ldots,P_k\subseteq P$, and a decomposition $E'=E'_1\cup \ldots E'_k$, for some $k=O(g\cdot |X|^3)$, such that for any $i\in \{1,\ldots,k\}$ the set $E'_i$ is $P_i$-coupled. Moreover, there exists a polynomial-time algorithm which given $G$, $g$, $X$, and $P$, outputs $P_1,\ldots,P_k$ and $E_1',\ldots,E_k'$.
For each triple $\tau = \{x_1,x_2,x_3\}\in {X \choose 3}$ let ${\cal P}_\tau = \{P_{\tau,i}\}_i^{k_{\tau}}$ be the collection of edge-disjoint paths of $P$ and $E'=E_{\tau,1}\cup\ldots\cup E_{\tau,k_{\tau}}$ the decomposition given by Lemma \[lem:interleaving\_number\], for some $k_{\tau}=O(g)$.
We construct a collection ${\cal P}=P_1,\ldots,P_k$ of subpaths of $P$ and a decomposition $E=E_1\cup \ldots\cup E_k$ as follows. Consider some $\{x_1,x_2\}\in {X\choose 2}$. For any $x_3\in X\setminus \{x_1,x_2\}$ the paths in ${\cal P}_{\{x_1,x_2,x_3\}}$ have at most $O(g)$ endpoints by Lemma \[lem:interleaving\_number\]. Therefore, for all possible choices for $x_3$, there are at most $O(g\cdot |X|)$ such endpoints. It follows that there exists a collection ${\cal P}_{\{x_1,x_2\}}$ of edge-disjoint subpaths of $P$ of $O(g\cdot |X|)$, such that each edge $e\in E(\{x_1,x_2\}, V(P))$ has an endpoint in some $Q\in {\cal P}_{\{x_1,x_2\}}$, and there is no edge with one endpoint in $X\setminus \{x_1,x_2\}$ and another endpoint in an interior vertex of some $Q\in {\cal P}_{\{x_1,x_2\}}$. We add all the paths in ${\cal P}_{\{x_1,x_2\}}$ to ${\cal P}$ and the corresponding subsets of $E(\{x_1,x_2\}, V(P))$ to the decomposition of $E'$. In total, we add at most $O(g\cdot |X|)$ paths to ${\cal P}$ that correspond to the pair $\{x_1,x_2\}$. Repeating over all possible choices for $\{x_1,x_2\}\in {X\choose 2}$, we obtain the desired decomposition with $k=O(g\cdot |X|^3)$.
Kissing decomposition
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We now give our first decomposition which is needed in a later proof. Let us begin with some definition. Intuitively, we would like to argue that in a $k$-apex graph of small genus, the edges between the apices and the planar piece can be partitioned into coupled sets that “interact” in a simple fashion; that is, a pair of coupled sets cannot intersect in an arbitrary way. The following definition formalizes this intuition.
Let $G$ be a graph and let $X\subseteq V(G)$ be such that $H=G\setminus X$ is planar and 2-connected. Fix a planar drawing ${\varphi}$ of $H$. Let $E_1,E_2\subseteq E(X,V(H))$ with $E_1\cap E_2 = \emptyset$. Let $F_1,F_2$ be distinct faces in ${\varphi}$. For any $i\in \{1,2\}$ let $P_i$ be a subpath of $F_i$ such that $E_i$ is $P_i$-coupled. Then we say that $(E_1,E_2)$ is *$(P_1,P_2)$-kissing* (w.r.t. ${\varphi}$) if at least one of the following conditions is satisfied:
[(1)]{} $V(P_1) \cap V(P_2) = \emptyset$.
[(2)]{} $V(P_1) \cap V(P_2) = \{v\}$ where $v$ is an endpoint of both $P_1$ and $P_2$.
[(3)]{} The paths $P_1$ and $P_2$ share both of their endpoints. Moreover, the following holds. The closed curve ${\varphi}(P_1)\cup {\varphi}(P_2)$ bounds a collection of zero or more disks ${\cal D}_1,\ldots,{\cal D}_k\subset \mathbb{R}^2$. Let $H_1,\ldots,H_k$ be the maximal 2-connected components of $H$ with ${\varphi}(H_i) \subset {\cal D}_i$. Then, we have $$E\left(X, V(P_1) \cup V(P_2) \cup \left(\bigcup_{i=1}^k V(H_i)\right)\right) = E_1 \cup E_2.$$ In other words, there is no edge in $E(X,V(H))$ with an endpoint in some interior vertex of some $H_i$. See Figure \[fig:kissing\] for an example.
Using the above definition, the main decomposition result of this subsection can now be stated as follows.
\[lem:kissing\_decomposition\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ be such that $H=G\setminus X$ is planar and 2-connected. Let $E'=E(X,V(H))$. There exists a planar drawing ${\varphi}$ of $H$ and a collection ${\cal F}=\{F_i\}_{i=1}^k$ of faces in ${\varphi}$, for some $k=O(g^2 + |X|^2)$, such that the following conditions are satisfied:
[(1)]{} For each $i\in \{1,\ldots,k\}$ there exists a multi-set of pairwise edge-disjoint subpaths $P_{i,1},\ldots,P_{i,k_i}$ of $F_i$, for some $k_i=O(g^6\cdot |X|^6 + g^2 \cdot |X|^{10})$, and a decomposition $$E'=\bigcup_{i=1}^k \bigcup_{j=1}^{k_i} E_{i,j},$$ such that for each $i\in \{1,\ldots,k\}$ and for each $j\in \{1,\ldots,k_i\}$ $E_{i,j}$ is $P_{i,j}$-coupled.
[(2)]{} For any $i<i'\in \{1,\ldots,k\}$, $j\in \{1,\ldots,k_i\}$, $j'\in \{1,\ldots,k_{i'}\}$, and for any $v\in V(P_{i,j}) \cap V(P_{i',j'})$, $E(v,X) \subseteq E_{i,j}$. In other words, the edges in $E_G(X,V(H))$ are assigned to the faces in ${\cal F}$ in a greedy fashion, giving priority to the faces $F_i\in {\cal F}$ with the smallest index $i$.
[(3)]{} For any $i,i'\in \{1,\ldots,k\}$ and for any $j\in \{1,\ldots,k_i\}$, $j'\in \{1,\ldots,k_{i'}\}$, with $(i,j)\neq (i',j')$, $(E_{i,j}, E_{i',j'})$ is $(P_{i,j},P_{i',j'})$-kissing.
Moreover, there exists a polynomial-time algorithm which given $G$, $g$, $X$, and ${\cal F}$, outputs $\{P_{i,j}\}_{i,j}$ and $\{E_{i,j}\}_{i,j}$.
Let $G'$ be the graph obtained by identifying $X$ into a single vertex $x^*$. By Lemma \[lem:contract\_U\] ${\mathsf{eg}}(G')\leq g+|X|-1$. By Lemma \[lem:face\_cover\] there exists a planar drawing ${\varphi}$ of $H$ and a ${\varphi}$-face cover ${\cal F}$ of $N_{G'}(x^*)$ with $|{\cal F}| = O(g^2+|X|^2)$. Note that ${\cal F}$ is also a ${\varphi}$-face cover of $N_G(X)$.
Let ${\cal F}=\{F_1,\ldots,F_k\}$. Let $E' = E(X, V(H))$. We define a partition $E'=E_1 \cup \ldots \cup E_k$ as follows. Let $E_1 = E_G(X, V(F_1))$, and for any $i\in \{2,\ldots,k\}$ let $$E_i = E(X, V(F_i)) \setminus \left( \bigcup_{j=1}^{i-1} E_j \right).$$ We have at most $k=O(g^2+|X|^2)$ different subsets $E_i$.
Let $i\in \{1,\ldots,k\}$. Since $H$ is 2-connected, it follows that $F_i$ is a cycle. Since we may assume that $G$ does not contain parallel edges, it follows that $|V(F_i)|\geq 3$. We may therefore decompose $F_i$ into three edge-disjoint subpaths $Q_{i,1}, Q_{i,2}, Q_{i,3}$, and we can also compute a decomposition $E_i = R_{i,1}\cup R_{i,2}\cup R_{i,3}$ such for each $i,j$ we have $R_{i,j}\subseteq E_i \cap E(X,Q_{i,j})$. For each $i,j$, by Lemma \[lem:interleaving\_number\_X\] we can compute a a collection $Q_{i,j,1},\ldots,Q_{i,j,k_{i,j}}$ of edge-disjoint subpaths of $Q_{i,j}$ and a decomposition $E_{i,j}=\bigcup_{r=1}^{k_{i,j}} E_{i,j,r}$, for some $k_{i,j} = O(g\cdot |X|^3)$ such that each $E_{i,j,r}$ is $Q_{i,j,r}$-coupled. For each $F\in {\cal F}$ we obtain a total number of at most $O(g\cdot |X|^3)$ subsets of $E'$.
It remains to enforce the kissing condition (3). There are at most $|{\cal F}| \cdot g\cdot|X|^3=O((g^2+|X|^2)\cdot g\cdot |X|^3) = O(g^3\cdot |X|^3 + g\cdot |X|^5)$ total subsets $E_{i,j,r}$ and corresponding paths $Q_{i,j,r}$. Let $U\subseteq V(H)$ be the set of all the endpoints of the paths $Q_{i,j,r}$. So $|U| = O(g^3\cdot |X|^3 + g\cdot |X|^5)$. For every path $Q_{i,j,r}$, we partition $Q_{i,j,r}$ into subpaths $Q_{i,j,r,1},\ldots,Q_{i,j,r,t}$, for some $t\leq |U|+1$ by cutting $Q_{i,j,r,1}$ along all vertices in $U$. We also partition each $E_{i,j,r}$ into subsets $E_{i,j,r,1},\ldots,E_{i,j,r,t}$ such that each $E_{i,j,r,t}$ is $Q_{i,j,r,t}$-coupled. Let ${\cal E}$ be the resulting collection of subsets $E_{i,j,r,t}$ of $E'$ and let ${\cal Q}$ be the resulting collection of paths $Q_{i,j,r,t}$. Then $|{\cal E}| = |{\cal Q}| = O(g^6 \cdot |X|^6 + g^2 \cdot |X|^{10})$. In particular, for every $F\in {\cal F}$ at most $O(g\cdot |X|^3 \cdot (g^3\cdot |X|^3 + g\cdot |X|^5)) = O(g^4\cdot |X|^6 + g^2 \cdot |X|^8)$ subsets of $E'$ in ${\cal E}$.
Consider now some $R\in {\cal E}$ and the corresponding path $Q\in {\cal Q}$ such that $R$ is $Q$-coupled. The path $Q$ was obtained as a subpath of some face $F\in {\cal F}$ in ${\varphi}$. Let $F'\in {\cal F}$ with $F'\neq F$. Since $H$ is 2-connected, there exists a minimal disk ${\cal D}\subset \mathbb{R}^2$ containing ${\varphi}(F) \cup {\varphi}(F')$. Let $L$ be the maximal subpath of $F$ such that the interior of the curve ${\varphi}(L)$ is completely contained in the interior of ${\cal D}$. We cut $Q$ along all endpoints of $L$ that are in the interior of $Q$. The result consists of at most three edge-disjoint subpaths $Q_1,Q_2,Q_3$ of $Q$. We accordingly partition $R$ into at most three subsets $R_1,R_2,R_3$ such that each $R_i$ is $Q_i$-coupled. Repeating the above process for all $F'\neq F \in {\cal F}$ we partition $Q$ into at most $3\cdot (|{\cal F}|-1)=O(g^2+|X|^2)$ subpaths, and $R$ into at most $3\cdot (|{\cal F}|-1)=O(g^2+|X|^2)$ subsets. Repeating the same process for all $R\in {\cal E}$ and all corresponding $Q\in {\cal Q}$, we obtain a partition ${\cal E'}$ of $E'$ with $|{\cal E}'| = O((g^2+|X|^2)\cdot (g^6 \cdot |X|^6 + g^2 \cdot |X|^{10})) = O(g^8\cdot |X|^6 + g^2\cdot |X|^{12})$, and a corresponding collection ${\cal Q}$ of paths such that each $R\in {\cal E}'$ is $Q$-coupled for some $Q$ in ${\cal Q}'$. Note that for each $F\in {\cal F}$ we have at most $O((g^2+|X|^2)\cdot (g^4\cdot |X|^6 + g^2 \cdot |X|^8)) = O(g^6\cdot |X|^6 + g^2 \cdot |X|^{10})$ subsets of $E'$ in ${\cal E}'$. It is immediate from the definition that for any $R,R'\in {\cal E}'$, $Q,Q'\in {\cal Q}'$ such that $R$ is $Q$-coupled and $R'$ is $Q'$-coupled we have that $(R,R')$ is $(Q,Q')$-kissing, concluding the proof.
Centipedes and butterflies
--------------------------
Having obtained the “kissing decomposition” of the previous subsection, we now proceed to define another decomposition into structures that we call “centipedes” and “butterflies”. Intuitively, our goal is to enforce additional structures on pairs of coupled edges sets, that will subsequently allow us to compute the desired embedding into a surface of small Euler genus. Let us now formally define these structures.
Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Let ${\varphi}$ be a planar drawing of $H$. Let $x_1,x_2\in X$. Let $F$ be a face in ${\varphi}$ and let $C$ be a subpath of $F$. Let $R\subseteq E(X, V(C))$ such that $R$ is $(C, \{x_1,x_2\})$-coupled. We say that $(C,R)$ is a *$({\varphi}, \{x_1,x_2\})$-centipede* (w.r.t. $X$), or just ${\varphi}$-centipede when $x_1$ and $x_2$ are clear from the context.
Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar and connected. Let $s,t\in V(H)$, $x_1,x_2\in X$. Let $H'$ be the graph obtained by cutting $H$ along $s$ and $t$. Let $C$ be a component of $H'$ and let $R\subseteq E_G(\{x_1,x_2\}, V(C))$. We say that $(C,R)$ is an *$\{x_1,x_2\}$-butterfly* (w.r.t. $X$), or just butterfly when $x_1$ and $x_2$ are clear from the context, if the following conditions are satisfied.
[(1)]{} For any $v\in V(C) \setminus \{s,t\}$, we have $E(v,X) = E(v,\{x_1,x_2\}) \subseteq R$.
[(2)]{} There exists a planar drawing ${\varphi}$ of $C$ such that $s$, $t$, and all the endpoints of edges in $R$ lie on the boundary face of ${\varphi}$.
We refer to $s$ and $t$ as the *endpoints* of $C$ (see Figure \[fig:butterfly\] for an example).
The following is the main lemma of this subsection. It starts with the kissing decomposition from the previous subsection, and modifies the decomposition into centipedes and butterflies.
\[lem:centipede-butterly-decomposition\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar and 2-connected. Let $E'=E_G(X,V(H))$. Then there exists a planar drawing ${\varphi}$ of $H$ and a collection ${\cal A}=\{(C_i,R_i)\}_{i=1}^a$, for some $a=O(g^9 \cdot |X|^6 + g^3\cdot |X|^{12})$, such that the following conditions are satisfied:
[(1)]{} For any $i\in \{1,\ldots,a\}$, $(C_i, R_i)$ is either a ${\varphi}$-centipede or a butterfly (w.r.t. $X$).
[(2)]{} For any $i\neq j\in \{1,\ldots,a\}$, $C_i$ and $C_j$ can intersect only on their endpoints.
[(3)]{} $E' = \bigcup_{i=1}^a R_i$ and for any $i\neq j\in \{1,\ldots,t\}$, $R_i\cap R_j = \emptyset$.
Moreover, there exists a polynomial-time algorithm which given $G$, $g$, and $X$, outputs ${\varphi}$ and ${\cal A}$.
Let ${\varphi}$ be the planar drawing of $H$, and ${\cal F}=\{F_i\}_{i=1}^k$ the collection of faces in ${\varphi}$, for some $k=O(g + |X|)$, satisfying the conditions of Lemma \[lem:kissing\_decomposition\]. Let ${\cal Q}=\{P_{i,j}\}_{i,j}$ be the multi-set of paths in $H$, let ${\cal E}=\{E_{i,j}\}_{i,j}$ be the decomposition of $E'$, and let ${\varphi}$ be the planar drawing of $H$ given by Lemma \[lem:kissing\_decomposition\].
We construct ${\cal A}$ inductively. Initially we set ${\cal A}=\emptyset$. We consider all paths in ${\cal Q}$ as being unmarked and proceed to examine all unmarked paths in an arbitrary order until all paths have been marked. Let $P_1 \in {\cal Q}$ be an unmarked path. Let $E_1 \in {\cal E}$ be the corresponding $P_1$-coupled set. If $|V(P_1)|=1$, then clearly $(P_1,E_1)$ is a butterfly. We add to $(P_1,E_1)$ to ${\cal A}$ and we mark $P_1$. Otherwise, suppose that $|V(P_1)|>1$, and therefore $P_1$ contains at least one edge. Suppose first that $P_1$ does not share any edges with the remaining unmarked paths. In that case, $(P_1,E_1)$ is a centipede. We add to $(P_1,E_1)$ to ${\cal A}$ and we mark $P_1$.
Finally, suppose that $P_1$ shares some edge with at least one other unmarked path. By Lemma \[lem:kissing\_decomposition\] it follows that there exists at most one unmarked path $P_2\neq P_1\in {\cal Q}_{i}$ that shares at least one edge with $P_1$. Let $E_2 \in {\cal E}$ be the corresponding $P_2$-coupled set. Then $(E_1,E_2)$ is $(P_1,P_2)$-kissing, and the paths $P_1$ and $P_2$ share their endpoints. Moreover, there exist faces $F_1,F_2$ of ${\varphi}$ such that for each $i\in \{1,2\}$, $P_i$ is a subpath of $F_i$. Also, there exist (not necessarily distinct) $x_1,x_2,x_3,x_4\in X$ such that $E_1$ is $(P_1,\{x_1,x_2\})$-coupled and $E_2$ is $(P_2,\{x_3,x_4\})$-coupled. By condition (2) of Lemma \[lem:kissing\_decomposition\] we may assume w.l.o.g. that $E(X, V(P_1)\cap V(P_2)) \subseteq E_1\cup E_2\subseteq E(X, V(P_1))$. Let $s,t$ be the common endpoints of $P_1$ and $P_2$. Let $H'$ be the graph obtained by cutting $H$ along $s$ and $t$. Let $C$ be the component of $H'$ that contains $P_1\cup P_2$. By the definition of kissing paths the closed curve ${\varphi}(P_1)\cup {\varphi}(P_2)$ bounds a collection of zero or more disks ${\cal D}_1,\ldots,{\cal D}_{\ell}\subset \mathbb{R}^2$. Let $H_1,\ldots,H_{\ell}$ be the maximal 2-connected components of $H$ with ${\varphi}(H_i) \subset {\cal D}_i$. Then $$E\left(X, V(P_1) \cup V(P_2) \cup \left(\bigcup_{i=1}^k V(H_i)\right)\right) = E_1 \cup E_2.$$ In other words, there is no edge in $E(X,V(H))$ with an endpoint in some interior vertex of some $H_i$. Let $C'$ be the graph obtained from $C$ by contracting every $H_i$ into a single vertex $h_i$. Let also $P'$ and $G'$ be the corresponding minors of $H$ and $G$. Note that $P'$ is a path. By Lemma \[lem:interleaving\_number\_X\] there exists a multi-set of edge-disjoint subpaths $P_1',\ldots,P_{k'}'\subseteq P'$, and a decomposition $E''=E''_1\cup \ldots E''_{k'}$, where $E''=E_{G'}(X, V(P'))$, for some $k'=O(g\cdot |\{x_1,x_2,x_3,x_4\}|^3)=O(g)$, such that for any $i\in \{1,\ldots,k'\}$ the set $E''_i$ is $P'_i$-coupled. We consider all the paths $P'_{\iota}$ in the above collection in an arbitrary order. We distinguish between the following two cases for each $P'_{\iota}$:
[Case 1:]{} Suppose that $P'_{\iota}$ does not contain any of the vertices $h_1,\ldots,h_{\ell}$. Then $P'_{\iota}$ is a subpath of $H$. It follows that $(P'_{\iota}, E''_{\iota})$ is a $\{x_1,x_2\}$-butterfly (in $G$, w.r.t. $X$). We add $(P'_{\iota}, E''_{\iota})$ to ${\cal A}$.
[Case 2:]{} Suppose that $P'_{\iota}$ contains some vertex $h_{\zeta}$. Since $P'_{\iota}$ is a subpath of $P'$ it follows that all the vertices in $\{h_1,\ldots,h_{\ell}\}$ that are contained in $P'_{\iota}$ must span a continuous interval of indices, say, $h_{\sigma},\ldots,h_{\tau}$. For each $r\in \{\sigma,\ldots,\tau\}$, the boundary of $H_r$ in ${\varphi}$ consists of a closed curve ${\varphi}(P_{\iota,r,1}) \cup {\varphi}(P_{\iota,r,2})$, where $P_{\iota,r,1}$ is a subpath of $P_1$ and $P_{\iota,r,2}$ is a subpath of $P_2$. Moreover, $E_{G'}(\{x_3,x_4\}, V(P')) \subseteq E_{G'}(\{x_3,x_4\}, \{h_{\tau},\ldots,h_{\sigma}\})$. Thus $E_(\{x_3,x_4\}, V(P')) \subseteq E(\{x_3,x_4\}, \{V(P_{\iota,\sigma,2}),\ldots,V(P_{\iota,\tau,2})\})$. We have the following sub-cases:
[Case 2.1:]{} If $\{s,t\} \cap \{h_{\sigma},h_{\tau}\}=\emptyset$ then let $R$ be the set of edges in $E(G)$ corresponding to the set $E''_{\iota}$. Let also $J$ be the subgraph of $C$ corresponding to $P'_{\iota}$. Then it follows that $(J, R)$ is a $\{x_1,x_2\}$-butterfly. We add $(P'_{\iota}, E''_{\iota})$ to ${\cal A}$.
[Case 2.2:]{} Suppose that $|\{s,t\} \cap \{h_{\sigma},h_{\tau}\}|=1$, and assume w.l.o.g. that $s=h_{\sigma}$. Let $W_1 = E_1 \cap E(X,V(P_{\iota,\sigma,1}))$. Then $(P_{\iota,\sigma,1}, W_1)$ is a $({\varphi}, \{x_1,x_2\})$-centipede. Note that $W_1$ might share edges with some $E''_{\iota'}$ for some $\iota'\neq \iota$. In that case, $(P_{\iota,\sigma,1}, W_1)$ might have already been added to ${\cal A}$ while considering $P'_{\iota'}$. If this is not the case, then we add $(P_{\iota,\sigma,1}, W_1)$ to ${\cal A}$. Similarly, let $W_2 = E_2 \cap E(X,V(P_{\iota,\sigma,2}))$. Then $(P_{\iota,\sigma,2}, W_2)$ is a $({\varphi}, \{x_3,x_4\})$-centipede. If $(P_{\iota,\sigma,2}, W_2)$ is not already in ${\cal A}$ then we add it to ${\cal A}$. Finally, let $R$ be the set of edges in $E(G)$ corresponding to the set $E''_{\iota}$, and let $J$ be the subgraph of $C$ corresponding to $P'_{\iota}$. Then $(J\setminus V(H_{\sigma}), R\setminus (W_1\cup W_2))$ is a $\{x_1,x_2\}$-butterfly. We add $(J\setminus V(H_{\sigma}), R\setminus (W_1\cup W_2))$ to ${\cal A}$.
[Case 2.3:]{} Suppose that $|\{s,t\} \cap \{h_{\sigma},h_{\tau}\}|=2$, and assume w.l.o.g. that $s=h_{\sigma}$ and $t=h_{\tau}$. As in case 2.2, we let $W_1 = E_1 \cap E(X,V(P_{\iota,\sigma,1}))$ and $W_2 = E_2 \cap E(X,V(P_{\iota,\sigma,2}))$. Then $(P_{\iota,\sigma,1}, W_1)$ is a $({\varphi}, \{x_1,x_2\})$-centipede and $(P_{\iota,\sigma,2}, W_2)$ is a $({\varphi}, \{x_3,x_4\})$-centipede. Similarly, let $W_1' = E_1 \cap E(X,V(P_{\iota,\tau,1}))$ and $W_2' = E_2 \cap E(X,V(P_{\iota,\tau,2}))$. Then $(P_{\iota,\tau,1}, W_1')$ is a $({\varphi}, \{x_1,x_2\})$-centipede and $(P_{\iota,\tau,2}, W_2')$ is a $({\varphi}, \{x_3,x_4\})$-centipede. We add to ${\cal A}$ any of the above centipedes that might have not been already added to ${\cal A}$. Finally, let $R$ be the set of edges in $E(G)$ corresponding to the set $E''_{\iota}$, and let $J$ be the subgraph of $C$ corresponding to $P'_{\iota}$. Then $(J\setminus (V(H_{\sigma}\cup V(H_{\tau}))), R\setminus (W_1\cup W_2\cup W_1'\cup W_2'))$ is a $\{x_1,x_2\}$-butterfly; we add it to ${\cal A}$.
This completes the description of the collection ${\cal A}$. It is immediate by the construction that conditions (1), (2), and (3) are satisfied. Moreover, for every path in ${\cal Q}$ we add at most $O(g)$ elements in ${\cal A}$. Thus $|{\cal A}| = O(g \cdot |{\cal Q}|) = O(g^9 \cdot |X|^6 + g^3\cdot |X|^{12})$, concluding the proof.
Algorithms for the 2-connected case
-----------------------------------
We now use the decomposition into centipedes and butterflies (Lemma \[lem:centipede-butterly-decomposition\]) to obtain an embedding for $k$-apex graphs with a 2-connected planar piece. We first need to show how to embed centipedes. This is done in the following Lemma.
\[lem:embedding\_centipedes\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar and $2$-connected. Let $E'=E(X, V(H))$. Let ${\varphi}$ be a planar drawing of $H$. Let ${\cal C}=\{C_i, R_i\}_{i=1}^k$ be a collection of ${\varphi}$-centipedes such that for any $i\neq j\in \{1,\ldots,k\}$ the paths $C_i$ and $C_j$ can intersect only on their endpoints. Suppose further that $E'=\bigcup_{i=1}^k R_i$ and for any $i\neq j\in \{1,\ldots,k\}$, $R_i\cap R_j = \emptyset$. Then there exists a polynomial time algorithm which given $G, g, X, {\varphi}$, and ${\cal C}$, outputs a drawing of $G$ into a surface of genus $O(k g^2)$.
For any $i\in \{1,\ldots,k\}$, $(C_i,R_i)$ is a ${\varphi}$-centipede. Let $s_i$, $t_i$ be the endpoints of the path $C_i$. Therefore, there exist distinct $x_i,y_i\in X$ such that $R_i$ is $(C_i,\{x_i,y_i\})$-coupled. Let ${\cal C}'$ be the subset of ${\cal C}$ containing all $(C_i,R_i)$ such that all edges in $R_i$ are incident to at most one vertex in $X$, that is either $|E(x_i,V(C_i))\cap R_i)|=0$ or $|E(y_i,V(C_i))\cap R_i)|=0$. Let also ${\cal C}''={\cal C}\setminus {\cal C}'$.
We may assume w.l.o.g. that every centipede $(C_i,R_i)\in {\cal C}''$ satisfies $|E(x_i,V(C_i))\cap R_i)|\geq 2$ and $|E(y_i,V(C_i))\cap R_i)|\geq 2$. This is because if, say, $|E(x_i,V(C_i))\cap R_i)|=1$ then we may remove one edge from $G$ and move $(C_i,R_i)$ to ${\cal C}'$. In total we remove at most $k$ edges. We can then compute a drawing for the resulting graph and extend it to the removed edges by adding at most $k$ additional handles.
We may assume w.l.o.g. that for every $(C_i,R_i)\in {\cal C}''$, $\{s_i,x_i\}\in R_i$ and $\{t_i,x_i\}\in R_i$. Otherwise, let $C_{i,1}$ be the minimal subpath of $C_i$ containing all the endpoints of edges in $R_i$ that are incident to $x_i$. Similarly, let $C_{i,2}$ be the minimal subpath of $C_i$ containing all the endpoints of edges in $R_i$ that are incident to $y_i$. Then, we may remove $(C_i, R_i)$ from ${\cal C}$ and add the centipede $(C_{i,1}, R_i \cap E(\{x_i,y_i\}, V(C_{i,1}))$ in ${\cal C}$. If $C_i\setminus C_{i,1}$ consists of a single path, then we also add the centipede $(C_i\setminus C_{i,1}, R_i \cap E(\{x_i,y_i\}, V(C_i\setminus C_{i,1}))$ in ${\cal C}$. Otherwise, if $C_i\setminus C_{i,1}$ consists of two paths $W$ and $W'$, then we add the centipedes $(W, E_i\cap E(\{x_i,y_i\}, V(W))$ and $(W', E_i\cap E(\{x_i,y_i\}, V(W'))$ to ${\cal C}$. This only increases the total number of centipedes by a constant factor, so it does not affect the assertion.
Similarly, we may further assume w.l.o.g. that for any $(C_i,R_i)\in {\cal C}'$, $\{s_i,x_i\}\in R_i$ and $\{t_i,x_i\}\in R_i$, by replacing $C_i$ with the minimal subpath of $C_i$ containing all the endpoints of the edges in $R_i$.
We next define a sequence of 2-connected planar graphs $\{H_i\}_{i=0}^k$ and corresponding planar drawings $\{{\varphi}_i\}_{i=1}^k$, with $H_0=H$ and ${\varphi}_0={\varphi}$. We will maintain the following inductive invariants: $$\begin{aligned}
\text{(I1)}~~& H_i\text{ is 2-connected}\\
\text{(I2)}~~& \text{for any } j\in \{1,\ldots,i\}\text{, there exists a } {\varphi}_i\text{-face cover } E_j\cap N_{H_i}(y_j) \text{ of size }O(g^2),\\
\text{(I3)}~~& \text{for any } j\in \{i+1,\ldots,k\}, (C_j,R_j)\text{ is a } {\varphi}_i\text{-centipede},\end{aligned}$$ which are both true for $i=0$.
Given $H_i$ and ${\varphi}_i$ for some $i<k$, we proceed to define $H_{i+1}$ and ${\varphi}_{i+1}$. Let $H_{i+1}$ be the graph obtained by adding to $H_i$ the vertex $x_{i+1}$ and all edges in $E_{i+1}$ that are incident to $x_{i+1}$. That is, $V(H_{i+1}) = V(H_i) \cup \{x_{i+1}\}$ and $E(H_{i+1}) = E(H_i) \cup (E_{i+1}\cup E(x_{i+1}, V(C_{i+1})))$. Note that invariant (I1) is maintained since there are at least two edges in $E_i$ that are incident to $x_i$. By invariant (I3) $(C_{i+1}, R_{i+1})$ is a ${\varphi}_i$-centipede. It follows that $C_{i+1}$ is a subpath of a face $W_i$ of ${\varphi}_i$, and thus $H_{i+1}$ is planar. By Lemma \[lem:face\_cover\] there exists a planar drawing $\psi_{i+1}$ of $H_{i+1}$ and a $\psi_{i+1}$-face cover ${\cal F}_{i+1}$ of $N_{H_{i+1}}(y_{i+1})$ of size $O(g^2)$. Let ${\cal Z}$ be the collection of connected components of $H_{i+1}\setminus W_i$. We define the embedding ${\varphi}_{i+1}$ of $H_{i+1}$ as follows. For every component $Z\in {\cal Z}$ we set ${\varphi}_{i+1}|_Z$ to be an embedding combinatorially equivalent to ${\varphi}_{i}|_Z$. By invariant (I1) $W_i$ is a cycle in $H_{i+1}$, and thus $\gamma=\psi_{i+1}(W_i)$ is a simple loop in the plane. Therefore, every $Z\in {\cal Z}$ is embedded inside one of the two connected components of $\mathbb{R}^2\setminus \gamma$. We extend ${\varphi}_{i+1}$ to each component $Z\in {\cal Z}$ by embedding it inside the same component of $\mathbb{R}^2\setminus \gamma$ as in $\psi_{i+1}$. This can be done by setting ${\varphi}_{i+1}|_{W_i}$ to be combinatorially equivalent to $\psi_{i+1}|_{W_i}$. Since the paths of any two centipedes can intersect only on their endpoints, it follows that for any $j\in \{1,\ldots,i\}$, the size of the minimum ${\varphi}_{i+1}$-face cover of $N_{H_{i+1}}(y_j)$ is equal to the size of the minimum ${\varphi}_{i}$-face cover of $N_{H_{i}}(y_j)$, and thus invariant (I2) is maintained. Similarly, since the paths of any two centipedes may intersect only on their endpoints, it follows that invariant (I3) is also maintained. This completes the construction of the planar graphs $\{H_i\}_{i=0}^k$ and the planar drawings $\{{\varphi}_i\}_{i=0}^k$.
By invariant (I2) for any $j\in \{1,\ldots,k\}$ there exists a ${\varphi}_k$-face cover ${\cal F}_j$ of $E_j\cap N_G(y_j)$ of size $O(g^2)$. Moreover, since the paths of any two centipedes in ${\cal C}$ can only intersect on their endpoints, and for any $i\in \{1,\ldots,k\}$, $\{x_i,s_i\}\in E_i$ and $\{x_i,t_i\}\in E_i$, it follows that we can choose the collection of face covers $\{{\cal F}_i\}_i$ such that for any $j\neq j'\in \{1,\ldots,k\}$, we have ${\cal F}_j \cap {\cal F}_{j'} = \emptyset$. Therefore, we can extend ${\varphi}_k$ to $G$ by adding for each $i\in \{1,\ldots,k\}$ at most $|{\cal F}_i| = O(g^2)$ handles. Therefore, we obtain an embedding of $G$ into a surface of Euler genus $O(k g^2)$, concluding the proof.
Using the above ingredients, we are now ready to obtain our algorithm for embedding $k$-apex graphs with a 2-connected planar piece.
\[lem:genus\_2-connected\] Let $G$ be a planar graph of genus $g$ and let $X\subset V(G)$ such that $H=G\setminus X$ is planar. Suppose that $H$ is 2-connected. Then there exists a polynomial-time algorithm that given $G$, $g$, and $X$, outputs a drawing of $G$ into a surface of Euler genus $O(g^{11}\cdot |X|^6 + g^5 \cdot |X|^{12})$.
By Lemma \[lem:centipede-butterly-decomposition\] we can compute a planar drawing ${\varphi}$ of $H$ and a collection ${\cal A} = \{(C_i,R_i)\}_{i=1}^a$, for some $a=O(g^9 \cdot |X|^6 + g^3\cdot |X|^{12})$, such that: (1) for any $i\in \{1,\ldots,a\}$, $(C_i, R_i)$ is either a ${\varphi}$-centipede or a butterfly (w.r.t. $X$), (2) for any $i\neq j\in \{1,\ldots,a\}$, $C_i$ and $C_j$ can intersect only on their endpoints, and (3) $E' = \bigcup_{i=1}^a R_i$ and for any $i\neq j\in \{1,\ldots,a\}$, $R_i\cap R_j = \emptyset$.
For any $i\in \{1,\ldots,a\}$, let $s_i$, $t_i$ be the endpoints of $C_i$. Let $H'$ be the graph obtained from $H$ by replacing $C_i$ by an edge $\{s_i,t_i\}$ and removing all edges in $R_i$, for each $i\in \{1,\ldots,a\}$ such that $(C_i,R_i)$ is a ${\varphi}$-butterfly. Let also $G'$ be the corresponding graph obtained from $G$ after performing the above operation. Clearly, $G'$ is a minor of $G$, and therefore ${\mathsf{eg}}(G') \leq {\mathsf{eg}}(G)$. Moreover, since $H$ is 2-connected, the graph $H'$ is also 2-connected.
Let ${\cal C}$ be the set containing all $(C_i,R_i)\in {\cal A}$ such that $(C_i,R_i)$ is a ${\varphi}$-centipede. By Lemma \[lem:embedding\_centipedes\] we can compute an embedding $\psi$ of $G'$ into a surface of Euler genus at most $O(g^2\cdot |{\cal C}|) = O(g^2\cdot |{\cal A}|) = O(g^2 a) = O(g^{11} \cdot |X|^6 + g^5\cdot |X|^{12})$.
Consider some $\{x_1,x_2\}$-butterfly $B_i=(C_i,R_i)\in {\cal A} \setminus {\cal C}$, for some $x_1,x_2\in X$. Let $J_i$ be the graph with $V(J_i) = V(C_i) \cup \{x_1\}$ and $E(J_i) = E(C_i) \cup (E_i\cap E(x_1,V(C_i)))$. By the definition of butterfly it follows that $C_i$ is a planar graph. Moreover, there exists a planar drawing $\zeta_i$ of $C_i$ such that $s_i$, $t_i$, and all the endpoints of edges in $R_i$ in $C_i$ lie on the outer face. It follows that $J_i$ is planar. By Lemma \[lem:face\_cover\] we can compute a planar embedding $\zeta_i'$ of $J_i$ and a $\zeta_i'$-face cover ${\cal F}_i$ of $R_i \cap E_G(x_2, V(C_i))$ of size $O(g^2)$. It follows that we can extend $\zeta_i'$ to $x_2$ by adding at most $O(g^2)$ handles. This results to an embedding $\xi_i$ of the graph $J_i'$ where $V(J_i')=V(C_i)\cup \{x_1,x_2\}$ and $E(J_i') = E(C_i) \cup E_i$ into a surface of Euler genus $O(g^2)$.
We can now compute an embedding of $G$ as follows. We begin with the embedding $\psi$ of $G'$. For every butterfly $B_i=(C_i,R_i)$ there exists an edge $e_i=\{s_i,t_i\}$ in $G'$. Let $J_i'$ be the graph constructed above, and the corresponding embedding $\xi_i$ of $J_i'$ into a surface of Euler genus $O(g^2)$. We remove $e_i$ and we identify the copies of $s_i$ and $t_i$ in $J_i'$ with their copies in $G'$, by increasing the Euler genus of the underlying surface by at most $O(g^2)$. Repeating the same process for all butterflies in ${\cal A}$ we obtain an embedding of $G$ into a surface of Euler genus $O(g^{11} \cdot |X|^6 + g^5\cdot |X|^{12} + |{\cal A}| \cdot g^2) = O(g^{11} \cdot |X|^6 + g^5\cdot |X|^{12} + (g^9 \cdot |X|^6 + g^3\cdot |X|^{12}) \cdot g^2) = O(g^{11}\cdot |X|^6 + g^5 \cdot |X|^{12})$.
Dealing with 1-separators {#sec:1-separators}
=========================
In this section we present some of the tools necessary to handle $k$-apex graphs when the planar piece is not $2$-connected.
We begin by giving the proof for Lemma \[lem:all3\].
The graph $G$ contains a $K_{3,r}$ minor, with $r=|U|$. By Lemma \[lem:K33\] it follows that $|U|=O(g)$.
We need the following two definitions.
Let $H$ be a planar graph. Let $v\in V(H)$ be a 1-separator of $H$. Let $C$ be a component of $H\setminus \{v\}$. The subgraph $H[C\cup \{v\}]$ is called a *$v$-petal*.
Let $H$ be a planar graph and let $v\in V(H)$ be a 1-separator of $H$. Let $C_1,\ldots,C_t$ be $v$-petals. Then, the subgraph $C_1\cup \ldots \cup C_t$ of $H$ is called a *$v$-propeller*.
The following two lemmas are needed for the main algorithm of this section.
\[lem:x1x2x3\] Let $G$ be a graph of Euler genus $g$ and let $X\subset V(G)$ such that $H=G\setminus X$ is planar. Let ${\varphi}$ a planar drawing of $H$. Suppose that no vertex in $V(H)$ is incident to at least three vertices in $X$. Let $x_1,x_2,x_3\in X$ be distinct vertices. Let $W$ be the set of 1-separators of $H$. Let $W' = W \cap N(x_1) \cap N(x_2)$. Let ${\cal L}$ be the set of connected components of the graph obtained by cutting $H$ along $W'$. Then the following two conditions hold.
[(1)]{} Let ${\cal L}'$ be the set of all components in ${\cal L}$ that do not contain any of the leaves of ${\cal T}$ and that contain at least one vertex incident to $x_3$. Then we have $|{\cal L}'| = O(g)$.
[(2)]{} Let ${\cal L}''$ be the set of components in ${\cal L}\setminus {\cal L}'$ that contain at least one vertex incident to $x_3$. Then, there exist distinct $w_1,\ldots,w_k\in W'$, with $k=O(g)$, and for each $i\in \{1,\ldots,k\}$ a $w_i$-propeller $H_i$, such that $$\bigcup_{C\in {\cal L}''} C = \bigcup_{i=1}^k H_i.$$
[(3)]{} There exists a $(H,{\varphi})$-splitting sequence of length $O(g)$ such that in the resulting graph every cluster in ${\cal L}'\cup \{V(H_1),\ldots,V(H_k)\}$ is contained in a distinct connected component.
\(1) Let $r=|{\cal L}'|$. Pick an arbitrary $C^*\in {\cal C}$ and consider ${\cal T}$ being rooted at $C^*$. Consider some $C\in {\cal L}'$. Pick some 1-separator $u_C\in C$ that connects $C$ to some descendant component in ${\cal T}$. Since all components in ${\cal L}'$ correspond to non-leaves of ${\cal T}$ it follows that for any $C\neq C'\in {\cal L}'$ $u_C \neq u_{C'}$. For any $C\in {\cal L}'$ there exists a vertex $z_C\in V(C)$ with $\{z_C,x_3\}\in E(G)$. Since $C^*$ is connected, it follows that there exists a path $P_{C^*} \subset C^*$ between $u_C$ and $z_C$. For any $C\neq C^*$ let $p_C\in V(C)$ be the vertex minimizing $d_G(p_C, C^*)$. Since $C$ is 2-connected it follows that $C\setminus p_C$ is connected. Therefore we can pick a path $P_C$ between $z_C$ and $u_C$ that is contained in $C\setminus p_C$. It follows that for any $C\neq C'\in {\cal L}'$ we have $V(P_C) \cap V(P_{C'}) = \emptyset$. We can now obtain a $K_{3,r}$ minor in $G$ by setting the left side to be $\{x_1,x_2,x_3\}$ and the right side containing a vertex for every $P_C$ with $C\in {\cal L}'$. By Lemma \[lem:K33\] it follows that $|U|=O(g)$, concluding the proof.
\(2) We greedily choose a sequence of pairs $w_i,H_i$, where $H_i$ is $w_i$-propeller in $H$ until all components in ${\cal L}''$ are covered, as follows. Suppose we have inductively chosen $w_1,\ldots,w_{i-1}$. We pick such an uncovered component $C\in {\cal L}''$. By construction $C$ contains some 1-separator $w_i \in W'$. We set $H_i$ to be the union of all uncovered components in ${\cal L}''$ that contain $w_i$. This completes the construction of $w_1,\ldots,w_k$ and $H_1,\ldots,H_k$. Clearly we have $\bigcup_{C\in {\cal L}''} C = \bigcup_{i=1}^k H_i$. It remains to obtain an upper bound on $k$. We can construct a $K_{3,k}$ minor in $G$ as follows. We set the left side to be $\{x_1,x_2,x_3\}$. For every $i\in \{1,\ldots,k\}$ let $U_i\subseteq H_i$ be an arbitrary $w_i$-petal. For every $i\in \{1,\ldots,k\}$ the right side of the $K_{3,k}$ minor contains a vertex obtained by contracting $U_i\setminus \{w_i\}$. This completes the construction of the $K_{3,k}$ minor. It follows by Lemma \[lem:K33\] that $k=O(g)$.
\(3) Since the components in ${\cal L}'$ correspond to subtrees of ${\cal T}$ (the biconnected component tree decomposition), it follows that there exist a $(H,{\varphi})$-splitting sequence of length $|{\cal L}'|-1$ that separates every pair of components in ${\cal L}'$. Moreover, for every propeller $H_i$ there exists a single $(H,{\varphi})$-splitting that separates $H_i$ from the rest of $H$. It follows that there exists a $(H,{\varphi})$-splitting sequence of length $|{\cal L'}|-1 + k = O(g)$ that separates every pair of clusters in ${\cal L}'\cup \{V(H_1),\ldots,V(H_k)\}$, concluding the proof.
We are now ready to give a proof of Lemma \[lem:2apices\_or\_simple1separators\].
If $|X|\leq 2$ then there is nothing to show. We may therefore assume that $|X|\geq 3$. We consider all possible triples $(x_1,x_2,x_3)\in X^3$.
Let $W$ be the set of 1-separators of $H$. Let $W' = W \cap N_{G}(x_1) \cap N_{G}(x_2)$. Let ${\cal L}$ be the set of connected components of the graph obtained by cutting $H$ along $W'$. Let ${\cal L}'$ be the set of all components in ${\cal L}$ that do not contain any of the leaves of ${\cal T}$ (from the biconnected component tree decomposition) and that contain at least one vertex incident to $x_3$. Let ${\cal L}''$ be the set of components in ${\cal L}\setminus {\cal L}'$ that contain at least once vertex incident to $x_3$.
Then, by Lemma \[lem:x1x2x3\] we have $|{\cal L}'| = O({\mathsf{genus}}(G)) = O(g)$. Moreover, there exist distinct $w_1,\ldots,w_k\in W'$, with $k=O(g)$, and for each $i\in \{1,\ldots,k\}$ a $w_i$-propeller $H_i$, such that $$\bigcup_{C\in {\cal L}''} C = \bigcup_{i=1}^k H_i.$$ Also, there exists a $({\varphi}, H)$-splitting sequence $\sigma_I$ of length $O(g)$ that separates every pair of clusters in ${\cal L}'\cup \{V(H_1),\ldots,V(H_k)\}$.
Setting $\sigma$ be the $(H,{\varphi})$-splitting sequence obtained by concatenating all sequences $\sigma_I$, $I\in X^3$, we obtain a sequence of length $O(g\cdot |X|^3)$ satisfying all conditions, concluding the proof.
Bounding the number of 2-connected components with at least three apices
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We next show that there is only a bounded number of 2-connected components that are incident to at least three apices.
\[lem:few\_2connected\_3apices\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Suppose that any $1$-separator $u$ in $H$ is incident to at most one vertex in $X$, that is $|N(u)\cap X| \leq 1$. Let ${\cal L}$ be the set of maximal $2$-connected components of $H$. Then there are at most $O(|X|^3 \cdot g^{2})$ components in ${\cal L}$ that are incident to at least three vertices in $X$.
Let $x_1,x_2,x_3\in X$ be distinct. Let ${\cal L}'$ be the set of all components in ${\cal L}$ that are incident to all three vertices $x_1,x_2,x_3$. We will first upper bound $|{\cal L}'|$. The assertion will then follow by considering all possible triples of vertices in $X$.
Let $t$ be the maximum number of components in ${\cal L}'$ that intersect the same component in ${\cal L}'$, that is $t=\max_{C'\in {\cal L}'}\left|\{C\in {\cal L}' : C\cap C' \neq \emptyset\}\right|$. Let $C'\in {\cal L}'$ intersect $t$ components in ${\cal L}'$. Let ${\cal W} = \{C\in {\cal L}' : C\cap C' \neq \emptyset\}$. For any $C\in {\cal W}$ the subgraph $C\cap C'$ consists of exactly one 1-separator in $H$. For any $j\in \{1,2,3\}$ let ${\cal W}_j$ be the set of components $C\in {\cal W}$ such that the 1-separator in $C\cap C'$ is incident to $x_j$, that is $${\cal W}_j = \{C\in {\cal W} : C\cap C' = \{v\} \text{ for some } v \text{ with } \{v,x_j\}\in E(G)\}.$$ Let also ${\cal W}_0 = {\cal W} \setminus \left(\bigcup_{\ell=1}^3 {\cal W}_{\ell} \right)$. We can construct a $K_{3,\Omega(|{\cal W}_{0}|)}$ minor in $G$ as follows. The left side consists of $\{x_1,x_2,x_3\}$ and the right side contains for each $C\in {\cal W}_0$ a vertex obtained by contracting $C\setminus C'$. We can also construct a $K_{3,\Omega(|{\cal W}_{1}|)}$ minor in $G$ as follows. The left side consists of $\{x_1',x_2,x_3\}$, where $x_1'$ is the vertex obtained by contracting $x_1\cup C'$. The right side contains for each $C\in {\cal W}_0$ a vertex obtained by contracting $C\setminus C'$. Similarly, we can construct a $K_{3,\Omega(|{\cal W}_{2}|)}$ minor and a $K_{3,\Omega(|{\cal W}_{3}|)}$ minor in $G$. By Lemma \[lem:K33\] we conclude that for each $\ell\in \{0,1,2,3\}$ we have $|{\cal W}_{\ell}| = O(g)$. Therefore, $t = |{\cal W}| \leq \sum_{\ell=0}^3 |{\cal W}_{\ell}| = O(g)$. Thus, every component in ${\cal L}'$ intersects at most $O(g)$ other components in ${\cal L}'$. It follows that there exists a collection ${\cal Q}$ of $t'=\Omega(|{\cal L}'|/g)$ components in ${\cal L}'$ that are pairwise vertex-disjoint. We can thus construct a $K_{3,t'}$ minor in $G$ where the left side is $\{x_1,x_2,x_3\}$ and for any $C\in {\cal Q}$ the right side contains a vertex obtained by contracting $C$. By Lemma \[lem:K33\] it follows that $t'=O(g)$ and therefore $|{\cal L}'| = O(t \cdot t') = O(g^2)$.
It follows that $|{\cal L}| \leq {n \choose 3} \cdot |{\cal L}'| = O(|X|^3 \cdot g^2)$, concluding the proof.
Isolating a connected collection of 2-connected components
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We also need the following lemma that will be used in the main algorithm of this section, Lemma \[lem:extending\_isolated\]. It allows us to extend an embedding of one 2-connected piece to an embedding of a component with a planar piece that consists of the connected union of maximal 2-connected components.
\[lem:isolating\_2-connected\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Let ${\cal Y}$ be a set of maximal 2-connected components of $H$ such that $C=\bigcup_{Y\in {\cal Y}} Y$ is connected. Let ${\cal C}$ be the set of maximal connected subgraphs of $H$ that do not contain any edges in $E(C)$. Then the following conditions are satisfied:
[(1)]{} Let ${\cal C}_1$ be the set of components $C' \in {\cal C}$ such that $|N_G(C' \setminus C) \cap X| \geq 2$. Then, $|{\cal C}_1| = O(g \cdot |X|^2)$.
[(2)]{} Let ${\cal C}_2$ be the set of components $C' \in {\cal C}$ such that $|N_G(C' \setminus C) \cap X| = 1$ and $G[C'\cup X]$ is non-planar. Then, $|{\cal C}_2| = O(g\cdot |X|)$.
[(3)]{} Let ${\cal C}_3$ be the set of components $C'\in {\cal C}$ such that $|N_G(C' \setminus C) \cap X| = 1$, $G[C'\cup X]$ is planar, and if we let $\{v\}=V(C)\cap V(C')$ and $\{u\}=N_G(C'\setminus C)\cap X$, then there exists no planar drawing of $G[C'\cup X]$ in which $v$ and $u$ are in the same face. Then, $|{\cal C}_3| = O(g \cdot |X|)$.
[(4)]{} Let ${\cal C}_4 = {\cal C} \setminus ({\cal C}_1 \cup {\cal C}_2 \cup {\cal C}_3)$. Then each $C'\in {\cal C}_4$ is an extremity (w.r.t. $X$).
\(1) By averaging, it follows that there exists a pair of distinct vertices $x_1,x_2\in X$ and a subset ${\cal C}_1'\subseteq {\cal C}_1$ with $|{\cal C}'_1| \geq |{\cal C}_1|/{|X| \choose 2}$ such that for every $C'\in {\cal C}_1$ we have $\{x_1,x_2\}\subseteq N(C'\setminus C)$. We can construct a $K_{3,|{\cal C}_1'|}$ minor in $G$ as follows. The left side consists of $\{x_1,x_2,x_3\}$ where $x_3$ is obtained by contracting $C$ into a single vertex. It follows that for every $C'\in {\cal C}_1'$, the graph $C'\setminus C$ is connected. For every $C'\in {\cal C}'_1$, the right side of the minor contains a vertex obtained by contracting $C'\setminus C$. It follows by Lemma \[lem:K33\] that $|{\cal C}'_1| = O(g)$, and therefore $|{\cal C}_1| = O(g \cdot |X|^2)$.
\(2) Let $G'$ be the graph obtained from $G$ by contracting $C$ into a single vertex $v_C$. By averaging, it follows that there exists $x\in X$ and ${\cal C}'_2 \subseteq {\cal C}_2$ with $|{\cal C}_2'| \geq |{\cal C}_2|/|X|$ such that for any $C'\in {\cal C}'_2$ $N(C'\setminus C) \cap X = \{x\}$ and $G'[C'\cup \{x\}]$ is non-planar. By Lemma \[lem:2sum\] we have $|{\cal C}'_2| = O({\mathsf{genus}}(G')) = O({\mathsf{genus}}(G)) = O(g)$. Therefore, $|{\cal C}_2| = O(g\cdot |X|)$.
\(3) Let $G'$ be the graph obtained from $G$ by contracting $C$ into a single vertex $v_C$. By averaging, it follows that there exists $x\in X$ and ${\cal C}'_3 \subseteq {\cal C}_3$ with $|{\cal C}_3'| \geq |{\cal C}_3|/|X|$ such that for any pair of distinct $C',C''\in {\cal C}_3'$ the graph $G'[C'\cup C''\cup \{x\}]$ is non-planar. By Lemma \[lem:2sum\] it follows that $|{\cal C}'_3| = O(g)$. Therefore, $|{\cal C}_3| = O(g\cdot |X|)$.
\(4) Follows immediately by the definition of ${\cal C}_1$, ${\cal C}_2$, ${\cal C}_3$, and ${\cal C}_4$.
We next show how the above result can be used in conjunction with the tools for dealing with extremities.
\[lem:extending\_isolated\] Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. Let ${\cal Y}$ be a set of maximal 2-connected components of $H$ such that $C=\bigcup_{Y\in {\cal Y}} Y$ is connected. Let ${\cal C}_4$ be the set of subgraphs of $H$ given by Lemma \[lem:isolating\_2-connected\]. Let $G'$ be the graph obtained by contracting each $C'\in {\cal C}_4$ into a single vertex $v_{C'}$. Then, given an embedding ${\varphi}'$ of $G'[C\cup X]$ into a surface of Euler genus $\gamma$ we can compute in polynomial time an embedding of $G\left[C\cup X \cup \bigcup_{C'\in {\cal C}_4} C'\right]$ into a surface of Euler genus $\gamma$.
Follows directly by Lemma \[lem:contracting\_extremities\].
Embedding $2$-apex graphs {#sec:2-apex}
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This section deals with the the case of $2$-apex graphs. We first show that any splitting of the graph into non-planar “parts”, has a small number of components.
\[lem:2-apex-components\] Let $G$ be a planar graph of Euler genus $g$ and let $X$, with $|X|\leq 2$, such that $H=G\setminus X$ is planar. Let $({\cal T},{\cal B})$ be a biconnected component tree decomposition of $H$ (that is, each $C\in {\cal B}$ is the union of maximal 2-connected components of $H$ and for each $C, C' \in H$ we have either $C\cap C'=\emptyset$ or $C\cap C'=\{v\}$, for some 1-separator $v$ of $H$). Suppose that for each $C\in {\cal B}$, $G[V(C) \cup X]$ is not planar. Then $|{\cal B}| = O(g^4)$.
We can assume that $|X|=2$, since otherwise we can add an isolated vertex to $X$. Let $X=\{x_1,x_2\}$. Let $${\cal B}_1 = \{C \in {\cal B} : G[C\cup \{x_1\}] \text{ is non-planar}\},$$ $${\cal B}_2 = \{C \in {\cal B} : G[C\cup \{x_2\}] \text{ is non-planar}\},$$ and ${\cal B}_3 = {\cal B} \setminus ({\cal B}_1 \cup {\cal B}_2)$. We will bound the size of each ${\cal B}_i$ independently.
Let us first argue that $|{\cal B}_1|=O(g^2)$. Let $|{\cal B}_1| = \ell$. Let ${\cal B}_1 = \{C_1,\ldots,C_\ell\}$. For any $C \in {\cal B}_1$, since $G[C\cup \{x_1\}]$ is non-planar, it follows that for any planar drawing $\psi_C$ of $C$, the size of the minimum ${\varphi}_C$-face cover of $N_G(x_1)\cap V(C)$ is at least two. For any $i\in \{1,\ldots,\ell\}$ let $H_i = \bigcup_{j=1}^{\ell} C_j$. We argue by induction on $i$ that for any $i\in \{1,\ldots,\ell\}$, for any planar drawing ${\varphi}_i$ of $H_i$, the size of the minimum ${\varphi}_i$-face cover of $N_G(x_1)\cap V(H_i)$ is at least $i+1$. The base case follows by the fact that in any planar drawing ${\varphi}_1$ of $H_1$ the size of the minimum ${\varphi}_1$-face cover of $N_G(\{x_1\}, V(H_1)) = N_G(\{x_1\}, V(C_1))$ is at least two. Suppose now that the assertion holds for some $i\in \{1,\ldots,\ell-1\}$. Consider some planar drawing ${\varphi}_{i+1}$ of $H_{i+1}$. Let also ${\cal F}_{i+1}$ be a minimum ${\varphi}_{i+1}$-face cover of $N_G(\{x_1\})\cap V(H_{i+1})$. The restriction of ${\varphi}_{i+1}$ on $C_{i+1}$ is a planar drawing ${\varphi}'_{i+1}$. Moreover, the ${\varphi}_{i+1}$-face cover ${\cal F}_i$ induces a ${\varphi}'_{i+1}$-face over ${\cal F}_i'$ of $N_G(x_1)\cap V(C_{i+1})$. Similarly, the restriction of ${\varphi}_{i+1}$ on $H_i$ is a planar drawing ${\varphi}_{i+1}''$, and the ${\varphi}_{i+1}$-face cover ${\cal F}_{i+1}$ induces a ${\varphi}_{i+1}''$-face cover of $N_G(\{x_1\}) \cap V(H_i)$. By the inductive hypothesis, we have $|{\cal F}_{i+1}''|\geq i+1$. Since $C\in {\cal B}_1$ we obtain by arguing as above that $|{\cal F}_{i+1}'|\geq 2$. Therefore, there exists a least some face in ${\cal F}_{i+1}'$ that is not the outer face of ${\varphi}_{i+1}'$. It follows that $|{\cal F}_{i+1}| \geq |{\cal F}_{i+1}'| + |{\cal F}_{i+1}''| - 1 \geq i + 1 + 2 - 1 = i+2$, as required. We conclude that for any planar drawing $\psi$ of $H_\ell$ the minimum face cover of $N_G(x_1)\cap V(H_\ell)$ is at least $\ell+1$. By Lemma \[lem:face\_cover\] we obtain $\ell=O(g^2)$, as required.
Similarly we can show that $|{\cal B}_2| = O(g^2)$.
It remains to bound $|{\cal B}_3|$. Suppose that $|{\cal B}_3| = t$. We may assume w.l.o.g. that there exists some ${\cal B}_3'\subseteq {\cal B}_3$, with $|{\cal B}_3'|=t'\geq t/2$, and such that for any $C\in {\cal B}_3$ the graph $G[C\cup \{x_1\}]$ is planar. Let ${\cal B}_3''$ be a maximal subset of pairwise vertex-disjoint components in ${\cal B}_3'$. Let $|{\cal B}_3''| = t''$. Let ${\cal B}_3''= \{X_1,\ldots,X_{t''}\}$. For any $i\in \{1,\ldots,t''\}$ let $Z_i=G[X_i\cup \{x_1\}]$, and recall that $Z_i$ is a planar graph. Since two graphs $Z_i$, $Z_j$ with $i\neq j$ can intersect only on $x_1$, it follows that the graph $Z=\bigcup_{i=1}^{t''} Z_i$ is planar. Since for any $i\in \{1,\ldots,t''\}$, $G[C\cup \{x_1,x_2\}]=G[Z_i\cup \{x_2\}]$ is non-planar, it follows that for any planar drawing $\psi$ of $Z_i$, the size of the minimum $\psi$-face cover of $N_G(x_2)\cap V(Z_i)$ is at least two. Arguing by induction, we can conclude that for any planar drawing $\psi$ of $Z$, the size of a minimum $\psi$-face cover of $N_G(x_2)\cap V(Z)$ is at least $t''+1$. By Lemma \[lem:face\_cover\] we obtain that $t''=O(g^2)$. On the other hand, there exists a collection ${\cal P}\subseteq {\cal B}_3$ with $|{\cal P}|=p\geq \Omega(|{\cal B}_3'| / |{\cal B}_3''|) = \Omega(t'/t'') = \Omega(t/t'')$, and some $v\in V(H)$ such that for any $C\in {\cal P}$, $v\in V(C)$. Let ${\cal P}'\subseteq {\cal P}$ be the set containing all $C\in {\cal P}$ such that $G[C\cup \{x_1\}]$ admits a planar drawing where $x_1$ and $v$ are in the same face. Let $|{\cal P}'|=p'$. Let also ${\cal P}'' = {\cal P} \setminus {\cal P}'$ and let $|{\cal P}''| = p''$. Then, the graph $W=G\left[\{x_1\} \cup \bigcup_{C\in {\cal P}'} V(C)\right]$ is planar. We can therefore argue by induction as above that for any planar drawing $\xi$ of $W$, the size of the minimum $\xi$-face cover of $N_G(x_2)\cap V(W)$ is at least $p''+1$. By Lemma \[lem:face\_cover\], we obtain $p''=O(g^2)$. Similarly, let $U=\bigcup_{C\in {\cal P}''} C$. The graph $U$ is planar since it is a subgraph of $H$. We can argue by induction as above that for any planar drawing $\zeta$ of $U$, the size of the minimum $\zeta$-face cover of $N_G(x_1)\cap V(U)$ is at least $p''+1$. By Lemma \[lem:face\_cover\] we obtain $p''=O(g^2)$. Therefore, $p = p'+p'' = O(g^2)$. Thus, $t = O(p\cdot t'') = O(g^4)$.
We thus obtain $|{\cal B}| \leq |{\cal B}_1| + |{\cal B}_2| + |{\cal B}_3| = O(g^4)$, concluding the proof.
We now use the previous lemma to give a decomposition theorem for 2-apex graphs.
\[lem:2-apex-decomposition\] Let $G$ be a planar graph of Euler genus $g$ and let $X\subseteq V(G)$, with $|X|=2$, such that $H=G\setminus X$ is planar. Let ${\cal C}$ be the set of maximal 2-connected components of $H$. Then there exists a decomposition ${\cal C}={\cal C}_0\cup \ldots\cup {\cal C}_{2t}$, for some $t=O(g^4)$, such that the following conditions are satisfied. For any $i\in \{0,\ldots,2t\}$ let $V_i = \bigcup_{C\in {\cal C}_i} V(C)$ and $H_i = H[V_i]$. Then:
[(1)]{} For any connected component $Z$ of $H_0$, we have that $G[V(Z)\cup X]$ is planar.
[(2)]{} For any $i\in \{1,\ldots,2t\}$, $H_i$ is connected. Moreover, $G[V_i\cup X]$ is planar or $|{\cal C}_i|=1$.
Moreover, there exists a polynomial-time algorithm which given $G$, $g$, and $X$, outputs ${\cal C}_0,\ldots,{\cal C}_{2t}$.
Let ${\cal L}$ be the set of maximal 2-connected components of $H$. We inductively define the sequence of pairwise disjoint sets ${\cal C}_1,\ldots,{\cal C}_{2t}$ of ${\cal C}$; the set ${\cal C}_0$ will be defined at the end of the induction. Consider some integer $i\geq 1$, and suppose that we have already defined ${\cal C}_1,\ldots,{\cal C}_{2i}$. We proceed to define ${\cal C}_{2i+1}$ and ${\cal C}_{2i+2}$. Let $\Gamma_i=H\left[{\cal C} \setminus \bigcup_{j=1}^{2i} C\right]$. If for each connected component $Z$ of $\Gamma_i$, the graph $G[V(Z)\cup X]$ is planar, then we set $t=i$, we terminate the sequence ${\cal C}_0,\ldots,{\cal C}_{2t}$; for each connected component $Z$ of $\Gamma_i$, we add a set ${\cal C}_Z$ to ${\cal C}_0$, where ${\cal C}_Z$ consists of all the components in ${\cal C}$ that are contained in $Z$. Otherwise, let $Z$ be a connected component of $\Gamma_i$ such that $G[V(Z)\cup X]$ is non-planar. Let ${\cal M}_i$ be a maximal (possibly empty) subset of ${\cal C} \setminus \bigcup_{j=1}^{2i} {\cal C}_j$ such that the graph $G\left[X\cup \left(\bigcup_{C\in {\cal M}_i} C\right)\right]$ is planar. By the maximality of ${\cal M}_i$ it follows that there exists come $C_i\in \left({\cal C}\setminus \bigcup_{j=1}^{2i} {\cal C}_i\right)\setminus {\cal M}_i$, such that $C_i$ intersects some $C\in {\cal M}_i$, and such that the graph $G\left[X\cup C_i \cup \left(\bigcup_{C\in {\cal M}_i} C\right)\right]$ is non-planar. We set ${\cal C}_{2i+1}={\cal M}_i$, and ${\cal C}_{2i+2} = \{C_i\}$. This completes the definition of ${\cal C}_0,\ldots,{\cal C}_{2t}$.
It is immediate by the construction of ${\cal C}_0$ that for any connected component $Z$ of $H_0$ we have that $G[V(Z)\cup X]$ is planar, and thus assertion (1) is satisfied. For any $i\in \{1,\ldots,t\}$, we have that $G[V_{2i+1}\cup X]$ is planar and $|{\cal C}_{2i+2}|=1$. Thus, assertion (2) is also satisfied. It remains to bound $t$. For any $i\in \{1,\ldots,t\}$ let $H_i' = H_{2i+1}\cup H_{2i+2}$. The collection of subgraphs ${\cal B}=\{H_1',\ldots,H_t'\}$ satisfies the conditions of Lemma \[lem:2-apex-components\]. It follows by Lemma \[lem:2-apex-components\] that $t=|{\cal B}| = O(g^4)$, concluding the proof.
Putting everything together, we obtain the following algorithm for 2-apex graphs.
Let $g={\mathsf{eg}}(G)$. We define a sequence of subgraphs $G_0\subset G_1 \subset \ldots \subset G_k$, for some $k=O(g^4)$ to be specified, with $G_k = G$. Let ${\cal L}$ be the set of maximal 2-connected components of $H$. Let ${\cal C}={\cal C}_0,\ldots,{\cal C}_{2t}$ be the sequence of subsets of ${\cal L}$ computed by Lemma \[lem:2-apex-decomposition\], for some $t=O(g^4)$. Let $k=2t$. For any $i\in \{0,\ldots,k\}$, let $H_i = \bigcup_{C\in {\cal C}_i} C$, $V_i=V(H_i)$, and $$G_i = G[V_0 \cup \ldots \cup V_i \cup X].$$
We now proceed to inductively compute a sequence ${\varphi}_0,\ldots,{\varphi}_k$, where each ${\varphi}_i$ is an embedding of $G_i$ into some surface ${\cal S}_i$. Let us first compute an embedding for $G_0$. By the definition of ${\cal C}_0$ in Lemma \[lem:2-apex-decomposition\] for any connected component $J$ of $H_0$, the graph $G[V(J)\cup X]$ is planar. Let $x_1\in X$. Clearly, for any connected component $J$ of $H_0$, the graph $G[V(J)\cup \{x_1\}]$ is planar. Therefore the graph $G[H\cup \{x_1\}]$ is obtained via 1-sums of planar graphs, and therefore it is planar. Thus the graph $G[H\cup X]$ is 1-apex. By Corollary \[cor:1-apex\] we can compute an embedding ${\varphi}_0$ of $G_0$ into a surface ${\cal S}_0$ of Euler genus $O(g^2)$.
Suppose now that ${\varphi}_i$ has been computed for some $i\in \{0,\ldots,k-1\}$. We proceed to compute ${\varphi}_{i+1}$. Let ${\cal L}$ be the set of maximal 2-connected components $H$. Note that since $i+1\geq 1$, by Lemma \[lem:2-apex-decomposition\] we have that $H_{i+1}$ is connected. Let ${\cal C}$ be the set of all maximal connected subgraphs of $H$ that do not contain any edges in $E[H_{i+1}]$. Let ${\cal C}={\cal C}_1 \cup \ldots \cup {\cal C}_4$ be the decomposition computed by Lemma \[lem:isolating\_2-connected\]. We have $|{\cal C}_1|+|{\cal C}_2| + |{\cal C}_3| = O(g\cdot |X|^2) = O(g)$, since $|X|=2$. Let $G_{i+1}'$ be the graph obtained by contracting each $C'\in {\cal C}_4$ into a single vertex $v_{C'}$. We shall first compute an embedding $\psi_{i+1}$ of $G'_{i+1}$ into a surface of small genus. Subsequently, using Lemma \[lem:extending\_isolated\] we will compute the embedding ${\varphi}_{i+1}$ given $\psi_{i+1}$.
Let us first compute an embedding $\xi_{i+1}$ of $G'[V_{i+1} \cup X]$ into a surface of small genus. By condition (2) of Lemma \[lem:2-apex-decomposition\] we have that $G[V_{i+1}\cup X]$ is planar or $|{\cal C}_{i+1}|=1$. In the former case, by the definition of ${\cal C}_4$ in Lemma \[lem:isolating\_2-connected\] the graph $G'[V_{i+1}\cup X]$ must also be planar; we set $\xi_{i+1}$ to be any planar drawing of $G'[V_{i+1}\cup X]$. In the latter case, the graph $G[V_{i+1}\cup X]$ has a $2$-connected planar piece. Since $G'$ is obtained from $G$ by contracting each cluster in ${\cal C}_4$ into a single vertex, it follows that the graph $G'[V_{i+1}\cup X]$ also has a 2-connected planar piece. Therefore, we can compute using Lemma \[lem:genus\_2-connected\] an embedding $\xi_{i+1}$ of $G'[V_{i+1}\cup X]$ into a surface of Euler genus $O(g^{11}\cdot |X|^6 + g^5 \cdot |X|^{12}) = O(g^{11})$. In either case, we obtain an embedding $\xi_{i+1}$ of $G'[V_{i+1}\cup X]$ into a surface ${\cal M}_{i+1}$ of Euler genus at most $O(g^{11})$.
Let ${\cal C}' = {\cal C}_1 \cup {\cal C}_2 \cup {\cal C}_3$. By the inductive hypothesis, ${\varphi}_i$ is an embedding of $G[V_0\cup\ldots\cup V_i \cup X]$ into a surface ${\cal S}_i$ of Euler genus $O(i\cdot g^{11})$. Since $G'$ is a minor of $G$ we can compute an embedding ${\varphi}_i'$ of $G'\left[X\cup \bigcup_{K\in {\cal C}'} K \right]$ into a surface ${\cal S}'_i$ of Euler genus $O(i\cdot g^{11})$.
For every $K\in {\cal C}'$, the graphs $K$ and $H_{i+1}$ share at most one vertex. Thus, the graphs $G'\left[X\cup \bigcup_{K\in {\cal C}'} K\right]$ and $G'[V_{i+1}\cup X]$ share at most $|X|+|{\cal C}'|= O(g)$ vertices (since $|X|=2$). Therefore, we can extend $\xi_{i+1}$ to $G'[V_0\cup \ldots \cup V_{i+1} \cup X]$ by adding at most $O(g)$ cylinders, each connecting a puncture in ${\cal M}_{i+1}$ to a puncture in ${\cal S}'_{i}$. It follows that the resulting embedding $\psi_{i+1}$ of $G'[V_0\cup \ldots \cup V_{i+1}\cup X]$ is into a surface of Euler genus at most ${\mathsf{eg}}({\cal S}_i') + {\mathsf{eg}}({\cal M}_{i+1}) + O(g) = {\mathsf{eg}}({\cal S}'_i) + O(g^3)$. By Lemma \[lem:extending\_isolated\] we can now compute an embedding ${\varphi}_{i+1}$ of $G_{i+1}$ into a surface of Euler genus at most ${\mathsf{eg}}(\psi_{i+1}) + O(g^{11}) = O((i+1) \cdot g^{11})$, as required.
Since $G=G_k$, it follows that we can compute an embedding of $G$ into a surface of Euler genus $O(k\cdot g^{11}) = O(g^{15})$, concluding the proof.
Embedding locally 2-apex graphs and their generalizations {#sec:2-apex_generalizations}
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This section deals with the following case: $G$ is a graph and let $X\subseteq V(G)$ and $H=G\setminus X$. For any 2-connected maximal component $C$, $C$ has at most two neighbors in $X$. We call such a graph [[*locally 2-apex*]{}]{}.
Formally, we have the following notation.
Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. We say that $G$ is *locally 2-apex* (w.r.t. $X$) if for every maximal 2-connected component $C$ of $H$, $C$ is connected to at most two vertices in $X$, that is $|N(C)\cap X|\leq 2$.
We next derive a decomposition of locally 2-apex graphs.
\[lem:locally-2-apex\_decomposition\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$, such that $H=G\setminus X$ is planar. Suppose that $G$ is locally 2-apex (w.r.t. $X$) and that every 1-separator $v$ of $H$ is incident to at most one vertex in $X$, that is $|N_G(v)\cap X| \leq 1$. Suppose further that the extremity number of $G$ is $M$. Then there exists a tree decomposition $({\cal T}, {\cal C})$ of $H$ and some ${\cal P}\subseteq {\cal C}$, such that the following conditions are satisfied:
[(1)]{} The intersection of any two distinct clusters in ${\cal C}$ is either empty or consists of a 1-separator of $H$.
[(2)]{} Any $C\in {\cal C}\setminus {\cal P}$ is incident to at most two vertices in $X$, that is $|N_G(V(C))\cap X| \leq 2$ (thus $G[C\cup X]$ is 2-apex).
[(3)]{} $|V({\cal T})| = O(g^4 \cdot |X|^8 + g^3 \cdot |X|^6 \cdot M + g^2 \cdot |X|^4 \cdot M^2)$.
Moreover, there exists a polynomial-time algorithm which given $G$, $g$, and $X$, outputs $({\cal T}, {\cal C})$ and ${\cal P}$.
We compute the tree decomposition $({\cal T}, {\cal C})$ greedily as follows. We begin with the tree ${\cal T}$ being empty and we inductively add vertices to ${\cal T}$. Let ${\cal L}$ be the set of maximal 2-connected components of $H$. Pick some maximal ${\cal Y}\subseteq {\cal L}$ such that the graph $C=H\left[\bigcup_{Y\in {\cal Y}} V(Y)\right]$ is connected and $|N_G(V(C)) \cap X| \leq 2$. We add $C$ to $V({\cal T})$ and we consider ${\cal T}$ to be the root of ${\cal T}$.
Suppose now that we have added some $C$ to $V({\cal T})$. Let ${\cal N}$ be the set of maximal connected subgraphs of $H$ that do not have any edges in the part of $H$ that has been added to the tree decomposition $({\cal T}, {\cal C})$ and intersect $C$; in particular, every such subgraph intersects $C$ at some 1-separator of $H$. Let ${\cal N}={\cal C}_1\cup {\cal C}_2\cup {\cal C}_3\cup {\cal C}_4$ be the decomposition of ${\cal N}$ given by Lemma \[lem:isolating\_2-connected\]. We have $|{\cal C}_1| + |{\cal C}_2| + |{\cal C}_3| = O(g\cdot |X|^2)$. For each $H'\in {\cal C}_1\cup {\cal C}_2\cup {\cal C}_3$, we recursively find a tree decomposition $({\cal T}',{\cal C}')$ for $H'$, rooted at some subgraph $C'\subset H$ that shares a 1-separator with $C$. We add ${\cal T}'$ to ${\cal T}$ by making $C'$ a child of $C$. Finally, we deal with the components in ${\cal C}_4$. Every component in ${\cal C}_4$ is a maximal extremity, and any two distinct $H',H''\in {\cal C}_4$ are edge-disjoint. Therefore, $|{\cal C}_4| \leq M$ by the assumption. We add every $H'\in {\cal C}_4$ to $V({\cal T})$ by making it a child of $C$. This completes the definition of $({\cal T}, {\cal C})$. It is immediate by the construction that every node in ${\cal T}$ has at most $\delta=|{\cal C}_1|+\ldots+|{\cal C}_4| = O(M+g\cdot |X|^2)$ children.
Let $h$ be the height of ${\cal T}$. Let $C_1,\ldots,C_{h+1}$ be a branch of ${\cal T}$ where $C_1$ is the root and $C_{h+1}$ is a leaf. It follows that there exist $$y_1<\ldots < y_{2t}\in \{1,\ldots,h+1\}$$ for some $t\geq \frac{h}{2{|X|\choose 2}} = \Omega(h/\delta)$ such that for any $i\in \{1,\ldots, t\}$, $$N_G(C_{y_1})\cap X = N_G(C_{y_{2i-1}})\cap X,$$ $$N_G(C_{y_2})\cap X = N_G(C_{y_{2i}})\cap X,$$ and $$|N_G(C_{y_1}) \cup N_G(C_{y_2}) \cap X| \geq 3.$$ We can therefore obtain a $K_{3,\lfloor t/2 \rfloor}$ minor in $G$ where the left side contains any three distinct vertices in $N_G(C_{y_1}) \cup N_G(C_{y_2}) \cap X$, and for each $i\in \{1,\ldots,\lfloor t/2 \rfloor\}$ the right side contains a vertex obtained by contracting $C_{y_{4t-3}} \cup C_{y_{4t-2}}$. It follows by Lemma \[lem:K33\] that $t = O(g)$ and thus $$\begin{aligned}
h &= O(t \cdot \delta) = O(g\cdot M + g^2\cdot |X|^2). \label{eq:h1}\end{aligned}$$
Let $k=|V({\cal T})$. For each $i\in \{0,\ldots,h\}$, let $X_i$ be the set of vertices in $T$ that are at distance exactly $i$ from the root. By averaging, there exists some $d\in \{1,\ldots,h\}$ such that $|X_d|\geq k/(h+1)$. Since each vertex in $T$ has at most $\delta$ children, it follows that there exist some $Y_{d-1}\subseteq X_{d-1}$ and some $Y_d \subseteq X_d$, with $|Y_{d-1}|=|Y_d| \geq |X|_d/\delta = \Omega\left(\frac{k}{h\cdot \delta}\right)$, such that each $C\in Y_{d-1}$ has a unique child in $Y_d$ and each $C'\in Y_d$ has a unique parent in $Y_{d-1}$. Furthermore, there must exist $Z_{d-1}\subseteq Y_{d-1}$ and $Z_d\subseteq Y_d$, with $|Z_{d-1}|=|Z_d|\geq \left(\frac{|Y_d|}{|X|^4}\right)=\Omega\left(\frac{k}{h\cdot \delta \cdot |X|^4}\right)$, such that each $C\in Z_{d-1}$ has a unique child in $Z_{d}$, each $C'\in Z_d$ has a unique parent in $Z_{d-1}$, for any $C,C'\in Z_{d-1}$, $$N_G(C)\cap X = N_G(C') \cap X,$$ and for any $C,C'\in Z_{d}$, $$N_G(C)\cap X = N_G(C') \cap X.$$ We can now obtain a $K_{3,r}$ minor in $G$, for some $r = |Z_d| = \Omega\left(\frac{k}{h\cdot \delta \cdot |X|^4}\right)$ as follows. For each $C_v\in Z_{d-1}$ with unique child $C_u\in Z_d$, the right size of $K_{3,r}$ contains a vertex obtained by contracting $C_v\cup C_u$. Let $C_{v'}\in Z_{d-1}$ with $C_{v'}\not=C_v$, and let $C_{u'}$ be the unique child of $C_v$ in $Z_d$. Let $U=N_G(V(C_{v'})) \cap N_G(V(C_{u'})) \cap X$. Then, $|U|\geq 3$ by the maximality of $C_{v'}$ and $C_{u'}$ (by the construction of ${\cal T}$). We set the left side of $K_{3,r}$ to be any three distinct vertices in $U$. By Lemma \[lem:K33\] we obtain $r=O(g)$. Thus $$\begin{aligned}
\frac{k}{h\cdot \delta \cdot |X|^4} &= O(g). \label{eq:h2}\end{aligned}$$ By and we get $k = O(g\cdot h \cdot \delta \cdot |X|^4) = O(g \cdot (g\cdot M + g^2\cdot |X|^2) \cdot (M+g\cdot |X|^2) \cdot |X|^4) = O(g^4 \cdot |X|^8 + g^3 \cdot |X|^6 \cdot M + g^2 \cdot |X|^4 \cdot M^2)$, as required.
Putting everything together, we obtain the following algorithm for locally 2-apex graphs.
\[lem:embedding\_locally\_2-apex\] Let $G$ be a planar graph of Euler genus $g$ and let $X\subseteq V(G)$, such that $H=G\setminus X$ is planar. Suppose that $G$ is locally 2-apex (w.r.t. $X$) and that every 1-separator $v$ of $H$ is incident to at most one vertex in $X$, that is $|N_G(v)\cap X| \leq 1$. Suppose further that the extremity number of $G$ is $M$. Then there exists a polynomial-time algorithm which given $G$, $g$, $M$, and $X$, outputs a drawing of $G$ into a surface of Euler genus $O(g^{19} \cdot |X|^8 + g^{18} \cdot |X|^6 \cdot M + g^{17} \cdot |X|^4 \cdot M^2)$.
Let $({\cal T}, {\cal C})$ be the tree decomposition of $H$ given by Lemma \[lem:locally-2-apex\_decomposition\]. Let ${\varphi}$ be a planar drawing of $H$. There exists a $(H, {\varphi})$-splitting sequence $\sigma$ of length $|V({\cal T})|-1$ such that in the graph $H'$ obtained by performing $\sigma$ on $H$ the set of fragments is exactly ${\cal C}$. Let also $G'$ be the corresponding graph obtained from $G$. For each $C\in {\cal}$ the graph $G'_C=G'[C\cup X]$ is 2-apex. By Lemma \[lem:embedding\_2-apex\] we can therefore compute an embedding of $G'_C$ into a surface of Euler genus $O(g^{15})$. Since for every $C,C'\in {\cal C}$ the graphs $G'_C$ and $G'_{C'}$ share at most two vertices, it follows that we can combine all these embeddings into an embedding of $G'=\bigcup_{C\in {\cal C}} G'_C$ into a surface of Euler genus $O(|{\cal C}|\cdot (2+g^{15})) = O(g^{15}\cdot (g^4 \cdot |X|^8 + g^3 \cdot |X|^6 \cdot M + g^2 \cdot |X|^4 \cdot M^2)) = O(g^{19} \cdot |X|^8 + g^{18} \cdot |X|^6 \cdot M + g^{17} \cdot |X|^4 \cdot M^2)$, as required.
Generalizations of locally 2-apex graphs
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In the previous section, we give an algorithm for locally 2-apex graphs with a small number of extremities. It turns out that the techniques presented in the previous subsection can be extended to a larger family, which we call [[*nearly locally 2-apex graphs*]{}]{}. This is necessary for our algorithm for embedding $k$-apex graphs.
Let $G$ be a graph and let $X\subseteq V(G)$ such that $H=G\setminus X$ is planar. We say that $G$ is *nearly locally 2-apex* (w.r.t. $X$) if there exists at most one maximal 2-connected component of $H$ that is incident to at least three vertices in $X$.
\[lem:embedding\_nearly\_locally\_2-apex\] Let $G$ be a graph of Euler genus $g$ and let $X\subseteq V(G)$, such that $H=G\setminus X$ is planar. Suppose that $G$ is nearly locally 2-apex. Suppose further that the extremity number of $G$ is $M$. Then there exists a polynomial-time algorithm which given $G$, $g$, $M$, and $X$, outputs a drawing of $G$ into a surface of Euler genus $O(g^{20}\cdot |X|^{12} + g^{20} \cdot |X|^{11} \cdot M + g^{19} \cdot |X|^8 \cdot M^2)$.
If $G$ is locally 2-apex, then we can compute the desired embedding using Lemma \[lem:embedding\_locally\_2-apex\]. We may therefore assume that $G$ is not locally 2-apex.
Let $C$ be the single maximal 2-connected component of $H$ that is incident to at least three vertices in $X$. Let ${\cal C}$ be the set of all other components of $H$ that intersect $C$. Let ${\cal C}=\bigcup_{i=1}^4 {\cal C}_i$ be the decomposition given by Lemma \[lem:isolating\_2-connected\]. We have $|{\cal C}_1| + |{\cal C}_2| + |{\cal C}_3| = O(g\cdot |X|^2)$. Moreover we can separate $C$ from every component in ${\cal C}_1\cup {\cal C}_2\cup {\cal C}_3$ via a $(H,{\varphi})$-splitting sequence of length $O(g\cdot |X|^2)$. Let $G'$ be the resulting graph after applying the splitting sequence. By Lemma \[lem:creating\_extremities\_splitting\_sequence\] the extremity number of $G'$ is at most $M'=M+O(g\cdot |X|^3)$.
We contract each component $C'\in {\cal C}_4$ into a single vertex. Let $G''$ be the resulting graph. In this graph the planar piece $H''$ is 2-connected. We can therefore compute an embedding $\xi''$ for $G''[C\cup X]$ using Lemma \[lem:genus\_2-connected\] into a surface of genus $O(g^{11} \cdot |X|^6 + g^5 \cdot |X|^{12})$. Given this embedding $\xi''$ of $G''[C\cup X]$, we can compute using Lemma \[lem:extending\_isolated\] an embedding $\xi'$ of $G'\left[C\cup X \cup \bigcup_{C'\in {\cal C}_4} C'\right]$ into a surface of Euler genus $O(g^{11} \cdot |X|^6 + g^5 \cdot |X|^{12})$.
Next, consider some component $C'\in {\cal C}_1\cup {\cal C}_2 \cup {\cal C}_3$. The graph $G[V(C')\cup X]$ is locally 2-apex and has extremity number at most $M'$. Thus, using Lemma \[lem:embedding\_locally\_2-apex\], we can compute an embedding $\xi_{C'}$ of $G[V(C')\cup X]$ into a surface of Euler genus $O(g^{19} \cdot |X|^8 + g^{18}\cdot |X|^5 \cdot (M') + g^{17} \cdot |X|^4 \cdot (M')^2) = O(g^{19} \cdot |X|^{10} + g^{19} \cdot |X|^9 \cdot M + g^{18} \cdot |X|^6 \cdot M^2)$. We compute such an embedding for each $C'\in {\cal C}_1\cup {\cal C}_2\cup {\cal C}_3$.
Recall that the splitting sequence $\sigma$ has length $k=O(g\cdot |X|^2)$. Thus using Lemma \[lem:glueing\_fragmented\] we can combine the embeddings $\{\xi'\} \cup \{\xi_{C'}\}_{C'\in {\cal C}_1\cup {\cal C}_2 \cup {\cal C}_3}$ into an embedding $\xi$ of $G$ into a surface of Euler genus at most $O(k\cdot |X|) + {\mathsf{eg}}(\xi') + \sum_{C'\in {\cal C}_1\cup {\cal C}_2 \cup {\cal C}_3} {\mathsf{eg}}(\xi_{C'}) = O(g^{20}\cdot |X|^{12} + g^{20} \cdot |X|^{11} \cdot M + g^{19} \cdot |X|^8 \cdot M^2)$, concluding the proof.
Next we give an embedding for graphs with “simple” 1-separators. We argue that such graphs are in some sense similar to “almost” locally 2-apex graphs.
By Lemma \[lem:few\_2connected\_3apices\], there are at most $O(g^{2} \cdot |X|^{3})$ maximal 2-connected components of $H$ that are incident to at least three vertices in $X$. Therefore, there exists a planar drawing ${\varphi}$ of $H$ and some $({\varphi},H)$-splitting sequence $\sigma$ of length $k=O(g^2 \cdot |X|^3)$ that separates every pair of such components. Let $H'$ be the graph obtained by performing $\sigma$ on $H$, and let $G'$ be the corresponding graph obtained from $G$. It follows that for each fragment $C$ of $H$, $G[V(C)\cup X]$ is nearly locally 2-apex. By Lemma \[lem:creating\_extremities\_splitting\_sequence\] the extremity number of $G'$ is at most $M'=M+O(g^2 \cdot |X|^4)$. Thus, by Lemma \[lem:embedding\_nearly\_locally\_2-apex\] we can compute an embedding ${\varphi}_C$ of $G[V(C)\cup X]$ into a surface of Euler genus $O(g^{20}\cdot |X|^{12} + g^{20} \cdot |X|^{11} \cdot M' + g^{19} \cdot |X|^8 \cdot (M')^2) = O(g^{22} \cdot |X|^{15} + g^{20} \cdot |X|^{11} \cdot M)$. Let ${\cal F}$ be the set of fragments of $H'$. By Lemma \[lem:glueing\_fragmented\] we can combine all the embeddings $\{{\varphi}_C\}_{C\in {\cal F}}$ to obtain an embedding ${\varphi}$ of $G$ into a surface of Euler genus $O(k\cdot |X|) + \sum_{C\in {\cal F}} {\mathsf{eg}}({\varphi}_C) = O(g^2\cdot |X|^4) + O(g^2 \cdot |X|^3 \cdot (g^{22} \cdot |X|^{15} + g^{20} \cdot |X|^{11} \cdot M)) = O(g^{24} \cdot |X|^{18} + g^{22} \cdot |X|^{14} \cdot M)$, concluding the proof.
Face covers and embedding 1-apex graphs {#sec:1-apex}
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In this section we present our algorithm for embedding 1-apex graphs. Towards this end, we extend Mohar’s theorem on face covers for 1-apex graphs. We begin by recalling the notion of a face cover.
Let $H$ be a planar graph, and let ${\varphi}$ be a planar drawing of $H$. Let $U\subseteq V(H)$. A *${\varphi}$-face cover* for $U$ is a collection ${\cal F}=\{F_1,\ldots,F_t\}$ of faces of ${\varphi}$ such that $U\subseteq \bigcup_{i=1}^t V(F_i)$.
An EPTAS for minimum face cover on planar graphs has been obtained by Frederickson [@DBLP:journals/jacm/Frederickson91], and also follows by the more general bi-dimensionality framework of Demaine and Hajiaghayi [@DBLP:conf/soda/DemaineH05]. Moreover, the result of [@DBLP:conf/soda/DemaineH05] implies a EPTAS for the more general problem where one seeks to find a minimum face cover, over all possible planar drawings of the input graph. We remark that Frederickson gave a constant-factor approximation for this more general setting, which is also sufficient for our application. Below is a formal statement of the result we will use.
\[lem:face\_cover\_approx\] For any ${\varepsilon}>0$, there exists an algorithm with running time $f^{O(1/{\varepsilon})} n^{O(1)}$, for some function $f$, which given a planar graph $H$ and some $U\subseteq V(H)$, outputs a planar drawing ${\varphi}$ of $H$ and a ${\varphi}$-face cover of $U$ of size at most $(1+{\varepsilon}) k^*$, where $k^*$ is the minimum size of a ${\varphi}^*$-face cover of $U$, taken over all possible planar drawings ${\varphi}^*$ of $H$.
\[lem:face\_cover\_mohar\] Let $G$ be a graph of orientable genus $g$. Let $a\in V(G)$ such that the graph $H=G\setminus \{a\}$ is planar. Then, there exists a planar drawing ${\varphi}$ of $H$ and a ${\varphi}$-face cover of $N_G(a)$ of size $O(g)$.
Lemma \[lem:face\_cover\_mohar\] and Theorem \[lem:face\_cover\_approx\] together imply a $O(1)$-approximation for the orientable genus of 1-apex graphs. We shall obtain a similar algorithm for the non-orientable case. To that end, we shall next obtain a generalization of Lemma \[lem:face\_cover\_mohar\] for non-orientable embeddings. The following is implicit in the work of Mohar [@DBLP:journals/jct/Mohar01]. Mohar uses this argument to find a large collection of bouquets in a 3-connected planar graph, such that no two cycles in a bouquet have more than one vertex in common.
Let ${\cal F}$ be a collection of cycles in some graph $H$ and let $x\in V(H)$. Suppose that the intersection of any two distinct cycles in ${\cal F}$ is either $x$ or an edge incident to $x$. Then we say that ${\cal F}$ is a *bouquet* with *center* $x$.
\[lem:mohar\_face\_cover\_implicit\] Let $H$ be a planar graph and let ${\varphi}$ be a planar drawing of $H$. Let ${\cal F}$ be a collection of faces in ${\varphi}$ such that each $F\in {\cal F}$ is an induced cycle. Suppose that no two faces in ${\cal F}$ have more than one vertex in common. Then there exists ${\cal F}'\subseteq {\cal F}$, with $|{\cal F}'| \geq |{\cal F}|/10$, such that the faces in ${\cal F}'$ form a collection of bouquets in which no two cycles intersect more than in a vertex.
\[lem:SPQR\_apex\] Let $G$ be a graph genus $g$, and let $a\in V(G)$ such that $H=G\setminus \{a\}$ is planar and 2-connected. Let ${\varphi}$ be a planar drawing of $H$. Let ${\cal F}$ be a minimal collection of faces in ${\varphi}$ that cover $N_G(a)$. Then there exists ${\cal F}'\subseteq {\cal F}$, with $|{\cal F}'| = \Omega(|{\cal F}|/g)$, such that no two faces in ${\cal F}'$ have more than one vertex in common.
We first show that there can be at most $O(g)$ faces in ${\cal F}$ that all have at least two vertices in common. Let $u,v\in V(H)$, and let ${\cal J}\subseteq {\cal F}$ such that any $F\in {\cal J}$ contains both $u$ and $v$. By the minimality of ${\cal F}$ it follows that for any $F\in {\cal J}$ there exists $w_F\in V(F)$ with $a\in N_G(a)$, and such that $w_F$ is not contained in any other $F'\in {\cal J}$. We can therefore construct a $K_{3,|{\cal J}|}$ minor in $G$ as follows. The left side consists of $a$, $u$, and $v$, and the right side contains all the vertices $w_F$, $F\in {\cal J}$. It follows by Lemma \[lem:K33\] that $|{\cal J}| = O(g)$, as required. This establishes that there are at most $O(g)$ faces in ${\cal F}$ that all share any two vertices.
Next, arguing as in [@DBLP:journals/jct/Mohar01], it follows by the 4-color theorem that there exists ${\cal F}_0\subseteq {\cal F}$, with $|{\cal F}_0|\geq |{\cal F}|/4$, and such all cycles in ${\cal F}_0$ are pairwise edge-disjoint.
Let now $T$ be an SPQR tree of $H$ (for the definition of SPQR trees and further exposition we refer the reader to [@DBLP:conf/focs/BattistaT89]). The tree $T$ corresponds to a tree decomposition of $H$, where every edge in $T$ corresponding to a 2-separator in $H$. We consider $T$ a being rooted at some $r\in V(T)$. For any $v\in V(T)$ let $H_v$ be the union of all bubbles in the subtree of $T$ rooted at $v$; in particular $H_r=H$. We start with the collection of faces ${\cal F}_0$, which we initially consider to be active, and proceed to refine it until no two faces have more than one vertex in common. Along the way, we delete some of the faces and *charge* them to the faces that remain. We examine all vertices in $T$ in a bottom-up fashion starting from the leaves. Each leaf of $T$ is a $Q$-node; we ignore all such nodes. Consider now some internal node $v\in V(T)$, and suppose that we have already examined all the children of $v$. Let $u_1,\ldots,u_k$ be the children of $v$. Let ${\cal F}_v$ be the set of all active faces that contain at least one edge in $E(H_v)$. By planarity, for any $i\in \{1,\ldots,k\}$ there is a set of at most two faces ${\cal Q}_i\subseteq {\cal F}_v$ that intersect edges of $H_{u_i}$. Similarly, if $v\neq r$, then there exists a subset of at most two faces ${\cal R}\subseteq {\cal F}_v$ that intersect edges in $E(H_v)\setminus E(H_{v'})$, where $v'$ is the parent of $v$ in $T$; if $v=r$ then we set ${\cal R}=\emptyset$. For any $i\in \{1,\ldots,k\}$, if ${\cal Q}_i\cap {\cal R}=\emptyset$, and ${\cal Q}_i\neq \emptyset$, then we pick some $F\in {\cal Q}_i$, and we mark it as inactive. If there exists another face $F'\in {\cal Q}_i$, then we also mark is as inactive, and we delete it form the collection of faces. We charge $F'$ to $F$. If ${\cal Q}_i\cap {\cal R}=\{F\}$, then we consider the following two cases: (i) If ${\cal Q}_i=\{F\}$, then we do nothing. (ii) Otherwise, we have ${\cal Q}_i=\{F,F'\}$ for some $F'$; we mark both $F$ and $F'$, we delete $F$, and we charge $F$ to $F'$. If ${\cal Q}_i\cap {\cal R}=\{F,F'\}$, then we do nothing. Finally, we deal with the faces in ${\cal R}$. If ${\cal R}=\emptyset$, then we do nothing. Otherwise, we have ${\cal R}\neq \emptyset$, which implies $v\neq r$. Let $\{x,y\}$ be the 2-separator corresponding to the edge between $v$ and its parent in $T$. Let ${\cal Z}$ be the set of all faces in ${\cal F}_v\setminus {\cal R}$ that intersect both $x$ and $y$. We consider the following two cases: (i) If ${\cal Z}=\emptyset$, then we do nothing. (ii) Otherwise, we pick some $F''\in {\cal Z}$, we mark all faces in ${\cal Z}$ as inactive (including the faces in ${\cal R}$), we delete all faces in ${\cal Z}\setminus \{F''\}$, and we charge all the deleted faces to $F''$. This concludes the description of the refinement process. Let ${\cal F}'$ be the resulting set of faces.
It remains to show that ${\cal F}'$ satisfies the assertion. We first argue that no two faces in ${\cal F}'$ share more than one vertex. Any two faces that share two vertices must both contain some 2-separator of $H$. After examine a vertex $v$ of the tree $T$, we maintain the invariant that for any $i\in \{1,\ldots,k\}$, either there exists at most one face in ${\cal Q}_i$ that is not deleted, or all faces in ${\cal Q}_i$ are active. Since all faces eventually become inactive, it follows that at the end of the refinement process, at most one of the faces in ${\cal Q}_i$ is not deleted. This establishes that no two faces in ${\cal F}'$ share more than one vertex.
Finally, we argue that $|{\cal F}'| = \Omega(|{\cal F}|/g)$. To that end, it suffices to show that $|{\cal F}'| = \Omega(|{\cal F}_0|/g)$. During the refinement process, whenever we charge some face $F$ to some face $F'$, we maintain the property that $F'$ is not deleted. Therefore, it suffices to show that we charge at most $O(g)$ faces to any face. Recall that for any two distinct $u,v\in V(H)$ there exist at most $O(g)$ faces in ${\cal F}$ that contain both $u$ and $v$. Consider some face $F'\in {\cal F}'$. We can charge at most one face to $F'$ when we consider some set ${\cal Q}_i$; however, if this happens, then we don’t consider $F'$ in any of the subsequent iterations. Moreover, we may charge $O(g)$ faces to $F'$ when we consider ${\cal R}$. Overall, we can charge at most $O(g)$ faces to $F'$, which concludes the proof.
The next lemma relates the Euler genus of a 1-apex graph with a 2-connected planar piece to the size of a minimum face cover of the neighborhood of the apex, taken over all possible planar embeddings of the planar piece.
\[lem:face\_cover\_2-connected\] Let $G$ be a graph of Euler genus $g$. Let $a\in V(G)$ such that the graph $H=G\setminus \{a\}$ is planar and 2-connected. Let $\tau$ be the size of a minimum ${\varphi}$-face cover of $N_G(a)$, taken over all possible planar drawings of $H$. If $\tau\geq 2$, then $\tau=O(g^2)$.
Let $\psi$ be an embedding of $G$ into a surface of Euler genus $g$. Let ${\cal F}^*$ be a minimum ${\varphi}$-face cover of $N_G(a)$, taken over all possible planar drawings ${\varphi}$ of $H$. Let $k^* = |{\cal F}^*|$. Following Mohar [@DBLP:journals/jct/Mohar01], we let ${\cal F}_0$ be the set of faces in ${\varphi}$ that are not faces in $\psi$. We have that ${\cal F}_0$ is a ${\varphi}$-face cover of $N_G(a)$. Let ${\cal F}_1$ be a minimal subset of ${\cal F}_0$ that covers $N_G(a)$. By Lemma \[lem:SPQR\_apex\] there exists ${\cal F}_2\subseteq {\cal F}_1$ with $|{\cal F}_2|=\Omega(|{\cal F}_1|/g)$ and such that any two distinct faces in ${\cal F}_2$ share at most one vertex. By Lemma \[lem:mohar\_face\_cover\_implicit\] there exist ${\cal F}_3\subseteq {\cal F}_2$ with $|{\cal F}_3| \geq |{\cal F}_2|/10 = \Omega(|{\cal F}_1|/g)$ and such that the faces in ${\cal F}_3$ form a collection of bouquets in which any two cycles share at most one vertex.
Since $H$ is 2-connected, every face in ${\cal F}_3$ is an induced cycle. We shall modify ${\varphi}$, and the sets ${\cal F}_0,\ldots,{\cal F}_3$, until cycles in ${\cal F}_3$ become non-contractible, while preserving the minimality of the face cover ${\cal F}^*$. Suppose that there exists some cycle $C\in {\cal F}_3$ that is contractible. Let ${\cal D}$ be the disk bounded by $\psi(C)$. Since $\tau\geq 2$, it follows that $\psi(a)$ is not inside ${\cal D}$. Since $C$ is not a face in $\psi$, it follows that there exists some vertex $v\in V(H)\setminus V(C)$ such that $\psi(v)$ is in the interior of ${\cal D}$. Moreover, there exists some $X\subset H$, with $x\in V(X)$, and a 2-separator $\{x,x'\}\subseteq V(C)\cap V(X)$, such that $\psi(X)\subset {\cal D}$. The boundary of $X$ consists of two paths $X_1,X_2$ with endpoints $x$ and $x'$, where $X_1\subset C$. There exists some face $C'$ in ${\varphi}$ that contains $X_2$ as a subpath. We consider the following two cases:
[Case 1:]{} Suppose that $V(C') \cap N_G(a)=\emptyset$. Then, we delete $X\setminus C$ from $H$. Let ${\varphi}$ be the new drawing of the resulting graph $H$. Since $C'$ does not intersect the neighborhood of $a$, it follows that ${\cal F}^*$ remains a minimum face cover.
[Case 2:]{} Suppose that $V(C') \cap N_G(a)\neq \emptyset$. By the minimality of ${\cal F}^*$ it follows that $C$ must have a vertex in $N_G(a)$ that is not in $X_1$. That is, $N_G(a) \cap (V(C) \setminus V(X_1)) \neq \emptyset$. Then we modify ${\varphi}$ by performing a Whitney flip on $X$. Thus, the replace the path $X_1$ by $X_2$ in $C$, and we replace the path $X_2$ by $X_1$ in $C'$. Since both $C$ and $C'$ have vertices in $N_G(a)$ that are not in $X_1\cup X_2$, it follows that the new ${\cal F}^*$ remains a minimum face cover.
We repeat the above process until all cycles in ${\cal F}_3$ become contractible, every time recomputing the sets ${\cal F}_0, \ldots, {\cal F}_3$. We argue that the process terminates after finitely many steps. This is because in Case 1 above we delete at least one vertex from $H$, and in Case 2 we increase by at least two the number of vertices in $V(H)$ where the local rotations of ${\varphi}$ and $\psi$ agree (fixing an appropriate signature). When the process terminates we arrive at some planar embedding ${\varphi}$, and corresponding sets ${\cal F}^*$, and ${\cal F}_0,\ldots, {\cal F}_3$, where ${\cal F}^*$ is a minimum face cover, and all cycles in ${\cal F}_3$ are non-contractible.
Since ${\cal F}_1$ is minimal, it follows that every $C\in {\cal F}_3\subseteq {\cal F}_1$ must contain at least one vertex incident to $a$ that is not in the center of the bouquet containing $C$. Let $C\in {\cal F}_3$. We argue that there can be at most $2$ cycles in ${\cal F}_3$ homotopic to $C$ in $\psi$. We consider two cases:
[Case 1:]{} Suppose that the loop $\psi(C)$ is two-sided. Let $C',C''\in ({\cal F}_3\setminus {\cal F}_4)\setminus \{C\}$ such that $\psi(C')$ and $\psi(C'')$ are homotopic to $\psi(C)$. Any two of the loops in $\psi(C)$, $\psi(C')$, and $\psi(C'')$ either bound a cylinder, or a disk with two points of its boundary identified. It follows that removing $\psi(C)$, $\psi(C')$, and $\psi(C'')$ from the surface we obtain three connected components. Thus, there is no path in the surface between $\psi(a)$ and at least one of the $\psi(C)$, $\psi(C')$, and $\psi(C'')$, a contradiction.
[Case 2:]{} Suppose next that the loop $\psi(C)$ is one-sided. Arguing as above, let $C',C''\in {\cal F}'''\setminus \{C\}$ such that $\psi(C')$ and $\psi(C'')$ are homotopic to $\psi(C)$. Any two one-sided homotopic loops must intersect. Therefore there exists some bouquet with center $c$ that contains all of the cycles $C$, $C'$, and $C''$, and such that $\psi(C)$, $\psi(C')$, and $\psi(C'')$ intersect at $\psi(c)$. Any two of these loops bound a disk with two points of its boundary identified. Therefore we conclude as above that removing $\psi(C)$, $\psi(C')$, and $\psi(C'')$ from the surface we obtain three connected components, and thus there is no path in the surface between $\psi(a)$ and at least one of the $\psi(C)$, $\psi(C')$, and $\psi(C'')$, a contradiction.
We have therefore obtained that there at most two loops in ${\cal F}_3$ that are in the same homotopy class.
Let ${\cal Z}\subseteq {\cal F}_3$ be obtained by keeping at most one loop from every homotopy class. Clearly, $|{\cal Z}|\geq |{\cal F}_3|/2$. It follows that $\{\psi(C)\}_{C\in {\cal Z}}$ can be partitioned into pairwise disjoint collections of loops, where all loops in each collection intersect at exactly one base-point. It follows that $|{\cal Z}| = O(g)$, which implies $\tau = O(|{\cal F}_1|) = O(g\cdot |{\cal F}_3|) = O(g\cdot |{\cal Z}|) = O(g^2)$, concluding the proof.
Next, we obtain a generalization of Lemma \[lem:face\_cover\_2-connected\] for the case where the planar graph is not necessarily 2-connected.
\[lem:face\_cover\] Let $G$ be a graph of Euler genus $g$. Let $a\in V(G)$ such that the graph $H=G\setminus \{a\}$ is planar. Let $\tau$ be the size of a minimum ${\varphi}$-face cover of $N_G(a)$, taken over all possible planar drawings of $H$. If $\tau\geq 2$, then $\tau=O(g^2)$.
We may assume that $g\geq 1$, since otherwise the assertion is immediate. Let ${\cal C}$ be the set of maximal 2-connected components of $H$. For each $C\in {\cal C}$, let $\tau_C$ be the minimum size of a ${\varphi}_C$-face cover of $N_G(a)\cap V(C)$, taken over all planar drawings ${\varphi}_C$ of $C$.
Let ${\cal C}_0 = \{C\in {\cal C} : N_G(C)\cap X=\emptyset\}$. Let $G'$ be the graph obtained from $G$ by contracting each all $C\in {\cal C}_0$ into single vertices. Let also $H'$ be the corresponding subgraph of $G$ obtained from $H$ after contracting each $C\in {\cal C}_0$. Let ${\cal T}$ be the tree of the biconnected component tree-decomposition of $H'$. We have $V({\cal T}) = {\cal C} \setminus {\cal C}_0$. Pick some arbitrary $C_R\in V({\cal T})$. Define the partition $V({\cal T})={\cal C}_1\cup {\cal C}_2$, where ${\cal C}_1$ (resp. ${\cal C}_2$) contains all $C\in V({\cal T})$ such that the distance between $C_R$ and $C$ in ${\cal T}$ is odd (resp. even). Let $i\in \{1,2\}$. Let $G''_i$ be the graph obtained from $G'$ by contracting each cluster $C\in {\cal C}_{3-i}$ into a vertex $v_C$. Let also $H''_i$ be the corresponding subgraph of $G''_i$ obtained form $H'_i$. Every 1-separator in $H''_i$ is adjacent to $a$. For each $C\in {\cal C}_i$ let $G''_{i,C}=G''_i[V(C)\cup \{a\}]$. Therefore by Lemma \[lem:2sum\] we get ${\mathsf{eg}}(G''_i) = \Omega\left(\sum_{C\in {\cal C}_i} {\mathsf{eg}}(G''_{i,C} \right)$. For each $C\in {\cal C}_i$, let $\tau_C'$ be the minimum size of a ${\varphi}_C$-face cover of $N_{G''_i}(\{a\}) \cap V(C)$, taken over all planar drawings ${\varphi}_C$ of $C$. Since $N_{G}(\{a\}) \cap V(C) \subseteq N_{G''_i}(\{a\}) \cap V(C)$, it follows that $\tau_C'\leq \tau_C$. Let ${\cal C}_{i,1}$ be the set of all $C\in {\cal C}_i$ such that $G''_{i,C}$ is non-planar. Let also ${\cal C}_{i,2}={\cal C}_i \setminus {\cal C}_{i,1}$. By Lemma \[lem:face\_cover\_2-connected\] we have that for each $C\in {\cal C}_{i,1}$, $\tau_C' = O(({\mathsf{eg}}(G''_{i,C}))^2)$. Thus $\sum_{C\in {\cal C}_{i,1}} \tau_C \leq \sum_{C\in {\cal C}_{i,1}} \tau_C' = O\left(\sum_{C\in {\cal C}_{i,1}}({\mathsf{eg}}(G''_{i,C}))^2\right) = O(({\mathsf{eg}}(G))^2)$. Thus $\sum_{i=1}^2 \sum_{C\in {\cal C}_{i,1}} \tau_C = O(({\mathsf{eg}}(G))^2)$.
For any $C\in {\cal C}\setminus {\cal C}_0$ fix a planar drawing ${\varphi}_C$ that minimizes the size of a ${\varphi}_C$-cover ${\cal F}_C$ of $N_G(a)\cap V(C)$. Moreover, for any $C\in {\cal C}_0$, let ${\varphi}_C$ be a planar drawing of $C$ where all 1-separators of $H$ in $C$ are in the outer face. We can combine all these drawings into a planar drawing ${\varphi}$ of $H$ as follows. Let ${\cal S}$ be the tree of biconnected component tree-decomposition of $H$. We perform a traversal of ${\cal S}$ starting from some arbitrary $C_r\in V({\cal S})\setminus {\cal C}_0$. We start by setting ${\varphi}={\varphi}_{C_r}$. We also maintain a face cover of $N_G(a)\cap \Gamma$, where $\Gamma$ is the current graph on which ${\varphi}$ is defined. The first time we visit some $C\in V({\cal S})$ other than $C_r$, we extend the current planar drawing ${\varphi}$ to $C$ as follows. Let be $v$ be a 1-separator of $H$ that is in $C$, and for which ${\varphi}(v)$ is already defined (such a 1-separator always exists between $C$ and component $C'\in V({\cal S})$ we traversed before visiting $C$). If there exists a face in the current ${\varphi}$-face cover that contains $v$, then we pick such a face $F_{C}$; otherwise we set $F_{C}$ to be an arbitrary face of ${\varphi}$ containing $v$. Similarly, if there exists a face in ${\varphi}_C$ that contains $v$, then we let $R_C$ be such a face; otherwise we let $R_C$ be an arbitrary face of ${\varphi}_C$ containing $v$. We extend ${\varphi}$ to $C$ be placing $C$ inside $F_{C}$, and by placing the rest of the current graph $\Gamma$ inside $R_C$. It is immediate by induction on the traversal that the resulting face cover has size at most $\sum_{C \in {\cal C}_{1,1}\cup {\cal C}_{2,1}} \tau_C = O(({\mathsf{eg}}(G))^2)$, concluding the proof.
The following is the main result of this section, and follows immediately from Lemma \[lem:face\_cover\] and Theorem \[lem:face\_cover\_approx\]. We remark that even though the embedding obtained by Corollary \[cor:1-apex\] is into some orientable surface ${\cal S}$, the genus of ${\cal S}$ is small compared to the Euler genus of $G$. In particular, this implies that the orientable and non-orientable genus of any $1$-apex graph are polynomially related.
\[cor:1-apex\] There exists a polynomial-time algorithm which given a 1-apex graph $G$ computes a drawing of $G$ into an orientable surface of orientable genus $O(({\mathsf{eg}}(G))^2)$.
Let $a\in V(G)$ be an apex of $G$, that is such that $H=G\setminus \{a\}$ is planar. Let $U=N_G(\{a\})$. Let $k^*$ be the size of the minimum ${\varphi}^*$-face cover of $U$, taken over all possible planar drawings ${\varphi}^*$ of $H$. By Lemma \[lem:face\_cover\] we have $k^*=O(({\mathsf{eg}}(G))^2)$. By Lemma \[lem:face\_cover\_approx\] we can compute a planar drawing ${\varphi}$ of $H$ and a ${\varphi}$-face cover ${\cal F}$ of $U$ of size at most $O(k^*)=O(({\mathsf{eg}}(G))^2)$. We can extend ${\varphi}^*$ to $G$ by one handle $C_F$ for every $F\in {\cal F}$ connecting a puncture in the interior of ${\varphi}(F)$ to a puncture in a neighborhood of ${\varphi}(a)$. For each $v\in U$ covered by $F$ we map the edge $\{v,a\}$ to a path from ${\varphi}(v)$ to ${\varphi}(a)$ along $C_F$. This results in an embedding of $G$ into an orientable surface of genus at most $O(({\mathsf{eg}}(G))^2)$, as required.
Computing flat grid minors {#sec:flat_grids}
==========================
In this section we present our algorithm for computing flat grid minors (Sub-problem 1). We begin by recalling the following result from [@DBLP:conf/focs/ChekuriS13] for computing planarizing sets.
\[lem:large\_tw\_planar\_piece\] Let $G$ be a graph of Euler genus $g\geq 1$, and treewidth $t\geq
1$. There is a polynomial time algorithm to compute a set $X\subseteq V(G)$, with $|X| = O(gt \log^{5/2} n)$, and a planar connected component $\Gamma$ of $G \setminus X$ containing the $(r'\times r')$-grid as a minor, with $r' = \Omega\left(
\frac{t}{g^{3} \log^{5/2}n} \right)$. (The algorithm does not require a drawing of $G$ as part of the input.)
We will also need the following $O(1)$-approximation algorithm for computing grid minors in planar graphs.
\[lem:planar\_grid\_minor\_approx\] Let $r>0$, and let $G$ be a planar graph containing a $(r \times
r)$-grid minor. Then, on input $G$, we can compute in polynomial time a $(\Omega(r)\times \Omega(r))$-grid minor in $G$.
We first establish the following auxiliary lemma for finding a large $K_{2,r}$ minor in a grid.
\[lem:K2r\_grid\] Let $r\geq 1$ and let $H$ be the $(r\times r)$-grid. Let $\partial H$ denote the boundary cycle of $H$. Let $A\subseteq V(H) \setminus V(\partial H)$. Then $H$ contains as a minor the graph $\Gamma=K_{2,\ell}$, for some $\ell\geq |A|/3$. Furthermore, let $U_1$ and $U_2$ be the left and right sides of $\Gamma$ respectively, where $|U_1| = 2$, $|U_2|=\ell$. Then there exists a minor mapping $\mu : V(\Gamma)\to 2^{V(H)}$, such that for each $v\in U_2$, $\mu(v)\cap A\neq \emptyset$.
For any $i,j\in \{1,\ldots,r\}$ let $v_{i,j}$ be the vertex in the $i$-row and $j$-th column of $H$. For any $i\in \{1,\ldots,r\}$ let $R_i$ be the $i$-th row of $H$, that is $R_i=\bigcup_{j=1}^r \{v_{i,j}\}$, and let $C_i$ be the $i$-th column of $H$, that is $C_i = \bigcup_{j=1}^r \{v_{j,i}\}$. For any $t\in \{0,1,2\}$ we define a pair of vertex-disjoint subtrees $T_t$, $T_t'$ of $H$. Let $$T_t = H\left[R_1 \cup \left(\bigcup_{i=0}^{\lceil r/3 \rceil-1} (C_{3i+1+t}\setminus R_r) \right) \right]$$ and $$T_t' = H\left[R_r \cup \left(\bigcup_{i=0}^{\lfloor r/3 \rfloor-1} (C_{3i+3+t}\setminus R_1) \right) \right].$$ It is immediate that for any $t\in \{0,1,2\}$ the trees $T_t$ and $T_t'$ are vertex-disjoint. Moreover, for any $v\in A$ there exists a unique $t\in \{0,1,2\}$ such that $v\notin V(T_t)\cup V(T_t')$. It follows that there exists $t^*\in \{0,1,2\}$ such that $|A\setminus (V(T_{t^*}) \cup V(T_{t^*}'))| \geq |A|/3$.
Let $A'=A\setminus (V(T_{t^*}) \cup V(T_{t^*}'))$. For every $v_{i,j}\in A'$ $v_{i,j-1}\in V(T_{t^*})$ and $v_{i,j+1} \in V(T_{t^*}')$. Therefore, for every $v\in A$ there exists an edge between $v$ and some vertex in $T_{t^*}$ and an edge between $v$ and some vertex in $T_{t^*}'$. We can therefore construct a $K_{2,|A'|}$ minor in $H$ as follows. The right side is $A'$ and the left side contains a vertex obtained by contracting each one of $T_{t^*}$ and $T_{t^*}'$. This concludes the proof.
Using the above lemma we next show how to find large $K_{3,r}$ minors in a 1-apex graph.
\[lem:K3r\_apex\] Let $r\geq 1$ and let $H$ be the $(r\times r)$-grid. Let $\partial H$ denote the boundary cycle of $H$. Let $A\subseteq V(H) \setminus V(\partial H)$. Let $G$ be the graph obtained by adding a new vertex $a$ to $H$ and connecting it to all vertices in $A$. That is, $V(G)=V(H)\cup \{a\}$ and $E(G) = E(H) \cup \bigcup_{a'\in A} \{\{a,a'\}\}$. Then $G$ contains $K_{3,\ell}$ as a minor, for some $\ell\geq |A|/3$.
By Lemma \[lem:K2r\_grid\] we can compute a mapping $\mu : V(\Gamma) \to 2^{V(H)}$, where $\Gamma=K_{2,\ell}$ for some $\ell \geq |A|/3$. Furthermore, if we let $U_1$ and $U_2$ be the left and right sides of $\Gamma$ respectively, then for any $v\in U_2$ we have $\mu(v) \cap A \neq \emptyset$. Therefore, for each $v\in A$ there exists in $G$ an edge between $a$ and some vertex in $\mu(v)$. It follows that adding $a$ to $\Gamma$ we obtain a $K_{3,\ell}$ minor, as required.
We are now ready to prove the main result of this section, concerning the computation of flat grid minors.
We first use Lemma \[lem:large\_tw\_planar\_piece\] to find a set $X \subseteq V(G)$, with $|X| = O(gt \log^{5/2} n)$, and a planar connected component $C$ of $G\setminus X$, such that $C$ contains a $(r'\times r')$-grid minor, for some $r'=\Omega\left(\frac{t}{g^{3} \log^{5/2} n}\right)$. Using Lemma \[lem:planar\_grid\_minor\_approx\] we can compute a $(k\times k)$-grid minor $H$ in $C$, for some $k=\Omega(r')$. We may assume that $k\geq 3$, since otherwise the assertion holds trivially.
Fix a mapping $\mu:V(H) \to 2^{V(C)}$ for $H$. We can choose $\mu$ so that $\mu(H)=C$. Let $H'$ be the $((k-2)\times (k-2))$-grid obtained by removing the boundary cycle of $H$. Let $h=c\cdot g \cdot |X|$, for some constant $c>0$ to be specified. The grid $H'$ contains a collection of $h$ pairwise vertex-disjoint $(k'\times k')$-grids ${\cal H}=\{H_1,\ldots,H_h\}$, for some $k'=\Omega\left(\frac{k}{h^{1/2}}\right) = \Omega\left(\frac{t^{1/2}}{g^{4} \log^{15/4} n}\right)$.
Let $v\in X$ be incident to $\delta$ distinct grids in ${\cal H}$. By Lemma \[lem:K3r\_apex\] it follows that $G$ contains a $K_{3,\delta/3}$ minor. By Lemma \[lem:K33\] we obtain that $\delta=O(g)$. It follows that every $v\in X$ is incident to at most $\delta = O(g)$ distinct grids in ${\cal H}$. Thus $X$ is adjacent to at most $O(g\cdot |X|)$ vertices in $G \setminus X$. Therefore, for a sufficiently large constant $c$, we get that there exists $i\in \{1,\ldots,h\}$, such that $\mu(H_i)$ is not adjacent to $X$. It follows that the neighborhood of $\mu(H_i)$ is contained in $H$, which implies that $\mu(H_i)$ is flat, concluding the proof.
High-level overview of the proof of Lemma \[lem:CS\_summary\] {#sec:CS_summary}
=============================================================
For the sake of completeness of our exposition, we now give a high-level overview of the proof of Lemma \[lem:CS\_summary\]. For more details we refer the deader to [@DBLP:conf/focs/ChekuriS13].
We us start by recalling some definitions from [@DBLP:conf/focs/ChekuriS13]. Let $G$ be a graph, let $X\subseteq G$ be a subgraph, and let $C\subsetneq X$ be a cycle. We say that the ordered pair $(X,C)$ is a *patch* (of $G$). Let now $(X,C)$ be a patch of some graph $G$. Let ${\varphi}$ be a drawing of $G$ into a surface ${\cal S}$. We say that $(X, C)$ is a *${\varphi}$-patch* (of $G$), if there exists a disk ${\cal D} \subset {\cal S}$, satisfying the following conditions:
[(1)]{} $\partial {\cal D} = {\varphi}(C)$.
[(2)]{} ${\varphi}(G) \cap {\cal D} = {\varphi}(X)$.
Let ${\cal A}_1$, ${\cal A}_2$ be the polynomial-time algorithms satisfying conditions (1) and (2) of Lemma \[lem:CS\_summary\], respectively. That is, given a $n$-vertex graph $G$ of treewidth $t$ and an integer $g\geq 1$, algorithm ${\cal A}_1$ either correctly decides that ${\mathsf{eg}}(G)>g$, or outputs a flat subgraph $G'\subset G$, such that $X$ contains a $\left(\Omega(r)\times \Omega(r)\right)$-grid minor, for some $r=r(n,g,t)$. Also, given an $n$-vertex graph $G$, an integer $g$, and some $X\subset V(G)$ such that $G\setminus X$ is planar, algorithm ${\cal A}_2$ either correctly decides that ${\mathsf{eg}}(G) > g$, or it outputs a drawing of $G$ into a surface of Euler genus at most $\gamma$, for some $\gamma=\gamma(n, g, |X|)$.
We now describe a polynomial-time algorithm ${\cal A}_3$ that satisfies the conclusion of Lemma \[lem:CS\_summary\]. The input consists of a graph $G$ and some integer $g\geq 1$. The algorithm proceeds in the following steps:
[**Step 1: Computing a skeleton.**]{} Let $t$ be the treewidth of $G$. We first describe a polynomial-time procedure which either correctly decides that ${\mathsf{eg}}(G)>g$ or outputs a collection of pairwise non-overlapping patches $(X_1,C_1),\ldots,(X_{\ell},C_{\ell})$ of $G$, so that the following conditions are satisfied:
[(1)]{} If ${\mathsf{eg}}(G)\leq g$, then there exists a drawing ${\varphi}$ of $G$ into a surface of Euler genus $g$, such that for any $i \in \{1,\ldots, \ell\}$, $(X_i, C_i)$ is a ${\varphi}$-patch. We emphasize the fact that ${\varphi}$ is not explicitly computed by the algorithm.
[(2)]{} Let $t'$ be the treewidth of the graph $G\setminus \left(\bigcup_{i=1}^{\ell} (X_i \setminus C_i) \right)$. Then $r(n,g,t')=O(g)$.
If $r(n,g,t)=O(g)$, then we may simply output an empty sequence of patches. We may therefore assume that $r(n,g,t)=c\cdot g$, for some sufficiently large universal constant $c>0$ to be specified later. We inductively compute the desired sequence of ${\varphi}$-patches, for some embedding ${\varphi}$ of $G$ into a surface of Euler genus $g$, as follows. We remark that we do not compute ${\varphi}$; we are only guaranteed that some ${\varphi}$ satisfying the condition exists. Suppose that we have computed some $(X_1,C_1),\ldots,(X_i,C_i)$. Let $G_i=G\setminus \left(\bigcup_{i=1}^{\ell} (X_i \setminus C_i) \right)$. In particular, we have $G_0=G$. Let $c$ be a sufficiently large constant, to be specified later. If the treewidth $t_i$ of $G_i$ satisfies $r(n,g,t_i) \leq c\cdot g$, then we terminate the sequence of patches by setting $\ell=i$. Otherwise, we proceed to compute $(X_{i+1}, C_{i+1})$. Using algorithm ${\cal A}_1$ we either correctly decide that ${\mathsf{eg}}(G_i)>g$ (and therefore ${\mathsf{eg}}(G)>g$) or we compute a flat subgraph $G'_i\subset G_i$, such that $G_i'$ contains a $\left(c\cdot g \times c\cdot g\right)$-grid minor. Using Lemma \[lem:planar\_grid\_minor\_approx\] we can compute a $(c\cdot c'\cdot g\times c\cdot c'\cdot g)$-grid minor $H_i$ in $G_i'$, for some universal constant $c'>0$. By setting $c$ to be sufficiently large, the grid $H_i$ contains a sequence at least $g+3$ “planarly nested” cycles. It can be shown that in any embedding of $G$ into a surface of Euler genus $g$, the “inner-most” cycle must bound a disk. This allows us to compute a ${\varphi}$-patch $(X_{i+1}, C_{i+1})$. With some extra analysis, after “merging” some of the patches, we may ensure that the disks bounded by different patches do not intersect. This completes the computation of the desired ${\varphi}$-patches $(X_1,C_1),\ldots,(X_{\ell},C_{\ell})$, for some embedding ${\varphi}$.
Let $G'=G\setminus \left(\bigcup_{i=1}^{\ell} (X_i \setminus C_i) \right)$. We refer to $G'$ as the *skeleton*.
[**Step 2: Framing the skeleton.**]{} For every $i\in \{1,\ldots,\ell\}$, we have that $C_i$ is a cycle in $G'$. We add a $(|V(C_i)|\times 3)$-cylinder $K_i$ to $G'$ by identifying the outer face of $K_i$ with $C_i$. Let $G''$ be the resulting graph. We refer to $G''$ as the *framed skeleton*.
[**Step 3: Embedding the framed skeleton.**]{} By recursively removing balanced vertex-separators, we can compute some $X\subseteq V(G')$ with $|X|=O(t' g \log^{3/2}n)$, such that $G''=G'\setminus X$ is planar. Moreover, we may ensure that for each component $\Gamma$ of $G'$, the “framing” of $\Gamma$ (defined in an analogous manner) remains planar.
Using algorithm ${\cal A}_2$ we either correctly decide that ${\mathsf{eg}}(G')>g$ (and therefore ${\mathsf{eg}}(G)>g$), or we compute an embedding of $G'$ into a surface of Euler genus at most $\gamma(n, g, |X|) = \gamma(n, g, k)$, for some $k=O(t' g \log^{3/2}n)$.
[**Step 4: Extending the embedding to the original graph.**]{} Since $G''$ is obtained from $G'$ by removing $O(t' g \log^{3/2}n)$ vertices, it follows that the cycles $C_1,\ldots,C_{\ell}$ are partitioned in $G''$ into at most $\ell+O(t' g \log^{3/2}n)$ cycles and paths. We may therefore extend the embedding of $G''$ to $G'$ by adding at most $O(t' g \log^{3/2}n)$ new handles to the surface. At the end of the algorithm we obtain an embedding of $G'$ into a surface of Euler genus at most $\gamma(n, g, k) + O(t' g \log^{3/2}n)$, for some $k=O(t' g \log^{3/2}n)$.
This completes the high-level overview of the proof of Lemma \[lem:CS\_summary\].
[^1]: National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, Japan. `[email protected]`. Supported by JST ERATO Kawarabayashi Large Graph Project.
[^2]: Dept. of Computer Science and Engineering, and Dept. of Mathematics, The Ohio State University. Columbus, OH, 43210. `[email protected]`. Supported by NSF grant CCF 1423230.
[^3]: We consider multi-sets of edge-disjoint paths so that we may allow single-vertex paths to be repeated.
|
{
"pile_set_name": "ArXiv"
}
|
Multimode description of correlated two-photon state
====================================================
In this Appendix, we introduce a general model for quantum detection efficiency for multimode analysis in various quantum communication scheme. Based on this detection model with the spectral description of correlated two-photon state, we derive the effective density matrix conditioning on the detection events of entanglement swapping, polarization maximally entangled (PME) state projection, and quantum teleportation.
Quantum Efficiency of Detector
------------------------------
To account for quantum efficiency of detector and the affect of its own spectrum filtering, we introduce an extra beam splitter (B.S.) with a transmissivity $\eta(\omega,\omega_{0})$ [@det] before the detection event. $\ \eta$ models the quantum efficiency of the detectors in the microscopic level (response at frequency $\omega_{0}$) and the macroscopic level (time-integrated detection). One example of conditioning on the single click of the detector, the output density operator becomes $$\begin{aligned}
\hat{\rho}_{out} & =\int_{-\infty}^{\infty}d\omega_{0}\hat{\Pi}_{1}%
\text{Tr}_{ref}\big[\hat{U}_{BS}\hat{\rho}_{in}\hat{U}_{BS}^{\dag}%
\big]\hat{\Pi}_{1}\label{model}\\
\hat{\Pi}_{1} & \equiv\int_{-\infty}^{\infty}d\omega|\omega\rangle
\langle\omega|\\
\hat{U}_{BS} & \equiv\left(
\begin{array}
[c]{cc}%
\sqrt{1-\eta} & \sqrt{\eta}\\
\sqrt{\eta} & -\sqrt{1-\eta}%
\end{array}
\right)\end{aligned}$$ where $\text{Tr}_{ref}$ is the trace over the reflected modes $m_{3}^{\dag}, $ and the flat spectrum projection operator $\hat{\Pi}_{1}$ (only photon number is projected and no frequency resolution) is considered in the measurement process [@spectral]. In Figure \[detect\], $m_{1}^{\dag} $ is the incoming photon operator before the detection, $m_{3}^{\dag}$ is the reflected mode, and $m_{4}^{\dag}$ is now the detection mode with a modelling of spectral quantum efficiency and an effective quantum efficiency is defined as $$\int_{-\infty}^{\infty}\eta(\omega,\omega_{0})d\omega_{0}=\eta_{eff}(\omega).$$
\[ptb\]
[detect.png]{}
Multimode Description of Entanglement Swapping
----------------------------------------------
From Eq. (\[mode\]), we use single mode $\Phi(\omega)$ for Raman photon and a multimode description $f(\omega_{s},\omega_{i})$ for cascade photons and rewrite the effective state. Note that a symmetric setup is considered so the mode description is the same for both sides A and B in the scheme of entanglement swapping. $$\begin{aligned}
& |\Psi\rangle_{eff}=\eta_{1}(1-\eta_{2})\times\nonumber\\
& \int f(\omega_{s},\omega_{i})\hat{a}_{s}^{\dag,A}(\omega_{s})\hat{a}%
_{i}^{\dag,A}(\omega_{i})d\omega_{s}d\omega_{i}\int f(\omega_{s}^{\prime
},\omega_{i}^{\prime})\hat{a}_{s}^{\dag,B}(\omega_{s}^{\prime})\hat{a}%
_{i}^{\dag,B}(\omega_{i}^{\prime})d\omega_{s}^{\prime}d\omega_{i}^{\prime
}|0\rangle+\nonumber\\
& \eta_{2}(1-\eta_{1})\int\Phi(\omega)d\omega\hat{a}_{r}^{\dag,A}(\omega
)\hat{S}_{A}^{\dag}\int\Phi(\omega^{\prime})d\omega^{\prime}\hat{a}_{r}%
^{\dag,B}(\omega^{\prime})\hat{S}_{B}^{\dag}|0\rangle+\sqrt{\eta_{1}%
(1-\eta_{1})}\times\nonumber\\
& \sqrt{\eta_{2}(1-\eta_{2})}\int f(\omega_{s},\omega_{i})d\omega_{s}%
d\omega_{i}\times\hat{a}_{s}^{\dag,A}(\omega_{s})\hat{a}_{i}^{\dag,A}%
(\omega_{i})\int\Phi(\omega^{\prime})d\omega^{\prime}\hat{a}_{r}^{\dag
,B}(\omega^{\prime})\hat{S}_{B}^{\dag}|0\rangle+\nonumber\\
& \sqrt{\eta_{1}\eta_{2}(1-\eta_{1})(1-\eta_{2})}\int\Phi(\omega)d\omega
\hat{a}_{r}^{\dag,A}(\omega)\hat{S}_{A}^{\dag}\int f(\omega_{s}^{\prime
},\omega_{i}^{\prime})\hat{a}_{s}^{\dag,B}(\omega_{s}^{\prime})\hat{a}%
_{i}^{\dag,B}(\omega_{i}^{\prime})d\omega_{s}^{\prime}d\omega_{i}^{\prime
}|0\rangle.\nonumber\\
&\end{aligned}$$
With the B.S., we have $\hat{a}_{i}^{\dag,A}=\frac{\hat{m}_{1}^{\dag}+\hat
{m}_{2}^{\dag}}{\sqrt{2}}$, $\hat{a}_{i}^{\dag,B}=\frac{\hat{n}_{1}^{\dag
}+\hat{n}_{2}^{\dag}}{\sqrt{2}}$, $\hat{a}_{r}^{\dag,A}=\frac{\hat{m}%
_{1}^{\dag}-\hat{m}_{2}^{\dag}}{\sqrt{2}}$, $\hat{a}_{r}^{\dag,B}=\frac
{\hat{n}_{1}^{\dag}-\hat{n}_{2}^{\dag}}{\sqrt{2}}$, where $\hat{a}%
_{i}^{\dagger}$ is the creation operator for idler photon and $\hat{a}%
_{r}^{\dagger}$ is for Raman photon. The input density operator is $\hat{\rho}_{in}=|\Psi\rangle_{eff}\langle\Psi|$ and conditioning on the pair of single click ($\hat{m}_{1,2}^{\dag},\hat{n}_{1,2}^{\dag}$), we are able to generate maximally entangled singlet or triplet state $|\Psi\rangle
_{DLCZ}=\frac{S_{A}^{\dag}\pm S_{B}^{\dag}}{\sqrt{2}}|0\rangle_{A,B}$. Without loss of generality, we consider a triplet state along with a pair of clicks ($\hat{m}_{1}^{\dag},\hat{n}_{1}^{\dag}$) and use the model of quantum efficiency in Eq. (\[model\]) with tracing over the detection modes ($\hat{m}_{4}^{\dag},\hat{n}_{4}^{\dag}$). Note that $\hat{m}_{1}^{\dag
}=\sqrt{1-\eta}\hat{m}_{3}^{\dag}+\sqrt{\eta}\hat{m}_{4}^{\dag}$ and $\hat
{n}_{1}^{\dag}=\sqrt{1-\eta}\hat{n}_{3}^{\dag}+\sqrt{\eta}\hat{n}_{4}^{\dag}$ as we model the quantum efficiency in the previous Section. $$\begin{aligned}
\hat{\rho}_{out} & =\int_{-\infty}^{\infty}d\omega_{0}\text{Tr}%
_{m4,n4}\big\{\text{Tr}_{m3,n3}\big[\hat{U}_{BS}^{B}\hat{U}_{BS}^{A}\hat{\rho
}_{in}\hat{U}_{BS}^{\dag,A}\hat{U}_{BS}^{\dag,B}\big]\hat{M}_{4,4}\big\}\\
\hat{M}_{4,4} & \equiv(\hat{I}_{m4}^{\dag}-|0\rangle_{m4}\langle
0|)\otimes|0\rangle_{m2}\langle0|\otimes(\hat{I}_{n4}^{\dag}-|0\rangle
_{n4}\langle0|)\otimes|0\rangle_{n2}\langle0|\end{aligned}$$ where the unitary B.S. operator is denoted by both sides (A and B) and NRPD projection operators are used [@shapiro]. These operators project the state with single click of the detected mode without resolving the number of photons. $\ \hat{I}$ is identity operator. The un-normalized output density operator after tracing out these modes becomes $$\begin{aligned}
& \hat{\rho}_{out}=\frac{\eta_{1}^{2}(1-\eta_{2})^{2}}{4}\times\nonumber\\
& \int d\omega_{i}d\omega_{i}^{\prime}\eta_{eff}(\omega_{i})\eta_{eff}%
(\omega_{i}^{\prime})\big[\int f(\omega_{s},\omega_{i})\hat{a}_{s}^{\dag
,A}(\omega_{s})d\omega_{s}\int f(\omega_{s}^{\prime},\omega_{i}^{\prime}%
)\hat{a}_{s}^{\dag,B}(\omega_{s}^{\prime})d\omega_{s}^{\prime}\big]\nonumber\\
& |0\rangle\langle0|\big[\int f^{\ast}(\omega_{s}^{\prime\prime},\omega
_{i})\hat{a}_{s}^{A}(\omega_{s}^{\prime\prime})d\omega_{s}^{\prime\prime}\int
f^{\ast}(\omega_{s}^{\prime\prime\prime},\omega_{i}^{\prime})\hat{a}_{s}%
^{B}(\omega_{s}^{\prime\prime\prime})d\omega_{s}^{\prime\prime\prime
}\big]\nonumber\\
& +\frac{\eta_{1}\eta_{2}(1-\eta_{1})(1-\eta_{2})}{4}\bigg\{\int d\omega
_{i}\eta_{eff}(\omega_{i})\Big[\int f(\omega_{s},\omega_{i})dw_{s}\int
f^{\ast}(\omega_{s}^{\prime},\omega_{i})d\omega_{s}^{\prime}\nonumber\\
& \int|\Phi(\omega)|^{2}\eta_{eff}(\omega)d\omega\Big]\Big(\hat{a}_{s}%
^{\dag,A}(\omega_{s})\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{B}\hat
{a}_{s}^{A}(\omega_{s}^{\prime})+\nonumber\\
& \hat{a}_{s}^{\dag,B}(\omega_{s})\hat{S}_{A}^{\dag}|0\rangle\langle0|\hat
{S}_{A}\hat{a}_{s}^{B}(\omega_{s}^{\prime})\Big)+\int\int f(\omega_{s}%
,\omega_{i})d\omega_{s}\Phi^{\ast}(\omega_{i})\eta_{eff}(\omega_{i}%
)d\omega_{i}\times\nonumber\\
& \int\int f^{\ast}(\omega_{s}^{\prime},\omega_{i}^{\prime})d\omega
_{s}^{\prime}\Phi(\omega_{i}^{\prime})\eta_{eff}(\omega_{i}^{\prime}%
)d\omega_{i}^{\prime}\Big(\hat{a}_{s}^{\dag,A}(\omega_{s})\hat{S}_{B}^{\dag
}|0\rangle\langle0|\hat{S}_{A}\hat{a}_{s}^{B}(\omega_{s}^{\prime})+\nonumber\\
& \hat{a}_{s}^{\dag,B}(\omega_{s})\hat{S}_{A}^{\dag}|0\rangle\langle0|\hat
{S}_{B}\hat{a}_{s}^{A}(\omega_{s}^{\prime})\Big)\bigg\}+\hat{\rho}%
_{out}^{\prime}%\end{aligned}$$ where $\eta_{eff}(\omega)$ is introduced after integration of $\omega_{0}, $ and we denote it as an effective quantum efficiency for idler field $\omega_{i}$ or Raman photon at frequency $\omega$ (wavelength $780$ nm for D2 line of Rb atom). $\ \hat{\rho}_{out}^{\prime}$ includes the terms that won’t survive after the interference of telecom photons in the middle B.S. (conditioning on a single click of detector). They involve operators like $\hat{a}_{s}^{\dag,A}\hat{a}_{s}^{\dag,B}|0\rangle\langle0|\hat{a}_{s}^{A}%
\hat{S}_{B}$, $\hat{a}_{s}^{\dag,A}\hat{a}_{s}^{\dag,B}|0\rangle\langle
0|\hat{S}_{A}\hat{S}_{B}$ and $\hat{S}^{\dag,A}\hat{S}^{\dag,B}|0\rangle
\langle0|\hat{S}_{A}\hat{S}_{B}$.
The normalization factor is derived by tracing over the atomic degree of freedom. $$\begin{aligned}
& \text{Tr}(\hat{\rho}_{out})\equiv\mathcal{N}=\nonumber\\
& \frac{\eta_{1}^{2}(1-\eta_{2})^{2}}{4}\int d\omega_{s}d\omega_{i}\eta
_{eff}(\omega_{i})|f(\omega_{s},\omega_{i})|^{2}\int d\omega_{s}^{\prime
}d\omega_{i}^{\prime}\eta_{eff}(\omega_{i}^{\prime})|f(\omega_{s}^{\prime
},\omega_{i}^{\prime})|^{2}+\nonumber\\
& \frac{\eta_{1}\eta_{2}(1-\eta_{1})(1-\eta_{2})}{2}\int d\omega_{s}%
d\omega_{i}\eta_{eff}(\omega_{i})|f(\omega_{s},\omega_{i})|^{2}\int|\Phi
|^{2}(\omega)\eta_{eff}(\omega)d\omega+\nonumber\\
& \frac{\eta_{2}^{2}(1-\eta_{1})^{2}}{4}\int|\Phi|^{2}(\omega)\eta
_{eff}(\omega)d\omega\int|\Phi|^{2}(\omega^{\prime})\eta_{eff}(\omega^{\prime
})d\omega^{\prime}\label{normalization}%\end{aligned}$$ which will be put back when we calculate the heralding and success probabilities.
Next we interfere telecom photons with B.S. that $\hat{a}_{s}^{\dag,A}%
=\frac{\hat{c}_{1}^{\dag}+\hat{c}_{2}^{\dag}}{\sqrt{2}}$, $\hat{a}_{s}%
^{\dag,B}=\frac{\hat{c}_{1}^{\dag}-\hat{c}_{2}^{\dag}}{\sqrt{2}},$ and again a quantum efficiency $\eta(\omega,\omega_{0})$ for telecom photon is introduced. Use $\hat{c}_{1}^{\dag}=\sqrt{1-\eta}\hat{c}_{3}^{\dag}+\sqrt{\eta}\hat
{c}_{4}^{\dag}$ and trace over the reflected mode $\hat{c}_{3}^{\dag}$ conditioning on the click of $\hat{c}_{4}^{\dag}$ from NRPD. The effective density matrix becomes $$\begin{aligned}
\hat{\rho}_{out}^{(2)} & =\int_{-\infty}^{\infty}d\omega_{0}\text{Tr}%
_{c4}\big\{\text{Tr}_{c3}\big[\hat{U}_{BS}^{C}\hat{\rho}_{in}\hat{U}%
_{BS}^{\dag,C}\big]\hat{M}_{4}\big\}\nonumber\\
& \equiv\int_{-\infty}^{\infty}d\omega_{0}\hat{\rho}_{out}^{(2)}(\omega
_{0}),\\
\hat{\rho}_{out}^{(2)}(\omega_{0}) & \equiv\text{Tr}_{c4}\big\{\hat{\rho
}_{in}^{(2)}(\omega_{0})\big\}\\
\hat{M}_{4} & \equiv(\hat{I}_{c4}^{\dag}-|0\rangle_{c4}\langle0|)\otimes
|0\rangle_{c2}\langle0|,\end{aligned}$$ $$\begin{aligned}
& \hat{\rho}_{in}^{(2)}(\omega_{0})=\frac{\eta_{1}^{2}(1-\eta_{2})^{2}}%
{16}\int d\omega_{i}d\omega_{i}^{\prime}\eta_{eff}(\omega_{i})\eta
_{eff}(\omega_{i}^{\prime})\bigg\{\nonumber\\
& \int d\omega_{s}(1-\eta(\omega_{s}))f(s,i)f^{\ast}(s,i^{\prime})\int
d\omega_{s}^{\prime}f(s^{\prime},i^{\prime})\sqrt{\eta(\omega_{s}^{\prime}%
)}\hat{c}_{4}^{\dag}(\omega_{s}^{\prime})|0\rangle\langle0|\times\nonumber\\
& \int d\omega_{s}^{\prime\prime}\hat{c}_{4}(\omega_{s}^{\prime\prime}%
)\sqrt{\eta(\omega_{s}^{\prime\prime})}f^{\ast}(s^{\prime\prime},i)+\int
d\omega_{s}(1-\eta(\omega_{s}))f(s,i)f^{\ast}(s,i)\times\nonumber\\
& \int d\omega_{s}^{\prime}f(s^{\prime},i^{\prime})\sqrt{\eta(\omega
_{s}^{\prime})}\hat{c}_{4}^{\dag}(\omega_{s}^{\prime})|0\rangle\langle0|\int
d\omega_{s}^{\prime\prime\prime}\hat{c}_{4}(\omega_{s}^{\prime\prime\prime
})\sqrt{\eta(\omega_{s}^{\prime\prime\prime})}f^{\ast}(s^{\prime\prime\prime
},i^{\prime})+\nonumber\\
& \int d\omega_{s}^{\prime}(1-\eta(\omega_{s}^{\prime}))f(s^{\prime}%
,i^{\prime})f^{\ast}(s^{\prime},i^{\prime})\int d\omega_{s}f(s,i)\sqrt
{\eta(\omega_{s})}\hat{c}_{4}^{\dag}(\omega_{s})|0\rangle\langle
0|\times\nonumber\\
& \int d\omega_{s}^{\prime\prime}\hat{c}_{4}(\omega_{s}^{\prime\prime}%
)\sqrt{\eta(\omega_{s}^{\prime\prime})}f^{\ast}(s^{\prime\prime},i)+\int
d\omega_{s}^{\prime}(1-\eta(\omega_{s}^{\prime}))f(s^{\prime},i^{\prime
})f^{\ast}(s^{\prime},i)\times\nonumber\\
& \int d\omega_{s}f(s,i)\sqrt{\eta(\omega_{s})}\hat{c}_{4}^{\dag}(\omega
_{s})|0\rangle\langle0|\int d\omega_{s}^{\prime\prime\prime}\hat{c}_{4}%
(\omega_{s}^{\prime\prime\prime})\sqrt{\eta(\omega_{s}^{\prime\prime\prime}%
)}f^{\ast}(s^{\prime\prime\prime},i^{\prime})+\nonumber\\
& \int d\omega_{s}^{\prime}\sqrt{\eta(\omega_{s}^{\prime})}f(s^{\prime
},i^{\prime})\int d\omega_{s}\sqrt{\eta(\omega_{s})}f(s,i)\hat{c}_{4}^{\dag
}(\omega_{s})\hat{c}_{4}^{\dag}(\omega_{s}^{\prime})|0\rangle\langle
0|\times\nonumber\\
& \int d\omega_{s}^{\prime\prime}\sqrt{\eta(\omega_{s}^{\prime\prime})}%
f^{\ast}(s^{\prime\prime},i)\int d\omega_{s}^{\prime\prime\prime}\sqrt
{\eta(\omega_{s}^{\prime\prime\prime})}f^{\ast}(s^{\prime\prime\prime
},i^{\prime})\hat{c}_{4}(\omega_{s}^{\prime\prime})\hat{c}_{4}(\omega
_{s}^{\prime\prime\prime})\bigg\}+\nonumber\\
& \frac{\eta_{1}\eta_{2}(1-\eta_{1})(1-\eta_{2})}{8}\bigg\{\int d\omega
_{i}\eta_{eff}(\omega_{i})\int f(s,i)d\omega_{s}\int f^{\ast}(s^{\prime
},i)d\omega_{s}^{\prime}\times\nonumber\\
& \int d\omega|\Phi(\omega)|^{2}\eta_{eff}(\omega)\sqrt{\eta(\omega_{s})}%
\hat{c}_{4}^{\dag}(\omega_{s})\Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat
{S}_{B}+\hat{S}_{A}^{\dag}|0\rangle\langle0|\hat{S}_{A}\Big)\times\nonumber\\
& \hat{c}_{4}(\omega_{s}^{\prime})\sqrt{\eta(\omega_{s}^{\prime})}\int\int
f(s,i)d\omega_{s}\Phi^{\ast}(\omega_{i})\eta_{eff}(\omega_{i})d\omega
_{i}\times\nonumber\\
& \int\int f^{\ast}(s^{\prime},i^{\prime})d\omega_{s}^{\prime}\Phi(\omega
_{i}^{\prime})\eta_{eff}(\omega_{i}^{\prime})d\omega_{i}^{\prime}\sqrt
{\eta(\omega_{s})}\hat{c}_{4}^{\dag}(\omega_{s})\times\nonumber\\
& \Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{A}+\hat{S}_{A}^{\dag
}|0\rangle\langle0|\hat{S}_{B}\Big)\hat{c}_{4}(\omega_{s}^{\prime})\sqrt
{\eta(\omega_{s}^{\prime})}\bigg\}\end{aligned}$$ where a brief notation for spectrum $f(s,i)\equiv f(\omega_{s},\omega_{i})$ and quantum efficiency $\eta(\omega)\equiv\eta(\omega,\omega_{0})$. This quantum efficiency refers to the telecom photon. We proceed to trace over the detected modes and the density matrix can be simplified by interchange of variables in integration. $$\begin{aligned}
& \hat{\rho}_{out}^{(2)}(\omega_{0})=\frac{\eta_{1}^{2}(1-\eta_{2})^{2}}%
{8}\int d\omega_{i}d\omega_{i}^{\prime}\eta_{eff}(\omega_{i})\eta_{eff}%
(\omega_{i}^{\prime})\bigg\{\nonumber\\
& \int d\omega_{s}(1-\eta(\omega_{s},\omega_{0}))f(\omega_{s},\omega
_{i})f^{\ast}(\omega_{s},\omega_{i}^{\prime})\int d\omega_{s}^{\prime}%
f(\omega_{s}^{\prime},\omega_{i}^{\prime})f^{\ast}(\omega_{s}^{\prime}%
,\omega_{i})\eta(\omega_{s}^{\prime},\omega_{0})+\nonumber\\
& \int d\omega_{s}(1-\eta(\omega_{s},\omega_{0}))|f(\omega_{s},\omega
_{i})|^{2}\int d\omega_{s}^{\prime}|f(\omega_{s}^{\prime},\omega_{i}^{\prime
})|^{2}\eta(\omega_{s}^{\prime},\omega_{0})+\nonumber\\
& \frac{1}{2}\int d\omega_{s}^{\prime}\eta(\omega_{s}^{\prime},\omega
_{0})|f(\omega_{s}^{\prime},\omega_{i}^{\prime})|^{2}\int d\omega_{s}%
\eta(\omega_{s},\omega_{0})|f(\omega_{s},\omega_{i})|^{2}+\frac{1}{2}%
\times\nonumber\\
& \int d\omega_{s}^{\prime}\eta(\omega_{s}^{\prime},\omega_{0})f(\omega
_{s}^{\prime},\omega_{i}^{\prime})f^{\ast}(\omega_{s}^{\prime},\omega_{i})\int
d\omega_{s}\eta(\omega_{s},\omega_{0})f(\omega_{s},\omega_{i})f^{\ast}%
(\omega_{s},\omega_{i}^{\prime})\bigg\}|0\rangle\langle0|\nonumber\\
& +\frac{\eta_{1}\eta_{2}(1-\eta_{1})(1-\eta_{2})}{8}\bigg\{\int d\omega
_{i}\eta_{eff}(\omega_{i})\int\eta(\omega_{s},\omega_{0})|f(\omega_{s}%
,\omega_{i})|^{2}d\omega_{s}\times\nonumber\\
& \int d\omega|\Phi(\omega)|^{2}\eta_{eff}(\omega)\Big(\hat{S}_{B}^{\dag
}|0\rangle\langle0|\hat{S}_{B}+\hat{S}_{A}^{\dag}|0\rangle\langle0|\hat{S}%
_{A}\Big)+\nonumber\\
& \int\int\eta(\omega_{s},\omega_{0})f(\omega_{s},\omega_{i})d\omega_{s}%
\Phi^{\ast}(\omega_{i})\eta_{eff}(\omega_{i})d\omega_{i}\int f^{\ast}%
(\omega_{s},\omega_{i}^{\prime})\Phi(\omega_{i}^{\prime})\eta_{eff}(\omega
_{i}^{\prime})d\omega_{i}^{\prime}\times\nonumber\\
& \Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{A}+\hat{S}_{A}^{\dag
}|0\rangle\langle0|\hat{S}_{B}\Big)\bigg\}\end{aligned}$$ where the trace over two photon states requires the commutation relation of photon operators. $$\begin{aligned}
& \text{Tr}[\hat{m}_{4}^{\dag}(\omega_{s})\hat{m}_{4}^{\dag}(\omega
_{s}^{\prime})|0\rangle\langle0|\hat{m}_{4}(\omega_{s}^{\prime\prime})\hat
{m}_{4}(\omega_{s}^{\prime\prime\prime})]\nonumber\\
& =\langle0|\hat{m}_{4}(\omega_{s}^{\prime\prime})[\delta(\omega_{s}%
,\omega_{s}^{\prime\prime\prime})+\hat{m}_{4}^{\dag}(\omega_{s})\hat{m}%
_{4}(\omega_{s}^{\prime\prime\prime})]\hat{m}_{4}^{\dag}(\omega_{s}^{\prime
})|0\rangle\nonumber\\
& =\delta(\omega_{s},\omega_{s}^{\prime\prime\prime})\delta(\omega_{s}%
^{\prime\prime},\omega_{s}^{\prime})+\delta(\omega_{s},\omega_{s}%
^{\prime\prime})\delta(\omega_{s}^{\prime},\omega_{s}^{\prime\prime\prime}).\end{aligned}$$
The above is the general formulation for the un-normalized density matrix conditioning on three clicks of NRPD’s. We’ve included spectral quantum efficiency of the detector either for near-infrared ($\eta_{eff}$) or telecom wavelength ($\eta_{t}\equiv\int_{-\infty}^{\infty}\eta(\omega,\omega
_{0})d\omega_{0}$)
To proceed, we assume a flat and finite spectrum response ($\eta_{eff}%
(\omega)=\eta_{eff}$, $\eta_{t}(\omega)=\eta_{t}$) with the range $\omega
_{0}\in\lbrack\Omega-\Delta,\Omega+\Delta]$ centered at $\Omega$ (near-infrared or telecom) and $\omega\in\lbrack\omega_{0}-\delta,\omega
_{0}+\delta]$. The widths $2\Delta$ and $2\delta$ are large enough compared to our source bandwidth so these detection events do not give us any information of spectrum for our source. A perfect efficiency also means no photon loss during detection. Note that the integral involves multiplication of two telecom photon efficiency $\int_{-\infty}^{\infty}\eta(\omega
,\omega_{0})\eta(\omega^{\prime},\omega_{0})d\omega_{0}=\eta_{t}^{2}(\omega)$ that is valid if the source bandwidth is smaller than detector’s.
After the integration of $\omega_{0}$, we have $$\begin{aligned}
& \hat{\rho}_{out}^{(2)}=\frac{\eta_{1}^{2}(1-\eta_{2})^{2}}{8}\eta_{eff}%
^{2}\int d\omega_{i}d\omega_{i}^{\prime}\bigg\{(1-\eta_{t})\eta_{t}\int
d\omega_{s}f(\omega_{s},\omega_{i})f^{\ast}(\omega_{s},\omega_{i}^{\prime
})\times\nonumber\\
& \int d\omega_{s}^{\prime}f(\omega_{s}^{\prime},\omega_{i}^{\prime})f^{\ast
}(\omega_{s}^{\prime},\omega_{i})+(1-\eta_{t})\eta_{t}\int d\omega
_{s}|f(\omega_{s},\omega_{i})|^{2}\int d\omega_{s}^{\prime}|f(\omega
_{s}^{\prime},\omega_{i}^{\prime})|^{2}+\nonumber\\
& \frac{\eta_{t}^{2}}{2}\int d\omega_{s}^{\prime}|f(\omega_{s}^{\prime}%
,\omega_{i}^{\prime})|^{2}\int d\omega_{s}|f(\omega_{s},\omega_{i})|^{2}%
+\frac{\eta_{t}^{2}}{2}\int d\omega_{s}^{\prime}f(\omega_{s}^{\prime}%
,\omega_{i}^{\prime})f^{\ast}(\omega_{s}^{\prime},\omega_{i})\times\nonumber\\
& \int d\omega_{s}f(\omega_{s},\omega_{i})f^{\ast}(\omega_{s},\omega
_{i}^{\prime})\bigg\}|0\rangle\langle0|+\frac{\eta_{1}\eta_{2}(1-\eta
_{1})(1-\eta_{2})}{8}\eta_{t}\eta_{eff}^{2}\times\nonumber\\
& \bigg\{\int d\omega_{i}\int|f(\omega_{s},\omega_{i})|^{2}d\omega_{s}\int
d\omega|\Phi(\omega)|^{2}\Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}%
_{B}+\hat{S}_{A}^{\dag}|0\rangle\langle0|\hat{S}_{A}\Big)+\nonumber\\
& \int\int f(\omega_{s},\omega_{i})d\omega_{s}\Phi^{\ast}(\omega_{i}%
)d\omega_{i}\int f^{\ast}(\omega_{s},\omega_{i}^{\prime})\Phi(\omega
_{i}^{\prime})d\omega_{i}^{\prime}\nonumber\\
& \Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{A}+\hat{S}_{A}^{\dag
}|0\rangle\langle0|\hat{S}_{B}\Big)\bigg\}.\label{out}%\end{aligned}$$
Density Matrix of PME Projection and Quantum Teleportation
----------------------------------------------------------
In Chapter 5.4, we have the normalized density operator $\hat{\rho}%
_{out,n}^{(2),AB}$ of the DLCZ entangled state through entanglement swapping. With another pair of DLCZ entangled state, $\hat{\rho}_{out,n}^{(2),CD}$, the joint density operator for these two pairs constructs the polarization maximally entangled state (PME) projection and is interpreted as $$\begin{aligned}
& \hat{\rho}_{out,n}^{(2),AB}\otimes\hat{\rho}_{out,n}^{(2),CD}=\nonumber\\
& \frac{1}{(a+b)^{2}}\bigg\{a^{2}|0\rangle\langle0|+\frac{ab}{2}%
\Big[|0\rangle_{AB}\langle0|\Big(\hat{S}_{C}^{\dag}|0\rangle\langle0|\hat
{S}_{C}+\hat{S}_{D}^{\dag}|0\rangle\langle0|\hat{S}_{D}\nonumber\\
& +\lambda_{1}\hat{S}_{C}^{\dag}|0\rangle\langle0|\hat{S}_{D}+\lambda_{1}%
\hat{S}_{D}^{\dag}|0\rangle\langle0|\hat{S}_{C}\Big)+|0\rangle_{CD}%
\langle0|\Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{B}+\hat{S}%
_{A}^{\dag}|0\rangle\langle0|\hat{S}_{A}\nonumber\\
& +\lambda_{1}\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{A}+\lambda_{1}%
\hat{S}_{A}^{\dag}|0\rangle\langle0|\hat{S}_{B}\Big)\Big]+\frac{b^{2}}%
{4}\Big(\hat{S}_{C}^{\dag}|0\rangle\langle0|\hat{S}_{C}+\hat{S}_{D}^{\dag
}|0\rangle\langle0|\hat{S}_{D}\nonumber\\
& +\lambda_{1}\hat{S}_{C}^{\dag}|0\rangle\langle0|\hat{S}_{D}+\lambda_{1}%
\hat{S}_{D}^{\dag}|0\rangle\langle0|\hat{S}_{C}\Big)\otimes\Big(\hat{S}%
_{B}^{\dag}|0\rangle\langle0|\hat{S}_{B}+\hat{S}_{A}^{\dag}|0\rangle
\langle0|\hat{S}_{A}\nonumber\\
& +\lambda_{1}\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{A}+\lambda_{1}%
\hat{S}_{A}^{\dag}|0\rangle\langle0|\hat{S}_{B}\Big)\bigg\},\end{aligned}$$ which is used to calculate the success probability after post measurement \[a click from each side, the side of (A or C) and (B or D)\]. $a=\eta_{r}%
(2-\eta)\Big(1+\sum_{j}\lambda_{j}^{2}\Big),b=4$, and $\eta_{r}=\eta_{1}%
/\eta_{2}$, $\eta=\eta_{t}$, $\lambda_{j}$ is Schmidt number that is used to decompose the two-photon source from the cascade transition.
In DLCZ protocol, quantum teleportation uses the similar setup in PME projection and combines with the desired teleported state, $|\Phi
\rangle=(d_{0}\hat{S}_{I_{1}}^{\dag}+d_{1}\hat{S}_{I_{2}}^{\dag})|0\rangle$, which is represented by two other atomic ensembles $I_{1}$ and $I_{2}$. $\ $The requirement of normalization of the state is $d_{0}|^{2}+|d_{1}%
|^{2}=1$, and the density operator of quantum teleportation is $\hat{\rho
}_{QT}=|\Phi\rangle\langle\Phi|\otimes\hat{\rho}_{out,n}^{(2),AB}\otimes
\hat{\rho}_{out,n}^{(2),CD}$. Conditioning on clicks of $\hat{D}_{I_{1}}$ and $\hat{D}_{I_{2}}$, the effective density matrix for quantum teleportation is (using $\hat{S}_{I_{1}}^{\dag}=(\hat{D}_{I_{1}}+\hat{D}_{A})/\sqrt{2},$ $\hat{S}_{I_{2}}^{\dag}=(\hat{D}_{I_{2}}+\hat{D}_{C})/\sqrt{2}$ for the effect of beam splitter) $$\begin{aligned}
& \hat{\rho}_{QT,eff}=\Big[\frac{|d_{0}|^{2}}{2}(\hat{D}_{I_{1}}^{\dag
}|0\rangle\langle0|\hat{D}_{I_{1}})+\frac{|d_{1}|^{2}}{2}(\hat{D}_{I_{2}%
}^{\dag}|0\rangle\langle0|\hat{D}_{I_{2}})+\frac{d_{0}d_{1}^{\ast}}{2}(\hat
{D}_{I_{1}}^{\dag}|0\rangle\langle0|\hat{D}_{I_{2}})\nonumber\\
& +\frac{d_{0}^{\ast}d_{1}}{2}(\hat{D}_{I_{2}}^{\dag}|0\rangle\langle0|\hat
{D}_{I_{1}})\Big]\otimes\frac{1}{(a+b)^{2}}\bigg\{a^{2}|0\rangle
\langle0|+\frac{ab}{2}\Big[|0\rangle_{AB}\langle0|\nonumber\\
& \Big(\frac{\hat{D}_{I_{2}}^{\dag}|0\rangle\langle0|\hat{D}_{I_{2}}}{2}%
+\hat{S}_{D}^{\dag}|0\rangle\langle0|\hat{S}_{D}+\lambda_{1}\frac{\hat
{D}_{I_{2}}^{\dag}}{\sqrt{2}}|0\rangle\langle0|\hat{S}_{D}+\lambda_{1}\hat
{S}_{D}^{\dag}|0\rangle\langle0|\frac{\hat{D}_{I_{2}}}{\sqrt{2}}%
\Big)\nonumber\\
& +|0\rangle_{CD}\langle0|\Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}%
_{B}+\frac{\hat{D}_{I_{1}}^{\dag}|0\rangle\langle0|\hat{D}_{I_{1}}}{2}%
+\lambda_{1}\hat{S}_{B}^{\dag}|0\rangle\langle0|\frac{\hat{D}_{I_{1}}}%
{\sqrt{2}}+\lambda_{1}\frac{\hat{D}_{I_{2}}^{\dag}}{\sqrt{2}}|0\rangle
\langle0|\hat{S}_{B}\Big)\Big]\nonumber\\
& +\frac{b^{2}}{4}\Big(\frac{\hat{D}_{I_{2}}^{\dag}|0\rangle\langle0|\hat
{D}_{I_{2}}}{2}+\hat{S}_{D}^{\dag}|0\rangle\langle0|\hat{S}_{D}+\lambda
_{1}\frac{\hat{D}_{I_{2}}^{\dag}}{\sqrt{2}}|0\rangle\langle0|\hat{S}%
_{D}+\lambda_{1}\hat{S}_{D}^{\dag}|0\rangle\langle0|\frac{\hat{D}_{I_{2}}%
}{\sqrt{2}}\Big)\otimes\nonumber\\
& \Big(\hat{S}_{B}^{\dag}|0\rangle\langle0|\hat{S}_{B}+\frac{\hat{D}_{I_{1}%
}^{\dag}|0\rangle\langle0|\hat{D}_{I_{1}}}{2}+\lambda_{1}\hat{S}_{B}^{\dag
}|0\rangle\langle0|\frac{\hat{D}_{I_{1}}}{\sqrt{2}}+\lambda_{1}\frac{\hat
{D}_{I_{2}}^{\dag}}{\sqrt{2}}|0\rangle\langle0|\hat{S}_{B}%
\Big)\bigg\},\label{QT}%\end{aligned}$$ which is used to calculate the success probability for teleported state.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Schrödinger equation with a potential periodically varying in time is used to model adiabatic quantum pumps. The systems considered may be either infinitely extended and gapped or finite and connected to gapless leads. Correspondingly, two descriptions of the transported charge, one relating to a Chern number and the other to a scattering matrix, have been available for some time. Here we generalize the first one and establish its equivalence to the second.'
author:
- |
G. Bräunlich, G.M. Graf, G. Ortelli\
*Theoretische Physik, ETH-Zürich, CH–8093 Zürich*
title: Equivalence of topological and scattering approaches to quantum pumping
---
Introduction
============
Quantum pumps are driven devices connected to leads kept at a same voltage. Two descriptions of charge transport are available for pumps depending on time periodically and adiabatically. One has been proposed by Thouless [@Th] (see also [@ThN]), the other by Büttiker et al. [@BTP] (see also [@B]). We shall refer to them as the topological, resp. the scattering approaches and denote by $\langle Q_T\rangle$, resp. $\langle Q_{BPT}\rangle$ the charges transported during a cycle. Each one depends on a different idealization of the devices. In the first proposal the model is a non-interacting Fermi gas, infinitely extended in one dimension with the Fermi energy lying in a gap. The charge transported within a period appears as a Chern number, indicating that it is quantized. In the second approach the device is viewed as a compact object connected to leads containing free, gapless Fermi gases. Here, the transported charge is expressed in terms of the scattering matrix at Fermi energy and is quantized in special cases only.
At first sight charge transport is accounted for in rather different, if not opposing, ways: The spatial extent of the two devices is infinite, resp. finite, reflecting a microscopic, resp. macroscopic, perspective; more strikingly, in the first case transport is attributed to energies way below the Fermi energy, which lies in a spectral gap, while in the second the scattering matrix matters only at Fermi energy. In physical terms, the first description applies to insulators, the second to conductors, at least seemingly so.
Yet, the two points of view are mathematically related. This has been shown in [@GO] for the simpler case of a single channel, modeled as a real line, and of a potential which is periodic also in space. A comparison becomes possible after truncating the potential to finitely many periods, while the rest of the line gives raise to the leads. Then the spectral gap closes and the model becomes amenable to the scattering approach. There, the conditions for quantized transport are attained in the limit of many periods, and quantitative agreement between the two approaches was established.
In this article we generalize the equivalence result in two ways, thereby extending it to the natural setting of both approaches. First, the requirement of spatial periodicity [@Th] is dropped. Such a situation was considered in [@ThN], though by approximating a general (e.g. quasi-periodic) potential by a sequence of periodic ones with increasing periods. Only the approximants were associated to fiber bundles, based on the corresponding Brillouin zones. Here we propose a bundle and hence a Chern number applying directly to the infinite, non-periodic system. Second, we extend the correspondence [@GO] to a multi-channel setting.
As far as we know, the earliest statement concerning the equivalence is found in [@Chern], though only for a particular, exactly solvable, periodic, tight binding Hamiltonian. On more general terms we note that, albeit the topological approach predates the scattering approach, several ideas underlying the equivalence can be traced back to [@Th]. Experimental work which is thematically related is described e.g. in [@Shilton; @Leek; @Blu].
In Section \[section:2\] we state the results for charge transport based on the two approaches separately, and formulate the comparison, which is the main result, as Theorem \[thm:2\]. In Section \[section:3\] we describe the relevant fiber bundle, while Section \[section:4\] is devoted to proofs. An appendix provides a result in adiabatic perturbation theory.
Main results {#section:2}
============
We begin by describing the topological approach [@Th] in the case of $n$ channels. The Hamiltonian, acting on $L^2(\mathbb{R}_x, \mathbb{C}^n)$, is $$\label{eq:uno}
H(s) = -\frac{d^2}{dx^2} + V(x, s) \,,$$ where the potential $V=V(x, s)$ takes values in the $n\times n$ matrices, $M_n(\mathbb{C})$, is Hermitian, $V=V^*$, and periodic in time, $V(x,s + 2\pi)= V(x, s)$. For simplicity, let $V(\cdot,s)\in L^\infty(\mathbb{R}_x, M_n(\mathbb{C}))$ with $C^1$-dependence on $s\in S^1{\mathrel{\mathop:}=}\mathbb{R}/2\pi\mathbb{Z}$. Then, for any $z\in\rho(H(s))$ in the resolvent set, the Schrödinger equation $H(s)\varphi=z\varphi$ is in the limit-point case at $x=+\infty$ (see [@L] or [@CG; @LM]), meaning that as an ordinary differential equation it has $n$ linearly independent solutions which are square-integrable at $x=+\infty$. We may thus introduce a family of sets, parametrized by $z\in\rho(H(s))$ and $s\in S^1$, consisting of matrix-valued solutions $\psi(x)\in M_n (\mathbb{C})$ of the Schrödinger equation $$\label{eq:psi}
-\psi'' (x) + V(x,s) \psi(x) = z \psi(x) \,,$$ which are regular in the sense that for any $x\in\mathbb{R}$ $$\label{eq:regular}
\psi(x)a=0,\, \psi'(x)a=0
\;
\Rightarrow\; a= 0\,, \qquad (a\in\mathbb{C}^n)\,.$$ It is: $$\label{eq:psi+}
S^+_{(z,s)} = \{ \psi_+ | \psi_+ \textrm{ is a regular solution of }
(\ref{eq:psi}), L^2 \textrm{ at } x=+\infty \} \, .$$ As a matter of fact such solutions tend to zero pointwise as $x\to +\infty$, together with their first derivatives. Similarly, solutions $\tilde\psi(x)\in M_n (\mathbb{C})$ of the adjoint equation $$\label{eq:chi}
-\tilde\psi '' (x) + \tilde\psi(x) V(x,s) = z \tilde\psi(x)$$ act on row vectors $a\in\mathbb{C}^n$ as $a\tilde\psi(x)$, and we set $$\tilde{S}^-_{(z,s)} = \{ \tilde\psi_- | \tilde\psi_- \textrm{ is regular solution of } (\ref{eq:chi}), L^2 \textrm{ at } x=-\infty \} \, .$$ For later use we also introduce the families $S^-_{(z,s)}$, $\tilde{S}^+_{(z,s)}$ of solutions to (\[eq:psi\]), resp. (\[eq:chi\]) decaying at the opposite ends. For any two differentiable functions $\psi,\,\tilde\psi:\mathbb{R}\to M_n(\mathbb{C})$ we define the Wronskian $$\label{eq:wr}
W(\tilde\psi, \psi ; x) =
\tilde\psi (x) \psi '(x) - \tilde\psi ' (x) \psi (x) \in M_n(\mathbb{C})\, .$$ It is independent of $x$ if $\psi$ and $\tilde\psi$ are solutions of (\[eq:psi\]), resp. of (\[eq:chi\]), in which case it is simply denoted as $W(\tilde\psi_-, \psi_+)$. As will also be shown later, $\det W(\tilde\psi_-, \psi_+)\neq 0$ for $\psi_+\in S^+_{(z,s)}$, $\tilde\psi_-\in \tilde{S}^-_{(z,s)} $. We observe that $S^+_{(z,s)}$ carries a transitive right action of ${\mathrm{GL}}(n)\ni T$, $$\label{eq:ra}
\psi_+ (x) \mapsto \psi_+ (x) T \, ,$$ while $\tilde{S}^-_{(z,s)}$ carries a left action, $$\tilde\psi_- (x) \mapsto T \tilde\psi_- (x) \, .$$ We thus have a bijective relation between $\psi_+\in S^+_{(z,s)}$ and $\tilde\psi_-\in \tilde{S}^-_{(z,s)}$ such that $$W(\tilde\psi_-, \psi_+ ) =1 \, .
\label{eq:nrmlzt}$$ We assume that the Fermi energy $\mu>0$ lies in a spectral gap at all times $s$: $$\mu\in \rho(H(s))\,.
\label{eq:gap}$$ Let $P_0(s)$ be the spectral projection of $H(s)$ up to the Fermi energy and $U_\varepsilon (s, s_0)$ be the propagator for the non-autonomous Hamiltonian $H(\varepsilon t)$, where $s=\varepsilon t$. In the Appendix we prove, in the smooth case, $$\label{eq:Pzero}
U_\varepsilon (s, s_0) (P_0(s_0)+ \varepsilon P_1(s_0)) U_\varepsilon (s, s_0)^* =
P_0 (s) + \varepsilon P_1 (s) + O (\varepsilon^2 )\,, \qquad
(\varepsilon \rightarrow 0)$$ with $$\label{eq:Puno}
P_1 (s) = -\frac{1}{2\pi} \oint_\gamma R(z,s) \dot{R} (z,s) dz\,,$$ where $R(z,s) = ( H(s) - z)^{-1}$ and $\gamma$ is a complex contour encircling the part of the spectrum of $H(s)$ lying below $\mu$ and $\dot{} = \partial / \partial s$. Eq. (\[eq:Pzero\]) is the 1-particle density matrix which has evolved from that of the Fermi sea, $P_0(s_0)$, after a gentle start of the pump. In fact such a start may be obtained from (\[eq:uno\]) by means of a smooth substitution $s'\mapsto s$ with $s'\mapsto s_0$, ($s\le s_0$), and $s'=s$, ($s'$ large). Then, in the new variable, $P_1(s_0)=0$ by (\[eq:Puno\]).
The current across a fiducial point $x=x_0$ is the rate of change of the charge contained in $x>x_0$ and hence given by the operator $I={\mathrm{i}}[H(s), \theta(x-x_0)]$, which is independent of $s$. The charge transported in a cycle (of duration $2\pi\varepsilon^{-1}$) is, in expectation value and in the adiabatic limit, given as $$\label{eq:QT}
\langle Q_T\rangle{\mathrel{\mathop:}=}\oint {\operatorname{Tr}}(IP_1(s))ds\,,$$ because of $dt=\varepsilon^{-1}ds$, with ${\operatorname{Tr}}$ denoting the trace on $L^2(\mathbb{R}_x, \mathbb{C}^n)$. This definition rests on the fact that the leading contribution from persistent currents, $\varepsilon^{-1}\oint{\operatorname{Tr}}(IP_0(s))ds$, which is potentially divergent in the limit, actually vanishes. If $V$ were real, this would follow trivially from time reversal invariance; however our hypothesis does not imply this, except for $n=1$, and we shall argue otherwise.
The result of [@Th], generalized as described in the Introduction, is part (ii) of the following theorem.
\[thm:1\] Assume (\[eq:gap\]). Then
1. $${\operatorname{Tr}}(IP_0(s))=0\,.$$
2. $$\langle Q_T\rangle=\frac{{\mathrm{i}}}{2\pi}\oint_\gamma dz \oint_{S^1} ds\,
{\operatorname{tr}}\bigl( W(\frac{\partial \tilde\psi_-}{\partial z} , \frac{\partial \psi_+}{\partial s};x_0) - W( \frac{\partial \tilde\psi_-}{\partial s} , \frac{\partial \psi_+}{\partial z};x_0) \bigr) \,,
\label{eq:th}$$
where ${\operatorname{tr}}$ denotes the matrix trace and the solutions $\psi_+\in S^+_{(z,s)}$, $\tilde\psi_-\in \tilde{S}^-_{(z,s)}$ satisfying (\[eq:nrmlzt\]) are locally smooth in $(z,s)$. Except for these conditions, the trace is independent of $\psi_+$, $\tilde\psi_-$, and the integral is it of $x_0$, too. Moreover, the r.h.s. is the first Chern number of a bundle described in Section \[section:3\].
We next present the scattering description [@BTP] of charge transport. Consider again the Hamiltonian (\[eq:uno\]), but now with $V$ of compact support in $x$. As a result, (\[eq:gap\]) fails: $$\mu\in \sigma(H(s))
\label{eq:nogap}$$ for all $s$. We may thus introduce the scattering matrix $S(s)$ at Fermi energy $\mu>0$, $$S(s)=\begin{pmatrix}R&T'\\T&R'\end{pmatrix}\,,$$ where the blocks are $n\times n$ matrices determined by the asymptotic behavior of solutions of (\[eq:psi\]) with $z=\mu$. More precisely, $R$ and $T$ are defined in terms of a plane wave incident from the left, $$\label{eq:incident}
\psi(x)=\begin{cases}
1{\mathrm{e}^{{\mathrm{i}}kx}}+R{\mathrm{e}^{-{\mathrm{i}}kx}}\,,&(x<-r)\,,\\
T{\mathrm{e}^{{\mathrm{i}}kx}}\,,&(x>r)\,,
\end{cases}$$ with $r>0$ large enough and $k=\sqrt{\mu}$. Similarly $R'$ and $T'$ are defined in terms of a wave incident from the right.
The charge emitted from all channels of the left lead together, in a cycle and in the adiabatic limit, is [@BTP] $$\langle Q_{BPT}\rangle=\frac{1}{2\pi{\mathrm{i}}}\oint{\operatorname{tr}}((dS)S^*P)\,,
\label{eq:bpt}$$ where $dS=(dS/ds)ds$ and $P=\bigl(\begin{smallmatrix} 1&0\\0&0\end{smallmatrix}\bigr)$ is the projection onto the left channels. For the same situation the variance is [@ILL; @AEGS] $$\langle\langle Q_{BPT}^2\rangle\rangle=\frac{1}{(2\pi)^2}\int_{-\infty}^\infty
ds\oint ds'\frac{{\operatorname{tr}}[(S^*(s)PS(s)-S^*(s')PS(s'))^2]}{\sin^2{(s-s')}}\,.$$ In general, and in contrast to (\[eq:th\]), $\langle Q_{BPT}\rangle$ is not an integer. However, $\langle\langle Q_{BPT}^2\rangle\rangle$ vanishes iff the time dependence of $S$ is of the form $$S(s)=\begin{pmatrix}U_1(s)&0\\0&U_2(s)\end{pmatrix}S_0
\label{eq:qntz}$$ with $U_j(s)$ ($j=1,2$) and $S_0$ unitary matrices of order $n$, resp. $2n$. In this case $\langle Q_{BPT}\rangle$ is an integer, $$\langle Q_{BPT}\rangle=\frac{1}{2\pi{\mathrm{i}}}\oint{\operatorname{tr}}((dU_1)U_1^*)
=\frac{1}{2\pi{\mathrm{i}}}\oint d\log\det U_1\,,$$ given as the winding number of $\det U_1$.\
We do not give here the definition of $\langle Q_{BPT}\rangle$ which makes (\[eq:bpt\]) a theorem [@AEGSS]. Rather we focus on the relation between Eqs. (\[eq:th\]) and (\[eq:bpt\]). To this end we truncate the potential to a finite interval, $V(x,s)\chi_{[0,L]}(x)$, and denote its scattering matrix by $S_L(s)$. In the limit $L\to\infty$ the original physical situation is recovered and the two approaches agree, as stated in the following result.
\[thm:2\] Assume (\[eq:gap\]) for the infinite system.
1. The scattering matrix $S_L(s)$ at Fermi energy $\mu$ has a limit of the form $$\label{eq:cinque}
\lim_{L\to\infty} S_L(s)=\begin{pmatrix}R(s)&0\\0&R'(s)\end{pmatrix}\,.$$ In particular, the condition (\[eq:qntz\]) for quantization of $\langle Q_{BPT}\rangle$ is attained in the limit.
2. The winding number of $\det R(s)$ equals the Chern number on the r.h.s of Eq. (\[eq:th\]). In physical terms, $$\label{eq:equiv}
\langle Q_{BPT}\rangle=\langle Q_T\rangle\,.$$
We conclude this section by summarizing the idea of the proof of (\[eq:equiv\]). We may assume that the contour $\gamma$ in Eqs. (\[eq:Puno\], \[eq:th\]) crosses the real axis just twice, once below the spectrum and once at Fermi energy $\mu$. The torus of integration in (\[eq:th\]), which is denoted by $\mathbb{T}=\gamma\times S^1$, is the base space of a bundle which will admit a global section except at isolated points along the line $\{\mu\}\times S^1\subset\mathbb{T}$. Using Stokes’ theorem its Chern number can be expressed in terms of solutions of the Schrödinger equation at Fermi energy and, in turn, of the scattering matrix (\[eq:cinque\]). The main steps are given in more detail in the following lemma. There the r.h.s. of eq. (\[eq:th\]) is denoted by $C$, and $x_0$ is fixed. The orientation of the torus is the natural one, $d\gamma\wedge ds$.
\[mainlm\]
1. Any point $(z_*, s_*)\in \mathbb{T}$ where $\det\psi_+ (x_0)=0$ for some (and hence all) $\psi_+\in S^+_{(z_*,s_*)}$ has $z_*=\mu$. For a dense set of potentials $V=V^*$, the points $s_*$ are isolated in $S^1$ and $0$ is a simple eigenvalue of $\psi_+ (x_0)$; moreover, $$\label{eq:reg}
\det\psi'_+ (x_0)\neq0\,.$$ Density is meant w.r.t. the topology of the class of potentials specified below (\[eq:uno\]).
2. Let $\psi_{(z,s)}\in S^+_{(z,s)}$ be a section defined in a neighborhood in $\mathbb{C}\times S^1\supset \mathbb{T}$ of any of the above points $(z_*=\mu, s_*)$, which is analytic in $z$. Then the family of matrices $L(z,s)=\psi'_{(\bar z,s)}(x_0)^*\psi_{(z,s)}(x_0)$ has the reflection property $$\label{eq:refl}
L(z,s)=L(\bar z,s)^*\,.$$ Its eigenvalues are real for real $z$. There is a single eigenvalue branch $\lambda(z,s)$ vanishing to first order at $(\mu, s_*)$. Its winding number there is $$w_{s_*}=-{\operatorname{sgn}}\bigl(\frac{\partial\lambda}{\partial z}\frac{\partial \lambda}
{\partial s}\bigr)\Big|_{(z=\mu,s= s_*)}\,.$$
3. $$C=-\sum_{s_*}w_{s_*}\,.$$
4. At any of the points $(\mu,s_*)$ we have $$\frac{\partial\lambda}{\partial z}<0\,.$$
5. The unitary matrix $R(s)$ has eigenvalue $-1$ iff $\det\psi_{\mu,s}(0)=0$. More precisely, as $s$ increases past $s_*$, an eigenvalue of $R$ crosses $-1$ counterclockwise if $$\frac{\partial \lambda}{\partial s}\Big|_{(z=\mu,s= s^*)}<0\,.$$
As a result, $C=-\sum_{s_*}{\operatorname{sgn}}\bigl(\partial \lambda/\partial s\bigr)|_{(z=\mu,s= s_*)}$ is the number of eigenvalue crossings of $R(s)$ past $-1$, i.e., the winding number of $\det R$. Actually the equality is first established if the conditions on the potential of part (i) are satisfied, but the conclusion, Eq. (\[eq:equiv\]), extends by density.
A fiber bundle {#section:3}
==============
We describe the bundle $P$ and the connection underlying Eq. (\[eq:th\]). Let $\mathcal{C} = C^1 \bigl(\mathbb{R}, M_n (\mathbb{C}) \bigr)$ be the space of matrix valued $C^1$-functions on $\mathbb{R}$. Let $\pi:
P\to \mathbb{T}$ be the subbundle of $\mathbb{T}\times\mathcal{C}$ with base $\mathbb{T}=\gamma\times S^1$ and fibers $S^+_{(z,s)}\subset\mathcal{C}$: $$P=\{((z,s),\psi)\in \mathbb{T}\times \mathcal{C}\mid \psi\in S^+_{(z,s)}\}\,.$$ It is a principal bundle w.r.t. the right action (\[eq:ra\]) of ${\mathrm{GL}}(n)$. This includes that ${\mathrm{GL}}(n)$ is its structure group. Indeed, for any sufficiently small open set $U\subset\mathbb{T}$ there is $x\in\mathbb{R}$ with $$\det\psi_+(x) \neq 0$$ for all $\psi_+\in S^+_{(z,s)}$ and $(z,s)\in U$, see Lemma \[lm:prlm\] below. This provides a local trivialization $\phi$ with $$\phi^{-1}:\pi^{-1}(U)\to U\times {\mathrm{GL}}(n)\,,\quad
\psi_+\mapsto(z,s,\psi_+(x))\,.$$ The transition function $\phi_2^{-1}\circ\phi_1: {\mathrm{GL}}(n)\to{\mathrm{GL}}(n)$ is multiplication from the left by the matrix $\psi_+(x_2)\psi_+(x_1)^{-1}$, which is clearly independent of $\psi_+\in S^+_{(z,s)}$ and belongs to ${\mathrm{GL}}(n)$.
We will give an explicit expression for the Chern number $C$ of $P$, which differs somewhat from that used in [@Th]. We recall that $$\label{eq:C}
C = \frac{{\mathrm{i}}}{2\pi} \int_\mathbb{T} {\operatorname{tr}}\mathcal{F} \,,$$ where $\mathcal{F} = D \mathcal{A}$ is the curvature of any connection $\mathcal{A}$ on $P$. We recall that ${\operatorname{tr}}\mathcal{F}$ defines a 2-form on $\mathbb{T}$, and not just on $P$; for any two connections, $\mathcal{A}$ and $\mathcal{A}'$, the same is true for the 1-form ${\operatorname{tr}}(\mathcal{A} - \mathcal{A}')$, whence $C$ is independent of the choice of connection. We consider connections of the following form. Let $B: \mathcal{C} \times \mathcal{C} \rightarrow M_n (\mathbb{C})$ be a bilinear form on $\mathcal{C}$ satisfying $$\begin{aligned}
B(\tilde\psi , \psi T) &= B(\tilde\psi , \psi)T\,, \label{eq:B1} \\
B(T \tilde\psi , \psi) &= T B(\tilde\psi ,\psi) \label{eq:B2}\end{aligned}$$ ($\tilde\psi$, $\psi \in \mathcal{C}$, $T \in {\mathrm{GL}}(n)$). Moreover we assume that its restriction $$\label{eq:B3}
B: \tilde{S}^-_{(z,s)} \times S^+_{(z,s)} \rightarrow {\mathrm{GL}}(n)$$ takes values $B(\tilde\psi_- , \psi_+)$ in the regular matrices (as shown below, an example is (\[eq:wr\])). We may then consider the ${\mathrm{gl}}(n)$-valued 1-form on $P$ $$\mathcal{A}_{\psi_+} (\delta \psi_+) = B(\tilde\psi_- ,
\psi_+)^{-1} B(\tilde\psi_- , \delta \psi_+ ) \, ,\qquad(\delta
\psi_+\in TP)\, ,$$ which is well-defined being independent of the choice of $\tilde\psi_-
\in \tilde{S}^-_{(z,s)}$ by (\[eq:B2\]). It is a connection on $P$ since it enjoys the defining properties $$\begin{aligned}
\mathcal{A}_{\psi_+} (\psi_+ t) &= t \, , \qquad (t \in {\mathrm{gl}}(n))\, , \\
\mathcal{A}_{\psi_+ T} (\delta \psi_+ T) &= T^{-1} \mathcal{A}_{\psi_+} (\delta \psi_+) T \, , \qquad (T \in {\mathrm{GL}}(n))\end{aligned}$$ by (\[eq:B1\]). Given $\psi_+ \in S^+_{(z,s)}$ there is a unique $\tilde\psi_- \in \tilde{S}^-_{(z,s)}$ such that $B(\tilde\psi_- ,
\psi_+) = 1$, as can again be seen from (\[eq:B2\]). Then $\mathcal{A} = B(\tilde\psi_- , \delta \psi_+)$ and the trace of its curvature is $${\operatorname{tr}}\mathcal{F} = {\operatorname{tr}}\bigl( B(\frac{\partial\tilde\psi_-}{\partial z} , \frac{\partial \psi_+}{\partial s}) - B( \frac{\partial \tilde\psi_-}{\partial s} , \frac{\partial \psi_+}{\partial z}) \bigr) dz\wedge ds\, .$$ We will use the bilinear $$B(\tilde\psi , \psi) = W(\tilde\psi, \psi ; x) = \tilde\psi (x) \psi '(x) - \tilde\psi ' (x) \psi (x) \, ,$$ whose restriction (\[eq:B3\]) is seen to be independent of $x$ (though $\mathcal{A}$ may not be); then (\[eq:C\]) coincides with the r.h.s. of (\[eq:th\]), as announced in Theorem \[thm:1\]. It remains to verify $B(\tilde\psi_- , \psi_+) \in {\mathrm{GL}}(n)$. Any column vector solution $\varphi (x)$ of (\[eq:psi\]) is determined by $\varphi(0)$, $\varphi'(0) \in \mathbb{C}^n$. Similarly for any row vector $\tilde\varphi
(x)$ solving (\[eq:chi\]). Their Wronskian $$\label{eq:W}
W(\tilde\varphi , \varphi) = \tilde\varphi (0) \varphi'(0) -
\tilde\varphi'(0) \varphi (0) \, ,$$ which now takes values in $\mathbb{C}$, clearly defines a non-degenerate bilinear form on $\mathbb{C}^{2n}$. Given $\psi_\pm \in S^{\pm}_{(z,s)}$, any solution $\varphi$ can be expressed as $$\label{eq:gs}
\varphi (x) = \psi_+ (x) a_+ + \psi_- (x) a_-$$ with $a_\pm \in \mathbb{C}^n$, and $\varphi \equiv 0$ iff $a_\pm = 0$; similarly for $\tilde\varphi (x) = b_+ \tilde\psi_+ (x) + b_-
\tilde\psi_- (x)$. In terms of the coefficients $(b_+ , b_-)$, $(a_+ ,
a_-)$, the bilinear form (\[eq:W\]) is given by the matrix $$\left( \begin{array}{cc}
0 & W(\tilde\psi_+ , \psi_-) \\ W(\tilde\psi_- , \psi_+) & 0
\end{array} \right) \, ,$$ since $$\label{eq:nrmlzt3}
W(\tilde\psi_\pm , \psi_\pm ) = \lim_{x \rightarrow \pm \infty} W
(\tilde\psi_\pm , \psi_\pm ; x) = 0\,.$$ Hence $W(\tilde\psi_- , \psi_+)$ is regular.\
[**Remark.**]{} In [@Th] (and later in [@GO]) the case of a potential $V(x)$ of period $L$ was considered. In the case $n=1$ the bilinear used there was $$B(\tilde\psi , \psi) = \int_0^L dx \, \tilde\psi (x) \psi (x) \, .$$ Non-degeneracy of (\[eq:B3\]) amounts to $\int_0^L dx \, \psi_- (x) \psi_+ (x) \neq 0 $, where $\psi_-\in \tilde{S}^-_{(z,s)} = S^-_{(z,s)}$, $\psi_+\in S^+_{(z,s)}$ are unique up to non-zero multiples.
Proofs {#section:4}
======
Here we prove Theorems \[thm:1\] and \[thm:2\] stated in Section \[section:2\]. First however we should dwell on a little point of precision: The current, informally given as $$I={\mathrm{i}}[H, \theta(x)]=-{\mathrm{i}}\bigl\{\frac{d}{dx},\delta(x)\bigr\}\,,
\label{eq:cu1}$$ is not a well-defined operator on Hilbert space. (We suppressed $s$ from the notation and set $x_0=0$.) Instead, it should be understood as the map $D(H)\to D(H)^*$, $$I={\mathrm{i}}(\gamma_1^*\gamma_0-\gamma_0^*\gamma_1)\,,$$ where $\gamma_0, \gamma_1:D(H)\to\mathbb{C}^n$ with $\gamma_0\psi=\psi(0)$, $\gamma_1\psi=\psi'(0)$. Then (\[eq:cu1\]) is replaced by $${\mathrm{i}}[R(z), \theta(x)]=-R(z)IR(z)\,,
\label{eq:cu2}$$ which can be verified first as a quadratic form. This operator is of trace class because $(p^2+1)^{-1}\gamma_i^*\gamma_i (p^2+1)^{-1}$ is.
Given an operator $K:D(H)^*\to D(H)$ one may, pretending cyclicity, take $${\operatorname{Tr}}(IK):={\mathrm{i}}{\operatorname{tr}}(\gamma_0K\gamma_1^*-\gamma_1K\gamma_0^*)$$ as a definition. In fact, this is the trace of the finite rank operator $IK$ on the Banach space $D(H)^*$, see e.g. [@Si], Eq. (10.2). It yields $${\operatorname{Tr}}(IK):={\operatorname{tr}}(-{\mathrm{i}}\partial_1K(0,0) +{\mathrm{i}}\partial_2K(0,0))\,,
\label{eq:cu3}$$ where $K(x,y)$ is the integral kernel of $K$ and $\partial_1$ and $\partial_2$ indicate a derivative w.r.t. the first, resp. second argument. As a further motivation we note that expectation values of the current are naturally written as ${\operatorname{Tr}}(P_0IP_0)$ and ${\operatorname{Tr}}(P_0IP_1+P_1IP_0)$ in zeroth and first order in $\varepsilon$. Then $${\operatorname{Tr}}(P_0IP_0)={\mathrm{i}}{\operatorname{Tr}}\bigl(P_0(\gamma_1^*\gamma_0-\gamma_0^*\gamma_1)P_0\bigr)
={\mathrm{i}}{\operatorname{tr}}(\gamma_0P_0\gamma_1^*-\gamma_1P_0\gamma_0^*)\,,
\label{eq:cu4}$$ where cyclicity is now justified since $\gamma_i P_0$ is Hilbert-Schmidt; also, $P_0^2=P_0$ was used. Similarly, $${\operatorname{Tr}}(P_0IP_1+P_1IP_0)={\mathrm{i}}{\operatorname{tr}}(\gamma_0P_1\gamma_1^*-\gamma_1P_1\gamma_0^*)\,,$$ by $P_0P_1+P_1P_0=P_1$.\
[**Proof of Theorem \[thm:1\].**]{} i) The projection $P_0$ has the integral representation $P_0= -(2\pi {\mathrm{i}})^{-1} \oint_\gamma R(z) \, dz$. Since $\oint_\gamma R(z)^2 \, dz=0$ we may replace $R(z)$ therein by $R(z)-R(z)^2H=-zR(z)^2$: $$P_0= \frac{1}{2\pi {\mathrm{i}}} \oint_\gamma z R(z)^2 \, dz\,.$$ We then have, by (\[eq:cu4\], \[eq:cu2\]), $$\begin{aligned}
{\operatorname{Tr}}(P_0IP_0)&=\frac{1}{2\pi} \oint_\gamma z
{\operatorname{tr}}(\gamma_0R(z)^2\gamma_1^*-\gamma_1R(z)^2\gamma_0^*)\, dz\nonumber\\
&=\frac{1}{2\pi} \oint_\gamma z
{\operatorname{Tr}}\bigl(R(z)(\gamma_1^*\gamma_0-\gamma_0^*\gamma_1)R(z)\bigr) \, dz
=-\frac{1}{2\pi} \oint_\gamma z{\operatorname{Tr}}([R(z), \theta(x)]) \, dz\,,
\label{eq:cu5}\end{aligned}$$ and, by $zR(z)=HR(z)-1$, also ${\operatorname{Tr}}(P_0IP_0)={\mathrm{i}}{\operatorname{Tr}}[HP_0,\theta]$. As the stationarity of $P_0$ suggests, the current is independent of $x_0$. In fact, upon replacing $\theta(x)$ by $\tilde\theta(x)=\theta(x-x_0)-\theta(x)$ both terms in ${\operatorname{Tr}}((HP_0)\tilde\theta-\tilde\theta(HP_0))$ are separately trace class, whence the trace vanishes ([@Si], Corollary 3.8). We next turn to (\[eq:cu5\]): The commutator $A=[R(z), \theta(x)]$ has integral kernel $A(x,y)=G(x,y)(\theta(y)-\theta(x))$, where $G(x, x') = R(z)(x,x')$ is the Green function. By the stated independence we may average over $x_0$ instead of setting it to $0$, thus effectively smoothing $\theta$. We will see in (\[eq:grnfct\], \[eq:ids\]) below that $G(x,y)$ is continuous. Thus $A(x,x)=0$, implying ${\operatorname{Tr}}(P_0IP_0)=0$. Alternatively the conclusion may be reached without smoothing by resorting to Brislawn’s theorem ([@Si], Theorem A.2), according to which ${\operatorname{Tr}}A=\int dx\, \tilde A(x,x)$, where $\tilde A(x,y)$ is the Lebesgue value of $A(x,y)$. Here, $\tilde A(x,x)=0$.\
ii) By applying (\[eq:cu3\]) to $K=R(z,s) \dot{R} (z,s)$ in (\[eq:QT\], \[eq:Puno\]) we obtain for the transported charge $$\label{eq:QT2}
\langle Q_T \rangle = \frac{{\mathrm{i}}}{2 \pi} \oint ds \oint_\gamma \! dz \int dx \, {\operatorname{tr}}\bigl( \partial_1 G(0,x) \dot{G}(x,0) - G(0,x) \partial_2 \dot{G}(x,0) \bigr)\,.$$ We claim that the Green function can be expressed as $$\label{eq:grnfct}
G(x, x')= - \theta(x - x') \psi_+ (x) \tilde\psi_- (x') - \theta (x' - x) \psi_- (x) \tilde\psi_+ (x') \, ,$$ where we complemented the locally smooth choice of $\psi_+ \in S^+_{(z,s)}$, $\tilde{\psi}_- \in \tilde{S}^-_{(z,s)}$ satisfying (\[eq:nrmlzt\]) by that of a pair $\tilde{\psi}_+ \in \tilde{S}^+_{(z,s)}$, $\psi_- \in S^-_{(z,s)}$ with $$\label{eq:nrmlzt2}
W( \tilde\psi_+ , \psi_- ) = -1 \, .$$ Indeed, because of (\[eq:nrmlzt\], \[eq:nrmlzt2\]) and of (\[eq:nrmlzt3\]) the general column solution (\[eq:gs\]) has coefficients $$a_\pm=\pm W( \tilde\psi_\mp,\varphi)=\pm\tilde\psi_\mp(y)\varphi'(y)
\mp\tilde\psi'_\pm(y)\varphi(y)\,.\\$$ By inserting this in (\[eq:gs\]) and in its derivative w.r.t. $x$, and by setting $y=x$, we conclude from the arbitrariness of $\varphi(x)$ and $\varphi'(x)$ that $$\begin{aligned}
\psi_+ (x) \tilde\psi_- (x) - \psi_- (x) \tilde\psi _+ (x) &= 0\,,
\label{eq:ids} \\
\psi_+ (x) \tilde\psi_- ' (x) - \psi_- (x) \tilde\psi _+ ' (x) &=
-1\,,
\nonumber\\
\psi_+ ' (x) \tilde\psi_- (x) - \psi_- ' (x) \tilde\psi_+ (x) &= 1
\, .
\nonumber\end{aligned}$$ By means of these relations one verifies that $G$, as given by the r.h.s. of (\[eq:grnfct\]), satisfies $$\Bigl(-\frac{d^2}{dx^2}+V(x)-z\Bigr)G(x,x')=\delta(x-x')1\,;$$ together with $G(x,x')\to 0$, ($|x|\to\infty$), which exhibits it as the Green function. We then apply (\[eq:grnfct\]) in Eq. (\[eq:QT2\]): For $x\ge 0$ the integrand is $$\begin{gathered}
{\operatorname{tr}}\bigl( \partial_1 G(0,x) \dot{G}(x,0) - G(0,x)
\partial_2 \dot{G}(x,0) \bigr)= \\
{\operatorname{tr}}\bigl( \psi_- ' (0) \tilde\psi_+ (x)
( \dot{\psi}_+ (x) \tilde\psi_- (0) + \psi_+ (x)
\dot{\tilde\psi}_- (0) )
- \psi_-(0) \tilde\psi_+ (x)
( \dot{\psi}_+ (x) \tilde\psi_-' (0) + \psi_+ (x)
\dot{\tilde\psi}_-' (0) )\bigr)\\
= {\operatorname{tr}}\bigl( W(\dot{\tilde\psi}_- , \psi_- ) \,
\tilde\psi_+ (x) \psi_+ (x)\bigr) \,,\end{gathered}$$ where we used cyclicity of the trace and (\[eq:nrmlzt3\]). Here and henceforth the Wronskian is evaluated at $x=0$, unless otherwise stated. Together with a similar computation for $x\le 0$ we obtain $$\label{eq:QT3}
\langle Q_T \rangle= \frac{{\mathrm{i}}}{2 \pi} \oint \! ds \oint_\gamma \! dz \, {\operatorname{tr}}\bigl( W(\dot{\tilde\psi}_- , \psi_- ) \int_0^\infty \! dx \, \tilde\psi_+ (x) \psi_+ (x) +W( \dot{\tilde\psi}_+ , \psi_+ ) \int_{-\infty}^0 \! dx \, \tilde\psi_- (x) \psi_- (x) \bigr) \, .$$ We maintain that the same expression is obtained from a computation of $C$, the r.h.s. of (\[eq:th\]). That calls for one of $\partial \psi_+ /
\partial z$, $\partial \tilde{\psi}_- / \partial z$. Differentiating (\[eq:psi\]) w.r.t. $z$ we obtain $$\Bigl( -{\frac{d^2}{dx^2}}+ V(x,s) -z \Bigr)
\frac{\partial \psi_+}{\partial z} = \psi_+ \, ,$$ whose general solution with $\partial \psi_+ / \partial z \rightarrow 0$, ($x \rightarrow \infty$) is $$\frac{\partial \psi_+}{\partial z} (x)
=\psi_+ (x)F_+(x) -
\psi_- (x) \int_x^{\infty} \tilde\psi_+ (x') \psi_+ (x') dx' \, ,\label{eq:dz1}$$ where $F_+' (x) =dF_+/dx=-\tilde \psi_- (x)\psi_+ (x)$. Hence $F_+$ is determined up to an additive constant, which reflects the gauge freedom (\[eq:ra\]) of $\psi_+$. Eq. (\[eq:dz1\]) is verified by twice differentiating it w.r.t. $x$, the first derivative being $$\frac{\partial \psi_+'}{\partial z} (x)
=\psi_+ '(x)F_+(x) -
\psi_-' (x) \int_x^{\infty} \tilde\psi_+ (x') \psi_+ (x') dx' \, ,
$$ by using (\[eq:ids\]). In the same way we find $$ \frac{\partial \tilde\psi_-}{\partial z} (x) = F_-(x) \tilde\psi_- (x) - \Bigl( \int_{-\infty}^x \tilde\psi_- (x') \psi_- (x') dx' \Bigr) \tilde\psi_+ (x) \, ,$$ with $F_-' = -F_+'$. The arbitrariness of $F_\pm$ is constrained by (\[eq:nrmlzt\]), which implies $$\label{eq:dc1}
F_++F_-=0\,.$$ This is seen by differentiating the constraint w.r.t. $z$ and by using $$\begin{aligned}
W(\tilde\psi_-, \frac{\partial \psi_+}{\partial z} ; x)
&=W(\tilde\psi_-,\psi_+;x)F_+(x)
-W(\tilde\psi_-,\psi_-;x)\int_x^\infty \tilde\psi_+ (x) \psi_+ (x) \, dx
=F_+(x)\,,\\
W(\frac{\partial \tilde\psi_-}{\partial z} , \psi_+ ; x)&=F_-(x)\,.\end{aligned}$$ Similarly, differentiating the constraint w.r.t. $s$ yields $$\label{eq:dc2}
W( \dot{\tilde\psi}_- , \psi_+;x) +
W( \tilde\psi_- , \dot{\psi}_+;x) = 0\,.$$ We are now in position to compute $C$ and in particular $$\begin{aligned}
W(\frac{\partial \tilde\psi_-}{\partial s} ,
\frac{\partial \psi_+}{\partial z})
=&
\dot{\tilde\psi}_- (0) \bigl( \psi_+ ' (0) F_+(0) -
\psi_- ' (0) \int_0^\infty \tilde\psi_+(x) \psi_+ (x) \, dx \bigr)
\\
& - \dot{\tilde\psi}_- ' (0) \bigl( \psi_+ (0) F_+(0) - \psi_- (0) \int_0^\infty \tilde\psi_+ (x) \psi_+ (x) \, dx \bigr)
\\
=& W(\dot{\tilde\psi}_- , \psi_+)F_+(0)-W(\dot{\tilde\psi}_- , \psi_-)\int_0^\infty \tilde\psi_+ (x) \psi_+ (x) \, dx\,, \\
W( \frac{\partial \tilde\psi_-}{\partial z} ,
\frac{\partial \psi_+}{\partial s}) =&
F_-(0)W(\tilde\psi_- , \dot{\psi}_+)-
\Bigl(\int_{- \infty}^0 \tilde\psi_- (x) \psi_- (x) \, dx\Bigr)
W(\tilde\psi_+ , \dot{\psi}_+)\,, \end{aligned}$$ Taking the trace of difference of the two expressions, the first terms on the r.h.s. cancel because of (\[eq:dc1\], \[eq:dc2\]). The result is that $C$ agrees with the r.h.s. of (\[eq:QT3\]).
The stated independence of the trace follows from its cyclicity by joining the left and right actions (\[eq:ra\]) in such a way as to preserve (\[eq:nrmlzt\]); that of the integral is explained after Eq. (\[eq:C\]). $\blacksquare$\
[**Proof of Theorem \[thm:2\].**]{} i) We recall that the scattering matrix $S_L=
\bigl(\begin{smallmatrix} R_L&T'_L\\T_L&R'_L\end{smallmatrix}\bigr)$ is that of the potential truncated to the interval $[0,L]$. The left incident solution of (\[eq:psi\]) is given by the expressions (\[eq:incident\]) in the intervals $x\le 0$, resp. $x\ge L$. Its adjoint is a solution of (\[eq:chi\]) since $z=\mu$ is real. By the constancy of the Wronskian, $$W( 1 {\mathrm{e}^{-{\mathrm{i}}kx}} + R_L^* {\mathrm{e}^{{\mathrm{i}}kx}} , \psi_\pm;x=0) =
W(T_L^* {\mathrm{e}^{-{\mathrm{i}}kx}}, \psi_\pm;x=L) \, ,$$ and by $W(1 {\mathrm{e}^{ {\mathrm{i}}kx}},\psi_\pm;x) =
{\mathrm{e}^{{\mathrm{i}}kx}} (\psi_\pm'(x) - {\mathrm{i}}k\psi_\pm(x))$ we find $$\label{eq:sl}
\bigl(\psi_\pm'(0) + {\mathrm{i}}k\psi_\pm(0)\bigr) +
R_L^* \bigl(\psi_\pm'(0)-{\mathrm{i}}k\psi_\pm(0)\bigr)=
T_L^*{\mathrm{e}^{-{\mathrm{i}}kL}}\bigl(\psi_\pm'(L)+{\mathrm{i}}k\psi_\pm(L)\bigr) \, .$$ We have that $$\begin{aligned}
\label{eq:lim1}
\lim_{x\to +\infty}\psi_+'(x)+{\mathrm{i}}k\psi_+(x)&=0\,,\\
\label{eq:lim2}
\lim_{x\to +\infty}\bigl(\psi_-'(x)+{\mathrm{i}}k\psi_-(x)\bigr)^{-1}&=0\,.\end{aligned}$$ Indeed, the first limit just repeats the definition (\[eq:psi+\]) and the second may be rephrased to the effect that $$A(x){\mathrel{\mathop:}=}\bigl(\psi_-'(x)+{\mathrm{i}}k\psi_-(x)\bigr)^*\bigl(\psi_-'(x)+{\mathrm{i}}k\psi_-(x)\bigr)$$ is invertible with $\lim_{x\to +\infty}\|A(x)^{-1}\|=0$. We note that $$A(x)=\psi_-'(x)^*\psi_-'(x)+k^2\psi_-(x)^*\psi_-(x)\,,$$ since the cross term is $-{\mathrm{i}}kW(\psi_-^*,\psi_-)=0$ by (\[eq:nrmlzt3\]). If the claim were false, there would exist a sequence $x\to\infty$ and $a(x)\in\mathbb{C}^n$, ($\|a(x)\|=1$) such that $\|\psi_-'(x)a(x)\|+\|\psi_-(x)a(x)\|$ remains bounded. Together with (\[eq:lim1\]) this however contradicts the fact that $W(\psi_+^*, \psi_-)$ is regular. Having so established (\[eq:lim2\]), we multiply the $-$ version of (\[eq:sl\]) by ${\mathrm{e}^{{\mathrm{i}}kL}}(\psi_-'(L)-{\mathrm{i}}k\psi_-(L))^{-1}$ from the right, while keeping the $+$ version unchanged. As $L\to+\infty$ the two equations then go over to $$\begin{aligned}
\label{eq:s}
\bigl(\psi_+'(0) + {\mathrm{i}}k\psi_+(0)\bigr) +
R^* \bigl(\psi_+'(0)-{\mathrm{i}}k\psi_+(0)\bigr)&=0\,,\\
0&=T^*\,, \nonumber\end{aligned}$$ in the sense that the coefficients do. Since the latter system has a unique solution $(R^*, T^*)$, it is the limit of $(R_L^*, T_L^*)$.\
ii) As indicated at the end of Section \[section:2\], part (ii) is an immediate consequence of Lemma \[mainlm\].\
As a preliminary to the proof of Lemma \[mainlm\](i) we state:
\[lm:prlm\] Let $\psi_+\in S^+_{(z,s)}$ and $x\in \mathbb{R}$. Then $0$ is an eigenvalue of $\psi_+(x)$ iff $z$ is a Dirichlet eigenvalue for $H(s)$ on $[x,\infty)$, including multiplicities. These conditions can occur only for $z\in \mathbb{R}$ and for isolated $x$.
[**Proof.**]{} Solutions $\varphi=\varphi(x)$ with values in $\mathbb{C}^n$ of the differential equation $H(s)\varphi=z\varphi$ are square-integrable at $x=+\infty$ iff $\varphi(x)=\psi_+(x)a$ for some $a\in \mathbb{C}^n$. Hence the equivalence of the two conditions. They imply $z\in \mathbb{R}$ because the operator $H(s)$ with Dirichlet boundary conditions on $[x,\infty)$ is self-adjoint. To show that $x$ is isolated, we assume $x=0$ without loss and Taylor expand $\psi_+(x)$ at $x=0$ up to second order. Using (\[eq:psi\]) on the second derivative, we so obtain $$\begin{gathered}
\psi_+(x)^*\psi_+(x)=\\
P^\perp\bigl(\psi_+(0)^*\psi_+(0)+
x(\psi'_+(0)^*\psi_+(0)+\psi_+(0)^*\psi'_+(0))+x^2\psi_+(0)^*(V(0)-z)\psi_+(0)\bigr)P^\perp+\\
x^2\psi'_+(0)^*\psi'_+(0)+o(x^2)\,,\qquad
(x\to 0)\,,
$$ where an orthogonal projection $P^\perp=1-P$ onto $(\ker \psi_+(0))^\perp$ has been inserted for free as a result of $\psi_+(0)P=0$ and of $\psi'_+(0)^*\psi_+(0)=\psi_+(0)^*\psi'_+(0)$, which follows from (\[eq:refl\]) for $\bar z=z$. For small $x\neq 0$ the two terms are positive semidefinite, with the first one being definite on $(\ker \psi_+(0))^\perp$. Since $$\label{eq:dn}
\ker \psi_+(0)\cap\ker \psi'_+(0)=\{0\}$$ by (\[eq:regular\]), their sum is positive definite on all of $\mathbb{C}^n$. Hence $\psi_+(x)$ is regular. $\blacksquare$\
[**Proof of Lemma \[mainlm\].**]{} We keep $x_0=0$ throughout the proof.\
i) If at $(z_*,s_*)$ a matrix $\psi_+(0)$ is singular, that remains true under gauge transformations (\[eq:ra\]). By the previous lemma, $z_* \in \gamma$ is real and not below the spectrum of $H(s_*)$. It remains to prove the properties holding true for a dense set of potentials. Eigenvalue curves $f(s)$ of the Dirichlet Hamiltonian $H(s)$ on $[0,\infty)$ are continuously differentiable, even through crossings. By Sard’s theorem the set $\{\mu'\in\mathbb{R}\mid f(s_*)=\mu', f'(s_*)=0 \text{ for some
$s_*\in S^1$}\}$ has zero measure. Upon adding to $V(x,s)$ an arbitrarily small constant we may assume that $\mu$ is not in that set. In particular, the points $s_*$ are isolated, as claimed. We further perturb $V$ by $t W(x,s)$ where $t$ is small and $W=W(x,s)$ is an arbitrary Hermitian matrix from the same class as $V$. To first order in $t$, the splitting of a degenerate Dirichlet eigenvalue $\mu$ of $H(s_*)$ is $\mu+t\tilde \mu+o(t^2)$, ($t\to 0$), where the $\tilde \mu$ are obtained by solving the finite dimensional eigenvalue problem $$\label{eq:evlp}
P\Bigl(\int_0^\infty dx\, \psi_+(x)^*W(x,s_*)\psi_+(x)\Bigr)P a=
\tilde \mu P\Bigl(\int_0^\infty dx \,\psi_+(x)^*\psi_+(x)\Bigr)P a\, ,
\qquad (a \in \mathbb{C}^n)\,,$$ and $P$ is again the projection onto $\ker \psi_+(0)$. Since $\psi_+(x)$ is regular a.e., the matrix in brackets on the l.h.s. may take arbitrary Hermitian values, while that on the r.h.s. is positive definite on $\mathbb{C}^n$; the latter may then be set equal to $1$ by means of a gauge transformation. As a result, the eigenvalues $\tilde \mu$ are generically distinct and, since $f'(s_*)\neq 0$, the points $s_*$ split into non-degenerate ones. Moreover, points $s_*$ with $\det\psi'_+ (x_0)=0$ correspond to Neumann eigenvalues. They are also perturbed and split according to (\[eq:evlp\]), except that $P$ now is the projection onto $\ker \psi'_+(0)$. Because of (\[eq:dn\]) the coincidence between Dirichlet and Neumann eigenvalues is generically lifted.\
ii) If $\psi_{(z,s)}(x)$ is a solution of (\[eq:psi\]), then $\psi_{(\bar z,s)}(x)^*$ is a solution of (\[eq:chi\]). Hence $$L(\bar z,s)^*-L(z,s)=W(\psi_{(\bar z,s)}^*,\psi_{(z,s)};0)=0,$$ by (\[eq:nrmlzt3\]), proving the reflection property. The statement about the eigenvalue branch follows from (i). The winding number can be read off from the linearization $$\lambda(z,s)=\frac{\partial\lambda}{\partial z}\Big|_{(\mu,s_*)}\cdot(z-\mu)+
\frac{\partial\lambda}{\partial s}\Big|_{(\mu,s_*)}\cdot(s-s_*)
+O(|z-\mu|^2+|s-s_*|^2)\,,$$ where the derivatives are real.\
iii) In view of the right action (\[eq:ra\]) a section $\psi^0_+:(z,s) \mapsto \psi^0_{(z,s)}(x)$ may be defined on all of the torus by $\psi^0_{(z,s)}(0)=1$, except for the points $(\mu, s_*)$ of part (i). We use it outside of the union $\cup_{s_*}U_{s_*}$ of arbitrarily small neighborhoods of those points; inside we use a section $\hat{\psi}_+$ defined there. Using these local sections, the connection is expressed as a 1-form on the corresponding patches of the torus, e.g. $\psi^{0*}_+\mathcal{A}$ (with ${}^*$ exceptionally denoting the pull-back), and the trace of the curvature as a 2-form, ${\operatorname{tr}}D\mathcal{A}=d{\operatorname{tr}}\psi^{0*}_+\mathcal{A}$. Upon changing the patch we have $\hat{\psi}_+=\psi^0_+T$ with $T=T(z,s)\in{\mathrm{GL}}(n)$ and hence $\hat{\psi}^*_+\mathcal{A}=T^{-1}(\psi^{0*}_+\mathcal{A})T+T^{-1}(dT)$. So, using Stokes’ theorem on (\[eq:C\]), we express the Chern number as $$C=\frac{{\mathrm{i}}}{2\pi}\sum_{s_*} \oint_{\partial U_{s_*}}
{\operatorname{tr}}\hat{\psi}^*_+\mathcal{A}-{\operatorname{tr}}\psi^{0*}_+\mathcal{A}=
\frac{{\mathrm{i}}}{2\pi} \oint_{\partial U_{s_*}}d\log\det T\,.$$ We may here replace $T=\hat{\psi}_{(z,s)}(0)\psi^0_{(z,s)}(0)^{-1}=
\hat{\psi}_{(z,s)}(0)$ by $L(z,s)$, because of (\[eq:reg\]). In $U_{s_*}$ we have $L(z,s)=\lambda(z,s)P(z,s)+\tilde L(z,s)$, where $P(z,s)$ is a rank 1 projection and $\tilde L(z,s)$ is a regular linear map from $\ker P(z,s)$ to itself. Thus $\det L$ can be in turn replaced by $\det(\lambda P)=\lambda$ and the claim follows.\
iv) Let $u \in \mathbb{C}^n$ be the normalized eigenvector of $L(\mu,s_*)$ with eigenvalue $\lambda(\mu,s_*)=0$. Then $$\label{eq:nove}
\frac{\partial \lambda}{\partial z}\big|_{(\mu,s_*)} =
\bigl( u, \frac{\partial L}{\partial z}\big|_{(\mu,s_*)} u \bigr)
=\bigl( u, \, {\psi_+^{'*}}
\frac{\partial \psi_+}{\partial z} u \bigr) \, ,$$ since $\psi_+ u =0$ at $(z=\mu,s=s_*)$. There we may write $$\frac{\partial \lambda}{\partial z} =
\bigl( u, ({\psi_+^{*}}' \frac{\partial \psi_+}{\partial z} - \psi_+^* \frac{\partial^2 \psi_+}{\partial x \partial z})u \bigr)= - \bigl(
u, \, W(\psi_+^*, \frac{\partial \psi_+}{\partial z}; x=0) u \bigr)\, .$$ On the other hand we have $$W(\psi_+^*, \frac{\partial \psi_+}{\partial z};x) =
\int_x^{\infty} dx' \psi_+^* (z,x')\psi_+(z,x') > 0 \, ,$$ which follows by differentiating (\[eq:wr\]) w.r.t. $x$ and by using (\[eq:psi\]).\
v) The matrix $R$ in (\[eq:cinque\]) is determined by (\[eq:s\]) or, after multiplication with $R$, $$R\bigl(\psi_+'(0) + {\mathrm{i}}k\psi_+(0)\bigr) + \bigl(\psi_+'(0)-{\mathrm{i}}k\psi_+(0)\bigr)=0 \, .$$ This shows that $\psi_+(0)$ has eigenvalue 0 iff $R$ has eigenvalue $-1$: $\psi_+(0) u =0$ implies $(R+1) \psi_+'(0) u =0$; conversely $(R+1)v=0$ implies $R^*v=-v$ and then $\psi_+^*(0) v = 0$. Moreover $$\label{eq:dieci}
\dot{R} \bigl(\psi_+'(0)+{\mathrm{i}}k\psi_+(0)\bigr)+R\bigl(\dot{\psi}'_+(0)
+{\mathrm{i}}k \dot{\psi}_+(0)\bigr) + \dot{\psi}'_+(0) - {\mathrm{i}}k \dot{\psi}_+(0)=0 \, .$$ We compute the rate at which the eigenvalue crosses $-1$ as $$\dot{Z} = \frac{\bigl(\psi_+'(0)u, \, \dot{R} \psi_+'(0)u\bigr)}{\bigl(\psi_+'(0)u, \, \psi_+'(0)u\bigr)} \, ,$$ since the eigenprojection of the unitary $R$ is orthogonal. Multiplying (\[eq:dieci\]) with $\psi_+'(0) u$ from the left and with $u$ from the right we obtain, using $R^* \psi_+'(0)u=-\psi_+'(0)u$, $$\bigl( \psi_+'(0)u, \, \dot{R} \psi_+'(0)u\bigr)-2{\mathrm{i}}k\bigl(\psi_+'(0)u, \, \dot{\psi}_+(0)u\bigr)=0$$ and hence $$\dot{Z} \, \bigl( \psi_+'(0)u, \, \psi_+'(0) u \bigr) = 2 {\mathrm{i}}k \frac{\partial \lambda}{\partial s} \, .$$
Adiabatic evolution
===================
We consider the usual quantum mechanical, adiabatic setting in presence of a spectral gap: A family of operators $H(s)$ depending smoothly on $s$ and corresponding spectral projections $P_0(s)$ belonging to an interval $I(s)$ whose endpoints lie in the resolvent set $\rho(H(s))$. Let $U_\varepsilon (s,s_0)$ be the propagator for the non-autonomous Hamiltonian $H(s)$ with $s = \varepsilon t$. Then $$U_\varepsilon (s, s_0) (P_0(s_0)+ \varepsilon P_1(s_0)) U_\varepsilon (s, s_0)^* = P_0(s) + \varepsilon P_1 (s) + O (\varepsilon^2 )\,, \qquad
(\varepsilon \rightarrow 0)$$ with $P_1 (s)$ as given by Eq. (\[eq:Puno\]). This result is implicit in [@Th]. We give an alternate derivation which does not approximate the continuous spectrum by a quasi-continuum of discrete eigenvalues.\
[**Proof.**]{} In Eq. (\[eq:Pzero\]) $P_1(s)$ is uniquely determined [@Nen] by the conditions $$\label{eq:conds}
\begin{gathered}
{\mathrm{i}}{\dot P}_0(s)=[H, P_1 (s)]\,,\\
P_0(s)P_1 (s)+P_1 (s)P_0(s)=P_1 (s)\,,
\end{gathered}$$ which are obtained by differentiating the expansion w.r.t. $s$, respectively from the fact that it represents a projection. We omit $s$ from the notation in the rest of the proof. Eq. (\[eq:Puno\]) satisfies the first condition because of $$[H, P_1]= -\frac{1}{2\pi}\oint_{\gamma} [H-z, R(z) \dot{R} (z)]dz
=-\frac{1}{2\pi}\oint_{\gamma}
(\dot{R} (z) +R(z)^2\dot{H})dz\,,$$ where we expanded the commutator and used $\dot{R}=-R\dot{H}R$. The second contribution vanishes and the first yields the claim by $P_0= -(2\pi {\mathrm{i}})^{-1} \oint_\gamma R(z) \, dz$. The second condition (\[eq:conds\]) is equivalent to $P_0P_1P_0=0$, $(1-P_0)P_1(1-P_0)=0$, which are satisfied, too: we rewrite $\dot{R}$ as before and use the spectral representation $P=\int_I dP_\lambda$ to compute $$P_0P_1P_0= \int_ I \int_I (dP_\lambda) \dot{H} (d P_\mu)
\oint_\gamma dz \frac{1}{(\lambda - z)^2(\mu - z)} = 0 \, ;$$ similarly, $(1-P_0)P_1(1-P_0)=0$.$\blacksquare$\
We may add that in [@ASY], Eq. (2.6) and [@ASY2], Eq. (2.10a), as well as in [@Nen], Eq. (2.28), the expression $$\label{eq:asy}
P_1 (s) = -\frac{1}{2\pi} \oint_{\gamma (s)} R(z,s) [\dot{P}(s),P(s)] R(z,s) \, dz$$ is given. Its equality with (\[eq:Puno\]) can be verified independently of (\[eq:conds\]).
[**Acknowledgments.**]{} We thank M. Büttiker for drawing our attention to the multi-channel case, and S. Jansen and R. Seiler for discussions. We are grateful to the referee who contributed the above derivation of Eq. (\[eq:Puno\]) in replacement of a longer one.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider training machine learning models that are fair in the sense that their performance is invariant under certain sensitive perturbations to the inputs. For example, the performance of a resume screening system should be invariant under changes to the gender and/or ethnicity of the applicant. We formalize this notion of algorithmic fairness as a variant of individual fairness and develop a distributionally robust optimization approach to enforce it during training. We also demonstrate the effectiveness of the approach on two ML tasks that are susceptible to gender and racial biases.'
author:
- |
Mikhail Yurochkin\
IBM Research\
MIT-IBM Watson AI Lab\
`[email protected]`\
Amanda Bower$^\dagger$, Yuekai Sun$^\ddagger$\
Department of Mathematics$^\dagger$\
Department of Statistics$^\ddagger$\
University of Michigan\
`{amandarg,yuekai}@umich.edu`\
bibliography:
- 'YK.bib'
- 'AB.bib'
- 'MY.bib'
title: Training individually fair ML models with Sensitive Subspace Robustness
---
Introduction
============
Machine learning (ML) models are gradually replacing humans in high-stakes decision making roles. For example, in Philadelphia, an ML model classifies probationers as high or low-risk [@metz2020Algorithm]. In North Carolina, “analytics” is used to report suspicious activity and fraud by Medicaid patients and providers [@metz2020Algorithm]. Although ML models appear to eliminate the biases of a human decision maker, they may perpetuate or even exacerbate biases in the training data [@barocas2016Big]. Such biases are especially objectionable when it adversely affects underprivileged groups of users [@barocas2016Big]. In response, the scientific community has proposed many mathematical definitions of algorithmic fairness and approaches to ensure ML models satisfy the definitions. Unfortunately, this abundance of definitions, many of which are incompatible [@kleinberg2016Inherent; @chouldechova2017Fair], has hindered the adoption of this work by practitioners. There are two types of formal definitions of algorithmic fairness: group fairness and individual fairness. Most recent work on algorithmic fairness considers group fairness because it is more amenable to statistical analysis [@ritov2017conditional]. Despite their prevalence, group notions of algorithmic fairness suffer from certain shortcomings. One of the most troubling is there are many scenarios in which an algorithm satisfies group fairness, but its output is blatantly unfair from the point of view of individual users [@dwork2011Fairness].
In this paper, we consider individual fairness instead of group fairness. Intuitively, an individually fair ML model treats similar users similarly. Formally, an ML model is a map $h:\cX\to\cY$, where $\cX$ and $\cY$ are the input and output spaces. The leading notion of individual fairness is metric fairness [@dwork2011Fairness]; it requires $$d_y(h(x_1),h(x_2)) \le Ld_x(x_1,x_2)\text{ for all }x_1,x_2\in\cX,
\label{eq:metricFairness}$$ where $d_x$ and $d_y$ are metrics on the input and output spaces and $L\ge 0$ is a Lipschitz constant. The fair metric $d_x$ encodes our intuition of which samples should be treated similarly by the ML model. We emphasize that $d_x(x_1,x_2)$ being small does not imply $x_1$ and $x_2$ are similar in all respects. Even if $d_x(x_1,x_2)$ is small, $x_1$ and $x_2$ may differ in certain problematic ways, e.g. in their protected/sensitive attributes. This is why we refer to pairs of samples $x_1$ and $x_2$ such that $d_x(x_1,x_2)$ is small as *comparable* instead of similar.
Despite its benefits, individual fairness was dismissed as impractical because there is no widely accepted fair metric for many ML tasks. Fortunately, there is a line of recent work on learning the fair metric from data [@ilvento2019Metric; @wang2019Empirical]. In this paper, we consider two data-driven choices of the fair metric: one for problems in which the sensitive attribute is reliably observed, and another for problems in which the sensitive attribute is unobserved (see Appendix \[sec:dataDriveFairMetric\]).
The rest of this paper is organized as follows. In Section \[sec:fairnessThruRobustness\], we cast individual fairness as a form of robustness: robustness to certain sensitive perturbations to the inputs of an ML model. This allows us to leverage recent advances in adversarial ML to train individually fair ML models. More concretely, we develop an approach to audit ML models for violations of individual fairness that is similar to adversarial attacks [@goodfellow2014Explaining] and an approach to train ML models that passes such audits (akin to adversarial training [@madry2017Deep]). We justify the approach theoretically (see Section \[sec:theory\]) and empirically (see Section \[sec:computationalResults\]).
Fairness through (distributional) robustness {#sec:fairnessThruRobustness}
============================================
To motivate our approach, imagine an auditor investigating an ML model for unfairness. The auditor collects a set of audit data and compares the output of the ML model on comparable samples in the audit data. For example, to investigate whether a resume screening system is fair, the auditor may collect a stack of resumes and change the names on the resumes of Caucasian applicants to names more common among the African-American population. If the system performs worse on the edited resumes, then the auditor may conclude the model treats African-American applicants unfairly. Such investigations are known as **correspondence studies**, and a prominent example is @bertrand2004Are’s celebrated investigation of racial discrimination in the labor market. In a correspondence study, the investigator looks for inputs that are comparable to the training examples (the edited resumes in the resume screening example) on which the ML model performs poorly. In the rest of this section, we formulate an optimization problem to find such inputs.
Fair Wasserstein distances {#sec:fair_w}
--------------------------
Recall $\cX$ and $\cY$ are the spaces of inputs and outputs. To keep things simple, we assume that the ML task at hand is a classification task, so $\cY$ is discrete. We also assume that we have a fair metric $d_x$ of the form $$d_x(x_1,x_2)^2 \triangleq \langle x_1 - x_2,\Sigma(x_1 - x_2)\rangle^{\frac12},$$ where $\Sigma\in\symm_+^{d\times d}$. For example, suppose we are given a set of $K$ “sensitive” directions that we wish the metric to ignore; $d(x_1,x_2) \ll 1$ for any $x_1$ and $x_2$ such that $x_1 - x_2$ falls in the span of the sensitive directions. These directions may be provided by a domain expert or learned from data (see Section \[sec:computationalResults\] and Appendix \[sec:dataDriveFairMetric\]). In this case, we may choose $\Sigma$ as the orthogonal complement projector of the span of the sensitive directions. We equip $\cX$ with the fair metric and $\cZ \triangleq \cX\times\cY$ with $$d_z((x_1,y_1),(x_2,y_2)) \triangleq d_x(x_1,x_2) + \infty\cdot \ones\{y_1\ne y_2\}.$$
We consider $d_z^2$ as a transport cost function on $\cZ$. This cost function encodes our intuition of which samples are comparable for the ML task at hand. We equip the space of probability distributions on $\cZ$ with the fair Wasserstein distance $$\textstyle
W(P,Q) = \inf_{\Pi\in\cC(P,Q)}\int_{\cZ\times\cZ}c(z_1,z_2)d\Pi(z_1,z_2),$$ where $\cC(P,Q)$ is the set of couplings between $P$ and $Q$. The fair Wasserstein distance inherits our intuition of which samples are comparable through the cost function; the fair Wasserstein distance between two probability distributions is small if they are supported on comparable areas of the sample space.
Auditing ML models for algorithmic bias
---------------------------------------
To investigate whether an ML model performs disparately on comparable samples, the auditor collects a set of audit data $\{(x_i,y_i)\}_{i=1}^n$ and solves the optimization problem $$\textstyle
\max_{P:W(P,P_n) \le \eps} \textstyle \int_{\cZ}\ell(z,h)dP(z),
\label{eq:auditorsProblem}$$ where $\ell:\cZ\times\cH\to\reals_+$ is a loss function, $h$ is the ML model, $P_n$ is the empirical distribution of the audit data, and $\eps > 0$ is a small tolerance parameter. We interpret $\eps$ as a moving budget that the auditor may expend to discover discrepancies in the performance of the ML model. This budget forces the auditor to avoid moving samples to incomparable areas of the sample space. We emphasize that detects *aggregate* violations of individual fairness. In other words, although the violations that the auditor’s problem detects are individual in nature, the auditor’s problem is only able to detect aggregate violations. We summarize the implicit notion of fairness in in a definition.
An ML model $h:\cX\to\cY$ is $(\eps,\delta)$-distributionally robustly fair (DRF) WRT the fair metric $d_x$ iff $$\textstyle
\max_{P:W(P,P_n) \le \eps} \textstyle \int_{\cZ}\ell(z,h)dP(z) \le \delta.
\label{eq:correspondenceFairness}$$
Although is an infinite-dimensional optimization problem, it is possible to solve it exactly by appealing to duality. @blanchet2016Quantifying showed that the dual of is $$\begin{aligned}
\textstyle\sup_{P:W(P,P_n) \le \eps}\Ex_P\big[\ell(Z,h)\big] = \inf_{\lambda \ge 0}\{\lambda\eps + \Ex_{P_n}\big[\ell_\lambda^c(Z,h)\big]\}, \\
\textstyle\ell_\lambda^c((x_i,y_i),h) \triangleq \sup_{x\in\cX}\ell((x,y_i),h) - \lambda d_x^2(x,x_i).
\end{aligned}
\label{eq:auditorsProblemDual}$$ This is a univariate optimization problem, and it is amenable to stochastic optimization. We describe a stochastic approximation algorithm for in Algorithm \[alg:DRObjDual\]. Inspecting the algorithm, we see that it is similar to the PGD algorithm for adversarial attack.
starting point $\hlambda_1$, step sizes $\alpha_t > 0$ draw mini-batch $(x_{t_1},y_{t_1}),\dots,(x_{t_B},y_{t_B})\sim P_n$ $x_{t_b}^* \gets \argmax_{x\in\cX}\ell((x,y_{t_b}),h) - \lambda d_x^2(x_{t_b},x)$, $b\in[B]$ $\hlambda_{t+1} \gets \max\{0,\hlambda_t - \alpha_t(\eps - \frac{1}{B}\sum_{b=1}^Bd_x^2(x_{t_b},x_{t_b}^*))\}$
It is known that the optimal point of is the discrete measure $\frac1n\sum_{i=1}^n\delta_{(T_\lambda(x_i),y_i)}$, where $T_\lambda:\cX\to\cX$ is the *unfair map* $$T_\lambda(x_i) \gets \argmax_{x\in\cX}\ell((x,y_i),h) - \lambda d_x^2(x,x_i).
\label{eq:unfairMap}$$ We call $T_\lambda$ an unfair map because it reveals unfairness in the ML model by mapping samples in the audit data to comparable areas of the sample space that the system performs poorly on. We note that $T_\lambda$ may map samples in the audit data to areas of the sample space that are not represented in the audit data, thereby revealing disparate treatment in the ML model not visible in the audit data alone. We emphasize that $T_\lambda$ more than reveals disparate treatment in the ML model; it *localizes* the unfairness to certain areas of the sample space.
We present a simple example to illustrating fairness through robustness (a similar example appeared in [@hashimoto2018Fairness]). Consider the binary classification dataset shown in Figure \[fig:toy\]. There are two subgroups of observations in this dataset, and (sub)group membership is the protected attribute ( the smaller group contains observations from a minority subgroup). In Figure \[fig:baseline\_decision\] we see the decision heatmap of a vanilla logistic regression, which performs poorly on the blue minority subgroup. The two subgroups are separated in the horizontal direction, so the horizontal direction is the sensitive direction. Figure \[fig:baseline\_map\] shows that such classifier is unfair with respect to the corresponding fair metric, i.e. the *unfair map* leads to significant loss increase by transporting mass along the horizontal direction with very minor change of the vertical coordinate.
[.32]{} {width="\linewidth"}
[.32]{} {width="\linewidth"}
[.32]{} {width="\linewidth"}
\[fig:toy\]
#### Comparison with metric fairness
Before moving on to training individually fair ML models, we compare DRF with metric fairness . Although we concentrate on the differences between the two definitions here, they are more similar than different: both formalize the intuition that the outputs of a fair ML model should perform similarly on comparable inputs. That said, there are two main differences between the two definitions. First, instead of requiring the output of the ML model to be similar on all inputs comparable to a training example, we require the output to be similar to the training label. Thus DRF not only enforces similarity of the output on comparable inputs, but also accuracy of the ML model on the training data. Second, DRF considers differences between datasets instead of samples by replacing the fair metric on inputs with the fair Wasserstein distance induced by the fair metric. The main benefits of this modifications are (i) it is possible to optimize efficiently, (ii) we can show this modified notion of individual fairness generalizes.
Fair training with Sensitive Subspace Robustness
------------------------------------------------
We cast the fair training problem as training supervised learning systems that are robust to sensitive perturbations. We propose solving the minimax problem $$\inf_{h\in\cH}\sup_{P:W(P,P_n) \le \eps}\Ex_P\big[\ell(Z,h)\big] = \inf_{h\in\cH}\inf_{\lambda\ge 0}\lambda\eps + \Ex_{P_n}\big[\ell_{\lambda}^c(Z,h)\big],
\label{eq:SenSR}$$ where $\ell_\lambda^c$ is defined in . This is an instance of a distributionally robust optimization (DRO) problem, and it inherits some of the statistical properties of DRO. To see why encourages individual fairness, recall the loss function is a measure of the performance of the ML model. By assessing the performance of an ML model by its worse-case performance on hypothetical populations of users with perturbed sensitive attributes, minimizing ensures the system performs well on all such populations. In our toy example, minimizing implies learning a classifier that is insensitive to perturbations along the horizontal (i.e. sensitive) direction. In Figure \[fig:sensr\_decision\] this is achieved by the algorithm we describe next. To keep things simple, we assume the hypothesis class is parametrized by $\theta\in\Theta\subset\reals^d$ and replace the minimization with respect to $\cH$ by minimization with respect to $\theta$. In light of the similarities between the DRO objective function and adversarial training, we borrow algorithms for adversarial training [@madry2017Deep] to solve (see Algorithm \[alg:SenSR\]).
starting point $\hat{\theta}_1$, step sizes $\alpha_t,\beta_t > 0$ sample mini-batch $(x_1,y_1),\ldots,(x_B,y_B)\sim P_n$ $x_{t_b}^* \gets \argmax_{x\in\cX}\ell((x,y_{t_b}),\theta) - \hlambda_td_x^2(x_{t_b},x)$, $b\in[B]$ $\hlambda_{t+1} \gets \max\{0,\hlambda_t - \alpha_t(\eps - \frac{1}{B}\sum_{b=1}^Bd_x^2(x_{t_b},x_{t_b}^*))\}$ $\hat{\theta}_{t+1} \gets \hat{\theta}_t - \frac{\beta_t}{B}\sum_{b=1}^B\partial_\theta\ell((x_{t_b}^*, y_{t_b}),\hat{\theta}_t)$
#### Related work
Our approach to fair training is an instance of distributionally robust optimization (DRO). In DRO, the usual sample-average approximation of the expected cost function is replaced by $\hL_{\DRO}(\theta) \triangleq \sup_{P\in\cU}\Ex_P\big[\ell(Z,\theta)\big]$, where $\cU$ is a (data dependent) uncertainty set of probability distributions. The uncertainty set may be defined by moment or support constraints [@chen2007Robust; @delage2010Distributionally; @goh2010Distributionally], $f$-divergences [@ben-tal2012Robust; @lam2015Quantifying; @miyato2015Distributional; @namkoong2016Stochastic], and Wasserstein distances [@shafieezadeh-abadeh2015Distributionally; @blanchet2016Robust; @esfahani2015Datadriven; @lee2017Minimax; @sinha2017Certifying]. Most similar to our work is @hashimoto2018Fairness: they show that DRO with a $\chi^2$-neighborhood of the training data prevents representation disparity, i.e. minority groups tend to suffer higher losses because the training algorithm ignores them. One advantage of picking a Wasserstein uncertainty set is the set depends on the geometry of the sample space. This allows us to encode the correct notion of individual fairness for the ML task at hand in the Wasserstein distance.
Our approach to fair training is also similar to adversarial training [@madry2017Deep], which hardens ML models against adversarial attacks by minimizing adversarial losses of the form $\sup_{u\in\cU}\ell(z + u,\theta)$, where $\cU$ is a set of allowable perturbations [@szegedy2013Intriguing; @goodfellow2014Explaining; @papernot2015Limitations; @carlini2016Evaluating; @kurakin2016Adversarial]. Typically, $\cU$ is a scaled $\ell_p$-norm ball: $\cU = \{u:\|u\|_p \le \eps\}$. Most similar to our work is [@sinha2017Certifying]: they consider an uncertainty set that is a Wasserstein neighborhood of the training data.
There are a few papers that consider adversarial approaches to algorithmic fairness. [@zhang2018Mitigating] propose an adversarial learning method that enforces equalized odds in which the adversary learns to predict the protected attribute from the output of the classifier. @edwards2015Censoring propose an adversarial method for learning classifiers that satisfy demographic parity. @madras2018Learning generalize their method to learn classifiers that satisfy other (group) notions of algorithmic fairness. [@garg2019counterfactual] propose to use adversarial logit pairing [@kannan2018adversarial] to achieve fairness in text classification using a pre-specified list of counterfactual tokens.
Computational results {#sec:computationalResults}
=====================
In this section, we present results from using SenSR to train individually fair ML models for two tasks: sentiment analysis and income prediction. We pick these two tasks to demonstrate the efficacy of SenSR on problems with structured (income prediction) and unstructured (sentiment analysis) inputs and in which the sensitive attribute (income prediction) is observed and unobserved (sentiment analysis). We refer to Appendix \[supp:sensr\_implementation\] and \[sec:adultExperimentDetails\] for the implementation details.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In classical semi-infinite Coulomb fluids, two-point correlation functions exhibit a slow inverse-power law decay along a uniformly charged wall. In this work, we concentrate on the corresponding amplitude function which depends on the distances of the two points from the wall. Recently \[L. Šamaj, J. Stat. Phys. [**161**]{}, 227 (2015)\], applying a technique of anticommuting variables to a 2D system of charged rectilinear wall with “counter-ions only”, we derived a relation between the amplitude function and the density profile which holds for any temperature. In this paper, using the Möbius conformal transformation of particle coordinates in a disc, a new relation between the amplitude function and the density profile is found for that model. This enables us to prove, at any temperature, the factorization property of the amplitude function in point distances from the wall and to express it in terms of the density profile. Presupposing the factorization property of the amplitude function and using specific sum rules for semi-infinite geometries, a relation between the amplitude function of the charge-charge structure function and the charge profile is derived for many-component Coulomb fluids in any dimension.'
author:
- Ladislav Šamaj
date: 'Received: / Accepted: '
title: Amplitude Function of Asymptotic Correlations Along Charged Wall in Coulomb Fluids
---
Introduction {#Sect.1}
============
The topic of interest in this paper is the equilibrium statistical mechanics of classical Coulomb fluids which consist in mobile charges and perhaps fixed surface or volume charge densities, the system as a whole being electroneutral. The charged entities interact by the Coulomb potential whose form depends on the manifold in which the system is formulated. For an infinite $d$-dimensional Euclidean space, the electrostatic potential $v$ at a point ${\bf r}\in \mathbb{R}^{d}$, induced by a unit charge at the origin ${\bf 0}$, is the solution of the Poisson equation $$\label{Poisson}
\Delta v({\bf r}) = - s_d \delta({\bf r}) ,$$ where $s_d$ is the surface area of the unit sphere in $d$ dimensions: $s_2=2\pi$, $s_3=4\pi$, etc. This $d$-dimensional definition of the Coulomb potential maintains generic properties of “real” 3D Coulomb systems with $v({\bf r})=1/r$, $r=\vert {\bf r}\vert$. In 2D, the solution of (\[Poisson\]), subject to the boundary condition $\nabla v({\bf r})\to 0$ as $r\to\infty$, reads as $v({\bf r}) = - \ln(r/L)$ where the scale $L$ is free. For $d\ge 3$, we have $v({\bf r})=1/r^{d-2}$.
In standard “dense” Coulomb fluids like the one-component plasma (jellium with a neutralizing bulk background) and the two-component plasma (Coulomb gas of $\pm$ charges), the number of mobile charges is proportional to the volume of the confining domain. Such systems exhibit good screening properties and their bulk two-point correlations have a short-ranged decay. There exist many exact sum rules which relate the particle one-body and two-body densities, in the bulk, semi-infinite and finite geometries, see review [@Martin88].
In “sparse” Coulomb systems of charged macromolecule surfaces, the number of identical mobile charges (coined as counter-ions) is proportional to the charged surface boundary from which they are released [@Attard96; @Hansen00; @Messina09]. The high-temperature (weak-coupling) limit is described by the Poisson-Boltzmann (PB) theory [@Andelman] and by its systematic improvement via the loop expansion [@Attard88; @Netz00; @Podgornik90]. The low-temperature (strong-coupling) limit is more controversial, the single-particle picture of counter-ions in the linear surface-charge potential appears in the leading strong-coupling order, see e.g. [@Boroudjerdi05; @Mallarino13; @Mallarino15; @Netz01; @Samaj11a]. In spite of the fact that sparse Coulomb fluids have poor screening properties, the standard sum rules hold in semi-infinite and finite geometries [@Samaj13; @Samaj14; @Samaj15].
In semi-infinite geometry of an electric double layer, the screening cloud around a particle sitting near hard wall is asymmetric and therefore two-point correlations decay slowly as an inverse-power law along the wall [@Jancovici82a; @Jancovici82b; @Usenko79]. The corresponding amplitude function, which depends on the distances of the two points from the wall, satisfies a sum rule [@Jancovici82b; @Jancovici95; @Samaj10]. A relation between the amplitude function and the dipole moment was found in Ref. [@Jancovici01]. The contribution of the long-ranged charge-charge correlations along a domain boundary, together with a bulk contribution, explains the dependence of the dielectric susceptibility tensor on the shape of the confining domain, in the thermodynamic limit, as required by macroscopic electrostatics [@Choquard86; @Choquard87; @Choquard89]. Interestingly, in all exactly solvable cases the amplitude function factorizes itself in the two point distances from the wall.
In a recent paper about 2D charged rectilinear wall with counter-ions only [@Samaj15], we used a technique of anticommuting variables [@Samaj95] to derive a relation between the amplitude function and the density profile which holds for any coupling (temperature) of the fluid regime. Moreover, using the Möbius conformal transformation of particle coordinates in the partition function for a disc geometry, an exact formula for the dielectric susceptibility tensor was derived. Since this tensor contains also long-ranged correlations along the wall, it is likely that a more detailed exploration of the Möbius conformal transformation might reveal another relation between the amplitude function and the density profile which is complementary to the one derived in Ref. [@Samaj15].
In this paper, using the formalism of anticommuting variables, we repeat the Möbius conformal transformation of particle coordinates on the level of the partition function and one-body density (Sect. 2). In this way, we derive a new relation between the amplitude function and the density profile (Sect. 3). This enables us to prove the factorization property of the amplitude function for any temperature, at least for the simplified 2D model of the charged line with counter-ions only. The amplitude function is subsequently expressed locally in terms of the density profile. In Sect. 4, a relation of our result to the sum rule obtained by Blum et al. [@Blum81] enables us to extend the analysis to one-component jellium. The generalization of the formalism to charge-charge structure function of many-component Coulomb fluids in any dimension is presented in Sect. 5. Here, presupposing the factorization property of the amplitude function, its explicit relation to the charge density profile is established. A short recapitulation and conclusions are drawn in Sect. 6.
2D charged rectilinear wall with counter-ions only
==================================================
We consider a system of $N$ identical pointlike particles of elementary charge $-e$ confined to a 2D domain $D$ of points ${\bf r}=(x,y)$. The system is studied within the canonical ensemble at the inverse temperature $\beta=1/(k_{\rm B}T)$. The particle interaction part of the energy reads $- e^2 \sum_{(i<j)=1}^N \ln\vert{\bf r}_i-{\bf r}_j\vert$, where the free length scale $L$ is set to unity. The one-body Boltzmann factor $w({\bf r}) = \exp[-\beta u({\bf r})]$ involves all external potentials (e.g. due to a neutralizing bulk or surface background) acting on particles. Introducing the coupling constant $\Gamma\equiv 2\gamma=\beta e^2$, the partition function is given by $$\label{part}
Z_N(\gamma) = \frac{1}{N!} \int_D \prod_{i=1}^N
\left[ {\rm d}^2 r_i\, w({\bf r}_i) \right]
\prod_{(i<j)=1}^N \vert {\bf r}_i-{\bf r}_j\vert^{2\gamma} ,$$ where we omit irrelevant constant prefactors.
The one-body density of particles at point ${\bf r}\in D$ is defined by $$n({\bf r}) = \langle \hat{n}({\bf r}) \rangle , \qquad
\hat{n}({\bf r}) = \sum_{i=1}^N \delta({\bf r}-{\bf r}_i) ,$$ where $\hat{n}({\bf r})$ is the microscopic density of particles and $\langle \cdots \rangle$ denotes the statistical average over canonical ensemble. The corresponding averaged charge density is simply $\rho({\bf r}) = -e n({\bf r})$. At two-particle level, one introduces the two-body densities $$n_2({\bf r},{\bf r}') = \left\langle \sum_{(i\ne j)=1}^N
\delta({\bf r}-{\bf r}_i) \delta({\bf r}'-{\bf r}_j) \right\rangle .$$ The one-body and two-body densities can be obtained from the partition function (\[part\]) in the standard way as the functional derivatives: $$\begin{aligned}
n({\bf r}) & = & w({\bf r}) \frac{1}{Z_N} \frac{\delta Z_N}{\delta w({\bf r})},
\\ n_2({\bf r},{\bf r}') & = & w({\bf r}) w({\bf r}') \frac{1}{Z_N}
\frac{\delta^2 Z_N}{\delta w({\bf r}) \delta w({\bf r}')} .\end{aligned}$$
The two-body densities $n_2({\bf r},{\bf r}')$ decouple to the product of densities $n({\bf r})$ and $n({\bf r}')$ at asymptotically large distances $\vert {\bf r}-{\bf r}'\vert\to\infty$. Therefore it is useful to introduce the (truncated) Ursell functions $$\label{Ursell}
U({\bf r},{\bf r}') = n_2({\bf r},{\bf r'}) - n({\bf r}) n({\bf r}')$$ which vanish at $\vert {\bf r}-{\bf r}'\vert\to\infty$. For one-component systems of particles of charge $-e$, the charge-charge structure function is defined as $$\begin{aligned}
S({\bf r},{\bf r}') & = & e^2 \left[
\langle \hat{n}({\bf r}) \hat{n}({\bf r}') \rangle
- n({\bf r}) n({\bf r}') \right] \nonumber \\ & = &
e^2 \left[ U({\bf r},{\bf r}') + n({\bf r}) \delta({\bf r}-{\bf r}') \right] .\end{aligned}$$ The structure and Ursell functions differ from one another by a term which is nonzero only if the two points merge, i.e. $$\label{USasymp}
U({\bf r},{\bf r}') = \frac{S({\bf r},{\bf r}')}{e^2} \qquad
\mbox{if ${\bf r}\ne {\bf r}'$.}$$ For any finite or infinite domain $D$, the structure function satisfies the zeroth-moment sum rule [@Martin88] $$\label{zeroth}
\int_D {\rm d}^2 r\, S({\bf r},{\bf r}') =
\int_D {\rm d}^2 r'\, S({\bf r},{\bf r}') = 0 .$$
Formalism of anticommuting variables
------------------------------------
The formalism of anticommuting variables for 2D one-component plasmas has been introduced in Ref. [@Samaj95] and developed further in Refs. [@Samaj00; @Samaj04; @Samaj11b; @Samaj13; @Samaj14]. For $\gamma$ a positive integer, the partition function (\[part\]) can be expressed in terms of two sets of anticommuting variables $\{ \xi_i^{(\alpha)},\psi_i^{(\alpha)} \}$ each with $\gamma$ components $(\alpha=1,\ldots,\gamma)$, defined on a discrete chain of $N$ sites $i=0,1,\ldots,N-1$, as follows $$\label{antipart}
Z_N(\gamma) = \int {\cal D}\psi {\cal D}\xi\, {\rm e}^{{\cal S}(\xi,\psi)} ,
\qquad {\cal S}(\xi,\psi) = \sum_{i,j=0}^{\gamma(N-1)} \Xi_i w_{ij} \Psi_j .$$ Here, ${\cal D}\psi {\cal D}\xi \equiv \prod_{i=0}^{N-1} {\rm d}\psi_i^{(\gamma)}
\cdots {\rm d}\psi_i^{(1)} {\rm d}\xi_i^{(\gamma)} \cdots {\rm d}\xi_i^{(1)}$ and the action ${\cal S}(\xi,\psi)$ involves pair interactions of composite operators $$\label{composite}
\Xi_i = \sum_{i_1,\ldots,i_{\gamma}=0\atop (i_1+\cdots+i_{\gamma}=i)}^{N-1}
\xi_{i_1}^{(1)} \cdots \xi_{i_{\gamma}}^{(\gamma)} , \qquad
\Psi_i = \sum_{i_1,\ldots,i_{\gamma}=0\atop (i_1+\cdots+i_{\gamma}=i)}^{N-1}
\psi_{i_1}^{(1)} \cdots \psi_{i_{\gamma}}^{(\gamma)} ,$$ i.e. the products of $\gamma$ anticommuting variables with the fixed sum of site indices. Using complex variables $z=x+{\rm i}y$ and $\bar{z}=x-{\rm i}y$, the interaction matrix is given by $$w_{ij} = \int_D {\rm d}^2 z\, z^i \bar{z}^j w(z,\bar{z}) ,
\qquad i,j=0,1,\ldots,\gamma(N-1) .$$ The one-body and two-body densities are expressible explicitly as $$\begin{aligned}
n({\bf r}) & = & w(z,\bar{z}) \sum_{i,j=0}^{\gamma(N-1)}
\langle \Xi_i \Psi_j \rangle z^i \bar{z}^j , \\
n_2({\bf r}_1,{\bf r}_2) & = & w(z_1,\bar{z}_1) w(z_2,\bar{z}_2)
\sum_{i_1,j_1,i_2,j_2=0}^{\gamma(N-1)}
\langle \Xi_{i_1} \Psi_{j_1} \Xi_{i_2} \Psi_{j_2} \rangle
z_1^{i_1} \bar{z}_1^{j_1} z_2^{i_2} \bar{z}_2^{j_2} , \phantom{aaa}\end{aligned}$$ where $\langle \cdots\rangle \equiv \int {\cal D}\psi {\cal D}\xi\,
{\rm e}^S \cdots/Z_N(\gamma)$ denotes averaging over the anticommuting variables.
Next we consider the disc domain $D=\{ {\bf r}, \vert {\bf r} \vert \le R \}$ with a constant line charge density $\sigma e$ on the disc circumference $r=R$. The requirement of the electroneutrality fixes the number of counter-ions with charge $-e$ to $N=2\pi R\sigma$. For this model, we have $w(z,\bar{z})\equiv w(r)=1$ [@Samaj15] and $$\label{wi}
w_{ij} = w_i\delta_{ij} , \qquad
w_i = 2\pi \int_0^R {\rm d}r\, r^{2i+1} = \frac{\pi}{i+1} R^{2(i+1)} .$$ The diagonalization of the action in composite operators $$\label{S0}
{\cal S}(\xi,\psi) = \sum_{i=0}^{\gamma(N-1)} \Xi_i w_i \Psi_i$$ implies that $\langle \Xi_i\Psi_j \rangle = \delta_{ij} \langle \Xi_i\Psi_i \rangle$, $\langle \Xi_{i_1}\Psi_{j_1} \Xi_{i_2}\Psi_{j_2}\rangle \ne 0$ only if $i_1+j_1=i_2+j_2$, etc. This fact simplifies the series representations of the one-body and two-body densities: $$\begin{aligned}
n(r) & = & \sum_{i=0}^{\gamma(N-1)} \langle \Xi_i \Psi_i \rangle
r^{2i} , \label{antione} \\
n_2({\bf r}_1,{\bf r}_2) & = & \sum_{i_1,j_1,i_2,j_2=0\atop (i_1+i_2=j_1+j_2)}^{\gamma(N-1)}
\langle \Xi_{i_1} \Psi_{j_1} \Xi_{i_2} \Psi_{j_2} \rangle
z_1^{i_1} \bar{z}_1^{j_1} z_2^{i_2} \bar{z}_2^{j_2} . \label{antitwo}\end{aligned}$$
Conformal transformation
------------------------
We consider the particles with complex coordinates $(z,\bar{z})$ inside the disc domain $D=\{ (z,\bar{z}), z\bar{z} \le R^2 \}$. The Möbius conformal transformation $$\label{conftrans}
z' = \frac{z + R a}{1+\frac{z\bar{a}}{R}} , \qquad
z = \frac{z'- R a}{1-\frac{z'\bar{a}}{R}}$$ (with a free complex parameter $a$ such that $a\bar{a} \ne 1$) transforms the particle coordinates in the disc domain $D$ to another domain $D'$ defined by the inequality $$( R^2 - z'\bar{z}') (1-a\bar{a}) \ge 0 .$$ If $a$ is chosen such that $a \bar{a} < 1$, the original disc domain $D$ is mapped onto itself, $D'=D$. Note that $a=0$ corresponds to the identity transformation.
### Partition function
Let us study the effect of the Möbius transformation of all particle coordinates on the partition function $$\label{part1}
Z_N(\gamma) = \frac{1}{N!} \int_D \prod_{i=1}^N {\rm d} z_i {\rm d}
\bar{z}_i\, \prod_{(i<j)=1}^N \vert z_i-z_j\vert^{2\gamma} .$$ Under the conformal transformation (\[conftrans\]), each surface element ${\rm d}z{\rm d}\bar{z}$ transforms as $${\rm d}z {\rm d}\bar{z} = \frac{(1-a\bar{a})^2}{\left( 1-\frac{z'\bar{a}}{R}
\right)^2 \left( 1-\frac{\bar{z}'a}{R} \right)^2} {\rm d}z' {\rm d}\bar{z}'$$ and each square of the distance between two particles transforms as $$\vert z_i-z_j\vert^2 = \frac{(1-a\bar{a})^2}{\left( 1-\frac{z'_i\bar{a}}{R}
\right) \left( 1-\frac{\bar{z}'_i a}{R} \right) \left( 1-\frac{z'_j\bar{a}}{R}
\right) \left( 1-\frac{\bar{z}'_j a}{R} \right)} \vert z'_i-z'_j\vert^2 .$$ The partition function (\[part1\]) can be written in terms of the transformed coordinates as follows $$\label{part2}
Z_N^a(\gamma) = \frac{1}{N!} \int_D \prod_{i=1}^N {\rm d} z'_i {\rm d}
\bar{z}'_i \left[ \frac{(1-a\bar{a})}{\left( 1-\frac{z'_i\bar{a}}{R} \right)
\left( 1-\frac{\bar{z}'_i a}{R} \right)} \right]^{\nu}
\prod_{(i<j)=1}^N \vert z'_i-z'_j\vert^{2\gamma} ,$$ where we use the notation $\nu\equiv \gamma(N-1)+2$. The transformed variables $z'$ and $\bar{z}'$ under integration can be replaced by the original ones $z$ and $\bar{z}$. We see that the effect of the conformal transformation consists in changing the circular one-body Boltzmann factor $w(r) = 1$ to the non-circular one $$w^a(z,\bar{z}) =
\left[ \frac{(1-a\bar{a})}{\left( 1-\frac{z\bar{a}}{R} \right)
\left( 1-\frac{\bar{z} a}{R} \right)} \right]^{\nu} .$$ The diagonal ${\cal S}$-action (\[S0\]) transforms itself into the non-diagonal one $$\label{Sa}
{\cal S}^a(\xi,\psi) = \sum_{i,j=0}^{\gamma(N-1)} \Xi_i w_{ij}^a \Psi_j ,$$ where $$\label{wij}
w_{ij}^a = \int_D {\rm d}^2 z\,
\left[ \frac{(1-a\bar{a})}{\left( 1-\frac{z\bar{a}}{R} \right)
\left( 1-\frac{\bar{z} a}{R} \right)} \right]^{\nu} z^i \bar{z}^j ,
\qquad i,j=0,1,\ldots,\gamma(N-1) .$$ The equivalence of the original partition function $Z_N(\gamma;\{ w_i\})$ with the transformed one $Z_N^a(\gamma;\{ w_{ij}^a\})$, $$\label{Z1}
Z_N(\gamma) = Z_N^a(\gamma) ,$$ can be expressed in terms of the integrals over anticommuting variables as $$\label{Z2}
\int {\cal D}\psi {\cal D}\xi\, \exp\left[ {\cal S}(\xi,\psi) \right] =
\int {\cal D}\psi {\cal D}\xi\, \exp\left[ {\cal S}^a(\xi,\psi) \right] .$$
### Particle density
Under the conformal transformation (\[conftrans\]), the density $n(z,\bar{z};\{ w_i\}) \equiv n(r)$ transforms itself to $n^a(z',\bar{z}';\{ w_{ij}^a\})$ according to $$n(z,\bar{z}) {\rm d}z {\rm d}\bar{z} = n^a(z',\bar{z}')
{\rm d}z' {\rm d}\bar{z}' .$$ Note that this relation, when integrated over the disk domain $D$, ensures the conservation of the total number of particles under the conformal transformation. Equivalently, $$n(z,\bar{z}) = \left[ \frac{\left( 1-\frac{z'\bar{a}}{R} \right)
\left( 1-\frac{\bar{z}' a}{R} \right)}{(1-a\bar{a})} \right]^2
n^a(z',\bar{z}') .$$ Within the formalism of anticommuting variables, the transformed particle density is expressible as $$n^a(z',\bar{z}') = w^a(z',\bar{z}') \sum_{i,j=0}^{\gamma(N-1)}
\langle \Xi_i\Psi_j \rangle^a (z')^i (\bar{z}')^j ,$$ where the symbol $\langle \cdots\rangle^a$ means the averaging with the ${\cal S}^a$-action (\[Sa\]). We conclude that $$\label{nr}
n(r) = \left[ \frac{(1-a\bar{a})}{\left( 1-\frac{z'\bar{a}}{R} \right)
\left( 1-\frac{\bar{z}' a}{R} \right)} \right]^{\gamma(N-1)}
\sum_{i,j=0}^{\gamma(N-1)} \langle \Xi_i\Psi_j \rangle^a (z')^i (\bar{z}')^j .$$
Derivation of sum rules
=======================
In this part, we use the above exact relations between the original and transformed partition functions and particle densities to derive certain sum rules.
Partition function
------------------
We start with the equality of the original and transformed partition functions, see Eqs. (\[Z1\]) and (\[Z2\]). First we expand the transformed interaction matrix (\[wij\]) in linear $a$, $\bar{a}$ and quadratic $a\bar{a}$ terms: $$w_{ij}^a = \delta_{ij} w_i + \frac{\nu a}{R} \delta_{i,j+1} w_i
+ \frac{\nu \bar{a}}{R} \delta_{i+1,j} w_{i+1}
+ a \bar{a} \delta_{ij} \left( \frac{\nu^2}{R^2} w_{i+1} -
\nu w_i \right) + \cdots ,$$ where $w_i$ are the original interaction strengths (\[wi\]). The corresponding expansion of the transformed action (\[Sa\]) around the original action (\[S0\]) reads as $$\begin{aligned}
{\cal S}^a & = & {\cal S} + \frac{\nu a}{R} \sum_i \Xi_{i+1} w_{i+1} \Psi_i
+ \frac{\nu \bar{a}}{R} \sum_i \Xi_i w_{i+1} \Psi_{i+1} \nonumber \\
& & + a \bar{a} \sum_i \Xi_i \left( \frac{\nu^2}{R^2} w_{i+1} -
\nu w_i \right) \Psi_i + \cdots . \label{Saexp}\end{aligned}$$ Inserting this expansion into Eq. (\[Z2\]) and expanding the exponential in $a$, $\bar{a}$ and $a\bar{a}$ terms, we obtain $$\begin{aligned}
Z_N(\gamma) & = & Z_N(\gamma) \Bigg[ 1 +
a \bar{a} \sum_i \langle \Xi_i \Psi_i \rangle
\left( \frac{\nu^2}{R^2} w_{i+1} - \nu w_i \right) \nonumber \\ & &
+ a \bar{a} \frac{\nu^2}{R^2} \sum_{i,j} w_{i+1} w_{j+1}
\langle \Xi_i \Psi_{i+1} \Xi_{j+1} \Psi_j \rangle + \cdots \Bigg] .\end{aligned}$$ The term proportional to $a\bar{a}$ must vanish. Simultaneously, there holds $$\sum_i w_i \langle \Xi_i\Psi_i \rangle = \int_D {\rm d}^2r\, n(r) = N$$ and $$\sum_i w_{i+1} \langle \Xi_i\Psi_i \rangle = \int_D {\rm d}^2r\, r^2 n(r) .$$ From the representation (\[antitwo\]) we get $$\sum_{i,j} w_{j+1} \langle \Xi_i \Psi_{i+1} \Xi_{j+1} \Psi_j \rangle r^{2(i+1)}
= \int_D {\rm d}^2r'\, {\bf r}\cdot {\bf r}' n_2({\bf r},{\bf r}') ,$$ where ${\bf r}\cdot {\bf r}' = (z \bar{z}' + \bar{z} z')/2$ denotes the scalar product of vectors ${\bf r}$ and ${\bf r}'$. Consequently, we end up with the sum rule $$\label{sum1}
\int_D {\rm d}^2 r \int_D {\rm d}^2 r'\, {\bf r}\cdot {\bf r}'
\langle \hat{n}({\bf r}) \hat{n}({\bf r}') \rangle
= \frac{R^2 N}{\gamma(N-1)+2} .$$ This sum rule has already been derived in connection with the calculation of the dielectric susceptibility tensor, see Eq. (6.10) of Ref. [@Samaj15]. The same equality holds for the truncated correlator $\langle \hat{n}({\bf r}) \hat{n}({\bf r}') \rangle - n(r) n(r')$ since $$\int_D {\rm d}^2r \int_D {\rm d}^2r'\, {\bf r}\cdot {\bf r}' n(r) n(r') = 0$$ after the integration of $\cos(\varphi-\varphi')$ over the angle $\varphi-\varphi'$ from $0$ to $2\pi$.
Particle density
----------------
In the density relation (\[nr\]), we expand up to terms linear in $a$ and $\bar{a}$ the transformed coordinates $$z' = z + Ra - \frac{z^2}{R} \bar{a} + \cdots , \qquad
\bar{z}' = \bar{z} + R\bar{a} - \frac{\bar{z}^2}{R} a + \cdots$$ and, with the aid of the $S^a$-expansion (\[Saexp\]), the transformed correlators $$\begin{aligned}
\langle \Xi_i\Psi_j\rangle^a & = & \delta_{ij} \langle \Xi_i\Psi_i\rangle
+ \delta_{i+1,j} \frac{\nu a}{R} \sum_k w_{k+1}
\langle \Xi_i\Psi_{i+1}\Xi_{k+1}\Psi_k \rangle \nonumber \\ & &
+ \delta_{i,j+1} \frac{\nu \bar{a}}{R} \sum_k w_{k+1}
\langle \Xi_{i+1}\Psi_i\Xi_k\Psi_{k+1} \rangle + \cdots . \end{aligned}$$ Thus we obtain $$\begin{aligned}
0 & = & \frac{\gamma(N-1)}{R} (\bar{z}a+z\bar{a}) n(r)
+ \frac{\nu a}{R} \sum_{ij} w_{j+1}
\langle \Xi_i\Psi_{i+1}\Xi_{j+1}\Psi_j \rangle z^i \bar{z}^{i+1} \nonumber \\ & &
+ \frac{\nu \bar{a}}{R} \sum_{ij} w_{j+1}
\langle \Xi_{i+1}\Psi_i\Xi_j\Psi_{j+1} \rangle z^{i+1} \bar{z}^i \nonumber \\ & &
+ (\bar{z}a+z\bar{a}) \sum_i \langle \Xi_i\Psi_i\rangle (z\bar{z})^i i
\left( \frac{R}{z\bar{z}} - \frac{1}{R} \right) .\end{aligned}$$ After simple algebra, this result leads to the equality $$\label{sum2}
\left[ \gamma(N-1)+2 \right] \int_D {\rm d}^2 r'\, {\bf r}\cdot {\bf r}'
\langle \hat{n}({\bf r}) \hat{n}({\bf r}') \rangle
= 2 r^2 n(r) - \frac{1}{2} r (R^2-r^2) \frac{\partial}{\partial r} n(r) .$$ Applying the integration $\int_D {\rm d}^2 r$ to this relation, it reduces to the previous sum rule (\[sum1\]), but the present sum rule is more informative.
Let the vector ${\bf r}$ be taken along the $x$-axis, ${\bf r}=(r,0)$, so that ${\bf r}\cdot {\bf r}' = r r' \cos\varphi'$. Substituting the correlator in (\[sum2\]) by its truncation $\langle \hat{n}({\bf r}) \hat{n}({\bf r}') \rangle - n(r) n(r')
\equiv S({\bf r},{\bf r}')/e^2$ and using the zeroth-moment sum rule (\[zeroth\]), we obtain $$\begin{aligned}
[\gamma(N-1)+2] \left[ \int_0^{2\pi} {\rm d}\varphi'
\int_0^R {\rm d}r'\, (r')^2 (1-\cos\varphi') \frac{S({\bf r},{\bf r}')}{e^2}
\phantom{aaaaaaa} \right. \nonumber \\
\left. + \int_D {\rm d}^2r'\, (R-r') \frac{S({\bf r},{\bf r}')}{e^2} \right]
= - 2 r n(r) + \frac{1}{2} (R-r) (R+r) \frac{\partial}{\partial r} n(r) .
\label{pom1}\end{aligned}$$ To go from the disc to the semi-infinite rectilinear geometry in the limit $R\to\infty$, we switch to the variables $x = R - r$ and $x' = R - r'$. Eq. (\[pom1\]) then becomes $$\begin{aligned}
\frac{[\gamma(N-1)+2]}{R} \left[ \frac{1}{2} \int_{-\pi}^{\pi}
{\rm d}\varphi \int_0^{\infty} {\rm d}x'\,
\left( 2 R \sin\frac{\varphi}{2} \right)^2 \frac{S(x,x';\varphi)}{e^2}
\phantom{aaaaa} \right. \nonumber \\
\left. + \int_{-\pi}^{\pi} {\rm d}(R\varphi) \int_0^{\infty} {\rm d}x'\, x'
\frac{S(x,x';\varphi)}{e^2} \right] = - \left[
x \frac{\partial}{\partial x} n(x) + 2 n(x) \right] . \label{eqno}\end{aligned}$$
For the disc geometry it was shown [@Jancovici02] that, as the radius $R\to\infty$, the Ursell function of two particles at finite distances $x$ and $x'$ from the disc boundary and with the angle $\varphi\ne 0$ between them behaves as $$\label{asympdisc}
U(x,x';\varphi) = \frac{S(x,x';\varphi)}{e^2}
\mathop{\simeq}_{R\to\infty} \frac{f(x,x')}{[2R\sin(\varphi/2)]^2} , \qquad
\varphi\ne 0 .$$ Since $(x'-x)\delta(x-x')=0$, we can also write $$\begin{aligned}
\int_0^{\infty} {\rm d}x'\, x' \frac{S(x,x';\varphi)}{e^2} & = &
\int_0^{\infty} {\rm d}x'\, (x'-x) \frac{S(x,x';\varphi)}{e^2} \nonumber \\
& = & \int_0^{\infty} {\rm d}x'\, (x'-x) U(x,x';\varphi) .\end{aligned}$$ Introducing the lateral distance $y = R \varphi$ for the rectilinear geometry, Eq. (\[eqno\]) becomes $$\begin{aligned}
2\pi\gamma\sigma \left[ \pi \int_0^{\infty} {\rm d}x'\, f(x,x')
+ \int_{-\infty}^{\infty} {\rm d}y \int_0^{\infty} {\rm d}x'\, (x'-x)
U(x,x';y) \right] \nonumber \\
= - \left[ x \frac{\partial}{\partial x} n(x) + 2 n(x) \right] .
\label{vys}\end{aligned}$$ There exists a 2D relation between asymptotic behavior and dipole moment seen from a fixed point with coordinate $x$ [@Jancovici01]: $$\label{rov1}
\int_{-\infty}^{\infty} {\rm d}y \int_0^{\infty} {\rm d}x'\, (x'-x) U(x,x';y)
= \pi \int_0^{\infty} {\rm d}x'\, f(x,x') .$$ Consequently, Eq. (\[vys\]) implies that $$\label{rov2}
\int_0^{\infty} {\rm d}x'\, f(x,x') = - \frac{1}{2\pi^2\Gamma\sigma}
\left[ x \frac{\partial}{\partial x} n(x) + 2 n(x) \right] .$$
In the previous paper [@Samaj15], we found that $$\label{rov3}
\pi f(x,0) = - \left[ x \frac{\partial}{\partial x} n(x) + 2 n(x) \right] .$$ Combining the last two equations, we finally arrive at the relation $$\label{crucial}
\int_0^{\infty} {\rm d}x'\, f(x,x') = \frac{1}{2\pi\Gamma\sigma} f(x,0)$$ containing only the function of interest $f(x,x')$.
This equation can be checked on two exactly solvable 2D cases of the present counter-ion system [@Samaj15]. The PB $\Gamma\to 0$ limit yields $f(x,x')$ of the long-range form [@Samaj13] $$\label{PBcounter}
f(x,x') = - \frac{2}{\pi^2\Gamma} \frac{b^4}{(x+b)^3(x'+b)^3} , \qquad
b = \frac{1}{\pi\Gamma\sigma} .$$ At $\Gamma=2$, we have $f(x,x')$ of the short-range form [@Jancovici84; @Samaj13] $$\label{Gamma2counter}
f(x,x') = - 4 \sigma^2 {\rm e}^{-4\pi\sigma x} {\rm e}^{-4\pi\sigma x'} .$$ It is easy to verify that both exact solutions fulfill our Eq. (\[crucial\]).
Properties of the amplitude $f$-function
----------------------------------------
Now we aim at showing fundamental properties of the $f$-functions following from Eq. (\[crucial\]).
It is known that in 2D the function $f(x,x')$ obeys the sum rule [@Jancovici82b; @Jancovici95; @Samaj10] $$\label{sumrule2}
\int_0^{\infty} {\rm d}x \int_0^{\infty} {\rm d}x'\, f(x,x')
= - \frac{1}{2\pi^2\Gamma} .$$ Applying the integration operation $\int_0^{\infty} {\rm d}x$ to Eq. (\[crucial\]), we get $$\int_0^{\infty} {\rm d}x\, f(x,0) = - \frac{\sigma}{\pi} .$$ Taking $x=0$ in (\[crucial\]) and using the symmetry $f(x,x')=f(x',x)$, the $f$-function with both points at the $x=x'=0$ boundary is given by $$f(0,0) = - 2\Gamma \sigma^2 .$$ As a check, the exactly solvable $\Gamma\to 0$ limit (\[PBcounter\]) and $\Gamma=2$ case (\[Gamma2counter\]) satisfy this relation.
The function $f(x,x')$ is assumed to be an analytic holomorphic function of its arguments. Therefore, when deriving both sides of the relation (\[crucial\]) with respect to $x$, one can interchange the integration and derivation [@Tutschke] to obtain $$\label{crucial1}
\int_0^{\infty} {\rm d}x'\, \frac{\partial f(x,x')}{\partial x}
= \frac{1}{2\pi\Gamma\sigma} \frac{\partial f(x,0)}{\partial x} .$$ The two Eqs. (\[crucial\]) and (\[crucial1\]) can be fulfilled simultaneously only if $$\frac{\partial f(x,x')}{\partial x} = h(x) f(x,x')$$ with some unknown function $h(x)$. Equivalently, $$\frac{\partial}{\partial x} \ln\left[ -f(x,x') \right] = h(x) .$$ Taking into account the symmetry relation $f(x,x') = f(x',x)$, this PDE has the unique solution $$\ln\left[ -f(x,x') \right] = \int {\rm d}x\, h(x) + \int {\rm d}x'\, h(x') .$$ Consequently, the function $f(x,x')$ factorizes as follows $$\label{factor}
f(x,x') = - g(x) g(x') , \qquad g(x) = \exp\left[ \int {\rm d}x\, h(x) \right] .$$ The factorization property of $f(x,x')$, seen in the $\Gamma\to 0$ limit (\[PBcounter\]) and at $\Gamma=2$ (\[Gamma2counter\]), thus extends to any value of $\Gamma$.
Due to the factorization property, the density profile $n(x)$ determines the function $f(x,x')$ as follows $$\label{finalgen}
f(x,x') = - \frac{1}{2\pi^2\Gamma\sigma^2}
\left[ x \frac{\partial n(x)}{\partial x} + 2 n(x) \right]
\left[ x' \frac{\partial n(x')}{\partial x'} + 2 n(x') \right] .$$ The prefactor is fixed by the sum rule (\[sumrule2\]) together with the equality $$\label{formula}
\int_0^{\infty} {\rm d}x\, \left[
x \frac{\partial n(x)}{\partial x} + 2 n(x) \right]
= \int_0^{\infty} {\rm d}x\, n(x) = \sigma ,$$ where we used the integration by parts for $x\partial n(x)/\partial x$, the known fact that $n(x)$ goes to 0 faster than $1/x$ as $x\to\infty$ and the electroneutrality condition.
Another approach to one-component systems
=========================================
Counter-ions only
-----------------
There exists an alternative way how to derive in the 2D case with counter-ions only the important relation (\[rov2\]). In 2D, the coupling constant $\Gamma=\beta e^2$ is dimensionless. The particle density $n$ has dimension \[length\]$^{-2}$ and the surface charge density $\sigma$ has dimension \[length\]$^{-1}$, so one can write $$n(x;\sigma) = \sigma^2 t(\sigma x) ,$$ where $t$ is an unknown function. For this scaling form of the density profile, we obtain the equality $$\label{derns}
\sigma \frac{\partial n(x)}{\partial \sigma}
= 2 \sigma^2 t(\sigma x) + \sigma^3 x t'(\sigma x)
= 2 n(x) + x \frac{\partial n(x)}{\partial x} .$$
Blum et al. [@Blum81] derived a sum rule which relates the variation of the particle density $n(x)$ with respect to the surface charge density to the dipole moment seen by a fixed particle. In 2D, the sum rule reads as $$\label{Blum2D}
\frac{\partial n(x)}{\partial \sigma} = - 2\pi\Gamma
\int_{-\infty}^{\infty} {\rm d}y \int_0^{\infty} {\rm d}x'\, (x'-x) U(x,x';y)$$ With the aid of the relations (\[rov1\]) and (\[derns\]), we recover Eq. (\[rov2\]).
We can go to higher dimensions $d$ within the present approach. The $d$-dimensional Blum counterpart of the 2D sum rule (\[Blum2D\]) is [@Blum81] $$\label{Blumd}
\frac{\partial n(x)}{\partial \sigma} = - s_d \beta e^2
\int_{-\infty}^{\infty} {\rm d}y \int_0^{\infty} {\rm d}x'\, (x'-x) U(x,x';y) .$$ The 2D relation between asymptotic behavior and dipole moment (\[rov1\]) takes in $d$ dimensions the form [@Jancovici01] $$\label{Uf}
\int_{-\infty}^{\infty} {\rm d}y \int_0^{\infty} {\rm d}x'\, (x'-x) U(x,x';y)
= \frac{s_d}{2} \int_0^{\infty} {\rm d}x'\, f(x,x')$$ so that $$\label{rovnica}
\frac{\partial n(x)}{\partial \sigma} = - \frac{s_d^2}{2} \beta e^2
\int_0^{\infty} {\rm d}x'\, f(x,x') .$$ This formula can be readily checked on the exactly solvable PB limit $\Gamma\to 0$ in any dimension $d$ [@Samaj13; @Samaj15]: $$\label{exactd}
n(x) = \frac{\sigma b}{(x+b)^2} , \qquad
f(x,x') = - \frac{8}{\beta e^2 s_d^2} \frac{b^4}{(x+b)^3 (x'+b)^3} ,$$ where $b=2/(\beta e^2\sigma s_d)$ is the Gouy-Chapmann length.
We cannot prove in general the factorization property (\[factor\]) of the function $f(x,x')$ in dimensions $d\ge 3$ since we miss a relation like the 2D one (\[rov3\]) derived for any temperature in Ref. [@Samaj15]. Let us suppose that the factorization property takes place, i.e. $f(x,x')= - g(x) g(x')$, and apply the present formalism to obtain $g(x)$. The generalization of the 2D sum rule (\[sumrule2\]) to any dimension $d$ reads as $$\label{sumruled}
\int_0^{\infty} {\rm d}x \int_0^{\infty} {\rm d}x'\, f(x,x')
= - \frac{2}{\beta e^2 s_d^2} .$$ Inserting the factorization assumption into this equation implies $$\label{r1}
\int_0^{\infty} {\rm d}x\, g(x) = \sqrt{\frac{2}{\beta e^2 s_d^2}} .$$ Then, considering $f(x,x')= - g(x) g(x')$ in Eq. (\[rovnica\]) leads to $$\label{r2}
g(x) = \sqrt{\frac{2}{\beta e^2 s_d^2}} \frac{\partial n(x)}{\partial \sigma} ,$$ i.e. for every distance $x$ from the wall the function $g(x)$ is expressible locally in terms of the density profile. Note that the relations (\[r1\]) and (\[r2\]) are fully consistent since the integration of Eq. (\[r2\]) over $x$ from $0$ to $\infty$ reduces to (\[r1\]) due to the electroneutrality condition $\int_0^{\infty} {\rm d}x\, n(x) = \sigma$. The factorized $d$-dimensional PB solution (\[exactd\]) with $$g(x) = \sqrt{\frac{8}{\beta e^2 s_d^2}} \frac{b^2}{(x+b)^3}$$ evidently fulfills Eq. (\[r2\]).
Jellium model
-------------
The relation (\[r2\]) in fact holds for an arbitrary one-component system whose $f(x,x')$-function factorizes into $- g(x) g(x')$. Here, we present the jellium model of mobile pointlike particles with charge $-e$ immersed in a homogeneous (bulk) background of density $n_0$ and charge density $e n_0$. The system is constrained to the $d$-dimensional Euclidean half-space of points ${\bf r}=(x,{\bf y})$ with ${\bf y}=(y_1,\ldots,y_{d-1})$, the coordinates $y_i\in (-\infty,\infty)$ and $x\ge 0$. There is a plane charged by a constant surface charge density $\sigma e$ at $x=0$. The density profile $n(x)$ and the function $f(x,x')$ were calculated exactly in two cases.
The high-temperature Debye-Hückel (linearized PB) theory in $d$ dimensions [@Jancovici82a] yields the density profile $$n(x,\sigma) = n(x,\sigma=0) + \sigma \kappa {\rm e}^{-\kappa x} ,$$ where $\kappa=\sqrt{s_d\beta e^2 n_0}$ is the inverse Debye length. The asymptotic $\vert {\bf y}\vert = y \to\infty$ decay of the Ursell function along the wall was found in the form $$U(x,x';y) \mathop{\simeq}_{y\to\infty} - \frac{2 n_0}{s_d} {\rm e}^{-\kappa x}
{\rm e}^{-\kappa x'} \frac{1}{y^d} ,$$ i.e. $$g(x) = \sqrt{\frac{2 n_0}{s_d}} {\rm e}^{-\kappa x} .$$ With regard to the equality $\partial n(x)/\partial\sigma = \kappa {\rm e}^{-\kappa x}$, it is easy to verify that this $g$-function satisfies Eq. (\[r2\]).
The other exactly solvable case is the 2D jellium at coupling $\Gamma=2$. The free-fermion method [@Jancovici82a] yields the density profile $$n(x,\sigma) = n_0 \frac{2}{\sqrt{\pi}} \int_{-\pi\sigma\sqrt{2}}^{\infty} {\rm d}t\,
\frac{1}{1+\phi(t)} {\rm e}^{-(t-x\sqrt{2})^2} ,$$ where $\phi$ denotes the error function $$\phi(t) = \frac{2}{\sqrt{\pi}} \int_0^t {\rm d}u\, {\rm e}^{-u^2} .$$ The asymptotic decay of the Ursell function along the wall $$U(x,x';y) \mathop{\simeq}_{y\to\infty} - n_0^2 \frac{2}{\pi} \frac{\exp\left\{
-2\left[ x^2+x'^2 + 2\pi\sigma(x+x') + 2\pi^2\sigma^2\right]\right\}}{
\left[ 1+\phi(-\pi\sigma\sqrt{2})\right]^2} \frac{1}{y^2}$$ implies the $g$-function of the form $$g(x) = n_0 \sqrt{\frac{2}{\pi}} \frac{{\rm e}^{-2(x+\pi\sigma)^2}}{
\left[ 1+\phi(-\pi\sigma\sqrt{2})\right]} .$$ After simple algebra it can be shown that $$g(x) = \frac{1}{2\pi} \frac{\partial n(x)}{\partial \sigma}$$ which is in agreement with our result (\[r2\]).
A generalization to many-component Coulomb systems
==================================================
Now let us consider a general Coulomb system which consists of $s$ species of particles $\alpha=1,\ldots,s$ with the corresponding charges $q_{\alpha} e$ ($q_{\alpha}$ is the valence and $e$ the elementary charge), plus perhaps a fixed background of density $n_0$ and charge density $\rho_0=e n_0$. As before, the particles are constrained to the $d$-dimensional Euclidean half-space of points ${\bf r}=(x,{\bf y})$ with $x\ge 0$. There is a plane charged by a constant surface charge density $\sigma e$ at $x=0$. The microscopic density of particles of species $\alpha$ is given by $\hat{n}_{\alpha}({\bf r}) = \sum_i \delta_{\alpha,\alpha_i}
\delta({\bf r}-{\bf r}_i)$, where $i$ indexes the charged particles. The total microscopic charge density reads as $\hat{\rho}({\bf r})
= \rho_0 + \sum_{\alpha} q_{\alpha}e \hat{n}_{\alpha}({\bf r})$. For the present geometry, the averaged charge density depends only on the $x$-coordinate, $\rho(x) = \langle \hat{\rho}({\bf r}) \rangle$. The charge-charge structure function, defined by $$\label{charge-charge}
S({\bf r},{\bf r}') \equiv
\langle \hat{\rho}({\bf r}) \hat{\rho}({\bf r}') \rangle
- \langle \hat{\rho}({\bf r}) \rangle \langle \hat{\rho}({\bf r}') \rangle ,$$ depends on coordinates $x$, $x'$ and on the lateral distance $y=\vert {\bf y}-{\bf r}'\vert$, $S(x,x';y)$. The asymptotic large-$y$ behavior is of the form $$\label{asym-charge-charge}
S(x,x';y) \mathop{\simeq}_{y\to\infty} \frac{F(x,x')}{y^d} .$$ For the previous one-component system of particles with charge $-e$, $F(x,x')$ is related to $f(x,x')$ by $F(x,x') = e^2 f(x,x')$. The counterpart of the one-component sum rule (\[sumruled\]) is $$\label{sumruleD}
\int_0^{\infty} {\rm d}x \int_0^{\infty} {\rm d}x'\, F(x,x')
= - \frac{2}{\beta s_d^2} .$$
According to Blume et al. [@Blum81], the many-component generalization of Eq. (\[Blumd\]) reads as $$\label{BlumD}
\frac{\partial \rho(x)}{\partial (e\sigma)} = - s_d \beta
\int_{-\infty}^{\infty} {\rm d}y \int_0^{\infty} {\rm d}x'\, (x'-x) S(x,x';y) .$$ The many-component generalization of the relation (\[Uf\]) reads [@Jancovici01] $$\label{UF}
\int_{-\infty}^{\infty} {\rm d}y \int_0^{\infty} {\rm d}x'\, (x'-x) S(x,x';y)
= \frac{s_d}{2} \int_0^{\infty} {\rm d}x'\, F(x,x') ,$$ so that $$\label{BlumDD}
\frac{\partial \rho(x)}{\partial (e\sigma)} = - \frac{s_d^2 \beta}{2}
\int_0^{\infty} {\rm d}x'\, F(x,x') .$$
Let us presuppose that the $F$-function factorizes as $$F(x,x') = - G(x) G(x') .$$ Taking into account the sum rule (\[sumruleD\]) and Eq. (\[BlumDD\]), we find the direct local relation between the $G$-function and the charge profile: $$\label{rr}
G(x) = \sqrt{\frac{2}{\beta}} \frac{1}{s_d}
\left\vert \frac{\partial \rho(x)}{\partial (e\sigma)} \right\vert .$$ Note that $G(x)$ is determined up to an irrelevant sign; for simplicity, we have chosen $G(x)>0$.
Exactly solvable cases
----------------------
In the Debye-Hückel high-temperature limit [@Jancovici82a], the charge density profile takes the form $$\rho(x,\sigma) = \rho(x,\sigma=0) - e \sigma \kappa {\rm e}^{-\kappa x} ,$$ where $\kappa=\sqrt{s_d\beta e^2 \sum_{\alpha} q_{\alpha}^2 n_{\alpha}}$ is the multi-component inverse Debye length. The asymptotic amplitude function $F(x,x')$ was found in the form $$F(x,x') = - \frac{2\kappa^2}{\beta s_d^2} {\rm e}^{-\kappa (x+x')} ,$$ implying $$G(x) = \sqrt{\frac{2}{\beta}} \frac{\kappa}{s_d} {\rm e}^{-\kappa x} .$$ Since $\partial \rho(x)/\partial(\sigma e) = - \kappa {\rm e}^{-\kappa x}$, it is trivial to verify that this $G$-function satisfies Eq. (\[rr\]).
Another exactly solvable case is the 2D two-component plasma (Coulomb gas) of $\pm e$ charges at coupling $\Gamma=2$ [@Cornu89; @Jancovici92]. The density profiles of $\pm e$ particles read as $$n_{\pm}(x,\sigma) = n_{\pm}(x,\sigma=0) + \frac{m^2}{2\pi}
\int_0^{\mp 2\pi\sigma} \frac{{\rm d}t}{\sqrt{m^2+t^2}-t}
{\rm e}^{-2\sqrt{m^2+t^2} x} ,$$ where $m$ is a rescaled fugacity which has dimension of an inverse length. Since $n_+(x,\sigma=0) = n_-(x,\sigma=0)$, the charge density $\rho(x) = e \left[ n_+(x) - n_-(x) \right]$ results in $$\rho(x) = - \frac{e}{\pi} \int_0^{2\pi\sigma} \sqrt{m^2+t^2}
{\rm e}^{-2\sqrt{m^2+t^2} x} .$$ Introducing the variable $k_0 = \sqrt{m^2+(2\pi\sigma)^2}$, we obtain that $$\frac{\partial \rho(x)}{\partial (e\sigma)} = - 2 k_0 {\rm e}^{-2k_0 x} .$$ Simultaneously, it holds [@Cornu89] $$F(x,x') = - \frac{k_0^2 e^2}{\pi^2} {\rm e}^{-2k_0 (x+x')} , \qquad
G(x) = - \frac{k_0 e}{\pi} {\rm e}^{-2k_0 x} .$$ Taking $\beta e^2 = 2$, Eq. (\[rr\]) is readily shown to be satisfied.
Conclusion
==========
This paper was motivated by the previous one [@Samaj15] where, using the technique of anticommuting variables for a 2D model of the charged wall with counter-ions only, a new relation was found between the amplitude function $f(x,x')$ (with $x'=0$) of the asymptotic decay of two-body densities along the wall and the particle density profile $n(x)$, see Eq. (\[rov3\]). Here in Sect. 2, using the Möbius conformal transformation of particle coordinates on the level of one-body density for the same model, the complementary relation (\[rov2\]) was derived. The combination of the two exact relations enabled us to prove the factorization property $f(x,x') = - g(x) g(x')$ and to express $g(x)$ in terms of the density profile.
For more-complicated many-component Coulomb fluids in any dimension, it is necessary to concentrate on the charge-charge structure function (\[charge-charge\]) with the asymptotic behavior (\[asym-charge-charge\]) and to look on the relation between the amplitude function $F(x,x')$ and the charge density profile $\rho(x)$. In all exactly solvable cases which are available in the high-temperature limit and at the 2D free-fermion coupling, the amplitude function $F(x,x')$ factorizes. There is no proof of the factorization property of the amplitude function at any temperature. In general, the statistical independence of two particles at asymptotically large distances is reflected by the nullity of the truncated correlation functions. In our semi-infinite problem, the distance between two particles goes to infinity along the wall, $y\to\infty$, but the distances of the particles from the wall $x,x'$ are finite. One can intuitively argue that the limit $y\to\infty$ automatically decouples the subspaces $x$ and $x'$ which is behind the factorization property of the amplitude function. Presupposing $F(x,x') = - G(x) G(x')$ for any Coulomb fluid, the combination of two sum rules (\[BlumD\]) and (\[UF\]) permits us to express $G(x)$ in terms of the charge density profile $\rho(x)$, see Eq. (\[rr\]).
As concerns future perspective, it would be desirable to find simplified models or new methods to prove the factorization property of the amplitude function for more general Coulomb fluids. A better comprehension of the form of the amplitude function might clarify the form of the dielectric susceptibility tensor for an arbitrarily shaped domain.
The support received from Grant VEGA No. 2/0015/15 is acknowledged.
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Šamaj, L.: Is the two-dimensional one-component plasma exactly solvable? J. Stat. Phys. [**117**]{}, 131–158 (2004)
Šamaj, L., Jancovici, B.: Charge and current sum rules in quantum media coupled to radiation II J. Stat. Phys. [**139**]{}, 432–453 (2010)
Šamaj, L., Trizac, E.: Counterions at highly charged interfaces: From one plate to like-charge attraction. Phys. Rev. Lett. [**106**]{}, 078301 (2011)
Šamaj, L., Trizac, E.: Counter-ions at charged walls: Two-dimensional systems. Eur. Phys. J. E [**34**]{}, 20 (2011)
Šamaj, L.: Counter-ions at single charged wall: Sum rules. Eur. Phys. J. E [**36**]{}, 100 (2013)
Šamaj, L., Trizac, E.: Counter-ions between or at asymmetrically charged walls: 2D free-fermion point. J. Stat. Phys. [**156**]{}, 932–947 (2014)
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Tutschke, W., Vasudeva, H.L.: An introduction to complex analysis: Classical and modern approaches. CRC Press, London (2004).
Usenko, A.S., Yakimenko, I.P.: Interaction energy of stationary charges in a bounded plasma. Sov. Tech. Phys. Lett. [**5**]{}, 549–550 (1979)
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{
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---
abstract: 'We show that an edge-dominating cycle in a $2K_2$-free graph can be found in polynomial time; this implies that every $\frac{1}{k-1}$-tough $2K_2$-free graph admits a $k$-walk, and it can be found in polynomial time. For this class of graphs, this proves a long-standing conjecture due to Jackson and Wormald (1990). Furthermore, we prove that for any $\epsilon>0$ every $(1+\epsilon)$-tough $2K_2$-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other similar results.'
author:
- Gao Mou
- 'Dmitrii V. Pasechnik'
bibliography:
- 'reftough.bib'
title: 'Edge-dominating cycles, $k$-walks and Hamilton prisms in $2K_2$-free graphs'
---
Introduction
============
A graph $G$ is called $\beta$-[*tough*]{}, for a real $\beta>0$, if for any $p\geq 2$ it cannot be split into $p$ components by removing less than $p\beta$ vertices. This concept, a measure of graph connectivity and “resilience” under vertex subsets removal, was introduced in 1973 by Chvátal [@chvatal1973tough], while studying Hamiltonicity of graphs. For a survey of results on graph toughness till 2006 see [@MR2221006].
In general, toughness of a graph is NP-hard to compute [@MR1074858]. Considerable work went into investigating this computational problem for various classes of graphs. In particular, recently, Broersma, Patel and Pyatkin proved [@broersma2014toughness] that toughness of a $2K_2$-free graph, i.e. a graph that does not contain an induced copy of the disjoint union of two edges, can be found in polynomial time.
Note that $2K_2$-free graphs are an interesting class from algorithmic complexity point of view; most classical algorithmic problems for them are hard, with a notable exception of the maximum weighted independent set problem [@balasyu1989], [@isgci Graphclass: $2K_2$-free]. In particular Hamiltonian cycle problem is NP-complete already for a subclass of $2K_2$-free graphs, the [*split*]{} graphs—graphs for which the set of vertices can be partitioned into a clique and an independent set [@Golum Exercise 6.2]. Due to the latter, for $2K_2$-free graphs it makes sense to study computational complexity of concepts which are generalisations of the Hamiltonian cycle problem, such as $k$-walk.
Let $p\times G$ denote the multigraph obtained from $G$ by taking each edge $p$ times. A $k$-[*walk*]{} is a spanning subgraph $W$ of $2k\times G$ such that each vertex of $W$ has even degree at most $2k$. In particular a graph has a $1$-walk if and only if it is $K_2$ (i.e. one edge) or Hamiltonian. [For a survey of results on walks in graphs till 2005 see [@kouider2005connected].]{} In 1990 Jackson and Wormald conjectured [@jackson1990k] that for any integer $k\ge2$ a $\frac{1}{k-1}$-tough graph $G$ admits a $k$-walk.
In this paper, we prove that Jackson and Wormald’s Conjecture is true under the assumption that $G$ is $2K_2$-free.
\[thm2\] For any integer $k\ge2$, every $\frac{1}{k-1}$-tough $2K_2$-free graph $G$ admits a $k$-walk. Moreover, the latter can be found in time polynomial in $|V(G)|$.
If for $k\geq 2$ we let the toughness value $\frac{1}{k-1}$ increase to $\frac{1}{k-2}$ then one does not need $2K_2$-freeness. Indeed, it is shown in [@jackson1990k] that every $\frac{1}{k-2}$-tough graph has a $k$-walk.
Clearly, if $G$ is Hamiltonian, then $G$ is 1-tough. More generally, if $G$ has a $k$-walk, then $G$ is $\frac{1}{k}$-tough [@jackson1990k]. However, the converse is not true already for $k=1$ (there even exist $2$-tough graphs which are not Hamiltonian, cf. [@bauer2000not]).
This more or less summarises the situation with $t$-tough graphs, $t\leq 1$. On the $t>1$ side a famous conjecture of Chvátal [@chvatal1973tough] claims that there exists a constant $\beta$ such that every $\beta$-tough graph is Hamiltonian. Towards this, Ellingham and Zha [@ellingham2000toughness] proved that every 4-tough graph has a 2-walk (cf. Theorem \[thm1\] below).
It was recently shown [@broersma2014toughness] that every 25-tough 2$K_2$-free graph on at least three vertices is Hamiltonian. Our Theorem \[thm2\] was inspired by this result. However, our approach is technically quite different.
Our next result concerns a structure that is half-way between 1- and 2-walks. The [*prism*]{} over a graph $G$ is the Cartesian product
[r]{}[.3]{}
(50,40)(10,10) (20,10) (40,10) (50,10) (30,20) (30,30) (10,10)[(1,0)[40]{}]{} (20,10)[(1,1)[10]{}]{} (30,20)[(1,-1)[10]{}]{} (30,20)[(0,1)[10]{}]{}
\[fignoprism\]
$G\square K_2$ of $G$ with the complete graph $K_2$. $G$ is called [*prism-Hamiltonian*]{} if $G\square K_2$ is Hamiltonian. If $G$ is Hamiltonian, then $G\square K_2$ is also Hamiltonian, but the converse does not hold in general, cf. [@kaiser2007hamilton]. As well, this property is stronger than having a $2$-walk: cf. figure on the right, where we have a $2K_2$-free graph with a $2$-walk, but without Hamiltonian prism.
\[thm1\] Every $(1+\epsilon)$-tough $2K_2$-free graph $G$ is prism-Hamiltonian, for any $\epsilon>0$. Moreover, a Hamiltonian cycle in the prism over $G$ can be found in time polynomial in $|V(G)|$.
It is worth mentioning that the toughness constant in Theorem \[thm1\] is much better than $\frac{3}{2}$, the lower bound on toughness of a $2K_2$-free graph needed for its Hamiltonicity, see [@broersma2014toughness Sect. 4].
To prove Theorem \[thm2\] and Theorem \[thm1\], we first prove a result on edge-dominating subgraphs (a subgraph $S$ of $G$ is called [*edge-dominating*]{} if each edge of $G$contains at least one vertex from $V(S)$).
\[addgen1\] Let $G$ be a $2K_2$-free graph. Then
1. $G$ admits an edge-dominating cycle (or an edge, or a vertex) $C$;
2. if $G$ contains a triangle, then $G$ admits an edge-dominating cycle $C$, with three successive vertices on $C$ forming a triangle in $G$.
Moreover, $C$ can be found in time polynomial in $|V(G)|$.
In fact, in 1983 Veldman [@veldman83] has proved the existence of edge-dominating cycles for $2K_2$-free graphs. However, his proof is based on contraposition, so it neither tells how to find $C$ in (1), nor allows to restrict $C$ as in (2).
In the remainder of the paper we provide the proofs, and then discuss related open questions.
Proof of Theorem \[addgen1\]
============================
The proof of the first part of Theorem \[addgen1\]
--------------------------------------------------
If $G$ is a tree, then, as it is $2K_2$-free, it must either have an edge-dominating vertex, or an edge-dominating edge. Otherwise, $G$ has a cycle, say $C=x_1x_2\cdots x_kx_1$, where $k\ge3$. If $C$ is edge-dominating, then we are done. Otherwise there must be an edge $v_1v_2$ (assume there are $t>0$ such edges), with neither $v_1$ nor $v_2$ being on $C$. Since $G$ is $2K_2$-free, $v_1$ and $v_2$ have at least two distinct neighbours on $C$. Let $x_1v_1\in E(G)$ without loss of generality;
1. if $x_2v_1\in E(G)$, then $C'=x_1v_1x_2x_3\cdots x_kx_1$ is a longer cycle;
2. if $x_2v_2\in E(G)$, then $x_1v_1v_2x_2x_3\cdots x_kx_1$ is a longer cycle;
3. if $x_2v_1,x_2v_2\not\in E(G)$, then applying $2K_2$-freeness to $v_1v_2$ and $x_2x_3$, we get either $x_3v_1\in E(G)$ or $x_3v_2\in E(G)$.
1. If $x_3v_2\in E(G)$, then $C'=x_1v_1v_2x_3\cdots x_kx_1$ is a longer cycle;
2. if $x_3v_2\not\in E(G)$, then $x_3v_1\in E(G)$.
1. if $x_2$ is adjacent to no vertex outside $C$, then use $C'=x_1v_1x_3\cdots x_kx_1$ instead of $C$. We know that $C$ and $C'$ have the same length, but $C'$ dominates all the edges that are dominated by $C$, and $C'$ also dominates $v_1v_2$, which is not dominated by $C$. So replacing $C$ by $C'$ decreases $t$.
2. Otherwise $x_2$ is adjacent to a vertex outside $C$, say $z$. As $x_2$ is not adjacent to $v_1$ or $v_2$, we have $z$ adjacent to either $v_1$ or $v_2$. If $zv_1\in E(G)$, then $C'=x_1v_1zx_2x_3\cdots x_kx_1$ is a longer cycle. Otherwise $C'=x_1v_1v_2zx_2x_3\cdots x_kx_1$ is a longer cycle.
Repeat the process above. At each iteration either $|V(C)|$ increases, or $t$ decreases. Thus the process will stop, with $t=0$, in at most $|E(G)|^2$ steps.
The proof of the second part of Theorem \[addgen1\]
---------------------------------------------------
The algorithmic procedure for the second part is almost the same, requiring only a minor modification described below.
Let $G$ contain a triangle $w_1w_2w_3$. If $w_1w_2w_3$ is edge-dominating, then there is nothing to prove. Otherwise, there is an edge, namely $u_1u_2\in E(G)$, with neither $u_1$ nor $u_2$ on $w_1w_2w_3$. Then, by the $2K_2$-freeness, we can connect $w_1w_2w_3$ and $u_1u_2$ together, to get a 5-cycle $C$, with $w_{\pi(1)}$, $w_{\pi(2)}$ and $w_{\pi(3)}$ successive on $C$ for some permutation $\pi$ of $\{1,2,3\}$.
If $C$ is edge-dominating, then we are done. Otherwise, we proceed by induction on $k:=|V(C)|$. Suppose $k\ge5$, and there are three successive vertices on $C$, namely $X'$, $X$ and $X''$ forming a triangle in $G$. We claim that then we can find a cycle $C'$ such that $C'$ dominates more edges than $C$ (perhaps all), and $X'$, $X$ and $X''$ are also successive on $C'$.
Let $v_1v_2\in E(G)$ such that neither $v_1$ nor $v_2$ is on $C$. By $2K_2$-freeness, $v_1$ and $v_2$ are adjacent to at least two of $\{X,X',X''\}$, and thus to at least one of $\{X',X''\}$. Suppose, without loss of generality, that $v_1X'\in E(G)$; label the vertices in $C$ in the following way: $X'$ is labeled by $x_1$. The neighbour of $x_1$ on $C$ distinct from $X$ is labeled by $x_2$. The other vertices on $C$ are labeled successively, see Figure \[labelcycle\].
(0,0)node\[left\][$X'=x_1$]{} circle(2pt); (1.8,0)node\[above right\][ $x_2$]{} circle(2pt); (3.5,-0.5)node\[above\][ $x_3$]{} circle(2pt); (0,0)–(1.8,0); (3.5,-0.5)–(1.8,0); (3.5,-0.5)–(4,-1); (4,-1)–(4.5,-1.5);
(-1.2,-1)node\[left\][$X=x_k$]{} circle(2pt); (0,0)–(-1.2,-1);
(-1.2,-2)node\[left\][$X''=x_{k-1}$]{} circle(2pt); (-1.2,-2)–(-1.2,-1); (-1.2,-2)–(-1,-2.2); (-1,-2.2)–(-0.6,-2.6);
(0,1)node\[below right\][ $v_1$]{} circle(2pt); (0,1)–(1.5,1); (1.5,1)node\[below right\][ $v_2$]{} circle(2pt); (0,1)–(0,0); (0,0)–(-1.2,-2);
Note that for $k\geq 5$ the operation used in the proof of the first part of Theorem \[addgen1\] of replacing $C$ by $C'$ (enlarging $|V(C)|$ or reducing $t$) does not touch the edges $x_{k-1}x_k$ and $x_kx_1$. Thus the triangle-forming vertices $X''$, $X$ and $X'$ are always successive on $C$ in our process. Then they are on the edge-dominating cycle we obtain there.
Proof of Theorem \[thm2\]
=========================
Combining Theorem \[addgen1\] and the following Lemma \[addtec\], we obtain Theorem \[thm2\].
\[addtec\] Let $k\geq 2$. If $G$ has an edge-dominating cycle $C$ (or an edge, or a vertex) and if $G$ is $\frac{1}{k-1}$-tough then $G$ admits a $k$-walk.
The induced subgraph $D=G-C$ is an independent set. For any $D_0\subset D$, by $\frac{1}{k-1}$-toughness, $D_0$ has at least $\lceil\frac{|D_0|}{k-1}\rceil$ neighbours in $C$. By Hall’s Theorem [@bomu08 Theorem 16.4], there is $E'\subset E(G)$ such that each $e\in E'$ has one vertex in $D$ and the other in $C$. Moreover, each vertex in $D$ is incident to exactly one edge in $E'$, while each vertex in $C$ is incident to at most $k-1$ edges in $E'$. Then the (doubled) edges in $E'$ and the edges in the edge-dominating cycle (respectively, the doubled edge) $C$ form a $k$-walk in $G$.
Proof of Theorem \[thm1\]
=========================
The following lemma is the key technique in the proof of Theorem \[thm1\].
\[keylem\] Let $G$ be $(1+\epsilon)$-tough, for some $\epsilon>0$.
1. If $G$ contains an edge-dominating cycle $C$ with even number of vertices, then the prism over $G$ is Hamiltonian.
2. If $G$ contains an edge-dominating cycle $C=v_1v_2\cdots v_{2p+1}v_1$ of odd length, and there are three vertices $v_1$, $v_{2q}$ and $v_{2q+1}$, for some $1\le q\le p$, inducing a triangle in $G$, then the prism over $G$ is Hamiltonian.
For the first part (see Figure \[fig:evenprism\]), denote $C=v_1v_2\cdots v_{2p}v_1$. The set $D=V(G)-V(C)$ of vertices outside $C$ is an independent set. By Hall’s Theorem and 1-toughness, there is a matching $M$ between $D$ and $C$. That means that for any vertex $u_j$ in $D$, there is a vertex $v_{i_j}$ on $C$ adjacent to $u_j$ in $M$.
Obviously, we have a Hamiltonian cycle in $\bar{C}$, the prism over $C$, namely $$v_1v'_1v'_2v_2\cdots v_{2p-1}v'_{2p-1}v'_{2p}v_{2p}v_1.$$ Now, we change every $v_{i_j}v'_{i_j}$ (or $v'_{i_j}v_{i_j}$) into $v_{i_j}u_ju'_jv'_{i_j}$ (or $v'_{i_j}u'_ju_jv_{i_j}$) to get a Hamilton cycle in $\bar{G}$.
(-2,0)node\[below left\][$v_{2p}$]{}–(-1,0);(-0.5,0)–(0.5,0);(1,0)–(2,0); (-2,0.6)node\[below left\][$v'_{2p}$]{}–(-1,0.6);(-0.5,0.6)–(0.5,0.6);(1,0.6)–(2,0.6); (0,0)node\[below left\][$v_{i_j}$]{} circle(1pt); (0,0.6)node\[below left\][$v'_{i_j}$]{} circle(1pt); (-2,0.6) circle(1pt); (2,0.6)node\[below left\][$v'_1$]{} circle(1pt); (-1.5,0.6)node\[below right\][$v'_{2p-1}$]{} circle(1pt); (1.55,0.6)node\[below left\][$v'_2$]{} circle(1pt); (2,0)node\[below left\][$v_1$]{} circle(1pt); (-2,0) circle(1pt); (1.55,0)node\[below left\][$v_2$]{} circle(1pt); (-1.5,0)node\[below right\][$v_{2p-1}$]{} circle(1pt); (-1,0)–(-0.5,0); (1,0)–(0.5,0); (-1,0.6)–(-0.5,0.6); (1,0.6)–(0.5,0.6); (1.1,0)node\[below left\][$v_3$]{} circle(1pt); (1.1,0.6)node\[below left\][$v'_3$]{} circle(1pt); (0.2,0.4)node\[below right\][$u_j$]{} circle(1pt); (0.2,1)node\[below right\][$u'_j$]{} circle(1pt); (2,0)–(2,0.6)–(1.55,0.6)–(1.55,0)–(1.1,0)–(1.1,0.6); (0,0)–(0.2,0.4)–(0.2,1)–(0,0.6); (0,0)–(0,0.6); (-1.5,0)–(-1.5,0.6)–(-2,0.6)–(-2,0); (-2,0)..controls(0,-0.5)..(2,0); (-2,0.6)..controls(0,0.9)..(2,0.6); at(0.2,-0.5)[$G$]{};at (-0.4,0.9)[$G'$]{};
For the second part, denote $C=v_1v_2\cdot v_{2p+1}v_1$ (see Figure \[fig2\]). The set $D=V(G)-V(C)$ of vertices outside $C$ is an independent set. By Hall’s Theorem, and $(1+\epsilon)$-toughness, there is a matching $M$ between $D$ and $C-\{v_1\}$. This means that for any vertex $u_j$ in $D$, there is a vertex $v_{i_j}$ in $C-\{v_1\}$ adjacent to $u_j$ in $M$.
Clearly, we have a Hamiltonian cycle in $\bar{C}$, namely $$v_1v_2v'_2v'_3v_3\cdots v_{2q-1}v_{2q}v'_{2q}v'_1v'_{2q+1}v_{2q+1}\cdots v_{2p+1}v_1.$$ Now, we change every $v_{i_j}v'_{i_j}$ (or $v'_{i_j}v_{i_j}$) into $v_{i_j}u_ju'_jv'_{i_j}$ (or $v'_{i_j}u'_ju_jv_{i_j}$) to get a Hamilton cycle in $\bar{G}$.
(-2,0)node\[below left\][$v_{2p+1}$]{}–(-1.4,0);(1,0)–(2,0); (-2,0.6)node\[below left\][$v'_{2p+1}$]{}–(-1.4,0.6);(1,0.6)–(2,0.6); (0.4,0)node\[below left\][$v_{i_j}$]{} circle(1pt); (0.4,0.6)node\[below right\][$v'_{i_j}$]{} circle(1pt); (-2,0.6) circle(1pt); (2,0.6)node\[below left\][$v'_1$]{} circle(1pt); (-1.5,0.6)node\[below right\][$v'_{2p}$]{} circle(1pt); (1.55,0.6)node\[below left\][$v'_2$]{} circle(1pt); (2,0)node\[below left\][$v_1$]{} circle(1pt); (-2,0) circle(1pt); (1.55,0)node\[below left\][$v_2$]{} circle(1pt); (-1.5,0)node\[below right\][$v_{2p}$]{} circle(1pt); (-1.4,0)–(-1.1,0);(-1.1,0)–(-0.4,0);(-1,0)node\[below right\][$v_{2q+1}$]{} circle(1pt); (1,0)–(0.5,0); (-1.4,0.6)–(-1.1,0.6);(-1.1,0.6)–(-0.4,0.6);(-1,0.6)node\[below right\][$v'_{2q+1}$]{} circle(1pt); (1,0.6)–(0.5,0.6); (1.1,0)node\[below left\][$v_3$]{} circle(1pt); (1.1,0.6)node\[below left\][$v'_3$]{} circle(1pt); (0.2,0.4)node\[below left\][$u_j$]{} circle(1pt); (0.2,1)node\[above left\][$u'_j$]{} circle(1pt); (2,0)–(1.55,0)–(1.55,0.6)–(1.1,0.6)–(1.1,0); (0.4,0)–(0.2,0.4)–(0.2,1)–(0.4,0.6); (0.4,0)–(0.4,0.6); (-1.5,0)–(-1.5,0.6)–(-2,0.6)–(-2,0); (-2,0)..controls(0,-0.5)..(2,0);
at(0.2,-0.5)[$G$]{};at (-0.4,0.9)[$G'$]{}; (-0.4,0)–(0.2,0);(0.2,0)–(0.5,0); (-0.4,0.6)–(0.2,0.6);(0.2,0.6)–(0.5,0.6); (-0.5,0)node\[below right\][$v_{2q}$]{} circle(1pt); (-0.5,0.6)node\[below right\][$v'_{2q}$]{} circle(1pt); (-0.5,0.6)..controls(0.7,0.85)..(2,0.6); (-1,0.6)..controls(0.3,0.97)..(2,0.6); (-0.5,0.6)–(-0.5,0); (-1,0.6)–(-1,0);
Now, suppose $G$ is a triangle-free $2K_2$-free graph. By [@broersma2014toughness Theorem 4], if $|V(G)|\geq 3$ then $G$ is Hamiltonian, and so prism-Hamiltonian. If $|V(G)|=2$, i.e. $G$ is a single edge, and obviously prism-Hamiltonian. Finally, if $G$ is not triangle-free then we are done by Theorem \[addgen1\] (2) and Lemma \[keylem\].
Concluding remarks
==================
Lemma \[addtec\] can be used to prove existence of 2-walks in classes of graphs wider than $2K_2$-free. For instance, it is immediate from [@veldman83 Corollary 3.2] that each 2-connected $3K_2$-free graph admits an edge-dominating cycle. From the latter and Lemma \[addtec\], it is easy to obtain the following.
\[2w3k2f\] Let $G$ be a 1-tough $3K_2$-free graph. Then $G$ admits a 2-walk.
It would be interesting to find out whether Theorem \[2w3k2f\] and similar results of this type can be made effective. Towards this end, we would like to propose the following
Let $\ell\geq 2$ be a fixed constant. Then for the $\ell-1$-connected $\ell K_2$-free graphs there is a polynomial time algorithm finding an edge-dominating cycle.
Of independent interest would be finding out whether more general results from [@veldman83], in particular Theorem \[thm:veld\], can be made algorithmic.
\[thm:veld\] [[@veldman83 Theorem 3].]{} Let $G$ be an $\ell-1$-connected graph such that for every induced $\ell K_2$-subgraph $H$ of $G$ one has the sum of degrees of vertices in $H$ at least $\frac{(\ell-1)(|V(G)-\ell +1)}{2}$. Then $G$ has an edge-dominating cycle.
Acknowledgements. {#acknowledgements. .unnumbered}
-----------------
The authors thank Nick Gravin and Edith Elkind for helpful comments on drafts of this text. Research supported by Singapore MOE Tier 2 Grant MOE2011-T2-1-090 (ARC 19/11).
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**Quantization Phenomena of critical Hamiltonians in 2D systems**
S. C. Chen$^{1}$, J. Y. Wu$^{2}$, C. Y. Lin$^{1}$, and M. F. Lin$^{1}$
$^{1}$Department of Physics, National Cheng Kung University, Tainan, Taiwan 701
$^{2}$Center of General Studies, National Kaohsiung Marine University, Kaohsiung 811, Taiwan
0.6 truecm
This review work addresses the recent advances in solving more comprehensive Hamiltonians. The generalized tight-binding model is developed to investigate the feature-rich quantization phenomena in emergent 2D materials. The mutli-orbital bondings, the spin-orbital interactions, the various geometric structures, and the external fields are taken into consideration simultaneously. Specifically, the IV-group layered systems, black phosphorus and MoS$_{2}$ exhibit the unique magnetic quantization. This is clearly indicated in three kinds of Landau levels (LLs), the orbital-, spin- and valley-dependent LL groups, the abnormal LL energy spectra, and the splitting, crossing and anticrossing behaviors. A detailed comparison with the effective-mass model is made. Some theoretical predictions have been confirmed by the experimental measurements.
0.6 truecm
**I. INTRODUCTION**
How to solve the Hamiltonian is one of the basic topics in physics science. It is very interesting to comprehend the diverse quantization phenomena due to the various Hamiltonians in condensed-matter systems, especially for the feature-rich magnetic quantization. Such Hamiltonians possess the complex effects coming from the multi-orbital bondings, the spin-orbital coupling (SOC), the magnetic field ($\mathbf{B}$=B$_{z}\widehat{z}$), the electric field ($\mathbf{E}$=E$_{z}\widehat{z}$), the interlayer hopping integrals, the number of layers, the stacking configurations, the curved surfaces, the hybridized structures, and the distinct dimensionalities. The generalized tight-binding model is developed to include the critical interactions simultaneously. The quantized energy spectra and wave functions can be computed very efficiently by the exact diagonalization method even for a rather large Hamiltonian with complex matrix elements. This model has been used to make systematic studies on the three-dimensional (3D) graphites [@chang2005; @Ho2011; @Wang2011; @Ho2012; @Chen2015; @Ho2014], 2D graphenes [@Y.K.Huang2014; @J.H.Ho2008; @C.Y.Lin2014; @Y.H.Lai2008; @C.Y.Lin2015; @T.N.Do2015], and 1D graphene nanoribbons [@Y.C.Huang2007; @Y.C.Huang2009; @H.C.Chung2016]. It is further extended to the mainstream layered materials, e.g., other group-IV systems [@S.C.Chen2015; @J.Y.Wu2015; @J.Y.Wu2014; @J.Y.Wu20152], and MoS$_{2}$ [@Y.H.Ho2015; @Y.H.Ho2014; @HoY.H.mos2]. Moreover, the generalized tight-binding model can directly combine with the single- and many-particle theories to study the other essential physical properties, such as, magneto-optical properties [@Y.C.Huang2008; @H.C.Chung2016; @Y.P.Lin2015; @R.B.Chen2014; @R.B.Chen2012; @Y.H.Ho2010; @Y.H.Ho20101] and Coulomb excitations [@J.Y.Wu2015; @J.Y.Wu2014; @J.Y.Wu20141; @J.Y.Wu2011; @Y.P.Lin20151]
On the other hand, the perturbation method is frequently used to investigate the low-energy electronic states and magnetic quantization. It is very suitable for the condensed-matter systems with monotonous band structures. For example, the effective-mass model can deal with the magnetic quantization in monolayer graphene [@YisongZheng2002; @S.G.Sharapov2004; @V.P.Gusynin2005; @M.O.Goerbig2011], AA- and AB-stacked few-layer graphenes [@C.P.Chang2011; @E.McCann2006; @MikitoKoshino2011; @H.Min2008; @S.H.R.Sena2011; @E.McCann2013], monolayer silicene and germanene [@Ezawa2012], MoS$_{2}$ [@M.Tahir2016], and black phosphorus [@PRodin2014]. This model will become too complex or cumbersome to magnetically quantize the multi-valley and/or multi-orbital electronic states, such as, the magnetic quantization for the oscillatory energy bands in ABC- and AAB-stacked graphenes [@C.Y.Lin2014; @T.N.Do2015], the seriously distorted Dirac-cone structure in sliding bilayer graphenes [@Y.K.Huang2014], the three constant-energy loops due to the significant sp$^{3}$ bondings in monolayer tinene [@S.C.Chen2015], and the mixed energy bands in hybridized carbon systems [@T.S.Li2008; @T.S.Li2010; @C.H.Lee2011; @C.H.Lee20111; @M.H.Lee2015]. Furthermore, it is very difficult to resolve the complex quantization phenomena in the presence of the non-uniform or the composite external fields [@Y.C.Ou2014; @Y.C.Ou2013; @Y.C.Ou2011; @Y.H.Chiu2010].
The layered condensed-matter systems have stirred a lot of experimental [RadisavljevicB2011,K.Hao2016,C.Zhang2016,HNLi2015,W-THsu2015,J.Qi2015]{} and theoretical studies [@C.P.Chang2011; @C.Y.Lin2015; @Y.H.Lai2008; @C.Y.Lin2014; @T.N.Do2015; @Y.K.Huang2014; @YisongZheng2002; @V.P.Gusynin2005; @T.Morimoto2013], mainly owing to the nano-scaled thickness and the specific symmetries. They are ideal 2D materials for studying the novel physical, chemical and material phenomena. Furthermore, such systems have shown high potentials for future technological applications, e.g., nano-electronics [@EngelM.2012; @KhanF.2016; @KumarA.2016; @LiMY2015; @Yu-TingWang2015; @A.Kasry2010; @Q.Xiang2012; @MCRechtsman2013], optoelectronics [@Koppens; @Bonaccorso; @Tseng; @LiuJ.B.; @Deng; @Yan; @Kocaman; @Tassin; @Vakil; @Deng1; @Deng2; @Abajo] and energy storage [@Shown; @Hsu; @Ibram; @Baughman; @Simon; @Stoller; @Chan; @Wang; @Bulusheva; @Bissett; @LiJ.Y.2016; @Sun; @Gwon; @Gwon1]. Few-layer graphenes have been successfully synthesized by the distinct experimental methods, such as, mechanical exfoliation [Hattendorf,Novoselov,Jayasena,Cooper,Noroozi,Dobbelin,Majee,Arao,Bracamonte,Song,Dou]{}, electrostatic manipulation of scanning tunneling microscopy (STM) [YinL.J.,XuP.2013jjap,XuP.2012,Kurys,XuP.2013sur]{}, and chemical vapor deposition [@YeS.2016; @NorimatsuW.; @WarnerJ.H.; @BiedermannL.B.; @BorysiukJ.; @ReinaA.; @LiuL.; @GomezT.; @ZhangX.2014; @ZhouZ.; @KimK.S.; @LiX.; @ZhouH.; @SuC.Y.; @BaeS.; @VijayaraghavanR.K.; @WuX.Y.; @BoscaA.; @LeeJ.K.; @QueY.]. Four kinds of typical stacking configurations, AAA [@BorysiukJ.; @LeeJ.K.], ABA [@BiedermannL.B.; @LiuL.; @ZhouZ.], ABC [@NorimatsuW.; @WarnerJ.H.] and AAB [@BiedermannL.B.; @QueY.], are clearly identified in the experimental measurements. It should be noticed that the STM tip can generate the continuous changes in the stacking configuration, e.g., the configuration transformation among the ABA, ABC and AAB stackings [@XuP.2012].
The essential electronic properties of planar graphenes are dominated by the $2p_{z}$-orbital hybridization, the hexagonal honeycomb symmetry, the stacking configuration and the number of layers. The main features of low-lying energy bands are further reflected in the rich magnetic quantization. The tri-layer AAA, ABA, ABC and AAB stackings, respectively, have the unusual energy dispersions: (1) the linearly intersecting bands (the almost isotropic Dirac-cone structures) [@C.P.Chang2011; @C.Y.Lin2015], (2) the parabolic bands and the linear bands [@Y.H.Lai2008], (3) the weakly dispersive bands, the sombrero-shaped bands, and the linear bands [@C.Y.Lin2014]; (4) the oscillatory bands, the sombrero-shaped bands, and the parabolic bands [@T.N.Do2015]. Such stacking systems exhibit the novel Landau levels (LLs), in which the rich magneto-electronic properties include the diverse B$_{z}$-dependent energy spectra, the asymmetric energy spectra about the Fermi level (E$_{F}$), the crossing and/or anti-crossing behaviors, the main and side modes, and the configuration- and E$_{z}$-created splitting states. Specifically, the configuration transformation between AA and AB stackings will induce the thorough destruction of the Dirac-cone structures. The three kinds of LLs, the well-behaved, perturbed and undefined LLs, are predicted to reveal in the changes from the linear to the parabolic bands
Few-layer germanene and silicene can be synthesized on distinct substrate surfaces, e.g., Si on Ag(111), Ir(111) & ZrB2 surfaces [@LiT.; @VogtP.; @B.Aufray; @A.Fleurence; @L.Meng]; Ge on Pt(111), Au(111) & Al(111) surfaces [@L.F.Li; @M.Derivaz; @Davila]. Germanene and silicene possess the buckled structures with a mixed $sp^{2}$-$sp^{3}$ bonding rather than a sp$^{2}$ bonding, since the relatively weak chemical bonding between the larger atoms cannot maintain a planar structure (Fig. 1(b)). These two systems have the significant SOC’s much stronger than that in graphene. The SOC’s can separate the Dirac-cone structures built from the dominating $3p_{z}$ or $4p_{z}$ orbitals; that is, the intrinsic systems are narrow-gap semiconductors (E$_{g}\sim $45 meV for Ge & E$_{g}\sim $5 meV for Si) [C.C.LiuPRB,C.C.LiuPRL]{}. Furthermore, the application of a uniform perpendicular electric field leads to the modulation of energy gap and the splitting of spin-related configurations [@Ezawa2012; @M.Ezawa2012; @N.D.Drummond2012]. The magneto-electronic properties are greatly enriched by SOC and E$_{z}$, including the modified B$_{z}$-dependent energy spectra, the spin-up- and spin-down-dominated states, and the E$_{z}$-generated crossing and anti-crossing behaviors.
Monolayer tinene is successfully fabricated on a substrate of bismuth telluride [@F.Zhu], while monolayer Pb system is absent in the experimental measurements up to now. The theoretical studies show that the single-layer Sn an Pb systems have rather strong sp$^{3}$ bondings and SOC’s [@C.C.LiuPRB; @Y.Xu2013]. Apparently, the complex chemical bondings from ($s,p_{x},p_{y},p_{z}$) orbitals need to be included in the low-energy model calculations. However, the low-lying electronic structures of graphene, silicene and germanene are mainly determined by the p$_{z}$ orbitals. The very pronounced mixing effects of multi-orbital bondings and SOC’s can create the $p_{z}$- and ($p_{x},p_{y}$)-dominated energy bands near E$_{F}$, indicating the existence of the multi-constant-energy loops. There exist two groups of low-lying LLs, with the different orbital components, spin configurations, localization centers, state degeneracy, and B$_{z}$- and E$_{z}$-dependencies. Specially, the LL splitting and anti-crossing behaviors strongly depend on the type of orbitals and the external fields. The competitive or cooperative relations among the orbital hybridizations, SOC, **B** and **E** are worthy of detailed investigations.
The group-V phosphorus possesses several allotropes in which black phosphorus (BP) is the most stable phase under normal experimental conditions [@PLiu]. Few-layer phosphorene is successfully obtained using the mechanical cleavage approach [@PLi2014; @PLiu2014], liquid exfoliation [@PBrent2014; @PYasaei2015; @Pkang2015], and mineralizer-assisted short-way transport reaction [@PLange2007; @PNilges2008; @PKopf2014]. Specially, the experimental measurements show that the BP-based field effect transistor has an on/off ratio of 105 and a carrier mobility at room temperature as high as 103 cm$^{2}/$Vs. BP is expected to play an important role in the next-generation electronic devices [@PLi; @PLiu2014]. Phosphorene exhibits a puckered structure related to the $sp^{3}$ hybridization of ($3s,3p_{x},3p_{y},3p_{z}$) orbitals. The deformed hexagonal lattice of monolayer BP has four atoms [@PRudenko], while the group-IV honeycomb lattice includes two atoms. The low-lying energy bands are highly anisotropic, e.g., the linear and parabolic dispersions near E$_{F}$, respectively, along the $\widehat{k_{x}}$ and $\widehat{k_{y}}$ directions. The anisotropic behaviors are further reflected in other physical properties, as verified by recent measurements on optical spectra and transport properties [@PLi; @PLow]. BP has a middle energy gap of $~\sim
1.5-2$ eV at the $\Gamma $ point, being quite different from the narrow or zero gaps of group-IV systems. The low-lying energy dispersions, which are dominated by $3p_{z}$ orbitals, can be described by a four-band model with the complicated multi-hopping integrals [@PRudenko]. The low-energy electronic structure is easily tuned by a perpendicular electric field, e.g., the monotonic increase of E$_{g}$ with E$_{z}$ in monolayer BP, and the transition from a semiconducting to a gapless system in bilayer BP. [PDolui,PLiu2015]{}. In sharp contrast with the group-IV monolayer systems, monolayer phosphorene presents the unique LLs, with the asymmetric energy spectrum about $E_{F}$, the reduced state degeneracy, and the spin-independent configuration. The important differences mainly come from the geometric structure, the orbital hybridization, and the SOC. The magnetic quantization is greatly diversified by the number of layers.
The transition metal dichalcogenide monolayers can be produced by the micromechanical cleavage [@K.S.Novoselov2005; @Z.Y.Yin2012; @H.Li2014; @MakK.F.2010], liquid-phase exfoliation [@J.N.Coleman2011; @K.G.Zhou2011] and chemical vapor deposition [@Y.H.Lee2012; @S.Najmaei2013; @B.Liu2015; @J.C.Shaw2014]. Due to the unusual electronic and optical properties, various technological applications have been proposed for these materials, such as, electronic [@RadisavljevicB2011; @Y.Yoon2011; @H.Wang2012; @Y.J.Zhang2012; @Q.H.Wang2012] and optoelectric [Q.H.Wang2012,D.Xiao2012,Y.J.Zhang2012PRB]{} devices. The high potentials in the field-effect transistors are supported by the room-temperature carrier mobility over 200 cm$^{2}/$Vs and the high on/off ratio of $\sim
10^{8}$ [@RadisavljevicB2011]. Furthermore, the experimental measurements show that they have a direct band gap in the visible frequency range [@MakK.F.2010; @A.Splendiani2010; @J.S.Ross2013] and the valley-dependent optical selection rules [@D.Xiao2012; @T.Cao2012]. The stronger SOC and the inversion symmetry breaking lead to the spin- and valley-dependent electronic states [@D.Xiao2012]. The MoS$_{2}$-related systems are very suitable for investigating the spintronics and valleytronics. Specifically, the single-layer MoS$_{2}$ is composed of staggered honeycomb-like lattice structures in which a single layer of Mo atoms is sandwiched by two sulfur layers. This semiconducting system has a direct energy gap of $\sim $1.59 eV [@HoY.H.mos2]. The low-lying electronic states near three valleys centered at the ($K,K^{^{\prime }}$) and $\Gamma $ points are dominated by the ($4d_{z^{2}},4d_{xy},4d_{x^{2}-y^{2}}$) orbitals of Mo atoms. The SOC can effectively destroy the spin degeneracy of energy bands, especially for the valence one contributed by the $4d_{xy}$ and $4d_{x^{2}-y^{2}}$ orbitals. The quantized LLs are characterized by the dominating orbitals and spin configurations, being enriched by the constant-energy loops in three valleys. The degeneracy of the $K$ and $K^{^{\prime }}$ valleys is further lifted by **B,** owing to the cooperation of the site-energy difference and the magnetic quantization [@HoY.H.mos2].
**II. GRAPHENE**
The Hamiltonian of the layered graphene, which is built from the $2p_{z}$-orbital tight-binding functions in a unit cell, is expressed as
$H=\underset{\left\langle ij\right\rangle \left\langle ll^{\prime
}\right\rangle }{\sum }-\gamma _{ij}^{ll^{\prime }}C_{il}^{+}C_{jl^{\prime
}}^{{}},\qquad (1)$ where $\gamma _{ij}^{ll^{\prime }}$ is the intralayer or interlayer hopping integral, $i$ the lattice site, and $l$ the layer index. $C_{il}^{+}$ ($C_{jl^{\prime }}^{{}}$) can create (annihilate) an electron at the $i$-th ($j $-th) site of the $l$-th ($l^{\prime }$-th) layer. A hexagonal unit cell has 2N carbon atoms for a N-layer graphene. Under a uniform perpendicular magnetic field, there are $4NR_{B}$ carbon atoms in an enlarged rectangular unit cell (Fig. 1(a)), since the vector potential ($\mathbf{A}=[0,B_{z}x,0]$) can induce a periodical Peierls phase. $R_{B}$ is the ratio between flux quantum ($\phi _{0}$=hc/e) and magnetic flux through a hexagon ($\phi $=3$\sqrt{3}$b$^{2}$B$_{z}$/2; b the C-C bond length), e.g., $R_{B}$=2$\times $10$^{3}$ at B$_{z}$=40 T. The quantized LLs are highly degenerate in the reduced first Brillouin zone with an area 4$\pi ^{2}\diagup $3$\sqrt{3}$b$^{2}R_{B}$. The (k$_{x}$=$0,$k$_{y}$=$0$) Hamiltonian, with the real matrix elements, is sufficient in calculating energy spectra and wavefunctions. Each LL wavefunction is the superposition of the $4NR_{B}$ tight-binding functions:
$\left\vert \Psi _{\mathbf{k}}\right\rangle =\underset{i,l}{\overset{}{\sum }}A_{i}^{l}\left\vert A_{i\mathbf{k}}^{l}\right\rangle +B_{i}^{l}\left\vert
B_{i\mathbf{k}}^{l}\right\rangle .\qquad (2)$ $A_{i}^{l}$ and $B_{i}^{l}$ are the probability amplitudes of the subenvelope functions in the two equivalent sublattices.
The hexagonal symmetry in monolayer graphene can create the low-lying isotropic Dirac-cone structure and thus the well-behaved LLs with the specific dependence on quantum number ($n^{c,v}$) and field strength. As to each (k$_{x}$,k$_{y}$) state, all the LLs have eight-fold degeneracy. This comes from the equivalent $K$ and $K^{^{\prime }}$ valley, the symmetry of $\pm \mathbf{B}$ and the spin degree of freedom. At (k$_{x}$=0,k$_{y}$=0), the state probabilities of the degenerate LLs are localized at the 1/6, 2/6, 4/6 and 5/6 positions of the enlarged unit cell. The (2/6,5/6) and (1/6,4/6) states, respectively, correspond to the magnetic quantization from the $K$ and $K^{^{\prime }}$ valleys [@Y.K.Huang2014]. The 2/6 localized LL wavefunctions, as shown in Fig. 2, have the normal probability distributions, being identical to those of a harmonic oscillator. Quantum number of each LL is characterized by the number of zero points in the dominating B sublattice. The $n^{c,v}$=0 LLs only come from the B sublattice. In general, the $n^{c,v}$ LL wavefunctions in the B sublattice are proportional to the ($n^{c,v}$+1) LL wavefunction in the A sublattice, directly reflecting the honeycomb symmetry. The same features are revealed in the 1/6 case under the interchange of two sublattices. Specifically, the low energy spectrum is characterized by E$^{c,v}$=$\pm $v$_{F}\sqrt{2\hbar
en^{c,v}B_{z}/c}$ (v$_{F}$ the Fermi velocity), consistent with that obtained from the effective-mass model [@YisongZheng2002; @V.P.Gusynin2005]. The square-root dependence is suitable at $\left\vert \text{E}^{c,v}\right\vert $$<$1 eV, since the linear bands gradually change into the parabolic bands in the increment of state energy. In addition, the high-energy LL spectrum can also be obtained by the generalized tight-binding model [@J.H.Ho2008; @C.Y.Lin2015]. Specifically, the dispersionless feature of 2D LLs is dramatically changed by the distinct dimensions, e.g., the 1D quasi-LLs and the 3D Landau subbands (discussed in conclusion).
The main features of LLs, energy spectra, spatial distribution modes and state degeneracy, are dramatically changed by the number of layers, the stacking configurations, and the perpendicular electric field. The LLs in the layered graphenes might exhibit the asymmetric energy spectra about the Fermi level, the non-square-root or non-monotonous dependence on $n^{c,v}$ and B$_{z}$, and the crossing or anti-crossing behaviors, mainly owing to the critical interlayer hopping integrals (Fig. 3(a)). Such interactions can induce three kinds of LLs with the distinct distribution modes: (1) the well-behaved LLs in a single mode, (2) the perturbed LLs with a main mode and side modes, and (3) the undefined LLs composed of many comparable modes (Fig. 4). The LL degeneracy will be reduced to half, when the z$\rightarrow $-z inversion symmetry is destroyed by the perpendicular electric field (Fig. 12(a)) or the specific stacking configuration. For example, the trilayer AAB-stacked graphene has an obvious splitting LL spectrum with observable spacings about 10 meV [@T.N.Do2015].
The ABC-stacked tetralayer graphene and the sliding bilayer graphene are chosen to see the geometry-enriched magnetic quantization. The former has four groups of LLs, in which the quantum numbers of the first, second, third and fourth groups (black, red, blue and green curves) are, respectively, obtained from the dominant (B$^{1}$, B$^{3}$, B$^{2}$, B$^{4}$) sublattices at the 2/6 center. Apparently, the valence and the conduction LLs are asymmetric about E$_{F}$ (Fig. 4). The LL energy spectrum exhibits the diverse B$_{z}$-dependences, indicating the sensitive changes of energy bands with wave vectors (Fig. 3(b)). In general, the first group of LLs has the monotonous dependence, i.e., their energies grow with the increasing B$_{z}$. However, the four LLs nearest to E$_{F}$, which mainly arise from the weakly dispersive energy bands dominated by the surface states (black curves), have distribution widths smaller than 8 meV even at rather high $B_{z}$ (Fig. 4(c)). The LLs, which are localized at two outmost graphene layers, are absent in the AA-, AB- and AAB-stacked graphenes [@C.Y.Lin2014; @T.N.Do2015].
Specifically, the second group of LLs exhibit the abnormal $n_{2}^{c,v}$ sequence and the unusual energy spectrum, as seen in the conduction and valence states. At rather small B$_{z}$, all the LLs have the reverse ordering of E$^{c}$($n_{2}^{c}$)$<$E$^{c}$($n_{2}^{c}-1$). They are initiated at a specific energy corresponding to the cusp $K$ point of the sombrero-shaped energy band (red curves). This clearly illustrates that LLs are quantized from the electronic states enclosed by the inner constant-energy loops. With the increase of B$_{z}$, the higher-$n_{2}^{c}$ LLs come to exist in the normal ordering, since they arise from the outer constant-energy loops related to parabolic dispersions. The completely normal ordering of E$^{c}$($n_{2}^{c}$)$<$E$^{c}$($n_{2}^{c}$+1) is revealed only at B$_{z}>$100 T, directly reflecting the fact that the electronic states under the cusp-shaped energy dispersions are only quantized into the $n_{2}^{c}$=0 LLs. The ordering of LLs is mainly determined by the competitive relation between the area covered by the cusp-shaped energy dispersions and the B$_{z}$-enhanced state degeneracy \[details in Ref. [C.Y.Lin2015]{}\].
The novel intragroup anticrossings appear frequently in the non-monotonous LL spectrum, as seen in the range of 0.29 eV$<$E$^{c}$($n_{2}^{c}$)$<$0.36 eV. In addition to a main mode, the specific interlayer hopping integrals, ($\beta _{3}$,$\beta _{2}$,$\beta _{5}$) (Fig. 3(a)), cause the $n_{2}^{c}$ LLs to possess certain side modes with the zero points of $n_{2}^{c}\pm $3I (I an integer) [@C.Y.Lin2014; @C.Y.Lin2015; @M.Inoue; @T.Morimoto2013]. The lower-$n_{2}^{c}$ perturbed LLs exhibit the distorted spatial distributions (Figs. 5(j)-5(k)); that is, they significantly deviate from the monolayer-like single modes (Fig. 2). For example, with the increase of B$_{z}$, the wave functions are drastically changed during the anticrossing of the $n_{2}^{c}$=0 and 3 LLs, as shown in Fig. 5. When the side mode, with three zero points in the $n_{2}^{c}$=0 LL (or without zero point in the $n_{2}^{c}$=3 LL), becomes comparable with their main mode, the same oscillation modes in these two LLs can prevent the direct crossing. Apparently, the intragroup LL anticrossings are derived from the magnetic quantization of the non-monotonous energy bands, e.g., the existence in the AAB-stacked graphenes and the absence in the AA- and AB-stacked graphenes [@T.N.Do2015]. It should be noticed that the LL anticrossings are also presented between any two distinct groups at sufficiently high $B_{z}$ and $\left\vert E^{c,v}\right\vert $ [@C.Y.Lin2014], i.e., there exist the intergroup LL anticrossings. Except for the regimes of these anticrossings, the third and the fourth groups of energy spectra have the normally continuous B$_{z}$-dependence (blue and green curves in Fig. 4). The similar anticrossing behaviors are shown in the AB and AAB stackings, but not in the AA stacking only with the single-mode LLs.
In addition to the well-behaved and the perturbed LLs, the undefiled LLs can be created during the transformation of stacking configuration. Specially, the stacking configuration could be changed by the electrostatic-manipulation STM [@XuP.2012; @XuP.2013jjap]. When the configuration of bilayer graphene is transformed from the AA to AB stacking by the shift along the armchair direction (Fig. 6(a)), two vertical Dirac cones gradually change into two pairs of parabolic bands. Each Dirac-cone structure is seriously distorted and thoroughly separated at the critical shift of $\sim $6b/8 (Fig. 6(b)). It is impossible to get the low-lying energy bands from the $K$-point expansion, and so does the LL quantization using the effective-mass model. The $\delta $=6b/8 stacking exhibits the eight-fold degenerate LLs, being the same with monolayer graphene. However, this bilayer system has a lot of undefined LLs, as indicated in the unusual B$_{z} $-dependent energy spectrum at $\left\vert \text{E}^{c,v}\right\vert >$ 0.3 eV (Fig. 6(c)). Each LL in the second group is composed of various zero points, and the irregular spatial distribution is very sensitive to the change in field strength. As a result, it displays the significant anticrossings with all the LLs in the first group.
**III. SILICENE, GERMANENE & TINENE**
For the IV-group inorganic layered systems, the sp$^{3}$ orbital bondings and the SOC’s are included in the critical Hamiltonians. In the bases of $\left\{ \left\vert p_{z}^{A}\right\rangle ,\left\vert p_{x}^{A}\right\rangle
,\left\vert p_{y}^{A}\right\rangle ,\left\vert s^{A}\right\rangle
,\left\vert p_{z}^{B}\right\rangle ,\left\vert p_{x}^{B}\right\rangle
,\left\vert p_{y}^{B}\right\rangle ,\left\vert s^{B}\right\rangle \right\}
{}\otimes \left\{ \uparrow ,\downarrow \right\} ,$ the nearest-neighbor Hamiltonian is expressed as
$H=\underset{\left\langle i\right\rangle ,o,m}{\sum }E_{o}C_{iom}^{+}C_{iom}^{{}}+\underset{\left\langle i,j\right\rangle
,o,o^{\prime },m}{\sum }\gamma _{oo^{\prime }}^{\mathbf{R}_{ij}}C_{iom}^{+}C_{jo^{\prime }m}^{{}}{}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\underset{\left\langle i\right\rangle ,p_{\alpha },p_{\beta }^{{}},m,m^{\prime }}{\sum
}\frac{\lambda _{\text{SOC}}}{2}C_{ip_{\alpha }m}^{+}C_{ip_{\beta }m^{\prime
}}^{{}}(-i\epsilon _{\alpha \beta \gamma }\sigma _{mm^{\prime }}^{\gamma
}),\qquad (3)\newline
$where $i(j)$, $o(o^{\prime })$, and $m(m^{\prime })$ stand for the lattice site, atomic orbital, and spin, respectively. The first and second terms are, respectively, the site energy (E$_{o}$) and the nearest-neighbor hopping integral ($\gamma _{oo^{\prime }}^{\mathbf{R}_{ij}}$). The latter is determined by the type of atomic orbitals, the translation vector $\mathbf{R}_{ij}$, and the angle $\theta $ between $\mathbf{R}_{ij}$ and $\widehat{z}$ (Fig. 1(c)). The details of interaction energies are given in Ref. [@C.C.LiuPRB]. The last term represents the SOC on the same atom where $\alpha
,\beta $ and $\gamma $, respectively, denote the $x$, $y$ and $z$ components, and $\sigma $ is the Pauli spin matrix. The SOC strength is, respectively, predicted to be $\lambda _{\text{SOC}}$=0.034, 0.196; 0.8 eV’s for (Si,Ge,Sn) [@C.C.LiuPRB]. The SOC between $\left\vert
p_{x}^{{}}\right\rangle $ and $\left\vert p_{y}^{{}}\right\rangle $ can create the splitting of states with opposite spin configurations, while that between $\left\vert p_{z}^{{}}\right\rangle $ and $\left\vert
p_{x}^{{}}\right\rangle $ ($\left\vert p_{y}^{{}}\right\rangle $) leads to the splitting of states and an interchange of spin configurations. Specially, the magnetic Hamiltonian of monolayer system is a 32$R_{B}\times $32$R_{B}$ Hermitian matrix with complex elements.
The SOC, buckled structure and orbital hybridizations in IV-group can induce the feature-rich energy bands and diversify the quantized LLs. Germanene and silicene have the similar band structures, in which the low-lying electronic states mainly come from the $4p_{z}$ and $3p_{z}$ orbitals, respectively. A small direct energy gap, which corresponds to the slightly separated Dirac points, is dependent on the strength of SOC. E$_{g}$ is, respectively, 45 meV and 5 meV for Ge and Si systems (inset in Fig. 7(a)). The first pair of valence and conduction bands have the doubly degenerate states associated with the spin-down- and spin-up-dominated equivalent configurations. It is sufficient to only discuss one of both configurations, as shown in Figs. 7(b) and 7(c) for germanene. Near the $K$ ($K^{\prime }$) point, the valence states are mainly determined by the $\left\vert 4p_{z}^{B};\downarrow
\right\rangle $ and $\left\vert 4p_{z}^{A};\downarrow \right\rangle $ ($\left\vert 4p_{z}^{B};\uparrow \right\rangle $ and $\left\vert
4p_{z}^{A};\uparrow \right\rangle $). Their contributions are very sensitive to the changes of wave vectors along $K\rightarrow $M ($K^{\prime
}\rightarrow \Gamma $) (solid curves in Fig. 7(b)). The similar behaviors are revealed in the conduction states under the interchange of the A and B sublattices (Fig. 7(c)). In addition, the ($4p_{x},4p_{y},4s$) orbitals can make important contributions to the middle-energy states close to the $\Gamma $ point.
The quantized LLs in monolayer germanene (silicene) are characterized by the subenvelope functions on the A and B sublattices with $sp^{3}$ orbitals and two spin configurations. All the low-lying LLs in monolayer germanene (silicene) belong to the well-behaved modes. They are eight-fold degenerate for each (k$_{x}$,k$_{y}$) state except the four-fold degenerate LLs of $n^{c,v}$=0. As to each localization center, there are two subgroups characterized by the up- and down-dominated configurations, as indicated in Fig. 8(a)-8(d) for the 2/6 states. The first and the second subgroups, respectively, have the $n^{c}$=0 conduction LL and the $n^{v}$=0 valence LL at E$^{c}$=23 meV and E$^{v}$=$-$20 meV. The former and the latter are caused by the spin-up and spin-down configurations in the dominating B sublattice, respectively. The other $n^{c,v}\neq $0 LLs in these two subgroups are doubly degenerate, and their wave functions are identical under the interchanges of spins and weights of A and B sublattices*.* There exist certain important differences between germanene and graphene. Germanene exhibits the significantly splitting $n^{c,v}$=0 LLs with the partial contributions from the A sublattice. The weight ratio between the A and B sublattices are quite different for the valence and conduction LLs. In addition to the dominating $4p_{z}$ orbitals, the contributions due to the ($4p_{x},4p_{y},4s$) orbitals are gradually enhanced as $\left\vert \text{E}^{c,v}\right\vert $ grows. However, the opposite is true for graphene (Fig. 2).
A perpendicular electric field applied to buckled systems can split energy bands and even induce the anti-crossing LL spectra. The destruction of the z=0 mirror symmetry causes one Dirac cone to become two splitting structures, when the gate voltage between two sublattices (V$_{z}$) grows from zero. The lower cone structure approaches to the Fermi level, and energy gap is vanishing at a critical V$_{z}$ where the linearly gapless Dirac-cone structure is recovered (the inset of Fig. 9(a)). The dependence of E$_{g}$ on V$_{z}$ is in the cusp form. Another cone structure is always away from E$_{F}$. The V$_{z}$-dependent cone structures are quantized into the unusual LL energy spectra (Fig. 9(a)). The K-valley-dependent (or the $K^{^{\prime }}$-valley-dependent) LLs are split according to the magnetic quantization of the lower and higher Dirac structures. The $n^{c,v}>$0 and $n^{c,v}$=0 LLs, respectively, have the four-fold and double degeneracy. The splitting LL energy spectrum, which corresponds to the lower Dirac cone, exhibits the non-monotonous V$_{z}$-dependence. As a result, the intra-group LL anticrossings occur frequently in the plentiful LL energy spectrum. In addition, the V$_{z}$-induced LL splittings and anticrossings are also presented in the layered graphenes except for the AA-stacked systems [S.J.Tsai2012CPL]{}.
The cooperation of the electric field and spin-orbital coupling can create the significant probability transfer between the spin-up and spin-down configurations and thus the frequent intra-group LL anticrossings. For example, the $n_{K}^{v}$=2 and $n_{K}^{v}$=3 LLs exhibit the dramatic changes in the spatial distributions within the critical range of 150 meV$<$V$_{z}<$350 meV (green and purple triangles in Fig. 9(b) ). At small V$_{z}$’s, these two $4p_{z}$-dominated LLs have the same quantum modes on the (A$_{\uparrow }$,B$_{\uparrow }$) and (A$_{\downarrow }$,B$_{\downarrow }$) sublattices (Fig. 9(c) and (d)). However, the weight of distinct spin configurations is very large and small for the former and the latter, respectively. With the variation of V$_{z}$, the electric field can induce the probability transfer between A$_{\uparrow }$ and B$_{\uparrow }$ (A$_{\downarrow }$ and B$_{\downarrow }$) sublattices. Furthermore, the intra-atomic SOC of $4p_{z}$ and ($4p_{x},4p_{y}$) orbitals leads to the significant distribution change on A$_{\uparrow }$ and A$_{\downarrow }$ (B$_{\uparrow }$ and B$_{\downarrow }$) sublattices. This means that the latter two orbitals play an important role in the anti-crossing behaviors, even if they have small weights. The comparable probability distributions on the spin-related sublattices are responsible for the anti-crossings of the $n_{K}^{v}$ and $n_{K}^{v}$+1 LLs (the $n_{K}^{c}$ and $n_{K}^{c}$+1 LLs). In addition, the direct crossing from the $n_{K^{^{\prime }}}^{v}$ and $n_{K^{^{\prime }}}^{v}$ +1 LLs (the $n_{K^{^{\prime }}}^{c}$ and $n_{K^{^{\prime }}}^{c}$+1 LLs) occurs simultaneously.
Tinene has more low-lying energy bands and diverse LLs, compared with germanene and silicene. A pair of slightly distorted Dirac cones appears near the K point, and there are parabolic energy bands initiated at the $\Gamma $ point (Fig. 10(a)). The latter, which mainly originate from the ($5p_{x},5p_{y}$) orbitals, are attributed to the stronger $sp^{3}$ bonding. State degeneracy at the $\Gamma $ point is further destroyed by the critical SOC between $5p_{x}$ and $5p_{y}$ orbitals so that one of the parabolic bands is very close to the Fermi level. The Dirac-cone structure is quantized into the first group of LLs (the black curves Fig. 11(a)), in which the main features are similar to those in germanene (Figs. 8(b) and 8(d)), such as, the p$_{z}$-orbital dominance, localization centers, state degeneracy, spin configurations, quantum modes on the A and B sublattices, and B$_{z}$-dependence of LL energy spectrum. However, the first group is in sharp contrast to the second group. Both $5p_{x}$ and $5p_{y}$ orbitals dominate the second group of LLs and make the almost same contributions (red and green curves in Fig. 10 (e)). Such LLs only have two equivalent centers of 1 and 1/2, and they are doubly degenerate (Fig. 10(d)). For each center, the splitting LLs are characterized by the up- and down-dominated configurations ($n_{\Gamma ,\uparrow }^{c,v}$ & $n_{\Gamma ,\downarrow
}^{c,v}$), being attributed to the significant effect of SOC. The spin-split LL energy spacing is observable, especially for the larger spacing in the conduction LLs. This spacing grows with the enhanced weight ration of two spin configurations on the same sublattice. Specifically, the A and B sublattices present the same quantum modes, since the nearest-neighbor hopping integrals near the $\Gamma $ point are roughly proportional to the square of wave vector. However, the hexagonal symmetry can generate the linear k-dependence in these atomic interactions near the $K$ point. As to the first group, this accounts for the mode difference of one between the A and B sublattices.
Tinene exhibits the rich B$_{z}$-dependent energy spectrum and density of states (DOS). Two group of LLs have the well-behaved modes, as indicated from the absence of anti-crossings and the existence of intergroup crossings (Fig. 11(a)). The LL state energies grow with the increase of field strength except the almost unchanged $n_{K}^{c,v}$=0 ones. As to the first and the second groups, the B$_{z}$-dependence is presented in the square-root and the linear forms, respectively (black and blue curves). This directly reflects the magnetic quantization from the linear and the parabolic energy dispersions. The spin-split energy spacings in the second group gradually become large, since the higher field strength creates more localized LL wave functions and enhances the spin-up or spin-down dominance. That is to say, the splitting energies are enlarged by the stronger effects of SOC. The main differences between two groups of energy spectra are further presented in DOS (Fig. 11(b)). A lot of strong peaks appear in the delta-function-like symmetric structure, in which their heights are proportional to state degeneracy. The single- and double-peak structure originate from the first and the second groups of LLs; furthermore, the former have the larger peak spacings. The main features of DOS peaks, structure, height, number and energy, could be verified from the experimental measurements using scanning tunneling spectroscopy (STS) [@LiG.H.2009; @MillerD.L.2009; @LuicanA.2011; @SongY.J.2010; @WangW.X.2015].
The V$_{z}$-dependent LL energy spectra are quite different among the IV-group layered systems. The splitting LLs cannot survive only in the AA-stacked graphenes, since the mirror symmetry is preserved even in the composite magnetic and electric fields [@S.J.Tsai2012CPL]. For monolayer silicene, the splitting energy spacings are very small, and the LL anti-crossings and crossings are absent (Fig. 12(a)). The weak SOC and the large v$_{F}$ (the strong energy dispersion) are responsible for the monotonous V$_{z}$-dependence. However, monolayer germanene and tinene frequently exhibit intragroup anti-crossings and crossings (Figs. 9(a) and 12(b)), in which two anti-crossing LLs have quantum number difference of $\Delta n$=1. Specifically, the latter have the intergroup crossings between the $5p_{z}$- and ($5p_{x},5p_{y}$)-dominated LLs. The V$_{z}$-induced intragroup anti-crossings are also observed in the non-AA-stacked graphenes, while they arise from two LLs with $\Delta n$=3I [@C.Y.Lin2014; @C.Y.Lin2015; @M.Inoue; @T.Morimoto2013]. In addition to V$_{z}$, $\Delta n $=1 and 3I are, respectively, determined by the significant SOC’s and certain interlayer hopping integrals.
**IV. BLACK PHOSPHORUS**
Monolayer phosphorene, with a puckered honeycomb structure, has a rectangular unit cell. There are four phosphorus atoms, in which two ones are situated at lower and upper subplanes (Fig. 13). The low-energy band structure is characterized by the $3p_{z}$-orbital hybridizations. The few-layer Hamiltonian is expressed as $$H=\sum_{\langle ij\rangle \langle ll^{\prime }\rangle }-t_{ij}^{ll^{\prime
}}C_{il}^{+}C_{jl^{\prime }},$$where $t_{ij}^{ll^{\prime }}$ represents the five intralayer (Fig. 13(a)) and four interlayer hopping integrals (Fig. 13(b); details in Ref. [PRudenko]{}). For monolayer system, the magnetic Hamiltonian is a $4R_{B}\times 4R_{B}$ Hermitian matrix.
The energy bands are greatly enriched by the complicated multi-hopping integrals. Monolayer phosphorene has a direct gap of $\sim 1.6$ eV near the $\Gamma $ point (Fig. 14(a)), being in sharp contrast with that dominated by the $K$ point in the group-IV systems. The first pair of energy bands nearest to $E_{F}$ is, respectively, linear and parabolic along $\Gamma X$ and $\Gamma Y$ directions. The valence (conduction) band is due to the linearly anti-symmetric (symmetric) superposition of the tight-binding functions on the upper and lower subplanes. As to bilayer phosphorene, two pairs of low-lying bands have parabolic dispersions, as shown in Fig. 14(b). The first and second pairs, respectively, correspond to the in-phase and out-of-phase combinations of two layers.
The quantized LLs in phosphorene are characterized by the subenvelope functions on the different subplanes and sublayers. They are localized at the 1/2 and 2/2 positions of the enlarged unit cell, corresponding to the magnetic quantization at the $\Gamma$ point. The well-behaved spatial distributions, as shown in Fig. 15(b), are similar to those of monolayer graphene. The $3p_{z}$-orbit quantization, with spin degree, is four-fold degenerate for each $(k_{x},k_{y})$ state. This is in sharp contrast to the eight-fold degeneracy in the group-IV systems, or the double degeneracy of the spin- and valley-dependent LLs in MoS$_2$. The LL degeneracy depends on the number of equivalent valleys and the existence of inversion symmetry ($z\rightarrow-z$ and $x\rightarrow-x$). There are two groups of valence and conduction LLs in bilayer phosphorene (the black and red lines in Fig. 15(a)). Both of them differ from each other in the initial energies and level spacings. The first and second groups, respectively, correspond to the in-phase and out-of-phase subenvelope functions on sublayers (Fig. 15(c)).
The highly asymmetric energy dispersion leads to the special dependence of LL energies on ($n^{c,v}$,$B_{z}$), as clearly indicated in Fig. 16. In monolayer and bilayer phosphorene, the low-lying LL energies cannot be described by a simple relation with $n^{c,v}B_{z}$, especially for the higher energy and field strength. This is different from the square-root dependence in monolayer group-IV systems (Fig. 11) [@J.H.Ho2008], and the linear dependence in AB-stacked graphene [@C.Y.Lin2014; @C.Y.Lin2015] and MoS$_{2}$ (Fig. 19). In general, the LL energies grow with the increment of $B_z$ monotonously. Only the intergroup LL crossings are revealed in bilayer system (Fig. 16(b)). However, the intragroup and the intergroup anticrossings are absent, since all the well-behaved LLs are quantized from the monotonous band structure in the energy-wave-vector space (Fig. 14).
**V. MoS$_2$**
A MoS$_{2}$ monolayer consists of three centered honeycomb structures, in which the middle Mo-atom lattice is sandwiched by two S-atom ones (Fig. 17 ). From the previous theoretical studies [@MoFP; @MoFP1; @MoFP2], the electronic states close to E$_{F}$ are predominantly contributed from the ($4d_{z^{2}},4d_{xy},4d_{x^{2}-y^{2}}$) orbitals of Mo atoms. The three-orbital tight-binding model is sufficient to describe the essential electronic properties. In the bases of $\left\{ \left\vert
4d_{z^{2}}\right\rangle ,\left\vert 4d_{xy}\right\rangle ,\left\vert
4d_{x^{2}-y^{2}}\right\rangle \right\} \otimes \left\{ \uparrow ,\downarrow
\right\} ,$ the Hamiltonian is given by
$H=\underset{\left\langle i\right\rangle ,o,m}{\sum }E_{o}C_{iom}^{+}C_{iom}^{{}}+\underset{\left\langle i,j\right\rangle
,o,o^{\prime },m}{\sum }\gamma _{oo^{\prime }}^{\mathbf{R}_{ij}}C_{iom}^{+}C_{jo^{\prime }m}^{{}}+\underset{\left\langle
i\right\rangle ,o,o^{\prime },m}{\sum }\frac{\lambda _{soc}}{2}C_{iom}^{+}C_{io^{\prime }m}^{{}}(L_{oo^{\prime }}^{z}\sigma
_{mm}^{z}),\qquad (4)$ where the first, second and third terms are, respectively, the site energy, the nearest-neighbor hopping integral and the on-site SOC ($\lambda _{soc}$=73 meV). These interaction energies could be found in [@MoS2TB]. The site energies are distinct for the $4d_{z^{2}}$ and ($4d_{xy}$,$4d_{x^{2}-y^{2}}$) orbitals, and this difference will result in the valley-dependent LLs. The SOC is only contributed by the z-component angular momentum ($L^{z}$) and spin moment ($\sigma _{{}}^{z}$). This interaction occurs between $\left\vert 4d_{x^{2}-y^{2}}\right\rangle $ and $\left\vert
4d_{xy}\right\rangle $ with the same spin configuration, while it is independent of $\left\vert 4d_{z^{2}}\right\rangle $. As to the magnetic Hamiltonian, 2$R_{B}$ Mo atoms in an enlarged unit cell can build a 12$R_{B}\times $12$R_{B}$ Hermitian matrix.
The multi-orbital bondings and the SOC cause monolayer MoS$_{2}$ to exhibit the unusual electronic structure. A direct energy gap of 1.59 eV at the $K$ or $K^{^{\prime }}$ point, as shown in Fig. 18(a), is dominated by the site energies of distinct orbitals. The significant orbital hybridizations lead to the strong wave-vector dependence. The electronic states of parabolic bands near E$_{F}$ are centered at the $K$, $K^{^{\prime }}$ and $\Gamma $ points. Furthermore, the SOC can create the spin-split energy bands, e.g., the largest splitting energy is 2$\lambda _{soc}$ at the $K$ and $K^{^{\prime }}$ points. Whether there exist the splitting spin-up and spin-down energy bands is dependent on the components of $4d_{x^{2}-y^{2}}$ and $4d_{xy}$ orbitals. The contributions of these two orbitals, as indicated in Figs. 18(b) and 18(c), are comparable in the splitting valence bands near the $K$ and $K^{^{\prime }}$ points. However, when the electronic states mainly come from one of them, or the $4d_{z^{2}}$ orbital, the spin splitting is very weak, e.g., the lower-lying conduction bands in Fig. 18(d). It is also noticed that the IV-group systems do not have the spin-split energy bands as a result of the mirror symmetry in A and B sublattices (Figs. 7(a) and 10(a)).
MoS$_{2}$ systems exhibit the novel magnetic quantization, since the valley- and spin-dependent LL subgroups could survive simultaneously. All the LLs have two degenerate localization centers, the 1/2 and 2/2 localization centers in an enlarged unit cell, e.g., the 1/2 localized LL wavefunctions shown in Figs. 19(b) and 20(b). The dominating modes have the well-behaved spatial probability distributions. Each mode is fully determined by the spin-up or spin-down configuration, but not a superposition of two opposite spins as revealed in IV-group systems (Figs. 8(b), 8(d) and 10(e)). Each LL group of monolayer MoS$_{2}$ only corresponds to the occupied LLs or the unoccupied LLs, while that in IV-group systems includes the valence and conduction ones. It is further split into LL subgroups under the destruction of the spin and/or valley degeneracy.
As to the valence LLs, they are magnetically quantized from the electronic states centered at the $\Gamma $, $K$ and $K^{^{\prime }}$ points (Fig. 19(a)). The $\Gamma $-dependent LL wavefunctions are independent of spin configuration (Fig. 19(b)), mainly owing to the $d_{z^{2}}$-orbital dominance and the almost vanishing SOC. The LL energies linearly grow with B$_{z}$ (the blue curves in Fig. 19(a)), directly reflecting the parabolic dispersion near the $\Gamma $ point. The similar B$_{z}$-dependence is revealed in the energy spectra of other LL subgroups. However, the spin-up and spin-down (spin-down & spin-up) LL subgroups, which come from the $K$ ($K^{^{\prime }}$) valley, are initiated at $-$0.792 eV and $-$0.938 eV, respectively. The spin-split LL subgroups are closely related to the $4d_{x^{2}-y^{2}}$- and $4d_{xy}$-dominated SOC’s, as indicated in Figs. 18(b) and 18(c). Specifically, the degeneracy of two valleys is clearly destroyed in the increase of B$_{z}$. That is to say, there also exist the $K
$- and $K^{^{\prime }}$-dependent LL subgroups, as shown in Fig. 19(a). The energy spacing is observable for a sufficiently high B$_{z}$, e.g., $\sim $15 meV between the $n_{K,\uparrow }^{v}$=0 and $n_{K^{^{\prime }},\downarrow
}^{v}$ LLs at B$_{z}$=40 T. By the detailed analysis, the site-energy differences in the B$_{z}$-enlarged unit cell is responsible for these LL subgroups [@HoY.H.mos2]. The coexistence of the spin- and valley-dependent LL subgroups is absent in IV-group systems. On the other hand, the lower-lying conduction LLs exhibit the significantly $K$- and $K^{^{\prime }}$-dependent subgroups, and the very weak splittings in the spin-dependent subgroups (Fig. 20).
Up to now, STS has served as a powerful experimental method for investigating the magneto-electronic energy of the layered graphenes. The measured tunneling differential conductance (dI/dV) is approximately proportional to DOS, and it directly reflects the structure, energy, number and height of the LL peaks. Part of theoretical predictions on the LL energy spectra are confirmed by STS measurements, such as, the $\sqrt{\text{B}_{z}}$-dependent LL energy in monolayer graphene [@LiG.H.2009; @MillerD.L.2009; @LuicanA.2011; @SongY.J.2010; @WangW.X.2015], the linear B$_{z}$-dependence in AB-stacked bilayer graphene [@RutterG.M.2011; @YinL.J.2016], the coexistent square-root and linear B$_{z}$-dependences in trilayer ABA stacking [@YinL.J.2015], and the 3D and 2D characteristics of the Landau subbands in AB-stacked graphite [@LiG.H.2007]. The other unusual magneto-electronic properties in the layered systems could be further verified using STS, including the normal and abnormal B$_{z}$-dependences in ABC-stacked graphenes, three kinds of LLs in sliding stacking systems, the SOC-induced spin-dominated LLs in germanene and silicene, two groups of low-lying LLs in tinene, the spin- and valley-dependent LLs in MoS$_{2}$, the special $n^{c,v} $- and B$_{z}$-dependence of LL energies in few-layer phosphorene, and the B$_{z}$- and V$_{z}$-dependent energy spectra with the LL splittings, crossings and anti-crossings. The STS examinations can provide the critical informations in the lattice symmetry, stacking configuration, SOC, and single- or multi-orbital hybridization.
**VI. CONCLUDING REMARKS**
In this review work, the generalized tight-binding model, based on the subenvelope functions of distinct sublattices, is developed to explore the feature-rich magneto-electronic properties of layered systems. Such model is suitable for the various symmetric lattices, the multi-layer structures, the low-symmetry stacking configurations, the distinct dimensions, the multi-orbital bondings, the coupling interactions of orbital and spin, the composite external fields, and the uniform and modulated fields [@Y.C.Ou2014; @Y.C.Ou2013; @Y.C.Ou2011; @Y.H.Chiu2010]. It is useful in understanding the essential physical properties, e.g., the diverse magneto-optical selection rules [@Y.C.Ou2014; @Y.C.Huang2007; @Y.H.Ho20101; @R.B.Chen2012; @Y.H.Ho2014] and the LL-induced plasmons [@J.Y.Wu20141; @Chen2015; @J.Y.Wu2011]. Moreover, this method could be further used to solve the new Hamiltonians of the emerging materials under the external fields.
The layered systems exhibit the unusual energy bands and the rich LLs in terms of the spatial distributions, orbital components, spin configurations, state degeneracy, and external-field dependences. The well-behaved, perturbed and undefined LLs are revealed in sliding graphenes, especially for the third ones mainly coming from the dramatic transformation between two high-symmetry stacking configurations. For the ABC- and AAB-stacked graphenes, the complicated interlayer hopping integrals result in the abnormal $n^{c,v}$ ordering and the non-monotonic B$_{z}$-dependence. The intragroup and intergroup LL anti-crossings occur frequently, clearly illustrating the strong competition of the constant-energy loops in the magnetic quantization. The SOC can create the up- and down-dominated LLs in silicene, germanene, tinene and MoS$_{2}$. Tinene and MoS$_{2}$ have the low-lying LLs composed of the different orbitals and spin configurations, mainly owing to the cooperation of the critical multi-orbital bondings and SOC. Concerning few-layer phosporene, the puckered structure induces the intralayer and the interlayer multi-hopping integrals and thus the special dependence of LL energies on ($n_{{}}^{c,v}$,B$_{z}$). The LL state degeneracy is reduced, when the inversion symmetry of z$\longrightarrow -$z (x$\longrightarrow -$x) is destroyed, or two equivalent valleys are absent. For example, there are four-fold degenerate LLs in AAB-stacked graphene and few-layer phosphorene, and doubly degenerate $n_{\Gamma }^{c,v}$ LLs in tinene. Specifically, MoS$_{2}$ exhibits the spin- and valley-dependent LL subgroups, in which they arise from the SOC and the site-energy differences in the B$_{z}$-enlarged cell, respectively. The LL splittings are easily observed in the presence of a perpendicular electric field except for the AA-stacked graphenes and monolayer slicene. Furthermore, they induce the frequent anti-crossings and crossings in the V$_{z}$-dependent energy spectra. The two anti-crossing LLs, which are, respectively, associated with the interlayer hopping integrals and the significant SOC, have the quantum number differences of $\Delta n$=1 and 3I. The above-mentioned diverse LL energy spectra are directly reflected in the structure, height, energy and number of the prominent DOS peaks; furthermore, they could be verified by the STS measurements.
Also, the different dimensions can diversify the magnetic quantization. The 3D graphites and 1D graphene nanoribbons are very different from the 2D graphenes in the quantized electronic properties. As to graphites, the periodical interlayer hopping integrals induce the Landau subbands with energy dispersions along $\widehat{k_{z}}$. The AA-, AB- and ABC-stacked graphites, respectively, possess one group, two groups, and one group of valence and conduction Landau subbands, in which the band widths are about 1 eV, 0.2 eV and 0.01 eV [@Ho2011; @Ho2014; @HoC.H.20114938; @HoC.H.2013NJP]. However, there are N groups of LLs in N-layer graphene systems. The AA- and ABC-staked graphites exhibit the monolayer-like wave functions, while the AB-stacked graphite displays the monolayer- and bilayer-like spatial distributions. In sharp contrast to the ABC-stacked graphenes, the anti-crossing phenomena in the B$_{z}$-dependent energy spectrum are absent in the rhombohedral graphite. On the other hand, the magneto-electronic properties of graphene nanoribbons are mainly determined by the rather strong competition between the magnetic quantization and the finite-width quantum confinement [@Y.C.Huang2007]. When ribbon widths are larger than magnetic lengths, 1D nanoribbons have many composite energy subbands. Each subband is composed of a dispersion-less quasi-LL (QLL) and parabolic dispersions along $\widehat{k_{x}}$. Such QLLs belong to the well-behaved modes localized at the ribbon center. Their magneto-optical selection rule is similar to that of monolayer graphene. In addition, a carbon nanotube could be regarded as a rolled-up graphene tubule. Except for a super-high magnetic field (B$_{z}>$10$^{5}$ T; [@AjikiH.1996]), it is impossible to generate the quantized QLLs in a curved cylinder because of the vanishing net magnetic flux.
**ACKNOWLEDGMENT**
This work was supported by the NSC of Taiwan, under Grant No. NSC 102-2112-M-006-007-MY3.
$^{\star }$e-mail address: [email protected]
$^{\star }$e-mail address: [email protected]
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0.6 truecm
**Figure Captions**
Figure 1: (a) Geometric structure of honeycomb graphene with an enlarged rectangular unit cell in B$_{z}\widehat{z}$, and (b) the buckled (silicene, germanene,tinene) with (c) the $sp^{3}$ bondings.
Figure 2: For monolayer graphene at B$_{z}$=40 T, (a) the low-lying valence and conduction LLs, and the probability distributions of the subenvelope functions at the (b) A and (B) sublattices. The unit of the x-axis is 2R$_{B} $, in which i represents the i-th A or B atom in an enlarged unit cell.
Figure 3: (a) Geometric structure and (b) energy bands of the ABC-stacked tetralayer graphene. $\gamma_i\,^{\prime }s$ denote the intralayer and interlayer hopping integrals.
Figure 4: The B$_{z}$-dependent LL energy spectrum of the ABC-stacked tetralayer graphene for (a) the first group, (b) the other three groups; (c) the four LLs nearest to E$_{F}$.
Figure 5: (a) The intragroup and intergroup LL anticrossing phenomena in the ABC-stacked tetralayer graphene. The spatial evolutions of subenvelope functions are shown in (b)–(i) for the second group of LLs, and (j)–(q) for the second and the third
groups of LLs.
Figure 6: (a) Geometric structure of sliding bilayer graphene along the armchair direction; (b) energy bands and (c) B$_{z}$-dependent LL spectrum at $\delta $=6b/8.
Figure 7: (a) Energy bands of monolayer germanene and silicene, and (b)&(c) orbital-decomposed state probabilities along the high-symmetry points. The inset of (a) is the band structure near the Dirac point.
Figure 8: For germanene at B$_{z}$= 15 T, (a) & (b) the up-dominated and (c) &(d) the down-dominated LL energies and spatial probability distributions, corresponding to the quantized $K$-valley states.
Figure 9: (a) The V$_{z}$-dependent LL energy spectra of germanene at B$_{z}$=15 T, and (b) the LL crossing and anticrossing within a certain range of E$^{v}$. (c) and (d) the drastic changes of probability distributions during the LL anticrossings. The inset of (a) is the band structure at B$_{z}$=0 and a critical V$_{z}$. The LL anticrossings are indicated by the red circles.
Figure 10: (a)-(c) Similar plot as Fig. 7, but shown for monolayer tinene. Also plotted in (b) are those from the quantized $\Gamma $-valley states.
Figure 11: (a) The B$_{z}$-dependent energy spectra of the first and second groups (black and blue curves) in tinene, and (b) density of states at B$_{z} $ = 30 T.
Figure 12: The V$_{z}$-dependent LL energy spectra at B$_{z}$=15 T for (a) tinene and (b) silicene.
Figure 13: (a) The geometric structure and the first Brillouin zone of monolayer phosphorene, and (b) the AB stacking structure of bilayer phosphorene. Also, the intralayer and interlayer hopping integrals are marked in (a) and (b), respectively.
Figure 14: Energy bands of (a) monolayer and (b) bilayer phosphorene.
Figure 15: (a) The LL energies of bilayer phosphorene at B$_{z}$=30 T and (b) the probability distributions. Also shown in (c) are the amplitudes of $n^{c}=1$ of the upper (solid curve) and lower (dashed curve) layers.
Figure 16: For monolayer and bilayer phosphorene, the $n^{c}$- and $B_{z}$-dependent LL energies are shown in (a) and (b) respectively. The black and purple dashed lines in (a) and (b) represent the linear dependence.
Figure 17: (a) Geometric structures for MoS$_{2}$ monolayer with an enlarged rectangular unit cell in B$_{z}\widehat{z}$ and (b) the structure of trigonal prismatic coordination.
Figure 18. (a) Energy bands of monolayer MoS$_{2}$, and (b)-(d) the orbital-decomposed state probabilities along the high-symmetry points.
Figure 19. (a) The B$_{z}$-dependent energy spectra of the valence LLs are, respectively, related to the quantized states near the ($K$,$K^{^{\prime }}$) and $\Gamma $ points (black and blue curves), in which the valley-dependent (spin-dependent) subgroups are represented by the solid and dashed curves (the red and yellow colors). The spatial probability distributions are shown in (b) at B$_{z}$=40 T.
Fig. 20. Same plot as Fig. 19, but shown for the low-lying conduction LLs.
![(a) Geometric structure of honeycomb graphene with an enlarged rectangular unit cell in B$_{z}\widehat{z}$, and (b) the buckled (silicene, germanene,tinene) with (c) the $sp^{3}$ bondings.[]{data-label="figure:1"}](fig1){width="90.00000%"}
![For monolayer graphene at B$_{z}$=40 T, (a) the low-lying valence and conduction LLs, and the probability distributions of the subenvelope functions at the (b) A and (B) sublattices. The unit of the x-axis is 2R$_{B} $, in which i represents the i-th A or B atom in an enlarged unit cell.[]{data-label="figure:2"}](fig2){width="80.00000%"}
![(a) Geometric structure and (b) energy bands of the ABC-stacked tetralayer graphene. $\gamma_i\,^{\prime }s$ denote the intralayer and interlayer hopping integrals.[]{data-label="figure:3"}](fig3){width="80.00000%"}
![The B$_{z}$-dependent LL energy spectrum of the ABC-stacked tetralayer graphene for (a) the first group, (b) the other three groups; (c) the four LLs nearest to E$_{F}$.[]{data-label="figure:4"}](fig4){width="90.00000%"}
![(a) The intragroup and intergroup LL anticrossing phenomena in the ABC-stacked tetralayer graphene. The spatial evolutions of subenvelope functions are shown in (b)-(i) for the second group of LLs, and (j)-(q) for the second and the third groups of LLs.[]{data-label="figure:5"}](fig5){width="100.00000%"}
![(a) Geometric structure of sliding bilayer graphene along the armchair direction; (b) energy bands and (c) B$_{z}$-dependent LL spectrum at $\delta $=6b/8.[]{data-label="figure:6"}](fig6){width="90.00000%"}
![(a) Energy bands of monolayer germanene and silicene, and (b)&(c) orbital-decomposed state probabilities along the high-symmetry points. The inset of (a) is the band structure near the Dirac point.[]{data-label="figure:7"}](fig7){width="80.00000%"}
![For germanene at B$_{z}$= 15 T, (a) & (b) the up-dominated and (c) &(d) the down-dominated LL energies and spatial probability distributions, corresponding to the quantized $K$-valley states.[]{data-label="figure:8"}](fig8){width="85.00000%"}
![(a) The V$_{z}$-dependent LL energy spectra of germanene at B$_{z}$=15 T, and (b) the LL crossing and anticrossing within a certain range of E$^{v}$. (c) and (d) the drastic changes of probability distributions during the LL anticrossings. The inset of (a) is the band structure at B$_{z}$=0 and a critical V$_{z}$. The LL anticrossings are indicated by the red circles.[]{data-label="figure:9"}](fig9){width="80.00000%"}
![(a)-(c) Similar plot as Fig. 7, but shown for monolayer tinene. Also plotted in (b) are those from the quantized $\Gamma $-valley states.[]{data-label="figure:10"}](fig10){width="100.00000%"}
![(a) The B$_{z}$-dependent energy spectra of the first and second groups (black and blue curves) in tinene, and (b) density of states at B$_{z} $ = 30 T.[]{data-label="figure:11"}](fig11){width="80.00000%"}
![The V$_{z}$-dependent LL energy spectra at B$_{z}$=15 T for (a) tinene and (b) silicene.[]{data-label="figure:12"}](fig12){width="90.00000%"}
![(a) The geometric structure and the first Brillouin zone of monolayer phosphorene, and (b) the AB stacking structure of bilayer phosphorene. Also, the intralayer and interlayer hopping integrals are marked in (a) and (b), respectively.[]{data-label="figure:13"}](fig13){width="80.00000%"}
![Energy bands of (a) monolayer and (b) bilayer phosphorene.[]{data-label="figure:14"}](fig14){width="80.00000%"}
![(a) The LL energies of bilayer phosphorene at B$_{z}$=30 T and (b) the probability distributions. Also shown in (c) are the amplitudes of $n^{c}=1$ of the upper (solid curve) and lower (dashed curve) layers.[]{data-label="figure:15"}](fig15){width="80.00000%"}
![For monolayer and bilayer phosphorene, the $n^{c}$- and $B_{z}$-dependent LL energies are shown in (a) and (b) respectively. The black and purple dashed lines in (a) and (b) represent the linear dependence.[]{data-label="figure:16"}](fig16){width="80.00000%"}
![(a) Geometric structures for MoS$_{2}$ monolayer with an enlarged rectangular unit cell in B$_{z}\widehat{z}$ and (b) the structure of trigonal prismatic coordination.[]{data-label="figure:17"}](fig17){width="100.00000%"}
![(a) Energy bands of monolayer MoS$_{2}$, and (b)-(d) the orbital-decomposed state probabilities along the high-symmetry points.[]{data-label="figure:18"}](fig18){width="70.00000%"}
![(a) The B$_{z}$-dependent energy spectra of the valence LLs are, respectively, related to the quantized states near the ($K$,$K^{^{\prime }}$) and $\Gamma $ points (black and blue curves), in which the valley-dependent (spin-dependent) subgroups are represented by the solid and dashed curves (the red and yellow colors). The spatial probability distributions are shown in (b) at B$_{z}$=40 T.[]{data-label="figure:19"}](fig19){width="70.00000%"}
![Same plot as Fig. 19, but shown for the low-lying conduction LLs.[]{data-label="figure:20"}](fig20){width="80.00000%"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Current models of magnetars require extremely strong magnetic fields to explain their observed quiescent and bursting emission, implying that the field strength within the star’s outer crust is orders of magnitude larger than the dipole component inferred from spin-down measurements. This presents a serious challenge to theories of magnetic field generation in a proto-neutron star. Here, we present detailed modelling of the evolution of the magnetic field in the crust of a neutron star through 3-D simulations. We find that, in the plausible scenario of equipartition of energy between global-scale poloidal and toroidal magnetic components, magnetic instabilities transfer energy to non-axisymmetric, kilometre-sized magnetic features, in which the local field strength can greatly exceed that of the global-scale field. These intense small-scale magnetic features can induce high energy bursts through local crust yielding, and the localised enhancement of Ohmic heating can power the star’s persistent emission. Thus, the observed diversity in magnetar behaviour can be explained with mixed poloidal-toroidal fields of comparable energies.'
author:
- 'Konstantinos N. Gourgouliatos, Toby Wood'
- Rainer Hollerbach
title: Magnetic field evolution in magnetar crusts through three dimensional simulations
---
n estimate of the magnetic field intensity in a neutron star can be obtained by assuming that the observed spin-down is caused by electromagnetic radiation from a dipolar magnetic field [@Deutsch:1955]. Neutron stars for which this estimate exceeds the QED magnetic field $4.4\times10^{13}$G are conventionally called magnetars, and typically exhibit highly energetic behaviour, as in the case of Anomalous X-ray Pulsars and Soft $\gamma$-ray Repeaters [@THOMPSON:1995; @THOMPSON:1996]. Puzzlingly, not all high magnetic field neutron stars exhibit energetic behaviour [@HABERL:2007], and conversely, “magnetar-like" activity has been observed in pulsars for which this field estimate is below [@REA:2010] or only marginally above the QED magnetic field [@AN:2013; @SCHOLZ:2012; @GAVRIIL:2002]. Furthermore, despite the high thermal conductivity of neutron stars’ solid outer crusts [@POTEKHIN:1999], observations of their thermal emission indicates that in some cases the surface is highly anisothermal [@GUILLOT:2015], with kilometre-sized “hot spots" thought to be produced by small-scale magnetic features [@Bernardini:2011]. Phase resolved spectroscopy has revealed that some magnetars have small-scale magnetic fields whose strength exceeds their large-scale component by at least an order of magnitude [@TIENGO:2013; @GUVER:2011], which can be correlated with outbursting events [@RODRIGUEZ:2015]. These observations all imply that the magnetic field structure in magnetars is more complicated, and varied, than the traditional picture of a simple inclined dipole.
The origin of the extreme magnetic fields in these objects, which are the strongest found in nature, is uncertain. Even if magnetic flux were exactly conserved during the star’s formation, the resulting field would not exceed $10^{13}$G. It seems likely then that the strong fields in magnetars must result from a combination of differential rotation and dynamo action prior to the formation of the crust [@SPRUIT:2008]. Dynamo models generally predict magnetic fields with features over a wide range of scales, and poloidal and toroidal components of comparable strength [@Mosta:2015]. However, once the crust forms, small-scale surface features in the magnetic field would decay by Ohmic dissipation much faster than the global field.
The evolution of the crustal magnetic field is mediated by the Hall effect, corresponding to advection by free electrons [@GOLDREICH:1992]. The Hall effect in magnetars typically operates on a shorter timescale than Ohmic dissipation, and might therefore explain the formation of small-scale magnetic features, whose dissipation could then power the star’s thermal radiation. However, axisymmetric simulations of the crustal magnetic field only generate such features if a toroidal magnetic field exceeding $10^{16}$G is assumed to reside within the crust [@PONS:2011; @GEPPERT:2014]. The toroidal field strength must significantly exceed that of the poloidal dipole, which is the only component that is measured from spin-down observations, presenting additional challenges to models of magnetic field generation. A possible resolution is the growth of localised patches of strong magnetic field via magnetic instabilities. Such instabilities have been demonstrated in local, plane-parallel models [@RHEINHARDT:2004; @WOOD:2014; @GOURGOULIATOS:2015], but not yet in a realistic global model.
Hall Evolution in Neutron Star Crusts
=====================================
Motivated by this puzzle we study the fully non-linear 3-D problem of the magnetic field evolution in a magnetar crust through numerical simulations, using a modified version [@WOOD:2015] of the PARODY code [@DORMY:1998; @AUBERT:2008], to determine the necessary conditions for the spontaneous generation of strong localised magnetic field out of a large-scale weaker one. The crust of the neutron star is treated as a solid Coulomb lattice, in which free electrons carry the electric current and the magnetic field evolution is described by Hall-MHD with subdominant Ohmic dissipation [@GOLDREICH:1992]. This process is described by the Hall-MHD induction equation: $$\begin{aligned}
\frac{\partial \bm{B}}{\partial t} = -\nabla \times \left(\frac{c}{{4 \pi \rm e}n_{\rm e}} \left(\nabla \times \bm{B}\right)\times \bm{B} +\frac{c^{2}}{4 \pi \sigma} \nabla \times \bm{B}\right)\,,
\label{HALL}\end{aligned}$$ where $\bm{B}$ is the magnetic induction, $n_{\rm e}$ the electron number density, $\sigma$ the electric conductivity, $c$ the speed of light and ${\rm e}$ the elementary charge. The first term in the right-hand-side of equation (\[HALL\]) describes the Hall effect and the second one Ohmic dissipation. We assume a neutron star radius $R_{*}=10$km, and a crust thickness of $1$km. We shall express the quantities in terms of the normalised radial distance $r=R/R_{*}$, where $R$ is the distance from the centre. The electron number density is taken to be $n_{\rm e}=2.5 \times10^{34}$ cm$^{-3} \left(\frac{1.0463-r}{0.0463}\right)^{4}$, the electric conductivity is $\sigma=1.8\times 10^{23}$s$^{-1} \left(\frac{1.0463-r}{0.0463}\right)^{8/3}$. Such profiles are good analytical fits of the form $\sigma \propto n_{\rm e}^{2/3}$ of more precise crust models [@CUMMING:2004] at temperatures $\approx 10^8\ {\rm K}$.
The code uses spherical harmonic expansions in latitude and longitude, and a discrete grid in radius. The linear Ohmic terms are evaluated using a Crank-Nicolson scheme, while for the non-linear Hall terms an Adams-Bashforth scheme is used. We have used the Meissner superconducting condition assuming that no magnetic field penetrates the crust-core surface, serving as the inner boundary of the domain [@Hollerbach:2004]. This is a simplifying assumption, as the core may be a Type-II superconductor and magnetically coupled to the crust. Nevertheless, the exact structure of the field in the core is still uncertain with some works suggesting that the crustal magnetic field is practically disconnected from the core \[c.f. Figs. 3 and 4 of [@Henriksson:2013], and Fig. 9 of [@Lander:2014]\]. With respect to the outer boundary we matched the magnetic field to an external vacuum field satisfying $\nabla \times \bm{B}=\bm{0}$. Given the lack of any other 3-D code or analytical solution of the Hall evolution, the code has been benchmarked against axially symmetric calculations [@GOURGOULIATOS:2014b; @GOURGOULIATOS:2014a] in various radial and angular resolutions with excellent quantitative agreement, and the free decay of Ohmic eigenmodes, in the absence of the Hall term, reproducing the analytical results. The results that we present here have a resolution of 128 grid-points in radius, and spherical harmonics $Y_\ell^m$ up to $\ell_{\rm max}, m_{\rm max}=60-80$. We have repeated some of the runs with the strongest magnetic fields, using twice the angular resolution, with $\ell_{\rm max}=m_{\rm max}=120$. We found good agreement in the overall magnetic field evolution of those runs with the lower resolution ones.
Simulations: Initial Conditions and Results
===========================================
Given the complexity of the processes taking place during the formation of a neutron star, our knowledge of its initial magnetic field is limited. We have therefore explored a wide range of initial conditions, based on three plausible scenarios regarding the magnetic field formation: the fossil field, the large-scale dynamo, and the small-scale dynamo. In the scenario of a fossil field [@BRAITHWAITE:2004], the magnetic field has relaxed dynamically to an axially symmetric twisted torus with comparable energy in its poloidal and toroidal components. The poloidal magnetic field has dipolar geometry and the toroidal magnetic field is strongest near the dipole equator; both components have $\ell=1$ symmetry, where $\ell$ is the spherical harmonic degree. The second scenario corresponds to a magnetic field formed through the combination of differential rotation and dynamo action in the proto-neutron star [@SPRUIT:2008; @Mosta:2015]; in this case the most likely geometry is an $\ell=2$ toroidal and a dipolar poloidal field. In these two scenarios the field is predominantly large-scale and axially symmetric. In the third scenario, turbulent convection in the proto-neutron star generates a small-scale dynamo magnetic field [@THOMPSON:2001]. In this case, the field comprises small loops with size comparable to that of the convection cells.
![Magnetic field lines in the simulation with an initially differentially twisted magnetic field consisting of an $\ell=1$ (dipole) poloidal and an $\ell=2$ toroidal field, starting with only $10^{-4}$ of the magnetic energy in the non-axisymmetric field, (run QU05-4, S.I. Table 3). The snapshot is at $t=15$kyr overplotted with the magnetic energy density $e_{m}=B^{2}/(8\pi)$ on part of the surface. The surface field is highly anisotropic, with small regions in which the magnetic energy density exceeds by at least an order of magnitude the average surface value. \[Figure:1\] ](FIG1.png){width="\columnwidth"}
We have performed 76 simulations integrating the Hall-MHD induction equation (\[HALL\]) in each of the three scenarios for the initial magnetic field in the star’s crust (see S.I. Tables 1-3). In the fossil field and large-scale dynamo scenarios we have used mixed poloidal and toroidal fields, varying the ratio of the toroidal to total magnetic energy and the overall normalisation of the magnetic field intensity. The intensity of the poloidal field on the surface takes values between $10^{13}$G to $5\times 10^{14}$G, except for the purely toroidal initial conditions.
The magnetic field is expressed in terms of two functions $V_p(r,\theta,\phi)$ and $V_{t}(r,\theta,\phi)$: $$\begin{aligned}
\bm{B} = \nabla \times \nabla \times (V_p\bm{r}) + \nabla \times (V_{t} \bm{r})\,.\end{aligned}$$ We express the fraction of the energy in the toroidal component of the magnetic field $e_{t}$, allowing the following values: $0$ (purely poloidal), $0.1$, $0.5$, $0.9$ and $1$ (purely toroidal). $B_{0}$ governs the overall amplitude and takes the values $0.5$, $1$, $2$ and $4$ expressed in units of $10^{14}$G; the normalisation is chosen so that the initial magnetic field energy for a simulation with amplitude $B_{0}$ is $2.25\times 10^{46} B_{0}^{2}$ erg. For the radial structure of the field, we have used two basic profiles: one where the latitudinal and azimuthal components decrease monotonically with radius; and a second one with a local maximum in the middle of the crust. We present them below, providing the exact expressions to allow reproducibility of our results.
The first family of potentials corresponding to dipole fields ($\ell=1$), is given by: $$\begin{aligned}
V_{ps}&=&B_{0}(1-e_{t})^{1/2}~ \frac{3^{1/2}}{r^2}\cos\theta(-913.2837868+16749.59757r^3\nonumber \\
&&-137700.2916 r^5 +292459.7609 r^6-269535.2857 r^7\nonumber \\
&&+120217.0825r^8-21277.28441r^9)\,,
\label{POTPS}\end{aligned}$$ $$\begin{aligned}
V_{ts}=B_{0}e_{t}^{1/2}3^{1/2}~88.96339451 \cos\theta(r-1)\,.
\label{POTTS}\end{aligned}$$ The poloidal part of this potential gives a dipole magnetic field supported by a uniformly rotating electron fluid corresponding to a Hall equilibrium [@GOURGOULIATOS:2013a], with the magnetic field strength at the pole being equal to $B_{0}$ for $e_{t}=0$. Under this profile $B_{\theta}$ and $B_{\phi}$ change monotonically with radius. The maximum value of the $B_{\phi}$ component under this profile occurs at the base of the crust on the equatorial plane, and is $B_{\phi, {\rm max}}=15B_{0}e^{1/2}_{t}$. Thus, for the strongest toroidal field simulated ($B_{0}=4$, $e_{t}=1$), the maximum azimuthal field is $6\times 10^{15}G$.
The second family of $\ell=1$ profiles is the following: $$\begin{aligned}
V_{pu}&=& B_{0}(1-e_{t})^{1/2}\frac{3^{1/2}}{r^{2}}\cos\theta (734.5987631-2333.649604r\nonumber \\
&&+2465.151852r^2-865.6887776r^3)\,,
\label{POTPU}\end{aligned}$$ $$\begin{aligned}
V_{tu}=B_{0}e_{t}^{1/2}3^{1/2}~2739.401879 \cos\theta (1-r)(r-0.9)\,.
\label{POTTU}\end{aligned}$$ These profiles have $B_{\theta}$ and $B_{\phi}$ components with an extremum close to the centre of the crust. The choice of normalisation is such that the magnetic energy in the crust is the same as the corresponding $V_{ps}$ and $V_{ts}$ profiles. The maximum value of the $B_{\phi}$ component under this profile occurs in the middle of the crust ($r=0.95$) on the equatorial plane and is $B_{\phi, {\rm max}}=12B_{0}e^{1/2}_{t}$. Thus, for the strongest toroidal field simulated ($B_{0}=4$, $e_{t}=1$), the azimuthal field has a maximum value of $4.8\times 10^{15}G$.
We also combined the poloidal dipole fields with $\ell=2$ toroidal fields. In such cases we have utilised the following profiles: $$\begin{aligned}
V_{ts,q}=\pm B_{0}e_{t}^{1/2}5^{1/2}~51.36303975 \left(\frac{3}{2}\cos^{2}\theta -\frac{1}{2}\right)(r-1)\,,
\label{POTTSQ}\end{aligned}$$ which is used in conjunction with $V_{ps}$, and $$\begin{aligned}
V_{tu,q}=\pm B_{0}e_{t}^{1/2}5^{1/2}~1581.594412 \left(\frac{3}{2}\cos^{2}\theta -\frac{1}{2}\right)(1-r)(r-0.9) \,,
\label{POTTUQ}\end{aligned}$$ which is used in conjunction with $V_{pu}$. In the simulations with the quadrupolar toroidal field we assumed equipartition in energy between the poloidal and the toroidal field ($e_{t}=0.5$) and varied $B_{0}=0.5, 1, 2, 4$ for the case of the positive sign, while we run simulations with $B_{0}=2$ only for the case of the negative sign. The maximum value of the azimuthal field assumed, using the $V_{ts,q}$ profile is $5\times 10^{15}G$ and occurs at the base of the crust at latitudes $\theta=\pi/4, 3\pi/4$; while for the case of of $V_{tu,q}$ is $3.8\times 10^{15}G$ and occurs at $r=0.95$ and $\theta=\pi/4, 3\pi/4$. In all these axially symmetric initial conditions, we have superimposed a non-axisymmetric random noise, containing $10^{-4}-10^{-8}$ of the total energy. Axially symmetric initial conditions correspond to equilibria with respect to the $\phi$ coordinate, so any growth of the non-axisymmetric part of the field may indicate the presence of a non-axisymmetric instability.
Finally, we have simulated a highly non-axisymmetric magnetic field. In this case we have populated the crust with a fully confined non-axisymmetric magnetic field, where $\sim 0.9$ of the total magnetic energy is in the non-axisymmetric part of the field, consisting of multipoles with $10\leq\ell \leq 40$. Similarly to the previous cases we have used four normalisation levels for the overall magnetic field strength.
We ran these simulations until the magnetic energy decayed to $0.01$ of its initial value. We also present a set of runs where the Hall term is switched off and the field evolves only via Ohmic decay.
Our basic conclusions are the following:
1. In systems where the initial azimuthal field is strong ($e_{t}\gtrsim 0.5$), and especially when the profiles $V_{pu}$ and $V_{tu}$ are used (Figs. \[Figure:1\] and \[Figure:2\]), the amount of energy in the non-axisymmetric part of the field increases exponentially. The growth rate is approximately proportional to the strength of the magnetic field (see S.I. Figure 1), i.e. for the case depicted in Figs. \[Figure:1\] and \[Figure:2\] the non-axisymmetric energy growth timescale is $\sim 1.5$kyr. The exponential growth stops if one of the following two conditions is fulfilled: either $0.5-0.9$ of the total magnetic energy has decayed, or the non-axisymmetric field contains about 50% of the total energy. While initially the $m=0$ mode was dominant, once the instability develops energy is transferred and a local maximum appears around $10\lesssim m \lesssim 20$ (see S.I. Figure 3), with some further local maxima at higher harmonics, leading to a characteristic wavelength for the instability of about $5$km at the equator, with finer structure appearing later giving kilometre-sized features. These features are well above the resolution of the simulation and appear both in the low and high resolution runs. This behaviour is indicative of the density-shear instability in Hall-MHD, where, based on analytical arguments [@WOOD:2014; @GOURGOULIATOS:2015], the wavelength of the dominant mode is $2 \pi L$ where $L$ is the scale height of the electron density and the magnetic field, which can be approximated by the thickness of the crust.
{width="0.7\columnwidth"} {width="0.7\columnwidth"} {width="0.7\columnwidth"} {width="0.7\columnwidth"} {width="0.7\columnwidth"} {width="0.7\columnwidth"}
2. At the start of each simulation, the structure of the magnetic field changes rapidly, on the Hall timescale, unless the field is initiated in a Hall equilibrium in which case the evolution occurs on the slower Ohmic timescale. This change is accompanied by rapid magnetic field decay, i.e. in the case of QU05-4 (S.I. Table 3) 50% of the energy decays within the first $20$kyr. We find that the early decay approximately scales with the magnetic field strength (Tables 1-3 S.I. column $t_{0.5}$), whereas the later decay has a much weaker dependence on the magnetic field strength.
3. Initial conditions with poloidal magnetic fields generate toroidal fields. The energy content of the toroidal field never becomes dominant irrespective of the strength of the initial field, (see S.I. Figure 2). Even if a toroidal component exists in the initial conditions, stronger fields tend to suppress it more efficiently than weaker ones, in favour of the poloidal component. Thus a strong toroidal field is unlikely to be generated by a poloidal field being out of equilibrium. Once the magnetic field has decayed substantially the amount of the energy in the toroidal field may become higher than that of the poloidal. However, at this stage the dominant effect is Ohmic decay rather than Hall drift. This happens after a few Myrs as it is illustrated in the last column of the tables in the S.I. for the simulations with $B_{0}=1$.
4. The evolution of the axisymmetric part of the magnetic field is in accordance with the results of previous axisymmetric numerical and semi-analytical studies. In particular the toroidal field interacts with the poloidal field winding it up, while magnetic flux is transferred in the meridional direction leading to the formation of zones, Fig. \[Figure:2\]. The drift towards the equator is due to the polarity of the toroidal field, an effect that has been discussed in detail in plane-parallel geometry in [@Vainshtein:2000] and the axially symmetric case in [@Hollerbach:2004].
5. Configurations initially dominated by the poloidal field, tend to remain axially symmetric, especially if the $V_{ps}$ profile is used. Their evolution is in general accordance with previous axially symmetric simulations that have described the interaction of the poloidal and toroidal part of the field.
6. Once about 90% of the magnetic energy has decayed the Hall effect saturates, and thereafter the magnetic energy decays on the Ohmic timescale. However, the remaining small-scale features decay at the same rate as the large-scale field, indicating that the Hall effect is still active. These results are indicative of the “Hall attractor” seen in axisymmetric simulations [@GOURGOULIATOS:2014b].
7. Initial conditions corresponding to highly non-axisymmetric structures undergo a weak inverse cascade (see runs Turb in the S.I.). The Hall effect transfers energy across the spectrum both to smaller and larger scales. Despite the great differences in the structure of the magnetic field compared to the axially symmetric case, the evolution follows a qualitatively similar pattern, with the rapid decay at early times, and saturation of the Hall effect later.
Discussion
==========
In all simulations we find that the magnetic field undergoes a major restructuring at the very beginning of its evolution. This process occurs on the Hall timescale, and is thus faster for stronger magnetic fields. We then identified two distinct behaviours. Initial conditions where the poloidal field was dominant remain axially symmetric, while cases with substantial, but not necessarily dominant, azimuthal fields triggered non-axisymmetric instabilities. As magnetic energy is conserved in Hall-MHD, this growth of the non-axisymmetric field occurs at the expense of the axisymmetric components.
In the model with initial dipole poloidal magnetic field of $4\times 10^{14}$G and a toroidal component containing the same amount of energy with the poloidal (Figs. \[Figure:1\] and \[Figure:2\]), magnetic spots of sizes $\sim 2$km appear. These spots contain energies up to $5\times 10^{43}$erg, while the dipole field at that time is $1-2\times 10^{14}$G, (Fig. \[Figure:5\]). The rapid decrease of the dipole component, with a drop by a factor of 2 within 10-20 kyrs, is caused by the latitudinal Hall drift towards the equator. This is more efficient when the stronger fields are in shallow depths, when the potentials $V_{tu}$ and $V_{tu,q}$ are used (runs DU05-4 and QU05-4), where the maximum value of the toroidal field occurs at $r=0.95$. On the contrary it takes ten times longer for the same decrease in the dipole component, in the case of the potentials $V_{ts}$ (i.e. run DS05-4), where the maximum of the toroidal field occurs at the base of the crust. Therefore, in the case of shallower fields the dipole field drops quickly, while there is still a large reservoir of energy deeper in the crust. This can be related to AXP 1E2259+586 and SGR 0418+5729 which despite their relative weak dipole field exhibit magnetar activity [@GAVRIIL:2002; @REA:2010]. The energy in the magnetic spots is sufficient to power strong individual magnetar bursts or sequences of weaker events [@Mereghetti:2009; @Horst:2012]. The energy in the equatorial zone of width $\sim~2$km is $\sim 2\times 10^{46}$erg, representing 20% of the total magnetic energy in the crust at that moment. Such energy is comparable to the energy released by a magnetar giant flare [@Mazets:1979; @Hurley:1999; @Palmer:2005]. Note though, that magnetars that have exhibited giant flares have spin-down dipole fields above $5\times 10^{14}$G, so the overall magnetic energy should be scaled by a factor of $\sim10$ compared to our simulations. These structures take $\sim10^{4}$yrs to develop, given the choice of an initial condition where only $10^{-4}$ of the total is in the non-axisymmetric part. Had we chosen initial conditions with more energy in the non-axisymmetric part, the development of the smaller scale structure would have been imminent and in agreement with the ages of the most active magnetars, which are in the range of a few $10^{3}$yrs. This was confirmed in the extreme case where no energy is in the axisymmetric part (see runs Turb in S.I. Table 3). Finally, we find that the Ohmic dissipation rate in these models provided enough thermal energy to cover the needs of young magnetars, (Fig. \[Figure:4\]), even though we have made a conservative choice of the magnetic dipole field. By contrast, in cases where the evolution remains predominantly axisymmetric, instead of magnetic spots the field forms magnetic zones, where the intensity is only a factor of $\sim3$ stronger than the large-scale dipole field. Eventually, once a significant amount of energy has decayed ($\sim90\%$ of the total energy) the magnetic evolution saturates, and any small-scale magnetic features become “frozen in” as in previous studies in Cartesian and axisymmetric simulations [@WAREING:2009b; @GOURGOULIATOS:2014b].
The diversity of observational manifestations of neutron stars has called for their Grand Unification [@Kaspi:2010] under a common theory, with their magnetic field being the key parameter [@Aguilera:2008b; @Pons:2007; @Pons:2009; @Vigano:2013]. From our 3-D simulation survey we confirm indeed, that the initial structure and magnitude of the magnetic field are critical to the later evolution. We also find that the 3-D evolution is in general qualitatively different from the axisymmetric one. While predominantly poloidal initial magnetic field tend to remain axially symmetric [@WOOD:2015], the inclusion of a strong, but not dominant, toroidal field is sufficient to break axial symmetry and has two major impacts, first: magnetic energy is converted faster to heat; second: small scale magnetic features of sizes appropriate for the hotspots observed in magnetars [@GUILLOT:2015] form spontaneously and persist for several $10^{4}$ yrs. This result is supportive of the importance of a second parameter, namely the toroidal magnetic field. However, contrary to the axisymmetric studies which suggested that most of the energy is concealed in the toroidal field in order to explain bursts [@PONS:2011] and the formation of magnetic spots [@GEPPERT:2014], a toroidal field containing the same amount of energy as the poloidal field is sufficient to power magnetars. This leads to a more economical magnetar theory compared to previous works. Moreover, it restores the theoretical consistency between the structure of the magnetic field in magnetars and the fact that the poloidal and toroidal fields inside a star need to be comparable [@Prendergast:1956; @Flowers:1977] a result that has been confirmed numerically in MHD equilibria and dynamo studies [@BRAITHWAITE:2004; @Mosta:2015]. While the magnetic evolution in the crust is now well understood, it is critical for future studies to address fully the role of the core and its coupling to the crust, as the existing core studies focus on axisymmetric structures [@Lander:2013; @Henriksson:2013; @Elfritz:2015]. Furthermore, it is important to assess the impact of a magnetic field penetrating the crust-core boundary in the overall stability and Hall drift timescale. Similarly, it is possible that the strong magnetic field may plastically deform the crust, impeding the Hall evolution as the magnetic field lines are no longer frozen into the electron fluid [@Beloborodov:2014]. Regarding the thermal evolution, while it was shown here that the Ohmic decay provides sufficient energy to power magnetar X-ray Luminosity, it is important to couple magnetic and thermal evolution [@Vigano:2013] in a single 3-D calculation. Such a calculation will consider the possibilities of heat transfer deeper into the crust, suppression of radiation because of the non-radial magnetic field components, or losses to neutrinos. Finally other parameters such as the neutron star mass and crust thickness need to be taken into account in a wider exploration of physical mechanisms.
![The maximum magnetic field strength (dashed line) at the surface of the star versus the dipole field in three simulations. In the two cases where the non-axisymmetric instability is triggered (red and blue lines), the strongest field on the surface of the star exceeds the dipole component by an order of magnitude at a very early time, as opposed to the model that remains axially symmetric (green lines). \[Figure:5\]](B_max.png){width="\columnwidth"}
![The Ohmic heating rate versus time for the same simulations as Figure \[Figure:5\] (red, green and blue), each initially containing $3.6\times 10^{47}$erg of magnetic energy. Also shown two simulations: one with an $\ell=1$ poloidal and an $\ell=2$ toroidal field (yellow line) where the latitudinal and azimuthal components decrease monotonically in the crust and the field remains mostly axisymmetric; and a simulation where the structure of the field is identical to that of the blue line, in which the Hall effect has been artificially suppressed (dashed line). The dots represent observations of X-ray luminosities for magnetars in the McGill Magnetar Catalogue [@Olausen:2014] plotted against their characteristic ages (or the age of the associated supernova remnant if available), see S.I. Table 4. \[Figure:4\]](L_xt.png){width="\columnwidth"}
Conclusions
===========
The observational implications of these results for magnetars are in three basic directions. First, we have shown that the Hall effect has the tendency to produce strong, small-scale magnetic fields. This process does not appeal to an external source of energy, as the Hall effect exactly conserves magnetic energy, but rather to redistribution of the existing magnetic energy. The local magnetic field strength can exceed $10^{15}$ G for a background dipole that is an order of magnitude weaker. Local magnetic fields of such strength are sufficient to reach the breaking stress [@ThompsonDuncan01; @Lander:2015] and produce bursting activity. Second, the enhanced Ohmic dissipation within the localised features generates sufficient heat to power their thermal X-ray luminosity. Therefore, the magnetic field will be used more efficiently to produce heat, making the overall energy requirements smaller, while the fact that the strong magnetic field is localised in smaller areas, gives a viable explanation for the existence of hotspots. Finally, the evolution of the magnetic field, under the Hall effect, saturates once the energy has decayed to $\sim 10\%$ of its initial value, within a few $10^{5}$yrs; in this case the neutron star still hosts a strong magnetic field $\sim 5 \times 10^{13}$G, but its slow magnetic evolution reduces the chances of bursts and flares. Therefore the most energetic behaviour occurs while the star is young, particularly for magnetars with very strong fields.
KNG and RH were supported by STFC Grant No. ST/K000853/1. The numerical simulations were carried out on the STFC-funded DiRAC I UKMHD Science Consortia machine, hosted as part of and enabled through the ARC HPC resources and support team at the University of Leeds. We acknowledge the use of the McGill Magnetar Catalogue [www.physics.mcgill.ca/$\sim$pulsar/magnetar/main.html]{}. We thank Andrew Cumming, Vicky Kaspi, Scott Olaussen, Yuri Levin, Anatoly Spitkovsky and Andrei Beloborodov for discussions and comments. We thank an anonymous referee for stimulating comments which improved the paper.
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Supplementary Information
=========================
We use the following naming convention: Simulations whose names start with DS correspond to combinations of $V_{ps}$ and $V_{ts}$ potentials, DU correspond to combinations of $V_{pu}$ and $V_{tu}$ potentials, QS correspond to combinations of $V_{ps}$ and $V_{ts,q}$ potentials, QU correspond to combinations of $V_{pu}$ and $V_{tu,q}$ and those with Turb correspond to highly non-axisymmetric structures. The two numbers following the letters correspond to $e_{t}$, the ratio of the initial toroidal energy to the total energy, with 00 corresponding to purely poloidal field and 10 to purely toroidal. The number following the hyphen is related to the increments of the normalisation factor $B_{0}$, while simulations ending $-0$ correspond to runs where the Hall term is switched off and the evolution is purely Ohmic. Note that the simulations QS05-2M and QU05-2M utilise the negative sign for the potential $V_{ts,q}$ and $V_{tu,q}$ respectively.
In the tables we provide the following information. The initial conditions are given in the first three columns (1-3); the following three columns (4-6) show the time it takes for the energy to decay to $0.5$ ($t_{0.5}$), $0.1$ ($t_{0.1}$) and $0.01$ ($t_{0.01}$) of its initial value; the rest of the table contains the values of the fraction of energy in the non-axisymmetric part of the field $E_{n}/E_{tot}$ (columns 7-10), and the ratio of the energy in the axisymmetric toroidal field over the total energy in the axisymmetric part of the field $E_{\phi}/E_{ax}$ (columns 11-14), at the instances determined by the decay levels mentioned before. Simulations appearing in Tables 1 and 2 start with setups corresponding to initial conditions of potentials $V_{ps}$, $V_{ts}$ (DS) and $V_{pu}$, $V_{tu}$ (DU) where the amount of energy in the non-axisymmetric part of the field is less than $10^{-7}$ of the total magnetic energy (except for the simulations that evolve only under the effect of Ohmic dissipation where the energy in the non-axisymmetric part of the field is $10^{-4}$ of the total energy). Simulations in Table 3 correspond to systems where the energy in the non-axisymmetric part is initially $\sim 10^{-4}$ of the total energy.
In the figures we illustrate the basic results of our simulations. In Fig. \[Figure:Non-Ax\] we plot the fraction of the energy in the non-axisymmetric part of the magnetic field. We find that (with the exception of the purely poloidal initial condition DS00-3) in all other cases the energy in the non-axisymmetric part grows exponentially. In Fig. \[Figure:Tor\] we plot the fraction of the energy in the axisymmetric toroidal component for various runs, showing that a poloidal field does not generate a dominant toroidal field, and it actually even suppresses the initial toroidal field. In Figs. (\[Figure:Ylm\_Q\]-\[Figure:Ylm\_Turb\]) we plot the energy decomposition in $\ell$ and $m$ modes of the spherical harmonics for various runs.
![Ratio of the energy in the non-axisymmetric part of the field over the total magnetic energy for six runs. In all runs except for the DS00-3, which corresponds to a poloidal Hall equilibrium supported by a uniformly rotating electron fluid, the amount of energy in the non-axisymmetric part of the field grows exponentially. The exponential growth is considerably faster when $V_{pu}-V_{tu}$ potentials are utilised (dotted lines) compared to $V_{ps}-V_{ts}$ and when the amount of energy in the toroidal field is larger. []{data-label="Figure:Non-Ax"}](FIG1_SI.pdf){width="0.5\columnwidth"}
![Ratio of the energy in the axisymmetric toroidal component of the field over the total energy in the axisymmetric part of the field for eight different models. The DU00-1,2,3,4 runs start with a purely poloidal field, with intensities doubling between consecutive runs. The magnetic field configuration generates a toroidal component, as it is out of Hall equilibrium, however, the toroidal component does not become dominant, with stronger fields suppressing it efficiently and remaining effectively poloidal. Note that once a significant fraction of the field has decayed and the Ohmic term becomes strong, the fraction of energy in the toroidal field rebounds and increases (see for instance the red solid curve). Once the energy is equally split between the toroidal and the poloidal components, in the runs DU05-1,2,3,4 plotted with dotted lines, stronger fields again tend to suppress the toroidal field. []{data-label="Figure:Tor"}](FIG2_SI.pdf){width="0.5\columnwidth"}
![Power spectrum of the $Y_{\ell}^{m}$ decomposition of the fields for run QU05-4, at times $t_{0}$, $t_{0.5}$, $t_{0.1}$ and $t_{0.01}$. Top panel: The $\ell$ spectrum, starting from a state where the energy is equally distributed in a poloidal $\ell=1$ field and a toroidal $\ell=2$ field. Early evolution excites higher modes with local maxima at characteristic scales, which are indicative of the zonal formations (please refer to Figure 3 of the main paper). Bottom panel: The $m$ spectrum; initially the energy is dominated by the axisymmetric mode $m=0$ and the system is in equilibrium in the azimuthal direction. Non-axisymmetric modes grow rapidly which is indicative of the unstable behaviour. []{data-label="Figure:Ylm_Q"}](FIG3A_SM.pdf "fig:"){width="0.5\columnwidth"} ![Power spectrum of the $Y_{\ell}^{m}$ decomposition of the fields for run QU05-4, at times $t_{0}$, $t_{0.5}$, $t_{0.1}$ and $t_{0.01}$. Top panel: The $\ell$ spectrum, starting from a state where the energy is equally distributed in a poloidal $\ell=1$ field and a toroidal $\ell=2$ field. Early evolution excites higher modes with local maxima at characteristic scales, which are indicative of the zonal formations (please refer to Figure 3 of the main paper). Bottom panel: The $m$ spectrum; initially the energy is dominated by the axisymmetric mode $m=0$ and the system is in equilibrium in the azimuthal direction. Non-axisymmetric modes grow rapidly which is indicative of the unstable behaviour. []{data-label="Figure:Ylm_Q"}](FIG3B_SM.pdf "fig:"){width="0.5\columnwidth"}
![Power spectrum as in Figure \[Figure:Ylm\_Q\] for run DU10-4. Top panel: The $\ell$ spectrum, starting from a state where all energy is contained in an $\ell=1$ toroidal mode. Hall evolution pushes energy into higher $\ell$’s especially early in the neutron star’s life. Bottom panel: The $m$ spectrum, where initially the energy is dominated by the axially symmetric component $m=0$, however the development of Hall instability generates features with local maxima at $m=7,14,17,21,28,31$.[]{data-label="Figure:Ylm_Tor"}](FIG5A_SM.pdf "fig:"){width="0.5\columnwidth"} ![Power spectrum as in Figure \[Figure:Ylm\_Q\] for run DU10-4. Top panel: The $\ell$ spectrum, starting from a state where all energy is contained in an $\ell=1$ toroidal mode. Hall evolution pushes energy into higher $\ell$’s especially early in the neutron star’s life. Bottom panel: The $m$ spectrum, where initially the energy is dominated by the axially symmetric component $m=0$, however the development of Hall instability generates features with local maxima at $m=7,14,17,21,28,31$.[]{data-label="Figure:Ylm_Tor"}](FIG5B_SM.pdf "fig:"){width="0.5\columnwidth"}
![Power spectrum as in Figure \[Figure:Ylm\_Q\] for run Turb-4. Top panel: The $\ell$ spectrum, starting from a state where the energy is distributed between $10\leq\ell \leq 40$. Energy is transferred both to smaller and larger scales, even to $\ell=1$, with the fraction of energy in this mode growing with time. Bottom panel: The $m$ spectrum, initially the energy is set to be equally distributed between $0\leq m \leq 10$ and then drops obeying a power-law. []{data-label="Figure:Ylm_Turb"}](FIG4A_SM.pdf "fig:"){width="0.5\columnwidth"} ![Power spectrum as in Figure \[Figure:Ylm\_Q\] for run Turb-4. Top panel: The $\ell$ spectrum, starting from a state where the energy is distributed between $10\leq\ell \leq 40$. Energy is transferred both to smaller and larger scales, even to $\ell=1$, with the fraction of energy in this mode growing with time. Bottom panel: The $m$ spectrum, initially the energy is set to be equally distributed between $0\leq m \leq 10$ and then drops obeying a power-law. []{data-label="Figure:Ylm_Turb"}](FIG4B_SM.pdf "fig:"){width="0.5\columnwidth"}
NAME $B_0$ $e_t$ $t_{0.5}$ $t_{0.1}$ $t_{0.01}$ $E_{n}/E_{tot}(t_0)$ $E_{n}/E_{tot}(t_{0.5})$ $E_{n}/E_{tot}(t_{0.1})$ $E_{n}/E_{tot}(t_{0.01})$ $E_{\phi}/E_{ax} (t_{0})$ $E_{\phi}/E_{ax} (t_{0.5})$ $E_{\phi}/E_{ax} (t_{0.1})$ $E_{\phi}/E_{ax} (t_{0.01})$
-------- ------- ------- ----------- ----------- ------------ ---------------------- -------------------------- -------------------------- --------------------------- --------------------------- ----------------------------- ----------------------------- ------------------------------
DS00-0 - 0 440 1585 3223 1.08E-04 2.28E-05 1.12E-05 6.05E-06 2.44E-06 6.56E-07 5.71E-07 5.73E-07
DS00-1 0.5 0 376 1111 2176 3.70E-07 1.65E-07 1.30E-07 2.55E-07 6.40E-12 9.79E-02 2.38E-01 4.33E-01
DS00-2 1 0 312 914 1841 9.57E-08 3.05E-08 7.06E-08 3.27E-07 0.00E+00 1.17E-01 1.44E-01 2.20E-01
DS00-3 2 0 236 812 1733 1.99E-08 9.76E-09 2.92E-08 4.53E-07 4.55E-13 8.48E-02 5.01E-02 6.07E-02
DS00-4 4 0 187 783 1712 4.99E-09 2.94E-09 1.08E-08 3.75E-06 1.38E-13 4.33E-02 1.11E-02 1.21E-02
DS01-0 - 0.1 448 1600 3249 1.08E-04 2.24E-05 1.09E-05 5.79E-06 9.98E-02 1.07E-01 1.17E-01 1.32E-01
DS01-1 0.5 0.1 376 1111 2168 3.71E-07 2.07E-07 2.31E-07 5.04E-07 9.97E-02 1.53E-01 2.71E-01 4.73E-01
DS01-2 1 0.1 305 883 1797 9.30E-08 4.87E-08 2.25E-07 1.38E-06 9.97E-02 1.47E-01 1.70E-01 2.50E-01
DS01-3 2 0.1 228 760 1672 2.49E-08 2.65E-08 2.81E-06 2.74E-06 9.97E-02 1.09E-01 6.77E-02 7.13E-02
DS01-4 4 0.1 171 730 1661 5.24E-09 1.20E-05 1.34E-04 1.80E-04 9.97E-02 6.74E-02 1.63E-02 1.47E-02
DS05-0 - 0.5 471 1656 3351 1.00E-04 2.12E-05 9.83E-06 4.89E-06 5.00E-01 5.19E-01 5.45E-01 5.80E-01
DS05-1 0.5 0.5 374 1088 2191 1.66E-06 6.89E-06 1.18E-05 2.03E-05 4.99E-01 4.23E-01 4.56E-01 6.93E-01
DS05-2 1 0.5 287 773 1610 4.11E-07 2.46E-06 2.51E-05 1.33E-04 5.00E-01 3.28E-01 3.40E-01 4.81E-01
DS05-3 2 0.5 197 558 1336 1.02E-07 2.53E-05 3.02E-03 2.17E-03 4.99E-01 2.42E-01 2.27E-01 1.94E-01
DS05-4 4 0.5 128 445 1305 2.50E-08 2.38E-03 7.45E-03 2.11E-02 4.99E-01 2.01E-01 1.07E-01 2.49E-02
DS09-0 - 0.9 497 1705 3441 1.09E-04 2.00E-05 8.95E-06 4.20E-06 9.00E-01 9.07E-01 9.15E-01 9.26E-01
DS09-1 0.5 0.9 420 1114 2458 3.72E-07 2.08E-05 2.69E-05 3.25E-05 9.00E-01 8.06E-01 8.32E-01 9.03E-01
DS09-2 1 0.9 289 717 1756 9.40E-08 1.21E-04 8.69E-04 1.73E-03 9.00E-01 6.94E-01 7.88E-01 8.96E-01
DS09-3 2 0.9 169 417 1108 1.07E-07 1.85E-02 2.49E-02 9.51E-02 9.00E-01 6.35E-01 6.82E-01 8.23E-01
DS09-4 4 0.9 92 233 650 4.25E-09 5.33E-02 9.90E-02 2.16E-01 9.00E-01 5.87E-01 5.48E-01 6.27E-01
DS10-0 - 1 497 1720 3464 1.09E-04 1.99E-05 8.73E-06 4.04E-06 1.00E+00 1.00E+00 1.00E+00 1.00E+00
DS10-1 0.5 1 448 1175 2496 3.71E-07 9.03E-04 6.13E-04 2.34E-04 1.00E+00 1.00E+00 1.00E+00 1.00E+00
DS10-2 1 1 297 776 1841 9.27E-08 1.57E-03 4.35E-03 1.27E-02 1.00E+00 1.00E+00 1.00E+00 1.00E+00
DS10-3 2 1 169 461 1234 2.32E-08 6.26E-03 2.43E-02 6.21E-02 1.00E+00 1.00E+00 1.00E+00 1.00E+00
DS10-4 4 1 92 259 794 4.55E-09 3.12E-02 5.61E-02 9.27E-02 1.00E+00 1.00E+00 9.99E-01 9.98E-01
: Simulations corresponding to $V_{ps}$ and $V_{ts}$ initial conditions. Times are expresses in $10^{3}$years.
\[tab:LPer\]
[lcccrrrrrrrrrr]{}
\
\[0.5ex\]
NAME & $B_0$ & $e_t$ & $t_{0.5}$ & $t_{0.1}$ & $t_{0.01}$ & $E_{n}/E_{tot}(t_0)$ & $E_{n}/E_{tot}(t_{0.5})$ &$E_{n}/E_{tot}(t_{0.1})$ & $E_{n}/E_{tot}(t_{0.01})$ & $E_{\phi}/E_{ax} (t_{0})$ & $E_{\phi}/E_{ax} (t_{0.5})$ & $E_{\phi}/E_{ax} (t_{0.1})$ & $E_{\phi}/E_{ax} (t_{0.01})$\
DU00-0 & - & 0 & 151 & 1096 & 2737 & 1.26E-04 & 6.83E-05 & 3.30E-05 & 1.63E-05 & 2.82E-06 & 2.19E-06 & 1.37E-06 & 1.17E-06\
DU00-1 & 0.5 & 0 & 90 & 637 & 1754 & 3.72E-07 & 2.37E-05 & 2.12E-05 & 1.14E-05 & 6.28E-10 & 1.88E-01 & 1.78E-01 & 3.69E-01\
DU00-2 & 1 & 0 & 67 & 530 & 1495 & 9.16E-08 & 9.89E-04 & 2.34E-04 & 3.16E-04 & 3.93E-10 & 1.98E-01 & 1.29E-01 & 2.32E-01\
DU00-3 & 2 & 0 & 46 & 474 & 1380 & 1.98E-08 & 1.32E-02 & 1.82E-03 & 1.04E-02 & 2.47E-10 & 2.58E-01 & 7.25E-02 & 9.30E-02\
DU00-4 & 4 & 0 & 28 & 404 &1206 & 7.34E-09 & 5.29E-02 & 3.58E-02 & 1.07E-02 & 1.56E-10 & 1.96E-01 & 3.51E-02 & 3.31E-02\
DU01-0 & - & 0.1 & 151 & 1103 & 2752 & 1.26E-04 & 6.83E-05 & 3.26E-05 & 1.58E-05 & 1.00E-01 & 1.00E-01 & 1.06E-01 & 1.20E-01\
DU01-1 & 0.5 & 0.1 & 90 & 648 & 1766 & 3.71E-07 & 2.14E-05 & 1.85E-05 & 1.30E-05 & 1.00E-01 & 2.49E-01 & 2.31E-01 & 4.26E-01\
DU01-2 & 1 & 0.1 & 67 & 530 & 1480 & 9.44E-08 & 6.17E-04 & 1.85E-04 & 2.31E-04 & 1.00E-01 & 2.45E-01 & 1.60E-01 & 2.64E-01\
DU01-3 & 2 & 0.1 & 46 & 443 & 1321 & 1.05E-07 & 3.16E-02 & 8.62E-03 & 3.24E-02 & 1.00E-01 & 2.27E-01 & 9.66E-02 & 1.05E-01\
DU01-4 & 4 & 0.1 & 28 & 351 & 1164 & 2.60E-08 & 8.90E-02 & 3.32E-02 & 1.34E-01 & 1.00E-01 & 2.02E-01 & 5.13E-02 & 3.77E-02\
DU05-0 & - & 0.5 & 151 & 1119 & 2816 & 1.26E-04 & 6.83E-05 & 3.14E-05 & 1.42E-05 & 5.00E-01 & 5.00E-01 & 5.17E-01 & 5.17E-01\
DU05-1 & 0.5 & 0.5 & 95 & 653 & 1869 & 3.71E-07 & 9.74E-05 & 1.10E-04 & 7.59E-05 & 5.00E-01 & 5.21E-01 & 4.95E-01 & 6.95E-01\
DU05-2 & 1 & 0.5 & 67 & 474 & 1418 & 9.20E-08 & 8.49E-03 & 4.01E-03 & 8.50E-03 & 5.00E-01 & 4.96E-01 & 3.65E-01 & 5.01E-01\
DU05-3 & 2 & 0.5 & 41 & 346 & 1119 & 1.04E-07 & 3.36E-02 & 1.98E-02 & 6.53E-02 & 5.00E-01 & 4.64E-01 & 2.22E-01 & 2.62E-01\
DU05-4 & 4 & 0.5 & 23 & 236 & 887 & 2.50E-08 & 1.15E-01 & 8.52E-02 & 1.84E-02 & 5.00E-01 & 4.57E-01 & 1.46E-01 & 1.34E-01\
DU09-0 & - & 0.9 & 151 & 1144 & 2880 & 1.26E-04 & 6.83E-05 & 3.01E-05 & 1.27E-05 & 9.00E-01 & 9.00E-01 & 9.06E-01 & 9.18E-01\
DU09-1 & 0.5 & 0.9 & 118 & 809 & 2132 & 3.72E-07 & 1.50E-03 & 7.90E-03 & 6.24E-03 & 9.00E-01 & 8.67E-01 & 8.33E-01 & 9.03E-01\
DU09-2 & 1 & 0.9 & 82 & 517 & 1462 & 9.40E-08 & 1.97E-02 & 4.90E-02 & 4.35E-02 & 9.00E-01 & 8.46E-01 & 7.58E-01 & 8.97E-01\
DU09-3 & 2 & 0.9 & 51 & 292 & 909 & 1.04E-07 & 5.47E-02 & 1.09E-01 & 1.37E-01 & 9.00E-01 & 8.21E-01 & 6.55E-01 & 8.46E-01\
DU09-4 & 4 & 0.9 & 28 & 159 & 522 & 2.68E-08 & 7.88E-02 & 1.68E-01 & 2.98E-01 & 9.00E-01 & 8.01E-01 & 5.58E-01 & 7.32E-01\
DU10-0 & - & 1 & 151 & 1144 & 2895 & 1.26E-04 & 6.83E-05 & 2.99E-05 & 1.24E-05 & 1.00E+00 & 1.00E+00 & 1.00E+00 & 1.00E+00\
DU10-1 & 0.5 & 1 & 146 & 742 & 2086 & 3.73E-07 & 1.60E-01 & 5.38E-02 & 1.79E-02 & 1.00E+00 & 1.00E+00 & 9.99E-01 & 1.00E+00\
DU10-2 & 1 & 1 & 102 & 451 & 1454 & 9.27E-08 & 3.43E-01 & 1.25E-01 & 4.68E-02 & 1.00E+00 & 9.99E-01 & 9.99E-01 & 9.98E-01\
DU10-3 & 2 & 1 & 54 & 236 & 891 & 1.04E-07 & 5.16E-01 & 2.22E-01 & 1.09E-01 & 1.00E+00 & 9.95E-01 & 9.97E-01 & 9.93E-01\
DU10-4 & 4 & 1 & 28 & 123 & 530 & 2.70E-08 & 6.29E-01 & 3.14E-01 & 1.21E-01 & 1.00E+00 & 9.90E-01 & 9.83E-01 & 9.73E-01\
\[tab:LPer\]
[l c c crrrrrrrrrr]{}
\
\[0.5ex\]
NAME & $B_0$ & $e_t$ & $t_{0.5}$ & $t_{0.1}$ & $t_{0.01}$ & $E_{n}/E_{tot}(t_0)$ & $E_{n}/E_{tot}(t_{0.5})$ &$E_{n}/E_{tot}(t_{0.1})$ & $E_{n}/E_{tot}(t_{0.01})$ & $E_{\phi}/E_{ax} (t_{0})$ & $E_{\phi}/E_{ax} (t_{0.5})$ & $E_{\phi}/E_{ax} (t_{0.1})$ & $E_{\phi}/E_{ax} (t_{0.01})$\
DS00-1 & 0.5 & 0 & 376 & 1111 & 2176 & 3.33E-04 & 1.50E-04 & 1.16E-04 & 2.28E-04 & 5.62E-09 & 9.79E-02 & 2.38E-01 & 4.33E-01\
DS05-1 & 0.5 & 0.5 & 371 & 1085 & 2191 & 1.09E-04 & 3.56E-04 & 7.67E-04 & 1.21E-03 & 4.99E-01 & 4.23E-01 & 4.56E-01 & 6.93E-01\
DS05-2 & 1 & 0.5 & 289 & 773 & 1608 & 1.08E-04 & 5.00E-04 & 4.46E-03 & 3.20E-02 & 5.00E-01 & 3.28E-01 & 3.40E-01 & 4.91E-01\
DS05-3 & 2 & 0.5 & 197 & 553 & 1311 & 1.08E-04 & 1.53E-03 & 2.38E-02 & 1.48E-01 & 4.99E-01 & 2.41E-01 & 2.25E-01 & 2.20E-01\
DS05-4 & 4 & 0.5 & 128 & 443 & 1250 & 1.09E-04 & 9.06E-03 & 2.37E-02 & 1.91E-01 & 4.99E-01 & 1.98E-01 & 1.12E-01 & 2.74E-02\
DU05-1 & 0.5 & 0.5 & 95 & 655 & 1846 & 1.26E-04 & 1.43E-02 & 2.85E-02 & 2.27E-02 & 5.00E-01 & 5.20E-01 & 4.84E-01 & 6.77E-01\
DU05-2 & 1 & 0.5 & 67 & 463 & 1388 & 1.26E-04 & 5.20E-02 & 5.09E-02 & 6.57E-02 & 5.00E-01 & 4.97E-01 & 3.66E-01 & 5.19E-01\
DU05-3 & 2 & 0.5 & 44 & 323 & 1060 & 1.26E-04 & 1.09E-01 & 9.77E-02 & 1.36E-01 & 5.00E-01 & 4.68E-01 & 2.37E-01 & 3.02E-01\
DU05-4 & 4 & 0.5 & 25 & 207 & 837 & 1.26E-04 & 1.76E-01 & 1.46E-01 & 2.43E-01 & 5.00E-01 & 4.42E-01 & 1.58E-01 & 1.56E-01\
QS05-0 & - & 0.5 & 471 & 1649 & 3336 & 1.00E-04 & 2.00E-05 & 9.60E-06 & 4.91E-06 & 5.00E-01 & 5.18E-01 & 5.41E-01 & 5.73E-01\
QS05-1 & 0.5 & 0.5 & 256 & 783 & 1800 & 1.01E-04 & 1.71E-04 & 9.82E-04 & 1.44E-03 & 4.99E-01 & 5.63E-01 & 7.27E-01 & 9.76E-01\
QS05-2 & 1 & 0.5 & 169 & 545 & 1288 & 1.00E-04 & 4.17E-04 & 2.56E-03 & 1.85E-02 & 5.00E-01 & 4.98E-01 & 5.12E-01 & 7.29E-01\
QS05-3 & 2 & 0.5 & 102 & 417 & 1190 & 1.01E-04 & 1.20E-03 & 3.00E-03 & 6.86E-02 & 4.99E-01 & 4.08E-01 & 2.49E-01 & 2.17E-01\
QS05-4 & 4 & 0.5 & 61 & 387 & 1267 & 1.01E-04 & 4.89E-03 & 3.89E-03 & 7.67E-02 & 4.99E-01 & 3.19E-01 & 6.66E-02 & 3.36E-02\
QU05-0 & - & 0.5 & 151 & 1119 & 2808 & 1.00E-04 & 5.58E-05 & 2.61E-05 & 1.23E-05 & 5.00E-01 & 5.00E-01 & 5.14E-01 & 5.46E-01\
QU05-1 & 0.5 & 0.5 & 84 & 571 & 1646 & 1.01E-04 & 4.68E-02 & 1.20E-01 & 1.02E-01 & 5.00E-01 & 2.37E-01 & 1.38E-01 & 1.86E-01\
QU05-2 & 1 & 0.5 & 61 & 389 & 1226 & 1.00E-04 & 1.89E-01 & 1.69E-01 & 2.08E-01 & 5.00E-01 & 2.47E-01 & 1.21E-01 & 1.51E-01\
QU05-3 & 2 & 0.5 & 38 & 259 & 952 & 1.01E-04 & 3.21E-01 & 2.23E-01 & 2.30E-01 & 5.00E-01 & 2.68E-01 & 1.12E-01 & 2.25E-01\
QU05-4 & 4 & 0.5 & 20 & 161 & 709 & 1.01E-04 & 3.67E-01 & 2.74E-01 & 1.59E-01 & 5.00E-01 & 2.87E-01 & 1.24E-01 & 2.61E-01\
QS05-2M & 2 & 0.5 & 200 & 612 & 1449 & 1.01E-04 & 9.22E-03 & 2.09E-03 & 4.89E-03 & 4.99E-01 & 1.37E-01 & 1.83E-01 & 1.49E-01\
QU05-2M & 2 & 0.5 & 36 & 238 & 940 & 1.01E-04 & 5.38E-02 & 3.87E-02 & 2.23E-01 & 5.00E-01 & 5.59E-01 & 3.34E-01 & 3.28E-01\
Turb-0 & - & - & 36 & 361 & 1231 & 9.31E-01 & 9.31E-01 & 9.27E-01 & 9.21E-01 & - & 3.27E-04 & 3.31E-04 & 5.98E-04\
Turb-1 & 0.5 & - & 15 & 154 & 622 & 9.31E-01 & 9.48E-01 & 9.56E-01 & 9.52E-01 & - & 6.35E-02 & 6.62E-02 & 6.62E-02\
Turb-2 & 1 & - & 10 & 113 & 468 & 9.31E-01 & 9.53E-01 & 9.62E-01 & 9.48E-01 & - & 6.76E-02 & 9.05E-02 & 8.28E-02\
Turb-3 & 2 & - & 8 & 77 & 335 & 9.31E-01 & 9.60E-01 & 9.65E-01 & 9.46E-01 & - & 7.81E-02 & 1.15E-01 & 1.48E-01\
Turb-4 & 4 & - & 5 & 46 & 225 & 9.31E-01 & 9.65E-01 & 9.66E-01 & 9.39E-01 & - & 8.95E-02 & 1.11E-01 & 1.53E-01\
\[tab:LPer\]
NAME $L_{x}$ $\tau_{c}$ (kyr) SNR Age (kyr)
----------------------- ---------- ------------------ ----------------------
CXOU J010043.1-721134 6.52E+34 6.8 -
4U 0142+61 1.05E+35 68 -
SGR 0418+5729 9.57E+29 36000 -
SGR 0501+4516 8.14E+32 15 4-7
SGR 0526-66 1.89E+35 3.4 $\sim $4.8
1E 1048.1-5937 4.94E+34 4.5 -
1E 1547.0-5408 1.31E+33 0.69 -
PSR J1622-4950 4.36E+32 4 $<$6
SGR 1627-41 3.62E+33 2.2 -
CXOU J164710.2-455216 4.55E+32 420 -
1RXS J170849.0-400910 4.20E+34 9 -
CXOU J171405.7-381031 5.59E+34 0.9 0.65$^{+2.5}_{-0.3}$
SGR J1745-2900 1.07E+32 4.3 -
SGR 1806-20 1.63E+35 0.24 -
XTE J1810-197 4.25E+31 11 -
Swift J1822.3-1606 3.98E+29 6300 -
Swift J1834.9-0846 8.44E+30 4.9 $\sim$100
1E 1841-045 1.84E+35 4.6 0.5-1
3XMM J185246.6+003317 6.03E+30 1300 -
SGR 1900+14 8.97E+34 0.9 -
1E 2259+586 1.73E+34 230 14(2)
PSR J1845-0258 1.85E+34 0.73 -
: Magnetar data used in Figure 5 of the main paper. The data are taken from the McGill Magnetar Catalogue (54). The error bars used in the plot are based in the distance uncertainties. With respect to age, the associated supernova remnant age is used in favour of the characteristic spin-down age $\tau_{c}$, if the former is known.
\[tab:LPer\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Based on the mechanism of neutral meson mixing, we predict the branching fraction of the semileptonic decay of $D_s^+\to \pi^0 \ell^+ \nu_\ell$ ($\ell=e$, $\mu$) using recently measured branching fraction of $D_s^+\to \eta e^+\nu_e$ by BESIII experiment . We also give a formula that can describe the neutral meson mixing of $\pi^0$, $\eta$, $\eta''$ and a pseudoscalar gluonium $G$ in a unified way. The predicted branching fraction of $D_s^+\to\pi^0 e^+\nu_e$ decay is ${\mathcal{B}}(D_s^+\to\pi^0 e^+\nu_e)=(2.65\pm 0.38)\times 10^{-5}$. It is important to search for the decay $D_s^+\to \pi^0 \ell^+ \nu_\ell$ in experiment, in order to understand the mechanism of $\pi^0$ production in $D_s^+\to\pi^0$ transition in semileptonic decay. We also estimate the branching fraction of $D_s^+\to \pi^0\tau^+\nu_\tau$, which is the only kinematically allowed semi-tauonic decay mode of the charmed meson, since the mass value of $D_s^+$ meson is just slightly above the threshold of $\pi^0\tau^+$ generation in the semi-tauonic decay.'
author:
- 'Hai-Bo Li'
- 'Mao-Zhi Yang'
title: ' **Semileptonic Decay of $D_s^+\to \pi^0 \ell^+ \nu_\ell$ Via Neutral Meson Mixing** '
---
[^1]
In principle neutral mesons with hidden flavors can mix via strong and electromagnetic interactions if these mesons carry the same quantum numbers, such as spin, parity and charge conjugation that are exactly conserved in strong and electromagnetic interactions. For vector mesons with $J^{PC}=1^{--}$, there are $\rho-\omega$ [@coon1; @mal1; @mal2; @con; @gard; @terasaki] and $\omega-\phi$ mixing [@bena1; @kucu; @bena2; @gronau1; @gronau2]. For pseudoscalar mesons of $J^{PC}=0^{-+}$, there are $\pi^0 -\eta$ [@coon1; @coon2], $\eta -\eta'$ [@feld1; @bena1; @feld2; @ric] and $\eta'$-gluonium mixing [@ric]. Meson mixing is an interesting phenomenon that can be used to explain some specific decay processes of heavy mesons. For example, both $\omega -\phi$ mixing and weak annihilation are used as the mechanism that leads to the semileptonic decay of $D_s^+\to \omega e^+\nu_e$ [@gronau2]. Due to the mixing of $\omega -\phi$ mesons, there is a small component of $s\bar{s}$ in the wave function of $\omega$ meson, therefore the transition of $D_s^+\to \omega$ can be induced by the $s\bar{s}$ component in $\omega$ via $D_s^+ \to (s\bar{s}) \ell^+ \nu_\ell$ transition. Meanwhile the weak annihilation mechanism refers to the effect that one $\omega$ meson is preradiated nonperturbatively from the $c\bar{s}$ system in strong interaction, then the $c\bar{s}$ system annihilates into $e^+\nu_e$ via the charged weak current. The theoretical analysis of Ref. [@gronau2] shows that, if the value of the branching fraction of $D_s^+\to \omega e^+\nu_e$ exceeds $2\times 10^{-4}$, the nonperturbative weak annihilation effect rather than $\omega -\phi$ mixing would be important in the decay.
We consider the semileptonic decay of $D_s^+\to \pi^0\ell^+\nu_\ell$ in this work. Similar to the decay process of $D_s^+\to \omega e^+\nu_e$, the process $D_s^+\to \pi^0\ell^+\nu$ can only occur via $\pi^0 -\eta$ mixing and the nonperturbative weak annihilation effects. For the weak annihilation effect, it occurs by the preradiation of a $\pi^0$ meson from the $c\bar{s}$ system in $D_s^+$ meson, followed by the weak transition of $c\bar{s}\to e^+\nu_e$. However, compared to $D_s^+\to \omega e^+\nu_e$ decay, the weak annihilation effect in $D_s^+\to \pi^0 e^+\nu_e$ is doubly suppressed because the nonperturbative radiation of $\pi^0$ is suppressed by not only the Okubo-Zweig-Iizuka (OZI) rule [@OZI1; @OZI2; @OZI3], but also isospin violation. So the weak annihilation effect in $D_s^+\to \pi^0 e^+\nu_e$ decay is relatively small. We can neglect the weak annihilation effect and only consider $\pi^0-\eta$ mixing contribution in the following analysis. We also give the estimation for the order of the branching fraction of $D_s^+\to \pi^0 \tau^+\nu_\tau$, which is the only semi-tauonic decay mode of $D_s^+$ meson and is highly suppressed by its tiny phase space, since the mass value of $D_s^+$ meson is just slightly above the threshold of $\pi^0\tau^+$ generation in the semileptonic decay. We hope that these semileptonic decays will be searched for in experiment, so that the decay dynamics can be further investigated.
Now that $\pi^0-\eta$, $\eta -\eta'$ and $\eta'-G $ can mix in each pair, then in principle they can mix in an enlarged unified way. The mixing between $\pi^0$, $\eta$, $\eta'$ and the gluonium $G$ should be described uniformly. Let us define $\pi^0_q=\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$, $\eta_q=\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$, $\eta_s=s\bar{s}$, and $G$ as the pure pseudoscalar gluonium. The physical mesons of $\pi^0$, $\eta$, $\eta'$ and a pseudoscalar $\eta_G$ should be the mixing states of these flavor bases $$\begin{pmatrix}
\pi^0\\ \eta \\ \eta' \\ \eta_G
\end{pmatrix}
=V
\begin{pmatrix}
\pi^0_q\\ \eta_q \\ \eta_s \\ G
\end{pmatrix}\label{eq1},$$ where $V$ is a $4\times 4$ unitary matrix that describes the mixing between the pseudoscalar bases $\pi^0_q$, $\eta_q$, $\eta_s$ and $G$. The matrix $V$ is treated as a real matrix here for simplicity. For the unitarity of the mixing matrix $V$, there should be 6 independent parameters in the $4\times 4$ matrix $V$. Let’s set $i=1,\; 2,\; 3,\; 4$ for $\pi^0_q$, $\eta_q$, $\eta_s$ and $G$, then the 6 free parameters can be denoted as $\theta_{12}$, $\theta_{23}$, $\theta_{34}$, $\theta_{13}$, $\theta_{14}$ and $\theta_{24}$, which can be viewed as mixing angles. About the mixing angles, the following statements should be given:
1\) The mixing angle $\theta_{12}$ is for the mixing between $\pi^0_q$ and $\eta_q$, which is isospin-violating, therefore the mixing angle $\theta_{12}$ should be small, here we denote it as $\theta_{12}\equiv\delta$;
2\) $\theta_{23}$ is the mixing angle for $\eta_q-\eta_s$, which is denoted as $\theta_{23}\equiv\phi$;
3\) $\theta_{34}$ is the angle for $\eta_s-G$ mixing, which is denoted as $\theta_{34}\equiv\phi_G$;
4\) $\theta_{13}$ is the angle for the mixing of $\pi^0_q-\eta_s$. The mixing between these two states is not only isospin-violating, but also with larger mass-gap between these two states. So the mixing angle should be tiny, which can be neglected, $\theta_{13}\sim 0$;
5\) $\theta_{14}$ is the mixing angle for $\pi^0_q-G$ mixing, which is also isospin-violating and with larger mass-gap between these two mixing states, so the mixing angle is also tiny and can be set to $\theta_{14}\sim 0$;
6\) $\theta_{24}$ is the angle for the mixing of $\eta_q-G$. We find that, if considering $\theta_{24}$ to be very small, then the component of $G$ in $\eta$ will be tiny, which is consistent with the result of QCD sum rule calculation that the coupling of the gluonium to $\eta$ is much smaller than its coupling to $\eta'$ [@def]. So we can take $\theta_{24}\sim 0$ in the following analysis for simplicity.
Then the nonzero mixing angles considered in our scenario are $$\theta_{12}\equiv\delta,\;\;\; \theta_{23}\equiv\phi,\;\;\; \theta_{34}\equiv\phi_G.$$ With these three nonzero mixing angles, we can write the sub-mixing matrices explicitly as $$V1=\begin{pmatrix}
\cos\delta & -\sin\delta &0 & 0\\
\sin\delta & \cos\delta &0 & 0\\
0 & 0 &1 & 0\\
0 & 0 &0 & 1
\end{pmatrix}$$ for $\pi^0-\eta_q$ mixing, $$V2=\begin{pmatrix}
1 & 0 &0 & 0\\
0 & \cos\phi & -\sin\phi & 0\\
0 & \sin\phi & \cos\phi & 0\\
0 & 0 &0 & 1
\end{pmatrix}$$ for $\eta_q-\eta_s$ mixing, and $$V3=\begin{pmatrix}
1 & 0 &0 & 0\\
0 & 1 &0 & 0\\
0 & 0 &\cos\phi_G & \sin\phi_G &\\
0 & 0 &-\sin\phi_G & \cos\phi_G &
\end{pmatrix}$$ for $\eta_s-G$ mixing. The total mixing matrix can then be taken as $$\begin{aligned}
V &=& V_3 V_1 V_2 \nonumber\\
&=& \begin{pmatrix}
\cos\delta & -\sin\delta \cos\phi & \sin\delta \sin\phi & 0\\
\sin\delta & \cos\delta \cos\phi & -\cos\delta \sin\phi & 0\\
0 & \cos\phi_G \sin\phi & \cos\phi_G \cos\phi & \sin\phi_G\\
0 & -\sin\phi_G \sin\phi & -\sin\phi_G \cos\phi & \cos\phi_G
\end{pmatrix} \label{v-matrix}\end{aligned}$$ Considering that $\delta$ should be tiny, so we can make the approximation: $\cos\delta\sim 1$, $\sin\delta\sim \delta$. Then Eq.(\[v-matrix\]) can be simplified as $$V\simeq\begin{pmatrix}
1 & -\delta \cos\phi & \delta \sin\phi & 0\\
\delta & \cos\phi & -\sin\phi & 0\\
0 & \cos\phi_G \sin\phi & \cos\phi_G \cos\phi & \sin\phi_G\\
0 & -\sin\phi_G \sin\phi & -\sin\phi_G \cos\phi & \cos\phi_G
\end{pmatrix} \label{v-matrix2}$$ Substituting Eq. (\[v-matrix2\]) into Eq. (\[eq1\]), we can obtain $$\begin{aligned}
&&\mid \pi^0\rangle =\mid \pi^0_q\rangle-\delta \cos\phi\mid \eta_q\rangle +\delta \sin\phi\mid \eta_s\rangle,
\label{pi-w}\\
&&\mid \eta\rangle =\delta\mid \pi^0_q\rangle+ \cos\phi\mid \eta_q\rangle - \sin\phi\mid \eta_s\rangle, \label{eta-w}\end{aligned}$$ $$\begin{aligned}
\mid \eta'\rangle &=& \cos\phi_G \sin\phi\mid \eta_q\rangle +\cos\phi_G \cos\phi\mid \eta_s\rangle\nonumber\\
&&+\sin\phi_G\mid G\rangle, \label{etap-w}\\
\mid \eta_G\rangle &=& -\sin\phi_G \sin\phi\mid \eta_q\rangle -\sin\phi_G \cos\phi\mid \eta_s\rangle\nonumber\\
&&+\cos\phi_G\mid G\rangle. \label{fG-w}\end{aligned}$$ Eq. (\[pi-w\]) implies that the physical neutral pion $\pi^0$ is dominantly $\pi^0_q$ which is a component with isospin 1, and there are small components of $\eta_q$ and $\eta_s$ mixed in $\pi^0$. Note $\delta$ is a tiny quantity. Eq. (\[eta-w\]) indicates that $\eta$ meson is mainly $\eta_q$ and $\eta_s$ with a small component of $\pi^0_q$ in it, and the gluonium component can be neglected. Eq. (\[etap-w\]) shows that $\eta'$ is a mixing state of $\eta_q$, $\eta_s$ and the gluonium component $G$. This is consistent with the expression given in Ref. [@ric]. Finally the state $\eta_G$ in Eq. (\[fG-w\]) is the orthogonal state of $\eta'$, which is also mixing state of $\eta_q$, $\eta_s$ and the gluonium component $G$. If the mixing angle $\phi_G$ is small, then $\eta_G$ is dominantly a gluonium state.
The mixing angle $\delta$ can be determined by the ratio of the branching fractions of $\eta'\to\pi^+\pi^-\pi^0$ and $\eta'\to\pi^+\pi^-\eta$ decays, where the former is $G$-parity violating, which can only occur via $\pi^0-\eta$ mixing [@gro]. The dominant decay mode of $\eta'$ is $\eta'\to\pi^+\pi^-\eta$. With the mixing scheme given in Eqs. (\[pi-w\]) and (\[eta-w\]), the decay amplitudes of $\eta'\to\pi^+\pi^-\pi^0$ and $\eta'\to\pi^+\pi^-\eta$ are $$\begin{aligned}
&&\;\;\;\;\langle \pi^+\pi^-\pi^0\mid H \mid \eta'\rangle \nonumber\\
&&= \langle\pi^+\pi^-\pi^0_q\mid H \mid \eta'\rangle
-\delta\cos\phi\langle\pi^+\pi^-\eta_q\mid H \mid \eta'\rangle \nonumber\\
&& \;\;+\delta\sin\phi\langle\pi^+\pi^-\eta_s\mid H \mid \eta'\rangle , \label{etap-pi}\end{aligned}$$ $$\begin{aligned}
&&\;\;\;\;\langle \pi^+\pi^-\eta\mid H \mid \eta'\rangle \nonumber\\
&&= \delta\langle\pi^+\pi^-\pi^0_q\mid H \mid \eta'\rangle
+\cos\phi\langle\pi^+\pi^-\eta_q\mid H \mid \eta'\rangle \nonumber\\
&& \;\;-\sin\phi\langle\pi^+\pi^-\eta_s\mid H \mid \eta'\rangle ,\label{etap-eta}\end{aligned}$$ where $H$ is the Hamiltonian that induces the $\eta'$ three-body decays. The matrix element $\langle \pi^+\pi^-\pi^0_q\mid H \mid \eta'\rangle$ is $G$-parity violating, so we can take $\langle \pi^+\pi^-\pi^0_q\mid H \mid \eta'\rangle\sim 0$. Then Eqs. (\[etap-pi\]) and (\[etap-eta\]) indicate $$\frac{\langle \pi^+\pi^-\pi^0\mid H \mid \eta'\rangle}{\langle \pi^+\pi^-\eta\mid H \mid \eta'\rangle }=-\delta.$$ So we can obtain the ratio of the decay branching fractions $$\begin{aligned}
&&\;\;\;\;\frac{{\mathcal{B}}(\eta'\to \pi^+\pi^-\pi^0)}{{\mathcal{B}}(\eta'\to\pi^+\pi^-\eta) }\nonumber\\
&&=\left| \frac{\langle \pi^+\pi^-\pi^0\mid H \mid \eta'\rangle}
{\langle \pi^+\pi^-\eta\mid H \mid \eta'\rangle}\right| ^2
\frac{\phi_s(\eta'\to \pi^+\pi^-\pi^0)}{\phi_s(\eta'\to\pi^+\pi^-\eta) }\nonumber\\
&&=\delta^2 \frac{\phi_s(\eta'\to \pi^+\pi^-\pi^0)}{\phi_s(\eta'\to\pi^+\pi^-\eta)} \label{delta-br},\end{aligned}$$ where $\phi_s(\eta'\to \pi^+\pi^-\pi^0(\eta))$ is the phase space volume of the decay mode $\eta'\to \pi^+\pi^-\pi^0(\eta)$. The ratio of the phase space can be calculated directly to be [@gro; @cheng] $$\frac{\phi_s(\eta'\to \pi^+\pi^-\pi^0)}{\phi_s(\eta'\to\pi^+\pi^-\eta) }=17.0. \label{phase-v}$$ The branching fraction of $\eta'\to \pi^+\pi^-\pi^0$ has been measured by CLEO collaboration in 2008 [@cleo] and BESIII collaboration in 2012 and 2017 [@BESIII1; @BESIII2]. The relative ratio of ${\mathcal{B}}(\eta'\to \pi^+\pi^-\pi^0)/{\mathcal{B}}(\eta'\to\pi^+\pi^-\eta)$ was analyzed based on the recent data of BESIII, and its value is determined to be $(8.8\pm 1.2)\times 10^{-3}$ as in Ref. [@fang]. With this ratio from experiment, and Eqs. (\[delta-br\]) and (\[phase-v\]), we can obtain $$\delta^2=(5.18\pm 0.71)\times 10^{-4}. \label{delta-value}$$
Next we shall go on to discuss the semileptonic decays of $D_s^+\to \pi^0 \ell^+\nu_\ell$ and $D_s^+\to \eta \ell^+\nu_\ell$. According to Eqs. (\[pi-w\]) and (\[eta-w\]), both of these decays occur via the $\eta_s$ component in $\pi^0$ and $\eta$ at tree level. For semileptonic decay of $D_s^+$ to a pseudoscalar $P$, the hadronic matrix element involved in the decay amplitude is $$\begin{aligned}
&&\;\;\;\; \langle P(p_2)\mid V_\mu\mid D_s(p_1)\rangle\nonumber\\
&&=F_+^{D_sP}(q^2)(p_1+p_2-\frac{m^2_{D_s}-m_P^2}{q^2})_\mu\nonumber\\
&&\;\;\;\;+F_0^{D_sP}(q^2)\frac{m^2_{D_s}-m_P^2}{q^2}q_\mu, \label{matrix-el}\end{aligned}$$ where $q=p_1-p_2$, and $p_{1,2}$ are the momenta of $D_s^+$ and $P$ mesons, respectively. $m_{D_s}$ and $m_P$ are the masses of $D_s^+$ and $P$, respectively. $F_{+}^{D_sP}(q^2)$ is the so-called vector form factor, and $F_{0}^{D_sP}(q^2)$ the scalar form factor. To avoid the divergence in the hadronic matrix element in Eq. (\[matrix-el\]) as $q^2\to 0$, there should be $$F_{+}^{D_sP}(0)=F_{0}^{D_sP}(0).$$ The differential decay width of a semileptonic decay $D_s^+\to Pl^+\nu$ can be calculated to be $$\begin{aligned}
\frac{d\Gamma}{dq^2}(D_s^+&\to&P \ell^+\nu_\ell)=\frac{G_F^2|V_{cs}|^2}{24\pi^3m^2_{D_s}q^4}(q^2-m_\ell^2)^2|\vec{p}_P|\nonumber\\
&& \times\left[ (1+\frac{m_\ell^2}{2q^2})m^2_{D_s}|\vec{p}_P|^2|F_+^{D_sP}(q^2)|^2\right. \nonumber\\
&&+\left.\frac{3m_\ell^2}{8q^2}(m^2_{D_s}-m^2_P)^2|F_0^{D_sP}(q^2)|^2\right], \label{width-ml}\end{aligned}$$ where $\vec{p}_P$ is the momentum of the pseudoscalar $P$ in the rest frame of $D_s^+$ meson. $G_F$ is the Fermi constant, $m_\ell$ the lepton mass, and $V_{cs}$ the Cabibbo-Kobayashi-Maskawa (CKM) matrix element. For $\ell=e$ and $\mu$, $m_\ell^2$ can be neglected. Then $$\frac{d\Gamma}{dq^2}(D_s^+\to Pl^+\nu)=\frac{G_F^2}{24\pi^3}|V_{cs}|^2|F_+^{D_sP}(q^2)|^2|\vec{p}_P|^3. \label{width-e}$$ For the $q^2$-dependent form factor $F_+^{D_sP}(q^2)$, we consider the modified pole model [@bech] $$F_+^{D_sP}(q^2)=\frac{F_+^{D_sP}(0)}{(1-\frac{q^2}{m^2_{\rm pole}})(1-\alpha\frac{q^2}{m^2_{\rm pole}})}, \label{ffq2}$$ where $\alpha$ is a free parameter, and $m_{pole}$ is fixed to be the mass of the vector state $D_s^{*+}$. The BESIII collaboration has measured the branching fraction of $D_s^+\to \eta^{(\prime)}e^+\nu_e$ and the $q^2$-dependent form factor $F_+^{D_s\eta^{(\prime)}}(q^2)$ [@BESIII3]. The parameter $\alpha$ was fitted for each decay mode. The value of $\alpha$ for $F_+^{D_s\eta}(q^2)$ is [@BESIII3] $$\alpha=0.304(44)(22).$$
According to Eqs. (\[pi-w\]) and (\[eta-w\]), both the transition matrix elements of $\langle \pi^0\mid V_\mu\mid D_s^+\rangle$ and $\langle \eta\mid V_\mu\mid D_s^+\rangle$ can be related to $\langle \eta_s\mid V_\mu\mid D_s^+\rangle$ by the following relation $$\begin{aligned}
\langle \pi^0\mid V_\mu\mid D_s^+\rangle &=& \delta \sin\phi \langle \eta_s\mid V_\mu\mid D_s^+\rangle , \\
\langle \eta\mid V_\mu\mid D_s^+\rangle &=& -\sin\phi \langle \eta_s\mid V_\mu\mid D_s^+\rangle,\end{aligned}$$ because both the transitions of $D_s^+\to \pi^0$ and $D_s^+\to \eta$ occur through the component $s\bar{s}$ mixed in $\pi^0$ and $\eta$ mesons. Then we can get the relations between the form factors $$\begin{aligned}
F_{+,0}^{D_s\pi}(q^2)&=&\delta\sin\phi F_{+,0}^{D_s\eta_s}(q^2),\\
F_{+,0}^{D_s\eta}(q^2)&=&-\sin\phi F_{+,0}^{D_s\eta_s}(q^2).\end{aligned}$$ Comparing both sides of the above two equations, we have the following relation $$\frac{F_{+,0}^{D_s\pi}(q^2)}{F_{+,0}^{D_s\eta}(q^2)}=-\delta. \label{ff-ratio}$$ Note that the form factors only explicitly depend on $q^2$, not on the masses of the initial and final mesons.
Using the expression of the differential decay width in Eq. (\[width-e\]) and considering the $q^2$-dependent form factor in Eq. (\[ffq2\]), we can get the ratio $$\begin{aligned}
&&\;\;\;\;\frac{{\mathcal{B}}(D_s^+\to\pi^0 e^+\nu_e)}{{\mathcal{B}}(D_s^+\to\eta e^+\nu_e)}=\delta^2\times\nonumber\\
&&\frac{\int_0^{(m_{D_s}-m_{\pi})^2}dq^2 |\vec{p}_\pi|^3/
[(1-\frac{q^2}{m^2_{D_s^{*+}}})(1-\alpha\frac{q^2}{m^2_{D_s^{*+}}})]^2}
{\int_0^{(m_{D_s}-m_{\eta})^2}dq^2 |\vec{p}_\eta|^3/
[(1-\frac{q^2}{m^2_{D_s^{*+}}})(1-\alpha\frac{q^2}{m^2_{D_s^{*+}}})]^2},\nonumber\\
&& \label{br-ratio}\end{aligned}$$ where $F_{+,0}^{D_s\pi}(0)/F_{+,0}^{D_s\eta}(0)=-\delta$ is used according to Eq. (\[ff-ratio\]). Using Eqs. (\[br-ratio\]), (\[delta-value\]) and the measured branching fraction by BESIII [@BESIII3] ${\mathcal{B}}(D_s^+\to\eta e^+\nu_e)=(2.323\pm 0.063\pm 0,063)\%$, we can obtain the branching fraction of $D_s^+\to\pi^0 e^+\nu_e$ decay $${\mathcal{B}}(D_s^+\to\pi^0 e^+\nu_e)=(2.65\pm 0.38)\times 10^{-5}, \label{br-prediction}$$ where the error mainly comes from the uncertainties of the parameters $\alpha$, $\delta^2$, and the error of the experimentally measured branching fraction of $D_s^+\to\eta e^+\nu_e$. The error caused by the uncertainty of the parameter $\alpha$ is about 1.3%, while the error caused by the uncertainty of $\delta^2$ is about 13.6%, and the uncertainty caused by the error of the experimental value of ${\mathcal{B}}( D_s^+\to\eta e^+\nu_e)$ is about 3.8%. The error caused by the other sources is tiny which can be ignored.
The prediction in Eq. (\[br-prediction\]) is based on the contribution of $\pi^0 -\eta$ mixing scheme given in Eqs. (\[pi-w\]) and (\[eta-w\]), and the possible weak annihilation contribution is neglected. As analyzed before, the weak annihilation contribution is doubly suppressed because it both violates isospin invariance and suppressed by the OZI rule. Therefore the weak annihilation contribution must be small. So the prediction or at least the order given in our prediction in Eq. (\[br-prediction\]) is reliable. Therefore, measurement of the branching fraction of $D_s^+\to\pi^0 e^+\nu_e$ in experiment can be used to test any sizable contribution from the weak annihilation effect.
Next we shall go on to consider the decay of $D_s^+\to\pi^0\tau^+\nu_\tau$, in which the mass of $D_s^+$ is just slightly above the threshold of $ \pi^0 \tau^+$ production. It is suppressed by both the small mixing amplitude of $\pi^0 -\eta$ and the limited phase space in the decay. In addition, the decay of $D_s^+\to\pi^0\tau^+\nu_\tau$ is the only kinematically allowed decay for the charmed mesons. Therefore, it is interesting to know the order of the decay rate of the process $D_s^+\to\pi^0\tau^+\nu_\tau$.
According to Eq. (\[width-ml\]), the decay width of $D_s^+\to P\tau^+\nu_\tau$ involves not only the form factor $F_+^{D_s P}(q^2)$, but also the form factor $F_0^{D_s P}(q^2)$. The form factor $F_0^{D_s P}(q^2)$ can not be measured in experiment through the semileptonic decay process of $D_s^+ \to P \ell^+ \nu_\ell$. Since there is no any information on the form factor $F_0^{D_s P}(q^2)$ in experiment up to now, we still use the modified pole model for the $q^2$-dependence of $F_0^{D_s P}(q^2)$. Specifically, for $P=\pi^0$, the form factor $F_0^{D_s \pi^0}(q^2)$ is taken as $$F_0^{D_s\pi}(q^2)=\frac{F_0^{D_s\pi}(0)}{(1-\frac{q^2}{m^2_{\rm pole}})(1-\beta\frac{q^2}{m^2_{\rm pole}})},
\label{ff0q2}$$ where $m_{\rm pole}$ should be a state of $c\bar{s}$ system with $J^P=0^+$, which can be taken as the mass of $D_{s0}^*(2317)^+$, $m_{D_{s0}^*(2317)}=2317.7\pm 0.6$ MeV according to PDG [@PDG], and $\beta$ is a free parameter.
Using Eqs. (\[width-ml\]) and (\[width-e\]), we can obtain the ratio of the branching fractions of the decay modes of $D_s^+\to\pi^0\tau^+\nu_\tau$ and $D_s^+\to\eta e^+\nu_e$ $$\frac{Br(D_s^+\to\pi^0\tau^+\nu_\tau)}{Br(D_s^+\to\eta e^+\nu_e)}=\delta^2\frac{a}{b}, \label{br-ratio-tau}$$ with $$\begin{aligned}
a&=&\int_{m_\tau^2}^{(m_{D_s}-m_\tau)^2}dq^2 \frac{(q^2-m_\tau^2)^2}{m^2_{D_s}q^4}|\vec{p}_\pi|\nonumber\\
&& \times\left[ (1+\frac{m_\tau^2}{2q^2})\frac{m^2_{D_s}|\vec{p}_P|^2}
{[(1-\frac{q^2}{m^2_{D_s^{*+}}})(1-\alpha\frac{q^2}{m^2_{D_s^{*+}}})]^2}\right. \nonumber\\
&&+\left.\frac{3m_\tau^2}{8q^2}\frac{(m^2_{D_s}-m^2_\pi)^2}
{[(1-\frac{q^2}{m^2_{D_{s0}^*(2317)}})(1-\beta\frac{q^2}{m^2_{D_{s0}^*(2317)}})]^2}\right]\end{aligned}$$ and $$b=\int_0^{(m_{D_s}-m_{\eta})^2}dq^2 \frac{|\vec{p}_\eta|^3}{
[(1-\frac{q^2}{m^2_{D_s^{*+}}})(1-\alpha\frac{q^2}{m^2_{D_s^{*+}}})]^2}$$ Here in deriving Eq. (\[br-ratio-tau\]), $F_+^{D_s\pi}(0)=F_0^{D_s\pi}(0)$ and $F_+^{D_s\pi}(0)/F_+^{D_s\eta}(0)=-\delta$ have been used. Since there is no any information on $\beta$ in experiment yet, $\beta$ is treated as a free parameter in this work. As an illustration, the branching fraction of $D_s^+\to\pi^0\tau^+\nu_\tau$ varying with the parameter $\beta$ is depicted in Fig. \[fig1\] according to Eq. (\[br-ratio-tau\]), where all the other parameters are input with their central values.
It can be known from Fig.\[fig1\] that the order of the branching fraction of $D_s^+\to\pi^0\tau^+\nu_\tau$ is about $10^{-9}$. If one assumes that the parameter $\beta$ is approximately the same order as $\alpha$, say $\beta$ can be 0.2$\sim$0.4, then the branching fraction of $D_s^+\to\pi^0\tau^+\nu_\tau$ will be $(2.7\sim 3.6)\times 10^{-9}$.
According to the future physics program at BESIII, about 6 fb$^{-1}$ integrated luminosity will be collected at the center-of-mass energy of 4180 MeV at BEPCII [@li2017; @Ablikim:2019hff]. For our predicted branching fraction of ${\mathcal{B}}(D_s^+\to\pi^0 e^+\nu_e)=(2.65\pm 0.38)\times 10^{-5}$, about a few signal events are expected to be reconstructed at BESIII experiment based on the double-tag technique [@Ablikim:2018jun]. We also hope that significant signal will be observed at Belle-II experiment [@Kou:2018nap] and the future super-tau-charm factory [@Luo:2018njj] and , which will collect about 100 times the amount of the current data set at BESIII. As for the decay of $D_s^+\to\pi^0 \tau^+\nu_\tau$, the decay rate is predicted to be $(2.7\sim 3.6)\times 10^{-9}$, and is not yet experimentally observable.
In summary, we study the mixing of $\pi^0-\eta-\eta'-\rm{Gluonium}$ and the mixing scheme is given in a unified way. Then the branching fractions of $D_s^+\to\pi^0 e^+\nu_e$ is predicted to be ${\mathcal{B}}(D_s^+\to\pi^0 e^+\nu_e)=(2.65\pm 0.38)\times 10^{-5}$, which can be searched for at the BESIII experiment and will be important observable at the future super-tau-charm factory. It will be interesting to search for the decay $D_s^+\to\pi^0 e^+\nu_e$, in order to understand the decay dynamics, namely to validate the $\pi^0-\eta$ mixing effect and the weak annihilation contribution. We also estimate the order of the branching fraction of $D_s^+\to\pi^0 \tau^+\nu_\tau$ decay, which is about $(2.7\sim 3.6)\times 10^{-9}$. This is the only allowed semi-tauonic decay mode in the charm sector.
This work is supported in part by the National Natural Science Foundation of China under Contracts No. 11875168, 11375088, 11935018, 11875054; the Chinese Academy of Sciences under Contract No. QYZDJ-SSW-SLH003.
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[^1]: Corresponding author
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In this work we study the class of algebras satisfying a duality property with respect to Hochschild homology and cohomology, as in [@VdB]. More precisely, we consider the class of algebras $A$ such that there exists an invertible bimodule $U$ and an integer number $d$ with the property $H^{\bullet}(A,M)\cong
H_{d-\bullet}(A,U\ot_AM)$, for all $A$-bimodules $M$. We will show that this class is closed under localization and under smash products with respect to Hopf algebras satisfying also the duality property.
We also illustrate the subtlety on dualities with smash products developing in detail the example $S(V)\#G$, the crossed product of the symmetric algebra on a vector space and a finite group acting linearly on $V$.
author:
- Marco Farinati
title: 'Hochschild duality, localization and smash products'
---
Introduction {#introduction .unnumbered}
============
The aim of this work is to study the class of algebras satisfying a duality property with respect to Hochschild homology and cohomology, as in [@VdB]. More precisely, we consider the class of algebras $A$ such that there exists an invertible bimodule $U$ and an integer number $d$ with the property $H^{\bullet}(A,M)\cong
H_{d-\bullet}(A,U\ot_AM)$, for all $A$-bimodules $M$. We will show that this class is closed under localization (theorem \[teoloc\]) and under smash products (theorem \[teogal\]). By localization we mean an algebra morphism $A\to B$ with the following two properties: $B\ot_AB\cong B$ as $B$-bimodule, and $B\ot_A-\ot_AB$ is exact. For smash product, the philosophy is the following: take $A$ an algebra in this class with dualizing bimodule $U$, and $H$ a Hopf algebra with dualizing bimodule $H$, then $A\#H$ has dualizing bimodule $U\#H$ (see remark \[rem:smash\] for the definition of $U\#H$).
There is a subtlety on dualities with smash products, so the last section is devoted to develop the simplest example illustrating this: the algebra $S(V)\#G$, the crossed product of the symmetric algebra on a vector space and a finite group acting linearly on $V$. Given al algebra $A$ with dualizing module $U_A\cong A$ and a Hopf algebra with dualizing bimodule isomorphic to $H$, theorem \[teogal\] says that $A\#H$ has a dualizing bimodule isomorphic to $U_A\#H$. The subtlety is that, eventhow the bimodule $U_A\cong A$ as $A$-bimodule, it may happens that $U_A\not\cong A$ as $H$-module, and so $U_A\#H\not\cong A\#H$ as $A\#H$-bimodule. In the example of $S(V)$ and $G\subset \GL(V)$, we show that the condition for $U_{S(V)}\cong S(V)$ as $G$-modules is that $G\subset \SL(V)$, and consequently, homology and cohomology will differ. In order to illustrate the duality, we compute the cohomology of this example in two different ways.
The example of section 3 was motivated by a question of Paul Smith, whether the methods used in [@AFLS] would apply to $S(V)\#G$. The answer to that question is yes, and this calculation has also motivated section 2. I am grateful to Jacques Alev to have transmitted this question to me. I also want to thank Mariano Suárez Álvarez for careful reading of this manuscript.
General notations {#general-notations .unnumbered}
-----------------
Fix a field $k$ of characteristic zero, unadorned $\ot$ and $\Hom$ will denote $\ot_k$ and $\Hom_k$. If $X$ is a graded vector space and $n\in\ZZ$, we will denote $X[n]$ the same vector space but with its degree shifted by $n$. For example, if $X$ is non-zero only in degree zero, then $X[n]$ is non-zero only in degree $n$.
For any $k$-algebra $B$ and $k$-symmetric bimodule $M$, the Hochschild homology and cohomology of $B$ with coeficients in $M$ are $\Tor_{\bullet}^{B^e}(B,M)$ and $\Ext^{\bullet}_{B^e}(B,M)$, respectively, where $B^e=B\ot B^{\op}$; they are denoted $H_{\bullet}(B,M)$ and $H^{\bullet}(B,M)$. In the special case where $M=B$, we will also write $HH_{\bullet}(B):=H_{\bullet}(B,B)$ and $HH^{\bullet}(B):=H^{\bullet}(B,B)$.
The word “module” will mean “left module”. All modules will be $k$-symmetric, so that $B$-bimodules is the same as $B^e$-modules. A $B$-bimodule $P$ is called [**invertible**]{} if there exists another bimodule $Q$ such that $P\ot_BQ\cong B$ and $Q\ot_BP\cong B$. The set of isomorphism classes of invertible $B$-bimodules which are $k$-symmetric is denoted by $\Pic_k(B)$.
Finally, in section 3 there is some abuse of notation with the symbol $\det$. Some times it denotes the usual determinant function, and some other times it denotes the 1-dimensional representation of $\GL(V)$, or its restriction to some $G\subset\GL(V)$. The meaning will be clear from the context.
The duality theorem of Van den Berg {#the-duality-theorem-of-van-den-berg .unnumbered}
===================================
In [@VdB], the author proves a theorem relating the Hochschild homology and cohomology of a certain class of algebras. We will state this theorem in a way convenient for our purposes:
\[teo:dual\] (Theorem 3 of [@VdB]). Let $A$ be a $k$-algebra which admits a finitely generated projective $A^e$-resolution (for instance, this is the case if $A^e$ is noetherian) . The following conditions are equivalent:
1. There exists an invertible $A$-bimodule $U_A$, and an integer $d$ such that $H^{\bullet}(A,M)\cong H_{d-\bullet}(A,U_A\ot_AM)$ for all $A^e$-modules $M$.
2. The projective dimension of $A$ as $A^e$-module is finite, and $\Ext_{A^e}^n(A,A^e)=0$ for all $n\geq 0$ except for $n=d$ where $U_A:=\Ext_{A^e}^n(A,A^e)$ is an invertible $A^e$-module.
Localization
============
The general framework of this section is the following: $A\to B$ is a $k$-algebra map such that
- The multiplication map induces an isomorphism of $B^e$-modules $B\ot_AB\cong B$.
- The functors $B\ot_A-$ and $-\ot_AB$ are exact.
We look for conditions on $B$ which, together with the assumption that $A$ satisfies Van den Bergh’s theorem, allow us to conclude that so does $B$.
\[lemapic\] Let $U\in \Pic(A)$ and $A\to B$ be such that $B\ot_AB\cong B$. If $U\ot_AB\cong B\ot_AU$ as $A^e$-modules, then
- $B\ot_AU \cong B\ot_AU\ot_AB$ as $B\ot A^{\op}$-modules;
- $U\ot_AB \cong B\ot_AU\ot_AB$ as $A\ot B^{\op}$-modules; and
- $B\ot_AU$ is a $B^e$-module in a natural way, $B\ot_AU \in \Pic(B)$, its inverse is $B\ot_AU^{-1}\ot_AB$, and $U^{-1}\ot_AB\cong B\ot_AU^{-1}$ as $A^e$-modules.
The first isomorphism is the composition: $$B\ot_A(U\ot_AB)\cong B\ot_A(B\ot_AU)=
(B\ot_AB)\ot_AU\cong B\ot_AU$$ The second one is similar.
Now let $U^{-1}$ be the inverse of $U$ in $\Pic(A)$, so that $U\ot_AU^{-1}\cong
U^{-1}\ot_AU\cong A$. Let us see that $B\ot_AU^{-1}\ot_AB$ is the inverse of $B\ot_AU$: $$\begin{array}{rl}
(B\ot_AU)\ot_B(B\ot_AU^{-1}\ot_AB)&\cong
(U\ot_AB)\ot_BB\ot_AU^{-1}\ot_AB\\
&\cong U\ot_AB\ot_AU^{-1}\ot_AB\\
&\cong B\ot_AU\ot_AU^{-1}\ot_AB\\
&\cong B\ot_AA\ot_AB\\
&\cong B\ot_AB\\
& \cong B,
\end{array}$$ and $$\begin{array}{rl}
(B\ot_AU^{-1}\ot_AB)\ot_B(B\ot_AU)&\cong
B\ot_AU^{-1}\ot_AB\ot_AU\\
&\cong B\ot_AU^{-1}\ot_AU\ot_AB\\
&\cong B\ot_AA\ot_AB\\
&\cong B\ot_AB\\
& \cong B.
\end{array}$$
A bimodule $U$ such that there is an isomorphism $B\ot_AU\cong
U\ot_AB$ of $A^e$-modules will be said to [**commute with $B$**]{}.
Let $g\in\Aut_k(A)$ be such that it admits an extension $\wt{g}\in\Aut_k(B)$, i.e. $\wt{g}(a)=g(a)$ for all $a\in A$. Then the element $Ag\in\Pic(A)$ commutes with $B$. In particular, $U=A$ commutes with $B$.
Let $g$ be such an element and consider $Ag\in\Pic(A)$. There is an isomorphism of $B\ot A^{\op}$-modules $$\begin{array}{rcl}
B\ot_AAg&\to& B\wt{g}\\
b\ot ag&\mapsto& ba\wt{g}
\end{array}$$ On the other hand, one can define an isomorphism of $A\ot B^{\op}$-modules $$\begin{array}{rcl}
Ag\ot_AB&\to& B\wt{g}\\
ag\ot a\wt{g}&\mapsto& a\wt{g}(b)\wt{g}
\end{array}$$ In particular, $Ag\ot_AB$ and $B\ot_AAg$ are isomorphic as $A^e$-modules.
Let $g\in\Aut_k(A)$ be such that there exists no element $\wt{g}\in\Aut_k(B)$ extending it. Then the bimodule $Ag$ doesn’t commutes with $B$.
Assume $B\ot_AAg\cong Ag\ot_AB$ as $A^e$-modules. From lemma \[lemapic\] it follows that $B\ot_AAg\in\Pic(B)$. But, as a left $B$-module, $B\ot_AAg\cong B$, and it is well-known that if an element $U\in\Pic(B)$ is such that ${}_BU\cong {}_BB$, then it is of the form $B\alpha$ for some $\alpha\in\Aut_k(B)$, the automorphism $\alpha$ being defined up to inner automorphism. In particular, for $a\in A$ one has that $
g(a)=u\alpha(a)u^{-1}$ for some $u\in\U(B)$. Denoting $\wt{g}:=u\alpha(-)u^{-1}$ we see that we have found an automorphism extending $g$, thus a contradiction.
Let $A\to B$ be such that $B\ot_AB\cong B$. If $M$ is a left $B$-module, then $M\cong B\ot_AM$ as a left $B$-module. If $N$ is another left $B$-module, then $\Hom_B(M,N)=\Hom_A(M,N)$.
Using the hypothesis on $B$, we see that $$M
\cong B\ot_BM
\cong (B\ot_AB)\ot_BM
\cong B\ot_A(B\ot_BM)
\cong B\ot_AM;$$ it follows then that $$\Hom_B(M,N)
\cong\Hom_B(B\ot_AM,N)
\cong\Hom_A(M,N).$$
\[teoloc\] Let $A\in VdB(d)$ with dualizing bimodule $U$, and $A\to B$ be a morphism of $k$-algebras such that
1. the functors $B\ot_A-$ and $-\ot_AB$ are exact;
2. the canonical map induced by multiplication $B\ot_AB\to B$ is an isomorphism; and
3. $B\ot_AU\cong U\ot_AB$ as $A^e$-modules.
Then $B\in VdB(d)$ with dualizing bimodule $B\ot_AU\cong B\ot_AU\ot_AB$.
Notice that if $U=A$, then condition 3 is automatically satisfied, and the dualizing bimodule associated to $B$ is $B$.
By theorem \[teo:dual\], it is enough to show that the projective dimension of $B$ as $B^e$-module is finite, that $B$ admits a resolution by means of finitely generated $B^e$-projectives. and that $\Ext_{B^e}^n(B,B^e)=B\ot_AU\ot_AB$ and it vanishes elsewhere.
Let $P_{\bullet}$ be a finite resolution of $A$ as $A^e$-modules, with $P_n$ projective and finitely generated as $A^e$-modules. Since $B\ot_A-$ and $-\ot_AB$ are exact, the complex $B\ot_AP_{\bullet}\ot_AB$ is a resolution of $B\ot_AA\ot_AB\cong B$, and so $B$ also has a finite resolution. The bimodules $B\ot_AP_n\ot_AB$ are clearly $B^e$-finitely generated and projective.
In order to compute $\Ext^{\bullet}_{B^e}(B,B^e)$ one can use this particular resolution, and consequently $$\Ext^{\bullet}_{B^e}(B,B^e)=
H^{\bullet}(\Hom_{B^e}(B\ot_AP_{\bullet}\ot_AB,B^e))
\cong H^{\bullet}(\Hom_{A^e}(P_{\bullet},B^e))$$ We claim that if $P$ is $A^e$-projective finitely generated, then $$\Hom_{A^e}(P_{\bullet},B^e)\cong
B\ot_A\Hom_{A^e}(P_{\bullet},A^e)\ot_AB$$ For that, consider the class of $A^e$-modules $P$ such that $\Hom_{A^e}(P_{\bullet},B^e)\cong
B\ot_A\Hom_{A^e}(P_{\bullet},A^e)\ot_AB$. This class is closed under direct summands and finite sums, so it is enough to show our claim that the module $A^e$ is in it, and that is clear. Using this isomorphism one gets $$H^{\bullet}(\Hom_{A^e}(P_{\bullet},B^e))
\cong H^{\bullet}(B\ot_A\Hom_{A^e}(P_{\bullet},B^e)\ot_AB)$$ and by flatness this is the same as $B\ot_A H^{\bullet}(\Hom_{A^e}(P_{\bullet},B^e))\ot_AB
=B\ot_AU[d]\ot_AB$.
\[a1\] We can take $A=A_1(k)=k\{x,y\}/\cl{[x,y]=1}$, $B=k\{x,x^{-1},y\}/\cl{[x,y]=1}$. This example is a particular case of the following:
[**Normal localization:**]{} Let $A$ be an algebra and $x\in A$ such that the set $\{1,x,x^2,x^3,\dots\}$ satisfies the Ore conditions. Take $B=A[x^{-1}]$. If $M$ is a right $A$-module, then as $k[x]$ modules we have an isomorphism $M\ot_AB\cong M\ot_{k[x]}k[x,x^{-1}]$. This shows that $A\to B$ is flat. It is also clear that $B\ot_AB\cong B$, in the same way as $k[x^{\pm1}]\ot_{ k[x]} k[x^{\pm1}]\cong k[x^{\pm1}]$.
Another generalization of example \[a1\] is the following situation: let $\O(X)$ be the algebra of functions on an affine variety $X$, and let $U$ be an affine open subset of $X$. Let $A=\Diff(X)$ be the algebra of algebraic differential operators on $X$ and similarly $B=\Diff(U)$. Since $B=\O(U)\ot_{\O(X)}\Diff(X)$, the map $A\to B$ is flat, and $B\ot_AB=B$. If $A$ satisfies the theorem of Van den Bergh, then so it does $B$.
Next section, we will study the behavior of the duality property with respect to smash products.
Smash products
==============
In this section $H$ is a hopf algebra such that $H\in VdB(d)$ with dualizing bimodule $U_H=H$, $A\in VdB(d')$ is an $H$-module algebra with dualizing bimodule $U_A$, and $B:=A\#H$. We will prove (see theorem \[teogal\]) that $B\in VdB(d+d')$, with dualizing bimodule $U_B=U_A\#H$ (see remark \[rem:smash\] for the definition of $U\#H$).
If $H$ is a Hopf algebra, then $H\in VdB(d)$ with dualizing bimodule $H$ if and only if $\Ext^{\bullet}_H(k,M)\cong \Tor_{\bullet-d}(k,M)$ for all left $H$-modules $M$.
Let $M$ be a left $H$-module, then $M_{\epsilon}$ is the $H^e$-module with right action defined by $m.h:=\epsilon(h)m$ for all $m\in M$ and $h\in H$. If $H\in VdB(d)$, it follows that $$\Ext^{\bullet}_H(k,M)=
H^{\bullet}(H,M_{\epsilon})\cong
H_{d-\bullet}(H,M_{\epsilon})
= \Tor_{\bullet-d}(k,M)$$ On the other direction, if $X$ is an $H^e$-module, then $X^{\ad}$ is the same underlying vector space but with left $H$ action defined by $h\cdot_{\ad}x:=h_1xS(h_2)$. With this structure (see for instance [@St]) one has $$H^{\bullet}(H,X)=
\Ext^{\bullet}_H(k,X^{\ad})\cong
\Tor_{\bullet-d}(k, X^{\ad})\cong
H_{d-\bullet}(H,M_{\epsilon})
= \Tor_{\bullet-d}(k,M)$$
Let $G$ be a finite group such that $\frac{1}{|G|}\in k$. The Reynolds operator $e=\frac{1}{|G|}\sum_{g\in G}g$ induces an isomorphism $M_G\cong M^G$ for any $G$-module $M$. This implies that $k[G]\in VdB(0)$ with $U_{k[G]}=k[G]$. This example can be easily generalized in the following direction:
Let $H$ be a semisimple unimodular Hopf algebra, so that $H$ admits a [*central*]{} integral $e\in H$ satisfying $$he=\epsilon(h)e,\ \qquad \epsilon(e)=1.$$ Then $H\in\VdB(0)$ with $U_H=H$. It is known (see Radford, [@R] theorem 4) that the Drinfel’d double of a finite dimensional hopf algebra is unimodular. If $K$ is a finite dimensional Hopf algebra and $D(K)$ is the Drinfel’d double, again by a result of Radford ([@R] proposition 7) $D(K)$ is semisimple if and only if $K$ is semisimple and cosemisimple. Taking $K=k[G]$ where $G$ is a non-commutative group with $|G|^{-1}\in k$, we get $H:=D(K)$ a non commutative not cocommutative semisimple unimodular Hopf algebra.
Let $H$ be a unimodular semisimple Hopf algebra, and let $e\in H$ be as above. We will show that $\Hom_H(k,M)\cong k\ot_HM$. If $M$ is a left $H$-module, then $$\Hom_H(k,M)\cong\{m\in M\ / \ hm=\epsilon(h)m\}=:M^H.$$ It is clear that every element of the form $em$ belongs to $M^H$ because $$h(em)=(he)m=\epsilon(h)em;$$ but if $m\in M^H$, then $$em=\epsilon(e)m=m,$$ so $M^H$ coincides with the image of the multiplication by $e$. Let us consider the map $$\begin{array}{rcl}
e:M&\to& M^H\\
m&\mapsto & em.
\end{array}$$ The elements of the form $hm-\epsilon(h)m$ belong to the kernel of this map, so it factors through $M_H:=M/\cl{hm-\epsilon(h)m}$. Now the map $M^H\to M_H$ defined by $m\mapsto \ra{m}$ defines an inverse, because in $M_H$, every element $m=\epsilon(e)m$ is equivalent to $em$. We have shown that $H\in VdB(0)$.
\[ejemplokx\] The algebra $H=k[x]$ is a Hopf algebra with $\Delta(x)=x\ot 1+1\ot x$. It belongs to the Class $VdB(d)$ with $U_H=H$.
Write $k[x]^e=k[x]\ot k+\cong =k[x,y]$, and consider the Koszul resolution $$0\to k[x,y] \to
k[x,y] \to k[x]\to 0$$ where the first map is the multiplication by $(x-y)$ and the second map is the evaluation $x=y$. Applying the functor $\Hom_{k[x,y]}(-,k[x,y])$ on obtain the complex
$$0\to \Hom_{k[x,y]}(k[x,y],k[x,y]) \to \Hom_{k[x,y]}(k[x,y],k[x,y]) \to 0$$ where the map is again multiplication by $x-y$. This complex identifies with $$0\to k[x,y] \to k[x,y] \to 0$$ but notice that now the grading increases to the right, so the homology is $k[x,y]/(x-y)\cong k[x]$ in degree one, zero elsewhere, and we conclude that $k[x]\in \VdB(1)$.
The algebra $k[x]$ admits a finitely generated $k[x]^e$-projective resolution; this fact implies a Künneth formula for Hochschild cohomology, and so the algebra $k[x_1,\dots,x_n]\in VdB(n)$, with $U_{k[x_1,\dots,x_n]}=
k[x_1,\dots,x_n]$.
The Hopf algebra $k[x_1^{\pm 1},\dots,x_d^{\pm1}]=k[\ZZ^n]$, belongs to the class $\VdB(d)$, because as an algebra, it is a localization of $k[x_1,\dots,x_d]$. Also $$U_ {k[x_1^{\pm 1},\dots,x_d^{\pm1}]}
=U_ {k[x_1,\dots,x_d}]\ot_{k[x_1,\dots,x_d]}
k[x_1^{\pm 1},\dots,x_d^{\pm1}]=
k[x_1^{\pm 1},\dots,x_d^{\pm1}]$$
\[rem:smash\] Let $A$ be an $H$-module algebra and $U\in \Pic_k(A)$ such that $U$ is also an $H$-module, with the compatibility property $$h(aub)=h_1(a)h_2(u)h_3(b)$$ for all $a,b \in A$, $h \in H$, and $u \in U$. Let $U^{-1}:=\Hom_A(U,A)$; this is also an $H$-module satisfying the same compatibility condition. If $U\#H$ is the abelian group $U\ot H$ with $A\#H$-bimodule structure given by $$(a\#h)(u\ot k):=
(ah_1(u)\ot h_2k)$$ $$(u\ot k)(a\#h)=
(uk_1(a)\ot k_2h),$$ then $U\#H\in\Pic_k(A\#H)$, and its inverse is $U^{-1}\#H$. If $M$ is left $A\#H$-module, then $$(U\#H)\ot_{A\#H}M\cong U\ot_AM$$ as $A\#H$-modules, where the $A\# H$-module structure on $U\ot_AM$ is the one induced by the obvious left $A$-structure and the diagonal $H$-structure.
We will only exhibit an isomorphism $U\#H\ot_{A\#H}U^{-1}\#H\to A\#H$. Let us denote by $\cl{\ ,\ }$ the evaluation map $U\ot_AU^{-1}\to A$; notice that $\cl{\ ,\ }$ is $H$-linear. For $u\in U$, $v\in U^{-1}$, $h$ and $k\in H$, define $$\begin{array}{rcl}
U\#H\ot_{A\#H}U^{-1}\#H &\to &A\#H\\
(u\ot h)\ot (v\ot k)&\mapsto & \cl{u,h_1(v)}h_2k.
\end{array}$$
\[teogal\]Let $H\in VdB(d)$ be a Hopf algebra with $U_H=H$. If $A$ is an $H$-module algebra with $A\in VdB(d)$, then $A\# H \in VdB(d+d')$ with $U_{A\# H}=U_{A}\#H$.
Let $B$ be $A\# H$. In [@St], the author shows that, for a $B$-bimodule $M$, there is a spectral sequence converging to $H^{\bullet}(B,M)$ whose second term is $\Ext^p(k,H^q(A,M))$. Similarly, there is a spectral sequence with $E^2$ term equal to $\Tor_p^H(k,H_q(A,M))$ converging to $H_{\bullet}(B,M)$.
Now consider $M=B^e$, and let us compute $H^{\bullet}(B,B^e)$. First, one notes the following isomorphism of left $A^e$-modules: $$B^e\cong A^e \ot V,$$ where $V$ is the vector space $H\ot H$.
Using Stefan’s spectral sequence, one has $$E_2^{pq}=\Ext_H^p(k,H^q(A,B^e))
=\Ext_H^p(k,H^q(A,A^e\ot V)).$$ Since $A\in VdB(d')$, it follows that $$\begin{array}{rcl}
H^{\bullet}(A,A^e\ot V))
&\cong& H_{d'-\bullet}(A,U\ot_AA^e\ot V)\\
&\cong& H_{d'-\bullet}(A,U\ot_AA^e)\ot V\\
&\cong& H^{\bullet}(A,A^e)\ot V\\
&\cong& U[d]\ot V.
\end{array}$$ This implies first that the spectral sequences degenerates at this step, and consequently, there is an isomorphism $$H^{\bullet}(B,B^e)\cong \Ext_H^{*-d'}(k,U\ot V)$$ Recall that $V=H\ot H^{op}$; we have to consider it as $H$-module with the adjoint action. Now we use the fact that $H\in VdB(d)$, with $U_H=H$, so $H^{\bullet}(H,X)\cong H_{d-\bullet}(H,X)$ for all $H$-bimodules $X$. In particular, for a left $H$-module $X$, one can consider the bimodule $X_{\epsilon}$, and this gives the formula $$\Ext_H^{\bullet}(k,X)
=H^{\bullet}(H,X_{\epsilon})
\cong H_{d-\bullet}(H,X_{\epsilon})
=\Tor^H_{d-\bullet}(k,X).$$ This formula implies that $$H^{\bullet}(B,B^e)\cong \Ext_H^{*-d'}(k,U_A\ot V)\cong
\Tor^H_{d'+d-\bullet}(k,U_A\ot V).$$
On the other hand, $H_{\bullet}(B,U_A\ot_AB^e))=
H_{\bullet}(B,(U_A\#H)\ot_BB^e))$ can be computed using a spectral sequence whose second term is $$\begin{array}{rcl}
\Tor_{\bullet}^H(k,H_{\bullet}(A,U_A\ot_AB^e))
&=&
\Tor_{\bullet}^H(k,H_{\bullet}(A,U\ot_A(A^e\ot V))\\
&=&\Tor_{\bullet}^H(k,U_A\ot V).
\end{array}$$ This spectral sequence collapes giving an isomorphism $$H_{\bullet}(B,U_A\ot_AB^e)
\cong \Tor_{\bullet}^H(k,U_A\ot V)$$ In particular, $$H^{\bullet}(B,B^e)\cong H_{d+d'-\bullet}(B,U\ot_AB^e)$$ and $$\begin{array}{rcl}
H^{d+d'}(B,B^e)
&=& H_{0}(B,U\ot_AB^e)\\
&=&H_{0}(B,(U\#H)\ot_BB^e)\\
&=&H_{0}(B,(U\#H)\ot B)\\
&=&U\#H.
\end{array}$$
With the notations of the above theorem, assume $U=A$ as $A$-bimodules and $H$-modules, then $$H^{\bullet}(B,M)\cong H_{d+d'-\bullet}(B,M)$$ for all $A\#H$-bimodules $M$.
Let $A\in VdB(d)$, $D\in\Der_k(A)$, and write the Ore extension $B=A[t,D]$. This algebra $B$ coincides with $A\#k[t]$ where the $k[t]$-module action on $A$ is given by $t.a=D(a)$, $B\in VdB(d+1)$. For $A=k[x]$ and $D=\frac{\partial}{\partial x}$ one obtains the known result that $A_1(k)\in VdB(2)$.
Let $0\neq q\in k$, then $B=k\{x^{\pm 1},y^{\pm 1}\}/\cl{yx=qxy}\in VdB(2)$. Indeed, this algebra is isomorphic to $k[x^{\pm 1}]\# k[y^{\pm 1}]$ where the $H$-module structure on $k[x^{\pm 1}]$ is given by $y.x=qx$.
Let $A$ be an algebra and $G$ a finite group of automorphism of $A$. If $A\in VdB(d)$, then $A\#G\in \VdB(d)$.
[**Warning:**]{} It can happen that $A$ is such that $U_A\cong A$ as $A$-bimodule, but $U_A\not\cong A$ as $H$-module. It is easy to show an example of this situation when $H=k[G]$.
One can first observe the following caracterization of the $A^e\#G$-structures on a $A$-bimodule isomorphic to $A$:
\[propstr\] Let $U$ be an $A^e$-bimodule isomorphic to $A$. The set of all possible $A^e\# G$-module structures on $U$, modulo $A^e\#G$-isomorphism, is parametrized by $H^1(G,\U\Z(A))$, the first cohomology of $G$ with coeficients in the (multiplicative) abelian group of units of the center of $A$.
Fix an isomorphism $A\cong U$ and let $u$ be the image of $1$ in $U$. Hence $U=Au=uA$, and moreover, $au=ua$ for all $a\in A$. One has to define a $G$-action on $U$ such that, for all $a,b\in A$ and $v\in U$, the following identity holds $$g(avb)=g(a)g(v)g(b).$$ Since the bimodule $U$ is generated by $u$, it is clear that it is only necesary to define $g(u)$. The element $g(u)$ must belong to $U$, so it is of the form $a_gu$ for some $a_g$ in $A$. But $$au=ua$$ for all $a\in A$, and applying $g$ one obtains $$ag(u)=g(u)a,\ \ \forall a\in A,$$ and so $$aa_gu=a_gua=a_gau.$$ It follows that $a_g$ must belong to the center of $A$. Also, every element of $U$ is of the form $$ag(u)=aa_gu,$$ so $a_g$ must be a unit. We have then shown that the assignment $g\mapsto a_g$ must be a map from $G$ into $\U(\Z(A))$.
If one wants associativity, the identity $$g(h(u))=(gh)(u),\ \forall g,h\in G.$$ is required, so $$\begin{array}{rcl}
g(h(u))&=&g(a_hu)\\
&=&g(a_h)a_gu\\
&=&(gh)(u)\\
&=&a_{gh}u.
\end{array}$$ But $u$ is a basis of $U$ with respect to the left $A$-structure, so $$g(a_h)a_g=a_{gh}.$$
On the other hand, it is clear that an assignment $g\mapsto a_g$ from $G$ into the units of center of $A$ satisfying the above cocycle condition defines a $G$-action compatible with the $A$-bimodule structure.
Now assume that $U$ has two $G$-actions that are isomorphic. Let us denote them by $g._1(u)=a_gu$, and $g._2(u)=b_gu$, and call $U_1$ and $U_2$ the bimodule $U$ with the first and the second $G$-structure, respectively.
If $\phi:U_1\to U_2$ is an isomorphism of $A^e\#G$-modules, then the image of $u$ is some element $\lambda u$, where $\lambda\in A$. Moreover, $\lambda$ is a unit because $\phi$ is an isomorphism, and $\lambda\in
\Z(A)$ because $\phi$ is $A^e$-linear.
Now $G$-linearity means that $$\begin{array}{rcl}
\phi(g._1u)&=&\phi(a_gu)\\
&=&\lambda a_gu,
\end{array}$$ but also $$\begin{array}{rcl}
\phi(g._1u)
&=&g._2\phi(u)\\
&=&g._2(\lambda u)\\
&=&g(\lambda)g._2 u\\
&=&g(\lambda)b_g u,
\end{array}$$ so we deduce $$b_g=\lambda g(\lambda^{-1})a_g,$$ and the two assignments differ by a coboundary.
Despite proposition \[propstr\], for an algebra $A\in VdB$, the dualizing bimodule $U$ is a very particular one, namely $U_A=Ext_{A^e}^d(A,A^e)$. The following is an example showing (without calculating $H^1(G,\U\Z(A))$ that $U$ is isomorphic to $A$ as $A^e$ bimodule, but not as $G$-module:
\[ejemplodet\] Let $V$ be a finite dimensional vector space, $A=S(V)$, and $G\subset \GL(V)$ a finite group. We claim that $$\Ext_{A^e}^{\bullet}(A,A^e)=A\ot\det{}^{-1}[d],$$ where $d=\dim(V)$, and $\det{}^{-1}$ is the dual of the determinant representation $\Lambda^d V$. Namely, $\det{}^{-1}$ is a one dimensional $k$-vector space, if $w\in\det{}^{-1}$ is a nonzero element, $g\in G$, and $a\in A$, then the $G$-action is given by $$g(a\ot w)=g(a)\det(g|_V)^{-1}\ot w.$$ We conclude that $U_A\cong A$ as $A^e\#G$-modules if and only if $G\subset \SL(V)$.
Let $g\in G$, and choose a basis $\{x_1,\dots, x_d\}$ of $V$ which diagonalizes $g$. Notice that $S(V)=\ot_{i=1}k[x_i]$, and this tensor product is $g$-equivariant with the diagonal action. The Künneth formula is $g$-equivariant, so we only need to prove the following lemma:
\[lemadet\] If $A=k[x]$ and $g$ is the automorphism of $A$ determined by $g(x)=\lambda x$, then $\Ext_{A^e}^{\bullet}(A,A^e)=A[1]$, and the action of $g$ is given by multiplication by $\lambda^{-1}$.
[*Proof of the lemma:*]{} It was shown in example \[ejemplokx\] that $k[x]\in\VDB(1)$, let us compute the $g$-action on $$H^1(k[x],k[x,y])=\Der(k[x],k[x,y])/\InnDer(k[x],k[x,y]).$$ If $D:k[x]\to k[x,y]$ is a derivation, then $D$ is determined by its value $D(x)$ on $x$, and this gives the isomorphism $$\begin{array}{rclr}
\Der(k[x],k[x,y])&\cong& k[x,y] \\
D&\mapsto &D(x).&
\end{array}
(\dag)$$
If $p\in k[x,y]$, the inner derivation $[p,-]$ takes in $x$ the value $$\begin{array}{rcl}
[p,x]&=&p(x,y)y-xp(x,y)\\
&=&(x-y)p(x,y).
\end{array}$$ This shows that, under the isomorphism (), $\InnDer\cong (x-y)k[x,y]$, obtaining $$\begin{array}{rcl}
H^1(A,A^e)
&=&\Der(A,A^e)/\InnDer(A,A^e)\\
&\cong&\dfrac{k[x,y]}{(x-y)k[x,y]}\\
&\cong & k[x].
\end{array}$$ In order to compute the action of $g$ on $H^1$ we recall that, if $D$ is a derivation, then $$g.D=g(D(g^{-1}(-)),$$ so $$\begin{array}{rcl}
(g.D)(x)&=& g(D(g^{-1}x))\\
&=&g(D(\lambda^{-1} x))\\
&=&\lambda^{-1}g(D(x)),
\end{array}$$ and if $D(x)\in k$ (this is always the case modulo an inner derivation) we get $$(g.D)(x)=\lambda^{-1} D(x).$$
Back to the example $A=S(V)$ and $G\subset\GL(V)$ a finite subgroup, we see that $S(V)\#G\in \VDB(\dim(V))$ but $U_{S(V)\#G}\cong S(V)\#G$ if and only if $G\subset\SL(V)$. This example shows a situation where $H^{\bullet}(B,M)=H_ {d-\bullet}(B,U\ot_BM)$ with $U\neq B$. In particular, $H^{\bullet}(B)\cong H_{\bullet}(B,U)$, which needs not be equal to $H_{d-\bullet}(B)$, and in fact it is different.
The example $S(V)\#G$
=====================
We finish with a computation of the homology and cohomology of $S(V)\#G$.
Let $k$ be a field, $V$ a finite dimensional $k$-vector space, $G$ a finite subgroup of $\GL(V,k)$, $A=S(V)$, and we will asume that $\frac1{|G|}\in k$. For simplicity we will also asume that $k$ has a primitive $|G|$-th root of 1. This condition is not really necessary because of the following reason: consider $\xi$ a primitive $|G|$-root of unity in the algebraic closure of $k$ and let $K$ be $k(\xi)$ the field generated by $k$ and $\xi$. One can view $G$ inside $\GL(V\ot K,K)$, and consider it acting on $A\ot K=S_K(V\ot K)$. A descend property of the Hochschild homology and cohomology with respect to this change of the base field assures that the dimension over $K$ of the (co)homology of the extended algebra is the same as the dimension over $k$ of the (co)homology of the original one.
If $g\in G$, $V^g=\{x\in V \ /\ g(x)=x\}$. As $g$-module, $V^g$ admits a unique complement in $V$, we will call it $V_g$. We have $V=V^g\oplus V_g$ as $g$-modules, and this decomposition is canonical.
Homology of $S(V)\#G$
---------------------
\[teo1\] With the notations as in the above paragraph, denote $\cl{G}$ the set of conjugacy classes of $G$, and for $g\in G$ let $\Z_g$ be the centralizer of $g$ in $G$, so that $\Z_g=\{h\in G\ /\ hg=gh\}$. The Hochschild homology of $S(V)\#G$ is given by: $$H_n(S(V)\#G)=
H_n(S(V),S(V)\#G)^G=
\bigoplus_{\cl{g}\in\cl{G} }(S(V^g)\ot \Lambda^n (V^g))^{\Z_g}$$ where $\Lambda^n(V^g)$ is the homogeneous component of degree $n$ of the exterior algebra on $V^g$.
With the hypothesis on the characteristic and the order of the group, the spectral sequence of [@St] gives the following isomorphism: $$\begin{array}{rcl}
H_n(S(V)\#G)&=&H_n(S(V),S(V)\#G)^G\\
&=&\bigoplus_{\cl{g}\in \cl{G} }H_n(S(V),S(V)g)^{\Z_g},
\end{array}$$ valid for any $k$-algebra of the type $A\#G$. Since $V=V^g\oplus V_g$, it follows that $$S(V)\cong S(V^g)\ot S(V_ g)$$ as algebras, and $$S(V)g\cong S(V^g)\ot S(V_ g)g$$ as $S(V)$-bimodules. Using the Künneth formula one gets $$H_n(S(V),S(V)g)^{\Z_g}
=\bigoplus_{p+q=n}
\left( H_p(S(V^g))\ot H_q(S(V_g),S(V_g)g) \right) ^{\Z_g}$$ By the Hochschild-Kostant-Rosenberg theorem, or directly by computing using a Koszul type resolution, one see that, if $W$ is a finite dimensional $k$-vector space, $$H_n(S(W))=\Omega^n(S(W))=S(W)\ot \Lambda^n W.$$
The homology with coeficients is computed in the following lemma:
\[lema:coef\] $H_{\bullet}(S(V_g),S(V_g)g)=k[0]$ with trivial $\Z_g$-action.
Let $h\in \Z_g$. One can diagonalize simultaneously $h$ and $g$ in $V_g$. If $\{x_1,\dots x_k\}$ is a basis of eigenvectors of both $h$ and $g$, then the algebra $S(V_g)$ is isomorphic to $$k[x_1,\dots x_k]=\bigotimes_{i=1}^kk[x_i]$$ and $$S(V_g)g=k[x_1,\dots x_k]g=\bigotimes_{i=1}^kk[x_i]g_i,$$ where $g_i$ acts on $x_i$ by multiplication of the corresponding eigenvalue og $g$. Notice also that $h$ acts on each $x_i$ by multiplication by some $\lambda_i'$, because $x_i$ is also an eigenvector of $h$.
Using the Künneth formula again, one gets: $$H_{\bullet}(S(V_g),S(V_g)g)=\bigotimes_i H_{\bullet}(k[x_i],k[x_i]g_i).$$ Let us now make the explicit computation for the algebra $k[x]$, $g$ acting by $x\mapsto \lambda x$, and $h$ acting by $x\mapsto \lambda'x$.
Consider, as in example \[ejemplokx\], the resolution of $k[x]$ as $k[x]$-bimodule $$0\to k[x]\ot k[x]\to k[x]\ot k[x]\to k[x]\to 0.$$ Here the first morphism is given by $p\ot q\mapsto px\ot q-p\ot xq$ and the second one is the multiplication map.
By tensoring with $k[x]g$ over $k[x]^e$, one gets the complex $$0\to k[x]g\to k[x]g\to 0$$ with differential $$pg\mapsto pgx-xpg=px(\lambda-1)g,$$ whose homology is $H_{\bullet}(k[x].k[x]g)$. The fact that $\lambda\neq 1$ implies that the differential is injective and the image equals $xk[x]g$, so $H_1=0$ and $H_0=k$. It is clear that $h$ acts trivially on $H_0$, and the proof of the lemma is complete.
The sum $$H_n(S(V),S(V)g)^{\Z_g}
=\bigoplus_{p+q=n}
\left(H_p(S(V^g))\ot H_q(S(V_g),S(V_g)g) \right) ^{\Z_g}$$ reduces to $$H_n(S(V),S(V))^{\Z_g}=
(S(V^g)\ot\Lambda^n(V^g) ) ^{\Z_g}$$ and the proof of the theorem is finished.
Let $k=\CC$, $V=\CC^2$, $G$ a finite subgroup of $\SL(2,\CC)$. Then $$\begin{array}{rcl}
H_0(S(V)\#G)&=&
S(V)^G\oplus \CC^{\#\{\cl{g}\neq 1\}}\\
H_1(S(V)\#G)&=&
(S(V)\ot V)^G\\
H_2(S(V)\#G)&=&
(S(V)\ot\Lambda^2(V))^G= S(V)^G\\
H_n(S(V)\#G)&=&0\ \forall n>2
\end{array}$$
Cohomology: direct computation
------------------------------
The formula $$H^n(S(V)\#G)=
H^n(S(V),S(V)\#G)^G=
\bigoplus_{\cl{g}\in \cl{G} }H^n(S(V),S(V)g)^{\Z_g}$$ is also valid. Using $S(V)=S(V^g)\ot S(V_ g)$, and the Künneth formula one gets $$\begin{array}{rcl}
H^n(S(V),S(V)g)^{\Z_g}
&=&\underset{p+q=n}{\bigoplus}
\left( H^p(S(V^g),S(V^g))\ot H^q(S(V_g),S(V_g)g) \right) ^{\Z_g}\\
&=&\underset{p+q=n}{\bigoplus}
\left( S(V^g)\ot \Lambda^p((V^g)^{\bullet})\ot H^q(S(V_g),S(V_g)g) \right) ^{\Z_g}.
\end{array}$$ Here we have used the isomorphism $$\begin{array}{rcl}
H^{\bullet}(S(W),S(W))
&=&\Lambda^{\bullet}_{S(W)}\Der(S(W))\\
&=&S(W)\ot\Lambda^{\bullet} W^*.
\end{array}$$ Now we need the analogue of the lemma \[lema:coef\] for cohomology, whose proof is the same as lemma \[lemadet\].
Let $A=k[x]$, $g,h$ the automorphisms determined by $g(x)=\lambda x$ and $h(x)=\mu x$, with $\lambda\neq 1$. Then $H^{\bullet}(A,Ag)=k[1]$, and the action of $h$ is given by multiplication by $\mu^{-1}$.
If we denote by $d_g=\dim_k(V_g)$ then $$H^{\bullet}(S(V_g),S(V_g)g)=\det|_{V_g}^{-1}[d_g].$$ This is an isomorphism of of $\Z_g$-modules.
From the fact that $g$ and $h$ commute, one can choose a basis $\{x_1,\dots,x_n\}$ of eigenvectors of both $g$ and $g$. The corollary follows from the Künnet formula, and the Lema above applied to $S(V)=\otimes_{i=1}^nk[x_i]$.
We have obtained the following formula:
\[teodirecto\] $$H^{\bullet}(S(V)\#G)=
\bigoplus_{\cl{g}\in\cl{G}}
\left(S(V^g)\ot\Lambda^{\bullet}((V^g)^{\bullet})\ot
\det|_{V_g}^{-1}[d_g]\right)^{\Z_g}.$$
Cohomology: computation using duality
-------------------------------------
Using theorem \[teogal\] for $H=k[G]$ (see example \[ejemplodet\]), we know that $$\begin{array}{rcl}
H^{\bullet}(A\#G)&=&H^{\bullet}(A\#G,(U_A\#G)\ot_{A\#G}A\#G)\\
&=&H_ {d-\bullet}(A\#G,U_A\#G)\\
&=&H_ {d-\bullet}(A\#G,(A\ot\det{}^{-1})\#G).
\end{array}$$ Using Stefan’s spectral, this is the same as $$\begin{array}{rcl}
H_ {d-\bullet}(A,(A\ot\det{}^{-1})\#G)^{G}
&=& \bigoplus_{\cl{g}\in\cl{G}}
H_ {d-\bullet}(A,(A\ot\det{}^{-1}(V)).g)^{\Z_g}\\
&=&\bigoplus_{\cl{g}\in\cl{G}}
(H_ {d-\bullet}(A,A.g)\ot\det{}^{-1}(V))^{\Z_g}.
\end{array}$$ Now the same techniques of writing $V=V^g\oplus V_g$ apply, and we obtain $$\begin{array}{rcl}
\underset{\cl{g}\in\cl{G}}{\bigoplus}
(H_ {d-\bullet}(A,A.g)\ot\det{}^{-1})^{\Z_g}
&=&
\underset{\cl{g}\in\cl{G}}{\bigoplus}
(H_ {d-\bullet}(S(V^g))\ot\det{}^{-1})^{\Z_g}\\
&=&
\underset{\cl{g}\in\cl{G}}{\bigoplus}
(S(V^g)\ot\Lambda^{d-\bullet}(V^g)\ot\det{}^{-1})^{\Z_g}.
\end{array}$$ The difference between this formula and that of Theorem \[teodirecto\], having $\det$ or $\det|_{V_g}$ is explained by the fact that in \[teodirecto\], one has also $\Lambda^{\bullet}((V^g)^{*})$, while here one has $\Lambda^{d-\bullet}(V^g)$. The multiplication map induces a morphism of $\Z_g$-modules $$\Lambda^{\bullet}(V^g)\ot\Lambda^{\dim(V^g)-\bullet}(V^g)
\to \Lambda^{\dim(V^g)}V^g=\det|_{V^g},$$ and as a consequence one has an isomorphism of $\Z_g$-modules $$\Lambda^{\bullet}(V^g)^*\cong \Lambda^{\dim(V^g)-\bullet}(V^g)
\ot\det|_{V^g}^{-1}.$$ So we get the same after noticing that $\det=\det|_{V^g}\ot\det|_{V_g}$.
Let $k=\CC$, $V=\CC^2$, $G$ a finite subgroup of $\SL(2,\CC)$. In this case, homology and cohomology is the same: $$\begin{array}{rcl}
H^0(S(V)\#G)&=&
S(V)^G\\
H^1(S(V)\#G)&=&
(S(V)\ot V)^G\\
H^2(S(V)\#G)&=&
S(V)^G\oplus \CC^{\#\{\cl{g}\neq 1\}}\\
H^n(S(V)\#G)&=&0\ \forall n>2
\end{array}$$
Let $G=C_2=\{1,t\}$ the cyclic group of order two. Let $k$ be a field of $\ch(k)\neq 2$, $A=k[x]$ with $t$ acting on $A$ by $x\mapsto -x$. Using theorem \[teo1\] one gets
$$\begin{array}{rcccl}
H_0(A\#G)&=&
A^G\oplus k&=&k[x^2]\oplus k\\
H_1(A\#G)&=&
(A\ot k.dx)^G&=&k[x^2]xdx \\
H_n(A\#G)&=&0&&\forall n>1
\end{array}$$ On the other hand, $$\begin{array}{rcccl}
H^0(A\#G)&=&
A^G&=&k[x^2]\\
H^1(A\#G)&=&
(A\ot k.\partial_x)^G \oplus
\left(
\frac{\Der(A,At)}{\InnDer(A,At)}\right)^{C_2}
&=&k[x^2]x\partial_x\oplus 0 \\
H_n(A\#G)&=&0&&\forall n>1
\end{array}$$ In this example, homology and cohomology are not the same. The cohomology is $k[x^2]$-free, while the homology has torsion.
In the above example, we see that the cohomology is a “part” of the homology. The same phenomenon happens in the following:
Let $W=k^{n}$, consider $S_n$ acting on $W$ by permutation of the coordinates, and let $$V=\{(1,1,\dots,1)\}^{\perp}:=\{(x_1,\dots,x_n)\in W\ / \
\sum_{i=1}^nx_i=0\}.$$ We claim that $$H^{\bullet}(S(V)\#S_n)=
H^{\bullet}(S(V), S(V)\#A_n)^{S_n},$$ where $A_n$ denote as usual the subgroup of even permutations.
In fact, we can prove an analogous formula in the following general setting:
Let $G\subset\GL(V)$ be a finite subgroup, $S:=G\cap\SL(V)=
\Ker(\det:G\to k^{\times})$, and $C:=\det(G)\subset k^{\times}$. Then $$H_{\bullet}(S(V)\#G)=
\bigoplus_{w\in C}
\left(
\bigoplus_{\cl{g}\in\cl{G},\ det(g)=w}
H_{\bullet}(S(V),S(V)g)^{\Z_g}
\right),$$ and each of this summands is non zero, while in cohomology, there are only the terms corresponding to $w=1$: $$H^{\bullet}(S(V)\#G)=
\bigoplus_{\cl{g}\in\cl{G},\ \det(g)=1}H^{\bullet}(S(V),S(V)g)^{\Z_g}.$$ In particular $$H^{\bullet}(S(V)\#G)=
H^{\bullet}(S(V), S(V)\#S)^G$$ and $$H^{\bullet}(S(V)\#G)\neq
H_{d-\bullet}(S(V)\#G).$$
The formula for the homology is just noticing that the set $\cl{G}$ can be split into smaller pieces, parametrized by the values of the determinant. To see that each summand is non-zero we make them explicit. Using theorem \[teo1\] we know that: $$H_{\bullet}(S(V),S(V)g)^{\Z_g}
=(S(V^g)\ot\Lambda^{\bullet}V^g)^{\Z_g}.$$ Even if $V^g=0$, one always has the element $1\in
(S(V^g)\ot\Lambda^{\bullet}V^g)^{\Z_g}$.
The interesting part is the formula for the cohomology. Recall from the duality formula that $$H^{\bullet}(S(V),S(V)g)\cong
\det{}^{-1} \otimes H_{d-\bullet}(S(V),S(V)g).$$ If one shows that $H_{\bullet}(S(V),S(V)g)$ is a trivial $g$-module, then, for $\det(g)\neq 1$ we will have $$\begin{array}{rcl}
\left(\det{}^{-1} \otimes H_{d-\bullet}(S(V),S(V)g)\right)^{\Z_g}
&\subseteq&
\left(\det{}^{-1} \otimes H_{d-\bullet}(S(V),S(V)g)\right)^{g}\\
&=&
(\det{}^{-1})^g \otimes H_{d-\bullet}(S(V),S(V)g)\\
&=&0.
\end{array}$$ So let us see that $H_{\bullet}(S(V),S(V)g)$ has trivial $g$-action. For that, write $V=V^g\oplus V_g$, then $H_{\bullet}(S(V),S(V)g) \cong
H_{\bullet}(S(V^g))\otimes H_{\bullet}(S(V),S(V)g))$. Clearly $H_{\bullet}(S(V^g))$ is a trivial $g$-module, and $H_{\bullet}(S(V),S(V)g))$ also has trivial $g$-action in virtue of lema \[lema:coef\].
The equality between homology and cohomology depends not only on $G$, but on the representation. For example, given an arbitarry finite subgroup $G\subset\GL(V)$, we can consider the action on $V$ and on $V^*$, and $G$ will act symplectically on $W=V\oplus V^*$. In this case we have $$G\hookrightarrow \Sp(W)\subset
\SL(W),$$ so that $$H^{\bullet}(S(W)\#G)=H_{\dim(W)-\bullet}(S(W)\#G).$$
[99]{} Alev, J.; Farinati, M.A.; Lambre, T.; Solotar, A.L.: Homologie des invariants d’une algèbre de Weyl sous l’action d’un groupe fini. J. Algebra 232, No.2, 564-577 (2000).
Radford, David E.: Minimal quasitriangular Hopf algebras. \[J\] J. Algebra 157, No.2, 285-315 (1993).
Stefan, Dragos: Hochschild cohomology on Hopf Galois extensions. J. Pure Appl. Algebra 103, No.2, 221-233 (1995).
Van den Bergh, Michel: A relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Am. Math. Soc. 126, No.5, 1345-1348 (1998); erratum ibid. 130, No.9, 2809-2810 (2002)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
There is a rapidly increasing interest in crowdsourcing for data labeling. By crowdsourcing, a large number of labels can be often quickly gathered at low cost. However, the labels provided by the crowdsourcing workers are usually not of high quality. In this paper, we propose a minimax conditional entropy principle to infer ground truth from noisy crowdsourced labels. Under this principle, we derive a unique probabilistic labeling model jointly parameterized by worker ability and item difficulty. We also propose an objective measurement principle, and show that our method is the only method which satisfies this objective measurement principle. We validate our method through a variety of real crowdsourcing datasets with binary, multiclass or ordinal labels.\
\
**Keywords:** crowdsourcing, human computation, minimax conditional entropy
author:
- 'Dengyong Zhou [^1]'
- 'Qiang Liu [^2]'
- 'John C. Platt [^3]'
- 'Christopher Meek [^4]'
- 'Nihar B. Shah[^5]'
bibliography:
- 'crowd.bib'
title: Regularized Minimax Conditional Entropy for Crowdsourcing
---
Introduction
============
In many real-world applications, the quality of a machine learning system is governed by the number of labeled training examples, but the labor for data labeling is usually costly. There has been considerable machine learning research work on learning when there are only few labeled examples, such as semi-supervised learning and active learning. In recent years, with the emergence of crowdsourcing (or human computation) services like Amazon Mechanical Turk[^6], the costs associated with collecting labeled data in many domains have dropped dramatically enabling the collection of large amounts of labeled data at a low cost. However, the labels provided by the workers are often not of high quality, in part, due to misaligned incentives and a lack of domain expertise in the workers. To overcome this quality issue, in general, the items are redundantly labeled by several different workers, and then the workers’ labels are aggregated in some manner, for example, majority voting.
The assumption underlying majority voting is that all workers are equally good so they have equal vote. Obviously, such an assumption does not reflect the truth. It is easy to imagine that one worker is more capable than another in some labeling task. More subtly, the skill level of a worker may significantly vary from one labeling category to another. To address these issues, @DawSke79 propose a model which assumes that each worker has a latent probabilistic confusion matrix for generating her labels. The off-diagonal elements of the matrix represent the probabilities that the worker mislabels an item from one class as another while the diagonal elements correspond to her accuracy in each class. The true labels of the items and the confusion matrices of the workers can be jointly estimated by maximizing the likelihood of the workers’ labels.
In the Dawid-Skene method, the performance of a worker characterized by her confusion matrix stays the same across all items in the same class. That is not true in many labeling tasks, where some items are more difficult to label than others, and a worker is more likely to mislabel a difficult item than an easy one. Moreover, an item may be easily mislabeled as some class rather than others by whoever labels it. To address these issues, we develop a minimax conditional entropy principle for crowdsourcing. Under this principle, we derive a unique probabilistic model which takes both worker ability and item difficulty into account. When item difficult is ignored, our model seamlessly reduces to the classical Dawid-Skene model. We also propose a natural objective measurement principle, and show that our method is the only method which satisfies this objective measurement principle.
The work is an extension of the earlier results presented in [@zhoplaby12; @ZhoLiuPlaMee14]. We organize the paper as follows. In Section \[sec:cat\], we propose the minimax conditional entropy principle for aggregating multiclass labels collected from a crowd and derive its dual form. In Section \[sec:reg\], we develop regularized minimax conditional entropy for preventing overfitting and generating probabilistic labels. In Section \[sec:objective\], we propose the objective measurement principle which also leads to the probabilistic model derived from the minimax conditional entropy principle. In Section \[sec:ordinal\], we extend our minimax conditional entropy method to ordinal labels, where we need to introduce a new assumption called adjacency confusability. In Section \[sec:implementation\], we present a simple yet efficient coordinate ascent method to solve the minimax program through its dual form and also a method for model selection. Related work are discussed in Section \[sec:related\]. Empirical results on real crowdsourcing data with binary, multiclass or ordinal labels are reported in Section \[sec:exp\], and conclusion are presented in Section \[sec:conclusion\].
Minimax Conditional Entropy Principle {#sec:cat}
=====================================
In this section, we present the minimax conditional entropy principle for aggregating crowdsourced multiclass labels in both its primal and dual forms. We also show that minimax conditional entropy is equivalent to minimizing Kullback-Leibler (KL) divergence.
Notation and Problem Setting
----------------------------
Assume that there are a group of workers indexed by $i,$ a set of items indexed by $j,$ and a number of classes indexed by $k$ or $c.$ Let $x_{ij}$ be the observed label that worker $i$ assigns to item $j, $ and $X_{ij}$ be the corresponding random variable. Denote by $Q(Y_j = c)$ the unobserved true probability that item $j$ belongs to class $c.$ A special case is that $Q(Y_j = c) = 1$ and $Q(Y_j = k) = 0$ for any other class $k \neq c. $ That is, the labels are deterministic. Denote by $P(X_{ij} = k |Y_j = c)$ the probability that worker $i$ labels item $j$ as class $k$ while the true label is $c. $ Our goal is to estimate the unobserved true labels from the noisy workers’ labels.
Primal Form
-----------
Our approach is built upon two four-dimensional tensors with the four dimensions corresponding to workers $i, $ items $j, $ observed labels $k, $ and true labels $c.$ The first tensor is referred to as the empirical confusion tensor of which each element is given by $$\widehat{\phi}_{ij}(c, k) = Q(Y_j = c) {\ensuremath{\mathbb{I}}}(x_{ij} = k)$$ to represent an observed confusion from class $c$ to class $k$ by worker $i $ on item $j.$ The other tensor is referred to as the expected confusion tensor of which each element is given by $$\phi_{ij}(c, k) = Q(Y_j = c) P(X_{ij} = k |Y_j = c)$$ to represent an expected confusion from class $c$ to class $k$ by worker $i $ on item $j.$
[|c|c|g|c|c|c|c|]{} & item 1 & item 2 & $\cdots$ & item $n$\
worker 1 & $x_{11}$ & $x_{12}$ &$\cdots$ & $x_{1n}$\
worker 2 & $x_{21}$ & $x_{22}$ &$\cdots$ & $x_{2n}$\
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$\
worker $m$ & $x_{m1}$ & $x_{m2}$ &$\cdots$ & $x_{mn}$\
[|c|c|g|c|c|c|c|]{} & item 1 & item 2 & $\cdots$ & item $n$\
worker 1 & $\pi_{11}$ & $\pi_{12}$ &$\cdots$ & $\pi_{1n}$\
worker 2 & $\pi_{21}$ & $\pi_{22}$ &$\cdots$ & $\pi_{2n}$\
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$\
worker $m$ & $\pi_{m1}$ & $\pi_{m2}$ &$\cdots$ & $\pi_{mn}$\
We assume that the labels of the items are independent. Thus, the entropy of the observed workers’ labels conditioned on the true labels can be written as $$\begin{aligned}
H(X|Y) = & -\sum_{j, c} Q(Y_j = c) \sum_{i,k} P(X_{ij} = k|Y_j = c) \log P(X_{ij} = k|Y_j = c). \nonumber\end{aligned}$$ Both the distributions $P$ and $Q$ are unknown here. To attack this problem, we first consider a simpler problem: estimate $P$ when $Q$ is given. Then, we proceed to jointly estimating $P$ and $Q$ when both are unknown.
Given the true label distribution $Q$, we propose to estimate $P$ which generates the workers’ labels by $$\label{eq:me}
\max_{P} \quad H(X|Y),$$ subject to the worker and item constraints (Figure \[fig:cfm\])
\[eq:mct\] $$\begin{aligned}
& \sum_j \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = 0, \ \forall i, k, c, \label{eq:mct1}\\
& \sum_i \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = 0, \ \forall j, k, c, \label{eq:mct2}\end{aligned}$$
plus the probability constraints
\[eq:prob\] $$\begin{aligned}
& \sum_k P(X_{ij} = k|Y_j = c) = 1, \ \forall i, j, c, \label{eq:prob1}\\
& \sum_c Q(Y_j = c) = 1, \ \forall j, \label{eq:prob2}\\
& Q(Y_j = c) \ge 0, \ \forall j, c. \label{eq:prob3}
\end{aligned}$$
The constraints in Equation enforce the expected confusion counts in the worker dimension to match their empirical counterparts. Symmetrically, the constraints in Equation enforce the expected confusion counts in the item dimension to match their empirical counterparts. An illustration of empirical confusion tensors is shown in Figure \[fig:empten\].
item 1 item 2 item 3 item 4 item 5 item 6
------------ -------- -------- -------- -------- -------- --------
worker $1$ $1$ $2$ $2$ $1$ $3$ $2$
worker $2$ $2$ $1$ $2$ $2 $ $1 $ $3 $
worker $3$ $1$ $1$ $1$ $2$ $2$ $3$
0.10in $$\widehat{\phi}_1 =
\begin{pmatrix}
1 & 1 & 0 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}, \quad
\widehat{\phi}_2 =
\begin{pmatrix}
1 & 1 & 0 \\
0 & 2 & 0 \\
1 & 0 & 1
\end{pmatrix}, \quad
\widehat{\phi}_3 =
\begin{pmatrix}
2 & 0 & 0 \\
1 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}$$
When both the distributions $P$ and $Q$ are unknown, we propose to jointly estimate them by $$\label{eq:mme}
\min_{Q} \max_{P} \quad H(X|Y),$$ subject to the constrains in Equation and . Intuitively, entropy can be understood as a measure of uncertainty. Thus, minimizing the maximum conditional entropy means that, given the true labels, the workers’s labels are the least random. Theoretically, minimizing the maximum conditional entropy can be connected to maximum likelihood. In what follows, we show how the connection is established.
Dual Form {#sec:mle}
---------
The Lagrangian of the maximization problem in can be written as $$\label{eq:lag1}
L = H(X|Y) + L _{\sigma} + L_{\tau}+ L_\lambda$$ with $$\begin{aligned}
L_{\sigma} = & \sum_{i, c, k} \sigma_i(c, k) \sum_j \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] \nonumber,\\
L_{\tau} = & \sum_{j, c, k} \tau_j(c, k) \sum_i \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] \nonumber, \\
L_{\lambda} = & \sum_{i,j, c} \lambda_{ijc}\bigg[\sum_{k} P(X_{ij} = k|Y_j = c) -1 \bigg], \nonumber\end{aligned}$$ where $\sigma_i(c, k), \tau_j(c, k)$ and $\lambda_{ijc}$ are introduced as the Lagrange multipliers. By the Karush-Kuhn-Tucker (KKT) conditions [@bova04], $$\frac{\partial L }{\partial P(X_{ij} = k|Y_j = c)} = 0,$$ which implies $$\log P(X_{ij} = k|Y_j = c) = \lambda_{ijc} -1 + \sigma_i(c, k) + \tau_{j}(c, k).$$ Combining the above equation and the probability constraints in eliminates $\lambda$ and yields $$\label{eq:model}
P(X_{ij} = k|Y_j = c) = \frac{1}{Z_{ij}} \exp[\sigma_i(c, k) + \tau_{j}(c, k)],
$$ where $Z_{ij}$ is the normalization factor given by $$Z_{ij} = \sum_k \exp[\sigma_i(c, k) + \tau_{j}(c, k)].$$ Although the matrices $[\sigma_i(c, k)]$ and $[\tau_{j}(c, k)]$ in Equation come out as the mathematical consequence of minimax conditional entropy, they can be understood intuitively. We can consider the matrix $[\sigma_i(c, k)]$ as the measure of the intrinsic ability of worker $i.$ The $(c, k)$-th entry measures how likely worker $i$ labels a randomly chosen item in class $c$ as class $k.$ Similarly, we can consider the matrix $[\tau_i(c, k)]$ as the measure of the intrinsic difficult of item $j.$ The $(c, k)$-th entry measures how likely item $j$ in class $c$ is labeled as class $k$ by a randomly chosen worker. In the following, we refer to $[\sigma_i(c, k)]$ as worker confusion matrices and $[\tau_i(c, k)]$ as item confusion matrices.
Substituting the labeling model in Equation into the Lagrangian in Equation , we can obtain the dual form of the minimax problem as (see Appendix \[app:dual1\]) $$\label{eq:dual1}
\max_{\sigma, \tau, Q} \quad \sum_{j,c}Q(Y_j = c) \sum_i \log P(X_{ij} = x_{ij}|Y_j = c).$$ It is obvious that, to be optimal, the true label distribution has to be deterministic. Thus, the dual Lagrangian can be equivalently expressed as the complete log-likelihood $$\begin{aligned}
& \log \bigg\{\prod_j \sum_{c} Q(Y_j = c) \prod_i P(X_{ij} = x_{ij}|Y_j = c)\bigg\}. \nonumber
$$ In Section \[sec:reg\], we show how to regularize the objective function in to generate probabilistic labels.
Minimizing KL Divergence {#sec:kl}
------------------------
Let us extend the two distributions $P$ and $Q$ to the product space $X \times Y.$ We extend the distribution $Q$ by defining $ Q(X_{ij} = x_{ij}) = 1, $ and $Q(Y)$ stays the same. We extend the distribution $P$ with $P(X, Y) = \prod_{ij}P{(X_{ij}|Y_j)} P(Y_j), $ where $P{(X_{ij}|Y_j)}$ is given by Equation , and $P(Y)$ is a uniform distribution over all possible classes. Then, we have
\[eq:minkl\] When the true labels are deterministic, minimizing the KL divergence from $Q$ to $P, $ that is, $$\begin{aligned}
\min_{P, Q} \bigg\{D_{\operatorname*{KL}} (Q\parallel P) = \sum_{X, Y}Q(X,Y) \log \frac{Q(X,Y)}{P(X,Y)}\bigg\},
\end{aligned}$$ is equivalent to the minimax problem in .
The proof is presented in Appendix \[sec:proofkl\]. A sketch of the proof is as follows. We show that, $$\begin{aligned}
D_{\operatorname*{KL}}(Q\parallel P) = & - \sum_{j, c} Q(Y_j = c) \sum_{i, k} P(X_{ij} = k|Y_j = c)\log P(X_{ij} = k|Y_j = c) \\
& + \sum_{Y} Q(Y)\log Q(Y) - \log P(Y). \\
\end{aligned}$$ By the definition of $P(X, Y), $ $P(Y)$ is a constant. Moreover, when the true labels are deterministic, we have $$\sum_{Y} Q(Y)\log Q(Y) = 0.$$ This concludes the proof of this theorem.
Regularized Minimax Conditional Entropy {#sec:reg}
=======================================
In this section, we regularize our minimax conditional entropy method to address two practical issues:
- **Preventing overfitting**. While crowdsourcing is cheap, collecting many redundant labels may be more expensive than hiring experts. Typically, the number of labels collected for each item is limited to a small number. In this case, the empirical counts in Equation may not match their expected values. It is likely that they fluctuate around their expected values although these fluctuations are not large.
- **Generating probabilistic labels**. Our minimax conditional entropy method can only generate deterministic labels (see Section \[sec:mle\]). In practice, probabilistic labels are usually more useful than deterministic labels. When the estimated label distribution for an item is close to uniform over several classes, we can either ask for more labels for the item from the crowd or forward the item to an external expert.
For addressing the issue of overfitting, we formulate our observation by replacing exact matching with approximate matching while penalizing large fluctuations. For generating probabilistic labels, we consider an entropy regularization over the unknown true label distribution. This is motivated by the analysis in Section \[sec:kl\].
Formally, we regularize our minimax conditional entropy method as follows. Let us denote the entropy of the true label distribution by $$H(Y) = -\sum_{j, c} Q(Y_j = c)\log Q(Y_j = c).$$ To estimate the true labels, we consider $$\label{eq:rmme}
\min_{Q} \max_{P} \ H(X|Y) - H(Y) - \frac{1}{\alpha}\Omega(\xi) - \frac{1}{\beta}\Psi(\zeta)
$$ subject to the relaxed worker and item constraints
\[eq:xizeta\] $$\begin{aligned}
& \sum_j \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = \xi_{i}(c, k), \ \forall i,s, \label{eq:rmctw}\\
& \sum_i \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = \zeta_{j}(c, k), \ \forall j,s, \label{eq:rmcti}\end{aligned}$$
plus the probability constraints in Equation . The regularization functions $\Omega$ and $\Psi$ are chosen as
\[eq:ref\] $$\begin{aligned}
& \Omega(\xi) = \frac{1}{2}\sum_{i}\sum_{c, k} \left[\xi_{i}(c, k)\right]^2, \label{eq:refunome}\\
& \Psi(\zeta) = \frac{1}{2} \sum_{j}\sum_{c, k} \left[\zeta_{j}(c, k)\right]^2. \label{eq:refunpsi}\end{aligned}$$
The new slack variables $\xi_{i}(c, k), \zeta_{j}(c, k)$ in Equation model the possible fluctuations. Note that these slack variables are not restricted to be positive. When there are a sufficiently large number of observations, the fluctuations should be approximately normally distributed, due to the central limit theorem. This observation motivates the choice of the regularization functions in to penalize large fluctuations. The entropy term $H(Y)$ in the objective function, which is introduced for generating probabilistic labels, can be regarded as penalizing a large deviation from the uniform distribution.
Substituting the labeling model from Equation into the Lagrangian of , we obtain the dual form (see Appendix \[app:dual2\]) $$\begin{aligned}
\max_{\sigma, \tau, Q} \quad & \sum_{j,c}Q(Y_j = c) \sum_i \log P(X_{ij} = x_{ij}|Y_j = c) + H(Y) - {\alpha} \Omega^*(\sigma) - {\beta}\Psi^*(\tau), \label{eq:dual2}
\end{aligned}$$ where $$\begin{aligned}
& \Omega^*(\sigma) = \frac{1}{2} \sum_{i}\sum_{c, k} \left[\sigma_{i}(c, k)\right]^2, \label{eq:cjomg}\\
& \Psi^*(\tau) = \frac{1}{2} \sum_{j}\sum_{c, k} \left[\tau_{j}(c, k)\right]^2. \label{eq:cjpsi}\end{aligned}$$ When $\alpha = 0$ and $ \beta = 0, $ the objective function in turns out to be a lower bound of the log marginal likelihood $$\begin{aligned}
& \log \bigg\{\prod_j \sum_{c} \prod_i P(X_{ij} = x_{ij}| Y_j = c)\bigg\} \nonumber \\
= & \log \bigg\{\prod_j \sum_{c} \frac{Q(Y_j = c)}{Q(Y_j = c)} \prod_i P(X_{ij} = x_{ij}| Y_j = c)\bigg\} \nonumber\\
\geq & \sum_{j,c}Q(Y_j = c) \sum_i \log P(X_{ij} = x_{ij}| Y_j = c) + H(Y). \nonumber\end{aligned}$$ The last step is based on Jensen’s inequality. Maximizing the marginal likelihood is more appropriate than maximizing the complete likelihood since only the observed data matters in our inference.
Finally, we introduce a variant of our regularized minimax conditional entropy. It is obtained by restricting the feasible region of the slack variables through $$\begin{aligned}
\sum_{c} \xi_i(c, c) = 0, \ \forall i. \label{eq:sumtozero}\end{aligned}$$ This is equivalent to $$\sum_{j,c} \left[\phi_{ij}(c, c) - \widehat{\phi}_{ij}(c, c)\right] = 0, \ \forall i.$$ It says that, the empirical count of the correct answers from each worker is equal to its expectation. According to the law of large numbers, this assumption is approximately correct when a worker has a sufficiently large number of correct answers. Note that this does not mean that the percentage of the correct answers from the worker has to be large. Let $K$ denote the class size. Under the additional constraints in Equation , the dual problem can still be expressed by except (see Appendix \[app:dual2\]) $$\Omega^*(\sigma) = \frac{1}{2} \sum_{i, c} \bigg(\left[\sigma_i(c, c) - \overline{\sigma_i(c, c)}\right]^2 + \sum_{k \neq c}\left[\sigma_i(c, k) - \overline{\sigma_i(c, k)}\right]^2 \bigg), \label{eq:comg} \\$$ where $$\overline{\sigma_i(c, c)} = \frac{1}{K}\sum_c\sigma_i(c, c), \quad \overline{\sigma_i(c, k)} = \frac{1}{K(K-1)} \sum_c \sum_{k \neq c} \sigma_i(c, k).$$ From our empirical evaluations, this variant is somewhat worse than its original version on most datasets. We include it here only for theoretic interest.
Objective Measurement Principle {#sec:objective}
===============================
In this section, we introduce a natural objective measurement principle, and show that the probabilistic labeling model in Equation is a consequence of this principle.
Intuitively, the objective measurement principle can be described as follows:
1. A comparison of labeling difficulty between two items should be independent of which particular workers were involved in the comparison; and it should also be independent of which other items might also be compared.
2. Symmetrically, a comparison of labeling ability between two workers should be independent of which particular items were involved in the comparison; and it should also be independent of which other workers might also be compared.
Next we mathematically define the objective measurement principle.
Assume that worker $i$ has labeled items $j$ and $j'$ in class $c.$ Denote by $E$ the event that one of these two items is labeled as $k, $ and the other is labeled as $c.$ Formally, $$E = \left\{{\ensuremath{\mathbb{I}}}(X_{ij} = k) + {\ensuremath{\mathbb{I}}}(X_{ij'} = k) = 1, \ {\ensuremath{\mathbb{I}}}(X_{ij} = c) + {\ensuremath{\mathbb{I}}}(X_{ij'} = c) = 1\right\}.$$ Denote by $A$ the event that item $j$ is labeled as $k$ and item $j'$ is labeled as $c.$ Formally, $$A = \left\{X_{ij} = k, \ X_{ij'} = c \right\}.$$ It is obvious that $A \subset E. $ Now we formulate the requirement (1) in the objective measurement principle as follows: $P(A|E)$ is independent of worker $i$. Note that $$\begin{aligned}
P(A|E) = \frac{P(X_{ij} = k|Y_j = c)P(X_{ij'} = c|Y_{j'} = c)}{P(X_{ij} = k|Y_j = c)P(X_{ij'} = c|Y_{j'} = c) + P(X_{ij} = c|Y_j = c)P(X_{ij'} = k|Y_{j'} = c)}.\end{aligned}$$ Hence, $P(A|E)$ is independent of worker $i$ if and only if $$\frac{P(X_{ij} = k|Y_j = c)P(X_{ij'} = c|Y_{j'} = c)}{P(X_{ij} = c|Y_j = c)P(X_{ij'} = k|Y_{j'} = c)}$$ is independent of worker $i$. In other words, given another arbitrary worker $i',$ we should have $$\frac{P(X_{ij} = k|Y_j = c)P(X_{ij'} = c|Y_{j'} = c)}{P(X_{ij} = c|Y_j = c)P(X_{ij'} = k|Y_{j'} = c)} = \frac{P(X_{i'j} = k|Y_j = c)P(X_{i'j'} = c|Y_{j'} = c)}{P(X_{i'j} = c|Y_j = c)P(X_{i'j'} = k|Y_{j'} = c)}.$$ Without loss of generality, we choose $i' = 0, \ j' = 0 $ as the fixed references. Then, $$\frac{P(X_{ij} = k|Y_j = c)}{P(X_{ij} = c|Y_j = c)} \propto \frac{P(X_{i0} = k|Y_{0} = c)}{P(X_{i0} = c|Y_{0} = c)}\frac{P(X_{0j} = k|Y_j = c)}{P(X_{0j} = c|Y_j = c)}.$$ By the fact that probabilities are nonnegative, we can write $$\begin{aligned}
{P(X_{i0} = k|Y_{0} = c)} = \exp[\sigma_i(c, k)], \quad {P(X_{0j} = k|Y_j = c)} = \exp[\tau_j(c, k)].
\end{aligned}$$ The probabilistic labeling model in Equation follows immediately. It is easy to verify that due to the symmetry between item difficulty and worker ability, we can instead start from formulating the requirement (2) in the objective measurement principle to achieve the same result. Hence, in this sense, the two requirements are actually redundant.
Extension to Ordinal Labels {#sec:ordinal}
===========================
In this section, we extend the minimax conditional entropy principle from multiclass to ordinal labels. Eliciting ordinal labels is important in tasks such as judging the relative quality of web search results or consumer products. Since ordinal labels are a special case of multiclass labels, the approach that we have developed in the previous sections can be used to aggregate ordinal labels. However, we observe that, in ordinal labeling, workers usually have an error pattern different from what we observe in multiclass labeling. We summarize our observation as the adjacency confusability assumption, and formulate it by introducing a different set of constraints for workers and items.
Adjacency Confusability
-----------------------
In ordinal labeling, workers usually have difficulty distinguishing between two adjacent ordinal classes whereas distinguishing between two classes which are far away from each other is much easier. We refer to this observation as adjacency confusability.
To illustrate this observation, let us consider the example of screening mammograms. A mammogram is an x-ray picture used to check for breast cancer in women. Radiologists often rate mammograms on a scale such as no cancer, benign cancer, possible malignancy, or malignancy. In screening mammograms, a radiologist may rate a mammogram which indicates possible malignancy as malignancy, but it is less likely that she rates a mammogram which indicates no cancer as malignancy.
Ordinal Minimax Conditional Entropy
-----------------------------------
In what follows, we construct a different set of worker and item constraints to encode adjacency confusability. The formulation leads to an ordinal labeling model parameterized with *structured* confusion matrices for workers and items.
We introduce two symbols $\Delta$ and $\nabla$ which take on arbitrary binary relations in $\{\ge, <\}. $ Ordinal labels are represented by consecutive integers, and the minima one is $0.$ To estimate the true ordinal labels, we consider $$\label{eq:omme}
\min_{Q} \max_{P} \ H(X|Y)$$ subject to the ordinal-based worker and item constraints
\[eq:omct\] $$\begin{aligned}
& \sum_{ c \Delta s} \sum_{k \nabla s } \sum_j \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = 0, \ \forall i,s \ge 1, \label{eq:omctw}\\
& \sum_{ c \Delta s} \sum_{k \nabla s } \sum_i \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = 0, \ \forall j,s \ge 1, \label{eq:omcti}\end{aligned}$$
for all $\Delta, \nabla \in \{\ge, <\}, $ and the probability constraints in . We exclude the case $s=0$ in which the constraints trivially hold.
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Let us explain the meaning of the constraints in Equation . To construct ordinal-based constraints, the first issue that we have to address is how to compare the observed label $x_{ij}$ and the true label $Y_j$ in an ordinal sense. For multiclass labels, as we have seen in Section \[sec:cat\], the label comparison problem is trivial: we only need to check whether they are equal or not. For ordinal labels, such a problem becomes tricky. Here, we propose an indirect comparison between two ordinal labels by comparing both to a *reference label* $s$ which varies through all possible values in a given ordinal label set (Figure \[fig:ord\]). Consequently, for every chosen reference label $s, $ we partition the Cartesian product of the label set into four disjoint regions $$\begin{aligned}
& \{(c, k)| c < s, k < s\}, \ \{(c, k)| c < s, k \ge s\}, \nonumber\\
& \{(c, k)| c \ge s, k < s\}, \ \{(c, k)| c \ge s, k \ge s\}. \nonumber\end{aligned}$$ A partition example is shown in Table 1 where the given label set is $\{0,1,2,3\}.$ Then, Equation defines a set of constraints for the workers by summing Equation over each region. Similarly, Equation defines a set of constraints for the items by summing Equation over each region.
From the discussion above, we can see that when there are more than two ordinal classes, the constraints in Equation are less restrictive than those in Equation . Consequently, as we see below, the labeling model resulted from Equation has fewer parameters. In the case in which there are only two ordinal classes, the sets of disjoint regions degenerate to pairs $(c,k)$ and, thus, the sets of constraints in Equations and are identical.
Next we explain why we construct the ordinal-based constraints in such a way. Let us write $$\begin{aligned}
\sum_{ c \Delta s} \sum_{k \nabla s } \sum_j \widehat{\phi}_{ij}(c, k) & = \sum_j \sum_{ c \Delta s} \sum_{k \nabla s } Q(Y_j = c) {\ensuremath{\mathbb{I}}}(x_{ij} = k) \nonumber \\
& = \sum_j \sum_{ c \Delta s} Q(Y_j = c) \sum_{k \nabla s } {\ensuremath{\mathbb{I}}}(x_{ij} = k)\nonumber\\
& = \sum_j Q(Y_j \Delta s) {\ensuremath{\mathbb{I}}}(x_{ij} \nabla s). \nonumber\end{aligned}$$ For example, when $\Delta = <$ and $\nabla = \ge, $ the above equation becomes $$\sum_j \sum_{ c < s} \sum_{k \ge s } \widehat{\phi}_{ij}(c, k) = \sum_j Q(Y_j < s) {\ensuremath{\mathbb{I}}}(x_{ij} \ge s).$$ This counts the items of which each belongs to a class less than $s$ but worker $i $ assigned a label larger or equal to $s.$
In general, for a comparison between an observed label and a reference label, there are two possible outcomes: the observed label is larger or equal to the reference label; or the observed label is smaller than the reference label. These are also the two possible outcomes for a comparison between a true label and a reference label. Putting these together, we have four possible outcomes in total. The constraints in Equation enforce expected counts of all the four kinds of outcomes in the worker dimension to match their empirical counterparts. Symmetrically, the constraints in Equation enforce expected counts of all the four kinds of outcomes in the item dimension to match their empirical counterparts.
0.13in 0.13in
The Lagrangian of the maximization problem in can be written as $$L = H(X|Y) + L_{\sigma} + L_{\tau} + L_{\lambda},
$$ with $$\begin{aligned}
L_{\sigma} = & \sum_{i, s} \sum_{\Delta, \nabla}\sigma^{{\Delta, \nabla}}_{is} \sum_{ c \Delta s} \sum_{k \nabla s } \sum_j \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right],\\
L_{\tau} = & \sum_{j, s} \sum_{\Delta, \nabla} \tau^{\Delta, \nabla}_{js} \sum_{ c \Delta s} \sum_{k \nabla s } \sum_i \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right], \\
L_{\lambda} = & \sum_{i,j, c} \lambda_{ijc}\bigg[\sum_k P(X_{ij} = k|Y_j = c) -1\bigg],
\end{aligned}$$ where $\sigma^{{\Delta, \nabla}}_{is}, \tau^{\Delta, \nabla}_{js}$ and $\lambda_{ijc}$ are the introduced Lagrange multipliers. By a procedure similar to that in Section \[sec:cat\], we obtain a probabilistic ordinal labeling model $$\label{eq:rating}
P(X_{ij} = k|Y_j = c) = \frac{1}{Z_{ij}} \exp[ \sigma_i (c, k) + \tau_j (c, k)], $$ where
\[eq:osigmatau\] $$\begin{aligned}
\sigma_{i}(c, k) = & \sum_{s \geq 1} \sum_{\Delta, \nabla} \sigma^{\Delta, \nabla}_{is}{\ensuremath{\mathbb{I}}}(c \Delta s, k \nabla s), \label{eq:osigma} \\
\tau_{j}(c, k) = & \sum_{s \geq 1} \sum_{\Delta, \nabla} \tau^{\Delta, \nabla}_{js}{\ensuremath{\mathbb{I}}}(c \Delta s, k \nabla s). \label{eq:otau}\end{aligned}$$
The ordinal labeling model in Equation is actually the same as the multiclass labeling model in Equation except the worker and item confusion matrices in Equation are now subtly structured through Equation . It is because of the structure that the ordinal labeling model has fewer parameters than the multiclass labeling model when there are more than two classes. In the case in which there are only two classes, the ordinal labeling model and the multiclass labeling model coincide as one would expect.
The regularized minimax conditional entropy for ordinal labels can be written as $$\label{eq:ormme}
\min_{Q} \max_{P} \ H(X|Y) - H(Y) - \frac{1}{\alpha}\Omega(\xi) - \frac{1}{\beta}\Psi(\zeta)
$$ subject to the relaxed worker and item constraints
\[eq:oxizeta\] $$\begin{aligned}
& \sum_{ c \Delta s} \sum_{k \nabla s } \sum_j \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = \xi_{is}^{\Delta, \nabla} , \forall i,s, \label{eq:ormctw}\\
& \sum_{ c \Delta s} \sum_{k \nabla s } \sum_i \left[\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right] = \zeta_{js}^{\Delta, \nabla}, \forall j,s, \label{eq:ormcti}\end{aligned}$$
for all $\Delta, \nabla \in \{\ge, <\}, $ and the probability constraints in Equation . When we choose $$\begin{aligned}
& \Omega(\xi) = \frac{1}{2}\sum_{i,s}\sum_{\Delta, \nabla} \left(\xi_{is}^{\Delta, \nabla}\right)^2, \\
& \Psi(\zeta) = \frac{1}{2}\sum_{j,s}\sum_{\Delta, \nabla} \left(\zeta_{js}^{\Delta, \nabla}\right)^2,\end{aligned}$$ the dual problem becomes $$\begin{aligned}
\max_{\sigma, \tau, Q} \quad & \sum_{j,c}Q(Y_j = c) \sum_i \log P(X_{ij} = x_{ij}|Y_j = c) + H(Y) - \alpha \Omega^*(\sigma) - \beta \Psi^*(\tau), \label{eq:odual}
\end{aligned}$$ where $$\begin{aligned}
& \Omega^*(\sigma) = \frac{1}{2}\sum_{i,s}\sum_{\Delta, \nabla} \left(\sigma_{is}^{\Delta, \nabla}\right)^2, \\
& \Psi^*(\tau) = \frac{1}{2}\sum_{j,s}\sum_{\Delta, \nabla} \left(\tau_{js}^{\Delta, \nabla}\right)^2.\end{aligned}$$
Ordinal Objective Measurement Principle
---------------------------------------
In this section, we adapt the objective measurement principle developed in Section \[sec:objective\] to ordinal labels.
Assume that worker $i$ has labeled items $j$ and $j'$ in class $c.$ For any class $k, $ we define two events. The first event is $$E = \left\{{\ensuremath{\mathbb{I}}}(X_{ij} = k) + {\ensuremath{\mathbb{I}}}(X_{ij'} = k) = 1, \ {\ensuremath{\mathbb{I}}}(X_{ij} = k+1) + {\ensuremath{\mathbb{I}}}(X_{ij'}= k+1) = 1\right\},$$ and the other event is $$A = \left\{X_{ij} = k, \ X_{ij'} = k+1 \right\}.$$ Note that $A \subset E. $ Now we formulate the objective measurement principle as follows: $P(A|E)$ is independent of worker $i$. Assume that the labels of the items are independent. Then, $P(A|E)$ can be written as $$\begin{aligned}
\frac{P(X_{ij} = k|Y_j = c)P(X_{ij'} = k+1|Y_{j'} = c)}{P(X_{ij} = k|Y_j = c)P(X_{ij'} = k+1|Y_{j'} = c) + P(X_{ij} = k+1|Y_j = c)P(X_{ij'} = k|Y_{j'} = c)}.\end{aligned}$$ Hence, $P(A|E)$ is independent of worker $i$ if and only if $$\frac{P(X_{ij} = k|Y_j = c)P(X_{ij'} = k+1|Y_{j'} = c)}{P(X_{ij} = k+1|Y_j = c)P(X_{ij'} = k|Y_{j'} = c)}$$ is independent of worker $i$. In other words, given another arbitrary worker $i',$ we should have $$\frac{P(X_{ij} = k|Y_j = c)P(X_{ij'} = k+1|Y_{j'} = c)}{P(X_{ij} = k+1|Y_j = c)P(X_{ij'} = k|Y_{j'} = c)} = \frac{P(X_{i'j} = k|Y_j = c)P(X_{i'j'} = k+1|Y_{j'} = c)}{P(X_{i'j} = k+1|Y_j = c)P(X_{i'j'} = k|Y_{j'} = c)}.$$ To introduce adjacency confusability, we further assume that, for any two classes $c, c' \ge k+1$ (or $c, c' < k+1$), $$\frac{P(X_{ij} = k|Y_j = c)P(X_{ij'} = k+1|Y_{j'} = c)}{P(X_{ij} = k+1|Y_j = c)P(X_{ij'} = k|Y_{j'} = c)} = \frac{P(X_{ij} = k|Y_j = c')P(X_{ij'} = k+1|Y_{j'} = c')}{P(X_{ij} = k+1|Y_j = c')P(X_{ij'} = k|Y_{j'} = c')}.$$ Then, by a procedure similar to that in Section \[sec:objective\], we reach the probabilistic ordinal labeling model described by Equation and .
Implementation {#sec:implementation}
==============
In this section, we present a simple while efficient coordinate ascent method to solve the minimax program through its dual form and also a practical procedure for model selection.
Coordinate Ascent
-----------------
The dual problem of regularized minimax conditional entropy for either multiclass or ordinal labels is nonconvex. A stationary point can be obtained via coordinate ascent (Algorithm \[alg:em\]), which is essentially Expectation-Maximization (EM) [@DemLaiRub77; @NeaHin98]. We first initialize the label estimate via aggregating votes in Equation . Then, in each iteration step, given the current estimate of the labels, update the estimate of the confusion matrices of the workers and items by solving the optimization problem in ; and, given the current estimate of the confusion matrices of worker and item, update the estimate of the labels through the closed-form formula in , which is identical to applying the Bayes’ rule with a uniform prior. The optimization problem in is strongly convex and smooth. Many algorithms can be applied here [@YuNest2004]. In our experiments, we simply use gradient ascent. Denote by $F$ the objective function in . For multiclass labels, the gradients are computed as $$\begin{aligned}
\frac{\partial F}{ \partial \sigma_i (c, k)} &= \sum_{j} Q(Y_j = c)\left[{\ensuremath{\mathbb{I}}}(x_{ij} = k) - P(X_{ij} = k|Y_j = c)\right] - \alpha \sigma_i (c, k), \nonumber \\
\frac{\partial F}{ \partial \tau_j (c, k)} &= \sum_{i} Q(Y_j = c)\left[{\ensuremath{\mathbb{I}}}(x_{ij} = k) - P(X_{ij} = k|Y_j = c)\right] - \beta \tau_j (c, k). \nonumber\end{aligned}$$ For ordinal labels, the gradients are computed as $$\begin{aligned}
\frac{\partial F}{ \partial \sigma_{is}^{\Delta, \nabla}} &= \sum_{c, k} {\ensuremath{\mathbb{I}}}(c \Delta s, k \nabla s)\sum_{j} Q(Y_j = c)\left[{\ensuremath{\mathbb{I}}}(x_{ij} = k) - P(X_{ij} = k|Y_j = c)\right] - \alpha \sigma_{is}^{\Delta, \nabla}, \nonumber \\
\frac{\partial F}{ \partial \tau_{js}^{\Delta, \nabla}} &= \sum_{c, k} {\ensuremath{\mathbb{I}}}(c \Delta s, k \nabla s)\sum_{i} Q(Y_j = c)\left[{\ensuremath{\mathbb{I}}}(x_{ij} = k) - P(X_{ij} = k|Y_j = c)\right] - \beta \tau_{js}^{\Delta, \nabla}. \nonumber\end{aligned}$$ It is worth pointing out that it is unnecessary to obtain the exact optimum at this intermediate step. We have observed that in practice, several gradient ascent steps here suffice for reaching a final good solution.
$\{x_{ij}\}, \alpha, \beta $\
$$\label{ag:mv}
Q(Y_j = c) ~\propto~ \sum_{i} {\ensuremath{\mathbb{I}}}(x_{ij} = c)$$
$$\begin{aligned}
& \{\sigma, \tau\} ~=~ \arg \max_{\sigma, \tau} \ \sum_{i, j, c} Q(Y_j = c)\log P(X_{ij} = x_{ij}|Y_j = c) - \alpha \Omega^*(\sigma) - \beta \Psi^*(\tau) \label{ag:mstep}\\[1.0ex]
& Q(Y_j = c)~\propto~\prod_i P(X_{ij} = x_{ij}|Y_j = c) \label{ag:estep}\end{aligned}$$
$Q$
Model Selection {#sec:cv}
---------------
The regularization parameters $\alpha$ and $\beta$ can be chosen as follows. If the true labels of a subset of items are known—such subsets are usually referred to as validation sets—we may choose the regularization parameters such that those known true labels can be best predicted. Otherwise, we suggest to choose the regularization parameters via $k$-fold likelihood-based cross-validation. Specifically, we first randomly partition the crowd labels into $k$ equal-size subsets, and define a finite set of possible choices for the regularization parameters. Then, for each possible choice of the regularization parameters,
1. Leave out one subset and use the remaining $k-1$ subsets to estimate the confusion matrices of the workers and items;
2. Plug the estimate into the probabilistic labeling model to compute the likelihood of the left-out subset;
3. Repeat the above two steps till each subset is left out once and only once;
4. Average the likelihoods that we have computed.
After going through all the possible choices for the regularization parameters, we choose the one which results in the largest average likelihood to run our algorithm over the full dataset. The cross-validation parameter $k$ is typically set to 5 or 10.
To simplify the model selection process, we suggest to choose $$\label{eq:gamma}
\begin{aligned}
&\alpha = \gamma \times (\text{number of classes})^2, \\
&\beta = \frac{\text{number of labels per worker}}{\text{number of labels per item}} \times \alpha.
\end{aligned}$$ In our experiments, we select $\gamma$ from $\{2^{-2}, 2^{-1}, 2^0, 2^1, 2^2\}.$ In our limited empirical studies, larger candidate sets for $\gamma$ did not give more gains. Two empirical observations motivate us to consider using the square of the number of classes in Equation . First, the square of the number of classes has the same magnitude as the number of parameters in a confusion matrix. Second, the label noise dramatically increases when the number of classes increases, requiring a super linearly scaled regularization.
Related Work {#sec:related}
============
In this section, we review some existing work that are closely related to our work.
**Dawid-Skene Model**. Let $K$ denote the number of classes. @DawSke79 propose a generative model in which the ability of worker $i$ is characterized by a $K \times K$ probabilistic confusion matrix $[p_i(c, k)]$ in which the diagonal element $p_i(c, c)$ represents the probability that worker $i$ correctly labels an arbitrary item in class $c$, and the off-diagonal element $p_i(c, k)$ represents the probability that worker $i$ mislabels an arbitrary item in class $c$ as class $k.$ Our probabilistic labeling model in Equation is reduced to the Dawid-Skene model when the item difficult terms $\tau_j(c, k)$ in our model disappear since we can then reparameterize $$p_i(c, k) = \frac{\exp[\sigma_i(c, k)]}{\sum_{k'} \exp[\sigma_i(c, k')]}.$$ In this sense, our model generalizes the Dawid-Skene model to incorporate item difficulty. To jointly estimate the workers’ abilities and the true labels in the Dawid-Skene model, in general, the marginal likelihood is maximized using the EM algorithm.
For binary labeling task, the probabilistic confusion matrix in the Dawid-Skene model can be written as $$\begin{pmatrix}
p_i & 1 - p_i\\
1 - q_i & q_i
\end{pmatrix},$$ where $p_i$ is the accuracy of worker $i$ in the first class, and $q_i$ the accuracy in the second class. Usually, this special case of the Dawid-Skene model is also referred to as the two-coin model [@RayYuZha10; @LiuPenIhl12; @CheLinZho13]. One may simplify the two-coin model by assuming $p_i = q_i$ [@GhoKalMca11; @KarOhSha11; @DalDasKum2013]. This simplification is accordingly referred to as the one-coin model.
@KarOhSha11 propose an inference algorithm under the one-coin model, and show that their algorithm achieves the minimax rate when the accuracy of every worker is bounded away from $0$ and $1, $ that is, with some fixed number $\epsilon > 0, $ $\epsilon < p_i < 1 - \epsilon.$ @LiuPenIhl12 show that the algorithm proposed by @KarOhSha11 is essentially a belief propagation update with the Haldane prior which assumes that each worker is either a hammer ($p_i = 1$) or adversary ($p_i = 0$) with equal probability.
@GaoZho14 show that under the one-coin model, the global optimum of maximum likelihood achieves the minimax rate. A projected EM algorithm is suggested and shown to achieve nearly the same rate as that of global optimum. @ZhaCheZhoJor14 show that the EM algorithm for the general Dawid-Skene model can achieve the minimax rate up to a logarithmic factor when it is initialized by spectral methods [@AnaGeHsu12] and the accuracy of every worker is bounded away from $0$ and $1.$
@RayYuZha10 extend the Dawid-Skene model by imposing a beta prior over the worker confusion matrices. Moreover, they jointly learn the classifier and the true labels by assuming that the true labels are generated by a logistic model. @LiuPenIhl12 develop full Bayesian inference via variational methods including belief propagation and mean field.
**Rasch model [@Ras61; @Ras68].** In educational tests, the Rasch model illustrates the response of each examinee of a given ability to each item in a test. In the model, the probability of a correct response is modeled as a logistic function of the difference between the person and item parameter which are locations on a continuous latent trait. Person parameters represent the ability of examinees while item parameters represent the difficulty of items.
Let $X_{ij} \in \{0, 1\}$ be a dichotomous random variable where $X_{ij} = 1$ denotes a correct response and $X_{ij} = 0$ an incorrect response to a given assessment item. Mathematically, the Rasch model is given by $$P(X_{ij} = 1) = \frac{\exp(\beta_i - \delta_j)}{1 + \exp(\beta_i - \delta_j)},$$ where $\beta_i$ is the ability of examinee $i$ and $\delta_i$ the difficulty of item $j.$ The larger an examinee’s ability relative to the difficulty of an item, the larger the probability of a correct response on that item. When the examinee’s ability on the latent trait is equal to the difficulty of the item, the probability of a correct response is $1/2.$
The Rasch model is a special item response theory (IRT) model [@LorNov68]. However, unlike other IRT models, the Rasch model satisfies the objective measurement principle pioneered by Rasch. Our work generalizes both the Rasch model and the objective measurement principle to multiclass labeling tasks. In addition, unlike the Rasch model, in our scenario, the true answers are unknown and have to be estimated.
**Polytomous Rasch model.** The Rasch model has been adapted to the applications in which responses to items are scored with successive integers such as rating scales. Let $X_{ij} = \{0, 1, \cdots, m\}.$ @AndRat1978 suggests $$P(X_{ij} = k) = \frac{\exp \sum_{s=0}^k[\beta_i - (\delta_j - \tau_s)]}{\sum_{k'=0}^m \exp \sum_{s=0}^{k'}[\beta_i - (\delta_j - \tau_s)]},$$ where $\beta_i$ is the location of person $i$ on a latent continuum, $\delta_j$ the difficulty of item $j$ on the same continuum, and $\tau_s$ the $s$-th threshold location of the rating scale which is in common to all the items. This model is usually referred to as the Rasch rating scale model. Later, the Rasch partial credit model developed by @MasARas1982 generalizes the Rasch rating scale model into $$P(X_{ij} = k) = \frac{\exp \sum_{s=0}^k(\beta_i - \tau_{js})}{\sum_{k'=0}^m \exp \sum_{s=0}^{k'}(\beta_i - \tau_{js})},$$ where $\tau_{js}$ is the $s$-th threshold location of item $i$ on a latent continuum. When $\tau_{js}$ can be decomposed as $\tau_{js} = \delta_j - \tau_s, $ these two models coincide. @uebersax1993latent and @MineiroOrdered11 apply the polytomous form of the Rasch model with minor changes to aggregate ordinal labels from a crowd.
**Probabilistic matrix factorization.** Let $X_{ij}$ be the label given by worker $i$ to item $j. $ Let $Y_j$ be the true label of item $j. $ @WRWBM09 model the labeling process by revising the Rasch model into $$P(X_{ij} = Y_j ) = \frac{1}{1 + \exp\left(-\dfrac{\beta_i}{\delta_j}\right)},$$ and refer to their model as GLAD (Generative Model of Labels, Abilities, and Difficulties). It is easy to see that GLAD violates the principle of invariant comparison. By using the per-worker confusion matrix in the Dawid-Skene model, @MineiroConfusion11 generalizes GLAD to multiclass labeling as $$P(X_{ij} = k|Y_j = c) \propto \exp\left[ \dfrac{\beta_{i}(c, k)}{\delta_j}\right].$$ [@WBBP10] parameterize workers and items with vectors and suggest $$P(X_{ij} = Y_i) = \Phi({w}_i^\top {z}_j - b_j),$$ where $\Phi(\cdot)$ is the cumulative standardized normal distribution, ${w}_i\in {\ensuremath{\mathbb{R}}}^d$ the unobserved worker parameter, and ${z}_j \in {\ensuremath{\mathbb{R}}}^d, b_j \in {\ensuremath{\mathbb{R}}}$ the unobserved item parameter. GLAD can be roughly thought of as a special case of this model with the dimension $d = 1 $ and $b_j = 0.$
**Other related work.** For other probabilistic modelling of crowdsourcing, we refer the readers to [@BacMinGui12; @TiaZhu12; @DaiLin13; @VenGui14]. For online decision making in crowdsourcing, we refer the readers to [@ShePro08; @abraham2013adaptive; @CheLinZho13; @SinKra13; @HoSliWor14; @AnaGoeNik14]. Regularized maximum entropy is studied in [@CheRos00; @LebLaf01; @KazTsu03; @AltSmo06; @DudPhiSch07]. @zhwumu97 propose a minimax entropy method for feature binding and selection, and apply it to texture modeling and obtain a new class of Markov random field models. @shaZhoDou14 propose a multiplicative payment mechanism to incentivize crowdsourcing workers to answer a question when they are sure and skip when they are not sure. They obtain extremely high quality crowdsourced data by using their mechanism.
Experiments {#sec:exp}
===========
In this section, we report empirical results of our method and some existing methods discussed in Section \[sec:related\]. Two error metrics are considered. One is the classification error rate for binary or multiclass data, and the other is the mean square error for ordinal data.
Datasets
--------
All datasets that we use are from real crowdsourcing tasks and publicly available.[^7] The details are as follows:
- **Bluebirds [@WBBP10]**. This dataset contains a set of 108 images which are labeled as indigo bunting or blue grosbeak by 39 crowdsourcing workers. Every worker labeled every image. The average error rate of the workers is $36.44\%, $ compared to the error rate of random guessing at $50\%.$
- **Price [@LiuSteIhl13]**. This dataset consists of 80 household items collected from stores such as Amazon and Costco. The prices of the products are estimated by 155 undergraduate students from UC Irvine. Seven price bins are created in this data collection: \$0$-$\$50, \$51$-$\$100, \$101$-$\$250, \$251$-$\$500, \$501$-$\$1000, \$1001$-$\$2000, and \$2001$-$\$5000. For each product, a student has to to decide which bin its price falls in. The average error rate of the students is $69.47\%$, compared to the error rate of random guessing at $85.71\%$. It may not be surprising that this dataset is systematically biased: all the students tend to underestimate the prices of the products.
- **RTE [@SnoCon08]**. For each crowdsourced question, the worker is presented with two sentences and asked to check if the second hypothesis sentence can be inferred from the first. This dataset contains 800 sentence pairs and 164 workers. Each sentence pair has 10 annotations. The average error rate of the workers is $15.87\%$, compared to the error rate of random guessing at $50\%$.
- **Temp [@SnoCon08]**. For each crowdsourced question, the worker is presented with a pair of verb events and asked to check if the event described by the first verb occurs before or after the second. This dataset contains 462 event pairs and 76 workers. Each event pair has 10 annotations. The average error rate of the workers is $16.30\%$, compared to the error rate of random guessing at $50\%$.
- **Age [@HanOttLiuJai14]**. Amazon mechanical turkers are asked to estimate the age of a person in a face image. This dataset contains 1002 images and 165 workers. Each image has 10 age estimates. Those estimates are integers not more than 100. We put them into 7 bins: \[1, 9\], \[10, 19\], \[20, 29\], \[30, 39\], \[40, 49\], \[50, 59\], \[60, 100\]. With respect to this partition, the average error rate of the workers is $44.64\%$, compared to the error rate of random guessing at $85.71\%$.
- **Web search [@zhoplaby12]**. This dataset contains 2665 query-URL pairs and 177 workers. Give a query-URL pair, a worker is required to provide a rating to measure how the URL is relevant to the query. The rating scale is 5-level: perfect, excellent, good, fair, or bad. On average, each pair was labeled by around 6 different workers, and each worker labeled around 90 pairs. More than 10 workers labeled only one query-URL pair. The ground truth labels used for evaluation are obtained via a consensus among a group of 9 search experts. The average error rate of the workers is $62.95\%, $ compared to the error rate of random guessing at $80\%.$
- **Web spam**. This dataset is provided by Microsoft web spam team. It contains 149 web pages and 18 workers. The workers are required to identify which web pages are spam. In average, each web page is labeled by around 13 workers. The ground truth labels used for evaluation are provided by web spam experts. The average error rate of the workers is $16.30\%$, compared to the error rate of random guessing at $50\%$.
Table \[table:summary\] shows a summary of these datasets.
\# classes \# items \# workers \# worker labels
------------ ------------ ---------- ------------ ------------------
Bluebirds $2$ $108$ $39$ $4212$
Price $7$ $80$ $155$ $12400$
RTE $2$ $800$ $164$ $8000$
Temp $2$ $462$ $76$ $4620$
Age $7$ $1002$ $165$ $10020$
Web search $5$ $2665$ $177$ $15567$
Web spam $2$ $149$ $18$ $1901$
: Summary of the real crowdsourcing datasets used in our experiments. []{data-label="table:summary"}
MV DS-EM DS-MF GLAD MMCE(M)
------------ --------- ------------------ ------------------ ----------------- ------------------
Bluebirds $24.07$ $10.19$ $10.19$ $12.04$ $\mathbf{8.33}$
Price $67.50$ $\mathbf{65.00}$ $67.50$ $68.75$ $67.50$
RTE $10.31$ $7.25$ $\mathbf{6.63}$ $7.00$ $7.50$
Temp $6.39$ $5.84$ $5.84$ $\mathbf{5.63}$ $\mathbf{5.63}$
Age $34.88$ $39.62$ $36.33$ $35.73$ $\mathbf{31.14}$
Web search $26.93$ $16.92$ $18.24$ $19.30$ $\mathbf{11.12}$
Web spam $19.80$ $13.42$ $\mathbf{12.75}$ $18.12$ $\mathbf{12.75}$
: Error rates $(\text{in} \ \%)$ of various methods on real datasets. []{data-label="table:errorrates"}
MV DS-EM DS-MF LTA MMCE(M) MMCE(O)
------------ --------- --------- --------- --------- ------------------ ------------------
Price $1.605$ $1.517$ $1.487$ $1.504$ $1.643$ $\mathbf{1.466}$
Age $0.730$ $0.852$ $0.739$ $0.696$ $\mathbf{0.605}$ $0.794$
Web search $0.930$ $0.539$ $0.559$ $0.481$ $0.419$ $\mathbf{0.384}$
: Mean square errors of various methods on ordinal datasets. []{data-label="table:meansqure"}
Probability Bin $(0, 0.5)$ $(0.5, 0.6)$ $(0.6, 0.7)$ $(0.7, 0.8)$ $(0.8, 0.9)$ $(0.9, 1)$
------------------- ------------ -------------- -------------- -------------- -------------- ------------
\# items $173$ $291$ $292$ $313$ $406$ $1178$
Error rate $0.416$ $0.381$ $0.199$ $0.080$ $0.020$ $0.001$
Mean square error $0.832$ $0.423$ $0.250$ $0.118$ $0.035$ $0.001$
: Positive correlation between probabilistic labels and errors. The results are from the regularized ordinal minimax conditional entropy method on the web dataset.[]{data-label="table:probcor"}
Methods
-------
We evaluate the following methods in our experiments:
- **Majority voting (MV)**. It is perhaps the simplest baseline.
- **Dawid-Skene model $+$ EM (DS-EM)**. Under the generative model by [@DawSke79], this method jointly estimates workers’ parameters and true labels by maximizing the likelihood of observed labels with the EM algorithm.
- **Dawid-Skene model $+$ mean field (DS-MF)**. This method performs variational Bayesian inference using the mean field (MF) algorithm [@LiuPenIhl12]. It assumes a Dirichlet prior parameterized by a vector $\alpha_k$ on the $k$-th row of the worker confusion matrix in the Dawid-Skene model with $\alpha_{k,k} = c_1$, and $\alpha_{k,l}= c_2$ for all $l\neq k$. The hyperparameters $\{c_1, c_2\}$ are selected by maximizing the marginal likelihood calculated by MF, and searched in a $10\times 10$ grid defined by $c_1 = c_2 \times \{10^{0}, 10^{0.1}, 10^{0.2}, \cdots, 10^1\}$ and $c_2 = \{10^{-1}, 10^{-0.8}, \cdots, 10^0, \cdots, 10^{0.8}, 10^1\}$.
- **GLAD**. We use the multiclass version of GLAD proposed by [@MineiroConfusion11] and also his open source implementation.
- **Latent trait analysis (LTA)**. It is a variant of the polytomous Rasch model proposed by [@MineiroOrdered11] with an open source implementation.
- **Regularized minimax conditional entropy for multiclass labels (MMCE(M))**. It is implemented with the Euclidian norm based regularization.
- **Regularized minimax conditional entropy for ordinal labels (MMCE(O))**. It is implemented with the Euclidian norm based regularization.
The regularization parameters in MMCE are chose through the cross-validation procedure described in Section \[sec:cv\].[^8]
Results
-------
Table \[table:errorrates\] shows the error rates of various methods on real crowdsourcing datasets. Our multiclass minimax conditional entropy method outperforms compared methods on most datasets. Table \[table:meansqure\] shows the mean square errors of various methods on three ordinal datasets. Our ordinal minimax conditional entropy method performs best on the price and web search datasets but performs poorly on the age dataset. Table \[table:probcor\] shows the correlation between probabilistic labels and errors for our ordinal minimax conditional entropy method on the web dataset. From the results, the labels estimated with larger probabilities are more likely to be correct. We observed similar behavior for our multiclass minimax conditional entropy method. We also evaluated our method with the regularization in Equation and observed that this variant somewhat hurts performance on most datasets.
Conclusion {#sec:conclusion}
==========
We have developed a minimax conditional entropy principle for aggregating noisy labels from crowdsourcing workers. Our formulation involves two probabilistic distributions. One is the distribution of the true labels of the items, and the other is the distribution under which the workers generate their labels for the items. Both the distributions are unknown. We jointly infer them by first maximizing the entropy of the observed labels of the workers conditioned on the true labels of the items over the distribution of generating workers’ labels, and then minimizing the maximum entropy over the distribution of the true labels of the items. Empirical results on real crowdsourcing datasets validate our approach.
We have considered aggregating multiclass and ordinal labels via minimax conditional entropy. The framework is general and should be extensible to many other labeling tasks in which the labels are structured in different ways, such as protein folding [@khatib2011algorithm], machine translation [@zaidan2011crowdsourcing], hierarchical classification [@KolSah], and speech captioning [@MurMilLas2013]. To achieve the extension, the constraints for workers and items need to be customized specific to each domain, and this probably results in differently structured confusion matrices.
Acknowledgements {#acknowledgements .unnumbered}
----------------
[We would like to thank Sumit Basu and Yi Mao for their early contribution to this work, Daniel Hsu, Xi Chen, Chris Burges for helpful discussions, and Gabriella Kazai for providing the web search dataset. ]{}
Dual Form of Minimax Conditional Entropy {#app:dual1}
========================================
To to derive the dual of minimax conditional entropy, we substitute the probabilistic model in Equation into the Lagrangian and obtain $$\begin{aligned}
L = & -\sum_{i, j, c} Q(Y_j = c)\sum_k P(X_{ij} = k| Y_j = c)\log \bigg\{\frac{1}{Z_{ij}}\exp[\sigma_i(c, k) + \tau_{j}(c, k)]\bigg\} \\
& + \sum_{i, c, k} \sigma_i(c, k) \sum_j Q(Y_j = c)\bigg[ P(X_{ij} = k |Y_j = c) - {\ensuremath{\mathbb{I}}}(x_{ij} = k) \bigg]\\
& + \sum_{j, c, k} \tau_j(c, k) \sum_i Q(Y_j = c) \bigg[P(X_{ij} = k |Y_j = c)- {\ensuremath{\mathbb{I}}}(x_{ij} = k) \bigg]\\
& + \sum_{i,j, c} \lambda_{ijc}\bigg[\sum_{k} P(X_{ij} = k|Y_j = c) -1 \bigg]\\
= & -\sum_{i, j, c} Q(Y_j = c)\sum_k P(X_{ij} = k| Y_j = c)[\sigma_i(c, k) + \tau_{j}(c, k)] + \sum_{i, j} \log Z_{ij}\\
& + \sum_{i, c, k} \sigma_i(c, k) \sum_j Q(Y_j = c)\bigg[ P(X_{ij} = k |Y_j = c) - {\ensuremath{\mathbb{I}}}(x_{ij} = k) \bigg]\\
& + \sum_{j, c, k} \tau_j(c, k) \sum_i Q(Y_j = c) \bigg[P(X_{ij} = k |Y_j = c)- {\ensuremath{\mathbb{I}}}(x_{ij} = k) \bigg]\\
= & -\sum_{i, j, c} Q(Y_j = c)\bigg(\sum_k {\ensuremath{\mathbb{I}}}(x_{ij} = k)[\sigma_i(c, k) + \tau_{j}(c, k)] - \log Z_{ij}\bigg)\\
= & -\sum_{i, j, c} Q(Y_j = c)\log P(X_{ij} = x_{ij}|Y_j = c).\end{aligned}$$
Proof of Theorem \[eq:minkl\] {#sec:proofkl}
=============================
Let us first check $\sum_{X, Y}Q(X,Y) \log {Q(X,Y)}. $ By definition, $$\sum_X Q(X) \log Q(X) = 0 , \quad Q(X|Y) = Q(X).$$ Hence, we have $$\begin{aligned}
\sum_{X, Y}Q(X,Y) \log {Q(X,Y)} & = \sum_{X, Y}[Q(X|Y)Q(Y)] \log [Q(X|Y)Q(Y)] \nonumber\\
& = \sum_{X, Y}[Q(X)Q(Y)] \log [Q(X)Q(Y)] \nonumber\\
& = \sum_X Q(X) \log Q(X) + \sum_Y Q(Y)\log Q(Y) \nonumber \\
& = \sum_Y Q(Y)\log Q(Y). \nonumber\end{aligned}$$ Next we check $\sum_{X, Y}Q(X,Y) \log {P(X,Y)}. $ Write $$\begin{aligned}
\sum_{X, Y} Q(X, Y) \log P(X, Y) = \sum_{X, Y} Q(X, Y) \log P(X|Y)+ \sum_{X, Y} Q(X, Y) P(Y).
\end{aligned}$$ Since $P$ is a uniform distribution over $Y,$ $P(Y)$ is a constant. Thus, $$\begin{aligned}
\sum_{X, Y} Q(X, Y) \log P(Y) & = \log P(Y) \sum_{X, Y} Q(X, Y) = \log P(Y), \nonumber\end{aligned}$$ which is still a constant. By Equation , we have $$\begin{aligned}
& \sum_{X, Y} Q(X, Y) \log P(X|Y) = \sum_{i,j,c, k} Q(X_{ij} = k, Y_j = c) \log P(X_{ij} = k| Y_j = c) \\
= & \sum_{i,j,c, k} Q(X_{ij} = k, Y_j = c) \log \bigg\{\frac{1}{Z_{ij}} \exp[\sigma_i(c, k) + \tau_{j}(c, k)]\bigg\} \\
= & \sum_{i,j,c, k} Q(X_{ij} = k, Y_j = c) \left[\sigma_i (c, k) + \tau_j (c, k)- \log Z_{ij}\right].
\end{aligned}$$ By Equation , we have $$\begin{aligned}
\sum_{i,j,c, k} Q(X_{ij} = k, Y_j = c) \sigma_i (c, k) = & \sum_{i,c,k} \sigma_i (c, k) \sum_{j}{\ensuremath{\mathbb{I}}}(X_{ij} = k) Q( Y_j = c) \\
= & \sum_{i,c,k} \sigma_i (c, k) \sum_j P(X_{ij} = k |Y_j = c) Q(Y_j = c) . \\
\end{aligned}$$ Similarly, by Equation , $$\sum_{i,j,c, k} Q(X_{ij} = k, Y_j = c) \tau_{j}(c, k) = \sum_{j,c,k} \tau_j (c, k) \sum_i P(X_{ij} = k|Y_j = c)Q(Y_j = c).$$ In addition, since $Z_{ij}$ does not depend on $k,$ $$\begin{aligned}
\sum_{i,j,c, k}Q(X_{ij} = k, Y_j = c) \log Z_{ij} = & \sum_{i,j} \sum_c \log Z_{ij} \sum_{k}Q(X_{ij} = k , Y_j = c) \\
= & \sum_{i,j}\sum_{c} Q(Y_j=c)\log Z_{ij} \\
= & \sum_{i,j}\sum_{ c} Q(Y_j=c)\sum_{k} {P(X_{ij} = k|Y_j = c)}\log Z_{ij} .
\end{aligned}$$ Putting all the pieces together, we have $$\begin{aligned}
D_{\operatorname*{KL}}(Q\parallel P) = & -\sum_{j, c} Q(Y_j = c) \sum_{i,k} P(X_{ij} = k|Y_j = c)[\sigma_i (k,c) + \tau_j (c, k) - \log Z_{ij}]\\
& + \sum_{Y} Q(Y)\log Q(Y)- \log P(Y)\\
= & - \sum_{j, c} Q(Y_j = c) \sum_{i, k} P(X_{ij} = k|Y_j = c)\log P(X_{ij} = k|Y_j = c) \\
& + \sum_{Y} Q(Y)\log Q(Y) - \log P(Y). \\
\end{aligned}$$ Note that, when the true labels are deterministic, $$\sum_{Y} Q(Y)\log Q(Y) = 0.$$ So, $$D_{\operatorname*{KL}}(Q\parallel P) = -\sum_{j, c} Q(Y_j = c) \sum_{i, k} P(X_{ij} = k|Y_j = c)\log P(X_{ij} = k|Y_j = c)- \log P(Y) .$$ This concludes the proof.
Dual Form of Regularized Minimax Conditional Entropy {#app:dual2}
====================================================
We derive the dual problem of regularized maximum conditional entropy with the sum-to-zero constraints in Equation . The dual derivation without the additional constraints can be obtained in a similar procedure. Let us write the Lagrangian as $$\label{eq:plag}
L = H(X|Y) - H(Y)- \frac{1}{\alpha}\Omega(\xi) - \frac{1}{\beta}\Psi(\zeta) + L_\sigma + L_\tau + L_{\lambda} + L _{\mu},$$ in which $$\begin{aligned}
& L_{\sigma} = \sum_{i, c, k} \sigma_i(c, k)\bigg[\xi_{i}(c, k) - \sum_{j}\left(\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right)\bigg],\\
& L_{\tau} = \sum_{j, c, k} \tau_j(c, k)\bigg[\zeta_{j}(c, k) - \sum_i \left(\phi_{ij}(c, k) - \widehat{\phi}_{ij}(c, k)\right)\bigg],\\
& L_{\lambda} = \sum_{i,j, c} \lambda_{ijc}\bigg[\sum_{k,c} P(X_{ij} = k|Y_j = c) -1 \bigg], \\
& L _{\mu} = \sum_{i}\mu_i \sum_{c} \xi_i(c, c).\end{aligned}$$ By the KKT conditions, maximizing $L$ with respect to $P$ results in $$\begin{aligned}
\frac{\partial L }{\partial P(X_{ij} = k|Y_j = c)} = -\log P(X_{ij} = k|Y_j = c)-1 + \lambda_{ijc} + \sigma_i(c, k) + \tau_{j}(c, k) = 0.\end{aligned}$$ As showed in Section \[sec:cat\], this leads to the probabilistic model in Equation . Similarly, maximizing $L$ with respect to $\xi$ results in $$\begin{aligned}
& \frac{\partial L}{\partial \xi_i (c, k)} = \sigma_i (c, k) - \frac{1}{\alpha} \xi_i (c, k) = 0, \ \forall c, k \neq c, \\
& \frac{\partial L}{\partial \xi_i (c, k)} = \sigma_i (c, c) - \frac{1}{\alpha} \xi_i (c, c) + \mu_i = 0, \ \forall c.\end{aligned}$$ So we have $$\begin{aligned}
& \xi_i (c, k) = \alpha \sigma_i (c, k), \ \forall c, k \neq c, \label{eq:pxi1}\\
& \xi_i (c, c) = \alpha [\sigma_i (c, c) + \mu_i], \ \forall c. \label{eq:pxi2}\end{aligned}$$ Moreover, maximizing $L$ with respect to $\zeta$ results in $$\begin{aligned}
\frac{\partial L}{\partial \zeta_i (c, k)} = \tau_i (c, k) - \frac{1}{\beta} \zeta_i (c, k) = 0, \ \forall c, k.\end{aligned}$$ Hence, $$\zeta_i (c, k) = \beta \tau_i (c, k), \ \forall c, k. \label{eq:pzeta}$$ Substituting , , , and into the Lagrangian , we have $$\begin{aligned}
L = & -\sum_{i, j, c} Q(Y_j = c) \log \bigg\{\frac{1}{Z_{ij}}\exp[\sigma_i(c, x_{ij}) + \tau_{j}(c, x_{ij})]\bigg\} - H(Y) \\
& - \frac{\alpha}{2} \sum_{i, c}\bigg\{\sum_{k \neq c} [\sigma_i(c, k)]^2 + [\sigma_i(c, c) + \mu_i]^2\bigg\}- \frac{\beta}{2} \sum_{i, c, k} [\tau_j(c, k)]^2 \\
& + \alpha \sum_{i, c} \bigg\{\sum_{k \neq c} [\sigma_i(c, k)]^2 + \sigma_i(c, c) [\sigma_i(c, c) + \mu_i]\bigg\} + \beta \sum_{i, c, k} [\tau_j(c, k)]^2 \\
& + \alpha \sum_{i, c}\mu_i(c, c) [\sigma_i(c, c) + \mu_i]\\
= & -\sum_{i, j, c} Q(Y_j = c)\log \bigg\{\frac{1}{Z_{ij}}\exp[\sigma_i(c, k) + \tau_{j}(c, k)]\bigg\} - H(Y) \\
& + \frac{\alpha}{2} \sum_{i, c}\bigg\{\sum_{k \neq c} [\sigma_i(c, k)]^2 + [\sigma_i(c, c) + \mu_i]^2\bigg\} + \frac{\beta}{2} \sum_{i, c, k} [\tau_j(c, k)]^2.\end{aligned}$$ By minimizing the Lagrangian over $\mu_i, $ we obtain $$\mu_i = - \overline{\sigma_i(c, c)}.$$ So, the dual problem can be expressed as $$\begin{aligned}
\min_{Q, \sigma, \tau} \quad & -\sum_{i, j, c} Q(Y_j = c)\log \bigg\{\frac{1}{Z_{ij}}\exp[\sigma_i(c, x_{ij}) + \tau_{j}(c, x_{ij})]\bigg\} - H(Y) \\
& + \frac{\alpha}{2} \sum_{i, c}\bigg\{\sum_{k \neq c} [\sigma_i(c, k)]^2 + \left[\sigma_i(c, c) - \overline{\sigma_i(c, c)}\right]^2\bigg\} + \frac{\beta}{2} \sum_{i, c, k} [\tau_j(c, k)]^2.\end{aligned}$$ Let us replace $\sigma_i(c, k) $ with $\sigma_i(c, k) + \nu_{i}. $ It is easy to verify that this dual problem can be equivalently written as $$\begin{aligned}
\min_{Q, \sigma, \tau, \nu} \quad & -\sum_{i, j, c} Q(Y_j = c)\log \bigg\{\frac{1}{Z_{ij}}\exp[\sigma_i(c, x_{ij}) + \tau_{j}(c, x_{ij})]\bigg\} - H(Y) \\
& + \frac{\alpha}{2} \sum_{i, c}\bigg\{\sum_{k \neq c} [\sigma_i(c, k) + \nu_{i} ]^2 + \left[\sigma_i(c, c) - \overline{\sigma_i(c, c)}\right]^2\bigg\} + \frac{\beta}{2} \sum_{i, c, k} [\tau_j(c, k)]^2.\end{aligned}$$ Minimizing the objective function over $\nu$ leads to $$\begin{aligned}
\min_{Q, \sigma, \tau} \quad & -\sum_{i, j, c} Q(Y_j = c)\log \bigg\{\frac{1}{Z_{ij}}\exp[\sigma_i(c, x_{ij}) + \tau_{j}(c, x_{ij})]\bigg\} - H(Y) \\
& + \frac{\alpha}{2} \sum_{i, c}\bigg\{\sum_{k \neq c} \left[\sigma_i(c, k) - \overline{\sigma_i(c, k)} \right]^2 + \left[\sigma_i(c, c) - \overline{\sigma_i(c, c)}\right]^2\bigg\} + \frac{\beta}{2} \sum_{i, c, k} [\tau_j(c, k)]^2.\end{aligned}$$
Coordinate Algorithm {#app:cm}
====================
To solve the dual problem $$\max_{\sigma, \tau, Q} \quad \sum_{j, c} Q(Y_j = c)\sum_i \log P(X_{ij} = x_{ij}|Y_j = c) + H(Y) - \alpha \Omega^*(\sigma) - \beta \Psi^*(\tau)$$ subject to the probability constraints $$\sum_c Q(Y_j = c) = 1, \forall j, \ Q(Y_j = c) \ge 0, \ \forall j, k,$$ we first split the variables into two groups and then alternatively update them. One group contains the parameters of workers and items in $P(X_{ij} = x_{ij}|Y_j = c), $ that is, $\{\sigma_{i}(c, k), \tau_j(c, k), \forall i, j, c, k\}$ and the other groups contains the unknown true labels $\{Q(Y_j = k), \forall j, k\}.$ When we update the variables in the first group, the variables in the second group take their current values. Then, the optimization problem becomes $$\max_{\sigma, \tau} \quad \sum_{j, c} Q(Y_j = c)\sum_i \log P(X_{ij} = x_{ij}|Y_j = c) - \alpha \Omega^*(\sigma) - \beta \Psi^*(\tau).$$ Instead, when we update the variables in the second group, the variables in the first group take their current values. We thus have the optimization problem $$\max_{Q} \quad \sum_{j, c} Q(Y_j = c)\sum_i \log P(X_{ij} = x_{ij}|Y_j = c) + H(Y)$$ subject to the above probability constraints. This constrained optimization problem can be solved with the Lagrangian dual $$L = \sum_{j, c} Q(Y_j = c)\sum_i \log P(X_{ij} = x_{ij}|Y_j = c) + H(Y) - \sum \lambda_{j} \bigg[\sum_c Q(Y_j = c) - 1\bigg],$$ where $\lambda_{j}$’s are the Lagrangian multipliers. By the KKT conditions, $$\frac{\partial L}{\partial Q(Y_j = c)} = \sum_i \log P(X_{ij} = x_{ij}|Y_j = c) - \log Q(Y_j = c) + 1 - \lambda_{j} = 0.$$ This implies $$Q(Y_j = c) \propto \prod_{i} P(X_{ij} = x_{ij}|Y_j = c).$$
0.2in
[^1]: Microsoft Research, Redmond, WA 98052. Email: [[email protected]]{}.
[^2]: Department of Computer Science, University of California at Irvine, Irvine, CA 92637. Email: [[email protected]]{}.
[^3]: Microsoft Research, Redmond, WA 98052. Email: [[email protected]]{}.
[^4]: Microsoft Research, Redmond, WA 98052. Email: [[email protected]]{}.
[^5]: Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA 94720. Email: [[email protected]]{}.
[^6]: [https://www.mturk.com]{}
[^7]: Some of the datasets can be found at [http://research.microsoft.com/en-us/projects/crowd/]{}
[^8]: Our code are available at [http://research.microsoft.com/en-us/projects/crowd/]{}
|
{
"pile_set_name": "ArXiv"
}
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[**A simultaneous decomposition of seven matrices over real quaternion algebra and its applications**]{}[^1]
[**Zhuo-Heng He, Qing-Wen Wang$^{*}$**]{}
Department of Mathematics, Shanghai University, Shanghai 200444. P.R. China
E-mail: [email protected] (Q.W. Wang), [email protected] (Z.H. He)
> **Abstract:** Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{n\times m}$ denote the set of all $n\times m$ matrices over $\mathbb{H}$. In this paper, we construct a simultaneous decomposition of seven general real quaternion matrices with compatible sizes: $A\in \mathbb{H}^{m\times n},
> B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n},G\in \mathbb{H}^{q_{3}\times n}$. As applications of the simultaneous matrix decomposition, we give solvability conditions, general solutions, as well as the range of ranks of the general solutions to the following two real quaternion matrix equations $BXE+CYF+DZG=A$ and $BX+WE+CYF+DZG=A,$ where $A,B,C,D,E,F,$ and $G$ are given real quaternion matrices. **Keywords:** Quaternion; Division ring; Matrix decomposition; Matrix equation; Rank; General solution**2010 AMS Subject Classifications: **[15A21, 15A22, 15A24, 15A33, 15A03]{}
**Introduction**
================
Throughout this paper, let $\mathbb{R},\mathbb{C},$ and $\mathbb{H}^{m\times
n}$ stand, respectively, for the real number field, the complex number field, and the set of all $m\times n$ matrices over the real quaternion algebra $$\mathbb{H}=\big\{a_{0}+a_{1}i+a_{2}j+a_{3}k\big|~i^{2}=j^{2}=k^{2}=ijk=-1,a_{0},a_{1},a_{2},a_{3}\in\mathbb{R}\big\}.$$ The rank of a quaternion matrix $A$ is defined to be the maximum number of columns of $A$ which are right linearly independent [@zhangfuzheng]. It is easy to see that for any nonsingular matrices $P$ and $Q$ of appropriate sizes, $A$ and $PAQ$ have the same rank [@zhangfuzheng]. Moreover, for $A\in \mathbb{H}^{m\times n},$ by [@TWH], there exist invertible matrices $P$ and $Q$ such that $$\begin{aligned}
PAQ=\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}\end{aligned}$$where $r=r(A),$ $I_{r}$ is the $r\times r$ identity matrix. The rank of a block real quaternion matrix $$\begin{aligned}
\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1,n}\\A_{21}&A_{22}&\cdots&A_{2,n}\\ \vdots & \vdots &\ddots &\vdots \\
A_{m,1}&A_{m,2}&\cdots&A_{m,n}\end{pmatrix}\end{aligned}$$is denoted by $r_{a_{11}a_{12}\cdots a_{1,n}|a_{21}a_{22}\cdots a_{2,n}|\cdots |a_{m,1}a_{m,2}\cdots a_{m,n}}.$ If the block real quaternion matrix has zero, we use “$0$" instead of zero in the subscript. For instance, the ranks of the following block real quaternion matrices $$\begin{aligned}
(B,~C,~D),~\begin{pmatrix}E\\F\\G\end{pmatrix},~\begin{pmatrix}D&0&A&B&0\\D&A&0&0&C\\0&E&0&0&0\\0&0&F&0&0\\0&F&0&0&0\end{pmatrix}\end{aligned}$$ are represented by $r_{bcd},~r_{e|f|g},~ r_{d0ab0|da00c|0e000|00f00|0f000}$, respectively. The set of all $n\times n$ invertible real quaternion matrices is denoted by $GL_{n}(\mathbb{H})$.
Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. It is well known that $\mathbb{H}$ is an associative and noncommutative division algebra. General properties of quaternion and real quaternion matrices can be found in [@zhangfuzheng]. Nowadays real quaternion matrices have always been at the heart of computer science, quantum physics, signal and color image processing, and so on (e.g. [@N.; @LE; @Bihan], [@J.C.K.Chu], [@S.; @De; @Leo], [@Took1]-[@Took4]).
In mathematics, engineering, signal, circuit and others, many problems can be transformed into the decomposition of multiple matrices (e.g. [@tree3], [@siamre], [@tree1]). In 1981, Paige and Saunders [@ccp] introduced the generalized singular value decomposition of two matrices with the same row number. In 1991, Zha [@tree1] gave a restricted singular values of a general matrix triplet $\begin{pmatrix}\begin{smallmatrix}A&B\\C&~\end{smallmatrix}\end{pmatrix}$ over $\mathbb{C}$. In 1991, Bart De Moor and G.H. Golub [@moor1] derived a generalization of the OSVD. Several applications were discussed in [@moor1]. Moreover, Bart De Moor and Paul Van Dooren [@moor3] presented the generalized singular value decompositions for $k$ general matrices $(A_{1},A_{2},\ldots,A_{k})$, where $A_{1}\in \mathbb{C}^{n_{0}\times n_{1}},A_{2}\in \mathbb{C}^{n_{1}\times n_{2}},\ldots,A_{k}\in \mathbb{C}^{n_{k-1}\times n_{k}}.$ In 1994, C.C. Paige and M.S. Wei [@ccpaige] introduced the history and generality of the CS Decomposition. In 2000, Delin Chu, Lieven De Lathauwer and Bart De Moor [@CHU5] proved that the restricted singular value decomposition of a general matrix triplet $\begin{pmatrix}\begin{smallmatrix}A&B\\C&~\end{smallmatrix}\end{pmatrix}$ can be computed using a CSD-based QR-type method. In 2011, Wang, van der Woude and Yu [@QWWangandyushaowen] derived the decomposition of three general matrices with the same row number over an arbitrary division ring. In 2012, Wang, Zhang and van der Woude [@zhangxia] gave a simultaneous decomposition concerning the general matrix quaternity $\begin{pmatrix}\begin{smallmatrix}A&B&C\\D&~&~\end{smallmatrix}\end{pmatrix}$ over an arbitrary division ring. Quite recently, He and Wang [@hezhuoheng] constructed a simultaneous decomposition of five general real quaternion matrices $\begin{pmatrix}\begin{smallmatrix}A&B&C\\D&~&~\\E&~&~\end{smallmatrix}\end{pmatrix}$. He and Wang [@hezhuoheng] gave the rang of ranks of the real quaternion matrix expression $A-BXD-CYE$ by using the simultaneous decomposition of five general real quaternion matrices $\begin{pmatrix}\begin{smallmatrix}A&B&C\\D&~&~\\E&~&~\end{smallmatrix}\end{pmatrix}$.
The remainder of the paper is organized as follows. In Section 2, we propose the simultaneous decomposition of the matrix array $$\begin{aligned}
\label{array1}
\begin{pmatrix}
A&B&C&D\\
E&~&~&~\\
F&~&~&~\\
G&~&~&~\end{pmatrix},\end{aligned}$$ where $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ are general real quaternion matrices. In Section 3, we discuss several applications of the simultaneous decomposition. In Section 3.2, we derive solvability conditions and the general solution to the real quaternion matrix equation $$\begin{aligned}
\label{system002}
BXE+CYF+DZG=A.\end{aligned}$$ In Section 3.3, we give the range of ranks of the general solution to the real quaternion matrix equation (\[system002\]). In Section 3.4, we present solvability conditions and the general solution to the real quaternion matrix equation $$\begin{aligned}
\label{system001}
BX+WE+CYF+DZG=A.\end{aligned}$$ In Section 3.5, we derive the range of ranks of the general solution to the real quaternion matrix equation (\[system001\]).
**A simultaneous decomposition of the general matrix array (\[array1\]) over $\mathbb{H}$**
===========================================================================================
We begin with the following lemma that is a basic tool for obtaining the main result.
\[lemma00\][@QWWangandyushaowen] Let $B\in \mathbb{H}^{m\times p_{1}},
C\in \mathbb{H}^{m\times p_{2}}$ and $D\in \mathbb{H}^{m\times p_{3}}$ be given. Then there exist $\widetilde{P}\in GL_{m}(\mathbb{H}),\widetilde{T_{1}}\in GL_{p_{1}}(\mathbb{H}),
\widetilde{T_{2}}\in GL_{p_{2}}(\mathbb{H})$ and $\widetilde{T_{3}}\in GL_{p_{3}}(\mathbb{H})$ such that $$\begin{aligned}
B=\widetilde{P} \widetilde{S_{B}}\widetilde{T_{1}}, \qquad C=\widetilde{P} \widetilde{S_{C}}\widetilde{T_{2}},\qquad D=\widetilde{P} \widetilde{S_{D}}\widetilde{T_{3}},\end{aligned}$$ where $$\begin{aligned}
(\widetilde{S_{B}},\widetilde{S_{C}},\widetilde{S_{D}})=\begin{pmatrix}\begin{matrix} I &0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
I & 0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&0 &I&0\\
0 & 0&0 &0&0&0\\
0&0&0&I&0&0\\
0&I & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0&I & 0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
I&0 & 0&0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}
\end{pmatrix}
\begin{matrix} m_{1}\\
m_{2}\\
m_{3}\\
m_{4}\\
m_{5}\\
m_{4}\\
m_{6}\\
m_{7}\\
m_{8}\\
m-r_{bcd}
\end{matrix},\end{aligned}$$ where $$\begin{aligned}
m_{4}+m_{6}+m_{7}=r_{bc}-r_{b},~m_{1}+m_{2}=r_{b}+r_{c}-r_{bc},~m_{8}=r_{bcd}-r_{bc},\end{aligned}$$ $$\begin{aligned}
m_{4}+m_{6}=r_{bc}+r_{bd}-r_{bcd}-r_{b},~m_{1}+m_{3}=r_{b}+r_{d}-r_{bd},~m_{3}+m_{4}=r_{bc}+r_{cd}-r_{bcd}-r_{c}.\end{aligned}$$
Now we give the main theorem of this section.
\[theorem01\] Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Then there exist $P\in GL_{m}(\mathbb{H}),~Q\in GL_{n}(\mathbb{H}),~T_{1}\in GL_{p_{1}}(\mathbb{H}),~
T_{2}\in GL_{p_{2}}(\mathbb{H}),~T_{3}\in GL_{p_{3}}(\mathbb{H}),~V_{1}\in GL_{q_{1}}(\mathbb{H}),~V_{2}\in GL_{q_{2}}(\mathbb{H}),$ and $V_{3}\in GL_{q_{3}}(\mathbb{H})$ such that$$\begin{aligned}
\label{equ021}
A=P S_{A}Q,B=P S_{B}T_{1}, C=P S_{C}T_{2},D=P S_{D}T_{3}, E=V_{1} S_{E}Q, F=V_{2} S_{F}Q,G=V_{3} S_{G}Q,\end{aligned}$$ where$$\begin{aligned}
\label{equ0022}
S_{A}=\begin{pmatrix}
A_{11} & \cdots& A_{19}& A_{1,10}&0\\
\vdots& \ddots& \vdots&\vdots&\vdots\\
A_{91}& \cdots&A_{99}&A_{9,10}& 0\\
A_{10,1}& \cdots&A_{10,9}&0&0\\0&\cdots&0 &0&I_{t}
\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\label{equ0023}
(S_{B},S_{C},S_{D})=\begin{pmatrix}\begin{matrix} I &0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
I & 0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&0 &I&0\\
0 & 0&0 &0&0&0\\
0&0&0&I&0&0\\
0&I & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0&I & 0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
I&0 & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}
\end{pmatrix}
\begin{matrix} m_{1}\\
m_{2}\\
m_{3}\\
m_{4}\\
m_{5}\\
m_{4}\\
m_{6}\\
m_{7}\\
m_{8}\\
m-r_{bcd}-t\\
t
\end{matrix},\end{aligned}$$ $$\begin{aligned}
\label{equ0024}
(S_{E}^{*},S_{F}^{*},S_{G}^{*})=\begin{pmatrix}\begin{matrix} I &0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
I & 0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&0 &I&0\\
0 & 0&0 &0&0&0\\
0&0&0&I&0&0\\
0&I & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0&I & 0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
I&0 & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}
\end{pmatrix}
\begin{matrix}
n_{1}\\
n_{2}\\
n_{3}\\
n_{4}\\
n_{5}\\
n_{4}\\
n_{6}\\
n_{7}\\
n_{8}\\
n-r_{e|f|g}-t\\t
\end{matrix},\end{aligned}$$ where $$\begin{aligned}
t=r_{abcd|e000|f000|g000}-r_{bcd}-r_{e|f|g},\end{aligned}$$ $$\begin{aligned}
\label{equh025}
m_{1}=r_{b}+r_{c}+r_{d}-r_{db0|d0c},
~
m_{2}=r_{db0|d0c}-r_{bc}-r_{d},\end{aligned}$$ $$\begin{aligned}
m_{3}=r_{db0|d0c}-r_{bd}-r_{c},
~
m_{4}=r_{bc}+r_{cd}+r_{bd}-r_{bcd}-r_{db0|d0c},\end{aligned}$$ $$\begin{aligned}
m_{5}=r_{bcd}-r_{cd},~
m_{6}=r_{db0|d0c}-r_{cd}-r_{b},~
~
m_{7}=r_{bcd}-r_{bd},~
m_{8}=r_{bcd}-r_{bc},\end{aligned}$$ $$\begin{aligned}
n_{1}=r_{e}+r_{f}+r_{g}-r_{gg|e0|0f},
~
n_{2}=r_{gg|e0|0f}-r_{e|f}-r_{g},\end{aligned}$$ $$\begin{aligned}
n_{3}=r_{gg|e0|0f}-r_{e|f}-r_{f},~
n_{4}=r_{e|f}+r_{f|g}+r_{e|g}-r_{e|f|g}-r_{gg|e0|0f},\end{aligned}$$ $$\begin{aligned}
\label{equh0210}
n_{5}=r_{e|f|g}-r_{f|g},~
n_{6}=r_{gg|e0|0f}-r_{f|g}-r_{e},~
~
n_{7}=r_{e|f|g}-r_{e|g},~
n_{8}=r_{e|f|g}-r_{e|f}.\end{aligned}$$ The block columns of $S_{A}$ are $(n_{1},n_{2},n_{3},n_{4},n_{5},n_{4},n_{6}.n_{7},n_{8},n-r_{e|f|g}-t,t).$ The block rows of $S_{A}$ are $(m_{1},m_{2},m_{3},m_{4},m_{5},m_{4},m_{6}.m_{7},m_{8},m-r_{bcd}-t,t).$
The proof is constructive. We establish the result through the following steps. First, we give equivalence canonical forms of the general matrix arrays $(B,~C,~D)$ and $\begin{pmatrix}\begin{smallmatrix}E\\F\\G\end{smallmatrix}\end{pmatrix}$. Second, we provide the simultaneous decomposition of the matrix array (\[array1\]).
Step 1. For the matrix arrays $(B,~C,~D)$ and $\begin{pmatrix}\begin{smallmatrix}E\\F\\G\end{smallmatrix}\end{pmatrix}$, we can find eight matrices $P_{1}\in GL_{m}(\mathbb{H}),$ $Q_{1}\in GL_{n}(\mathbb{H}),$ $W_{B}\in GL_{p_{1}}(\mathbb{H}),$ $
W_{C}\in GL_{p_{2}}(\mathbb{H}),$ $W_{D}\in GL_{p_{3}}(\mathbb{H}),$ $W_{E}\in GL_{q_{1}}(\mathbb{H}),$ $W_{F}\in GL_{q_{2}}(\mathbb{H}),$ $W_{G}\in GL_{q_{3}}(\mathbb{H})$ such that $$\begin{aligned}
P_{1}\begin{pmatrix}B&C&D\end{pmatrix}\begin{pmatrix}W_{B}&0&0\\0&W_{C}&0\\0&0&W_{D}\end{pmatrix}
=\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}\begin{matrix} I &0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
I & 0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&0 &I&0\\
0 & 0&0 &0&0&0\\
0&0&0&I&0&0\\
0&I & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0&I & 0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
I&0 & 0&0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}
\end{pmatrix}
\begin{matrix} m_{1}\\
m_{2}\\
m_{3}\\
m_{4}\\
m_{5}\\
m_{4}\\
m_{6}\\
m_{7}\\
m_{8}\\
m-r_{bcd}
\end{matrix},\end{aligned}$$ $$\begin{aligned}
\left[\begin{pmatrix}W_{E}&0&0\\0&W_{F}&0\\0&0&W_{G}\end{pmatrix}\begin{pmatrix}E\\F\\G\end{pmatrix}Q_{1}\right]^{*}
=\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}\begin{matrix} I &0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
I & 0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&0 &I&0\\
0 & 0&0 &0&0&0\\
0&0&0&I&0&0\\
0&I & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0&I & 0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
I&0 & 0&0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}
\end{pmatrix}
\begin{matrix}
n_{1}\\
n_{2}\\
n_{3}\\
n_{4}\\
n_{5}\\
n_{4}\\
n_{6}\\
n_{7}\\
n_{8}\\
n-r_{e|f|g}
\end{matrix}.\end{aligned}$$ Let $$\begin{aligned}
P_{1}AQ_{1}\triangleq
\begin{pmatrix}
A_{11}^{(1)} & \cdots& A_{1,10}^{(1)}\\
\vdots& \ddots& \vdots\\
A_{10,1}^{(1)} & \cdots&A_{10,10}^{(1)}
\end{pmatrix},\end{aligned}$$ where the symbol $\triangleq$ means “equals by definition”. For the matrix $A_{10,10}^{(1)}$, we can find $P_{2}\in GL_{m-r_{bcd}}(\mathbb{H}),$ $Q_{2}\in GL_{n-r_{e|f|g}}(\mathbb{H})$ such that $$\begin{aligned}
P_{2}A_{10,10}^{(1)}Q_{2}=\begin{pmatrix}0&0\\0&I_{t}\end{pmatrix},t\triangleq r(A_{10,10}^{(1)}).\end{aligned}$$ Then we have $$\begin{aligned}
\begin{pmatrix}I_{r_{bcd}}&0\\0&P_{2}\end{pmatrix}\begin{pmatrix}
A_{11}^{(1)} & \cdots& A_{1,10}^{(1)}\\
\vdots& \ddots& \vdots\\
A_{10,1}^{(1)} & \cdots&A_{10,10}^{(1)}
\end{pmatrix}\begin{pmatrix}I_{r_{e|f|g}}&0\\0&Q_{2}\end{pmatrix}\triangleq
\begin{pmatrix}
A_{11}^{(2)} & \cdots& A_{19}^{(2)}& A_{1,10}^{(2)}& A_{1,11}^{(2)}\\
\vdots& \ddots& \vdots&\vdots&\vdots\\
A_{91}^{(2)} & \cdots&A_{99}^{(2)}&A_{9,10}^{(2)}& A_{9,11}^{(2)}\\
A_{10,1}^{(2)} & \cdots&A_{10,9}^{(2)}&0&0\\A_{11,1}^{(2)} &\cdots&A_{11,9}^{(2)} &0&I_{t}
\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix} I_{r_{bcd}}&0\\0&P_{2} \end{pmatrix}P_{1}\begin{pmatrix}B&C&D\end{pmatrix}\begin{pmatrix}
W_{B}&0&0\\0&W_{C}&0\\0&0&W_{D} \end{pmatrix}
=\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}\begin{matrix} I &0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
I & 0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&0 &I&0\\
0 & 0&0 &0&0&0\\
0&0&0&I&0&0\\
0&I & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0&I & 0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
I&0 & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}
\end{pmatrix}
\begin{matrix} m_{1}\\
m_{2}\\
m_{3}\\
m_{4}\\
m_{5}\\
m_{4}\\
m_{6}\\
m_{7}\\
m_{8}\\
m-r_{bcd}-t\\
t
\end{matrix},\end{aligned}$$ $$\begin{aligned}
\left[\begin{pmatrix} W_{E}&0&0\\0&W_{F}&0\\0&0&W_{G} \end{pmatrix}\begin{pmatrix}
E\\F\\G \end{pmatrix}Q_{1}
\begin{pmatrix} I_{r_{e|f|g}}&0\\0&Q_{2} \end{pmatrix}\right]^{*}
=\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}\begin{matrix} I &0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&I&0&0\\
0 & 0&0&0&I&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
I & 0&0&0&0&0\\
0 & I&0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}~&
\begin{matrix} 0 &0&0&0 &I&0\\
0 & 0&0 &0&0&0\\
0&0&0&I&0&0\\
0&I & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0&I & 0&0&0&0\\
0 & 0&I&0&0&0\\
0 & 0&0 &0&0&0\\
I&0 & 0&0&0&0\\
0 & 0&0 &0&0&0\\
0 & 0&0 &0&0&0
\end{matrix}
\end{pmatrix}
\begin{matrix}
n_{1}\\
n_{2}\\
n_{3}\\
n_{4}\\
n_{5}\\
n_{4}\\
n_{6}\\
n_{7}\\
n_{8}\\
n-r_{e|f|g}-t\\t
\end{matrix}.\end{aligned}$$ Let $$\begin{aligned}
P_{3}=\begin{pmatrix}I_{r_{bcd}}&\begin{pmatrix}0&-A_{1,11}^{(2)}\\
\vdots&\vdots\\0&-A_{9,11}^{(2)}\end{pmatrix}\\0&I_{m-r_{bcd}}\end{pmatrix},~
Q_{3}=\begin{pmatrix}I_{r_{e|f|g}}&0\\
\begin{pmatrix}0&\cdots&0\\
-A_{11,1}^{(2)}&\cdots&-A_{11,9}^{(2)}\end{pmatrix}&I_{n-r_{e|f|g}}\end{pmatrix}.\end{aligned}$$ Then we obtain $$\begin{aligned}
P_{3}\begin{pmatrix}
A_{11}^{(2)} & \cdots& A_{19}^{(2)}& A_{1,10}^{(2)}& A_{1,11}^{(2)}\\
\vdots& \ddots& \vdots&\vdots&\vdots\\
A_{91}^{(2)} & \cdots&A_{99}^{(2)}&A_{9,10}^{(2)}& A_{9,11}^{(2)}\\
A_{10,1}^{(2)} & \cdots&A_{10,9}^{(2)}&0&0\\A_{11,1}^{(2)} &\cdots&A_{11,9}^{(2)} &0&I_{t}
\end{pmatrix}Q_{3}\triangleq
\begin{pmatrix}
A_{11} & \cdots& A_{19}& A_{1,10}&0\\
\vdots& \ddots& \vdots&\vdots&\vdots\\
A_{91}& \cdots&A_{99}&A_{9,10}& 0\\
A_{10,1}& \cdots&A_{10,9}&0&0\\0 &\cdots&0 &0&I_{t}
\end{pmatrix}.\end{aligned}$$ Let $$\begin{aligned}
P\triangleq P_{3}\begin{pmatrix}I_{r_{bcd}}&0\\0&P_{2}\end{pmatrix}P_{1},~
Q\triangleq Q_{1}\begin{pmatrix}I_{r_{e|f|g}}&0\\0&Q_{2}\end{pmatrix}Q_{3},\end{aligned}$$ $$\begin{aligned}
T_{1}=W_{C},~T_{2}=W_{D},~T_{3}=W_{E},~V_{1}=W_{E},~V_{2}=W_{F},~V_{3}=W_{G}.\end{aligned}$$ Hence, the matrices $P\in GL_{m}(\mathbb{H}),~Q\in GL_{n}(\mathbb{H}),~T_{1}\in GL_{p_{1}}(\mathbb{H}),~
T_{2}\in GL_{p_{2}}(\mathbb{H}),~T_{3}\in GL_{p_{3}}(\mathbb{H}),~V_{1}\in GL_{q_{1}}(\mathbb{H}),~V_{2}\in GL_{q_{2}}(\mathbb{H}),~V_{3}\in GL_{q_{3}}(\mathbb{H})$ satisfy the equation (\[equ021\]). It follows from $S_{A},S_{B},S_{C},S_{D},S_{E},S_{F},$ and $S_{G}$ in (\[equ0022\])-(\[equ0024\]) that $$\begin{aligned}
\begin{pmatrix}
1&1&1&1&1&0&0&0\\
1&1&0&1&0&1&1&0\\
1&0&1&1&0&1&0&1\\
1&1&1&2&1&1&1&0\\
1&1&1&2&1&1&0&1\\
1&1&1&2&0&1&1&1\\
1&1&1&2&1&1&1&1\\
1&0&1&1&0&1&0&1\end{pmatrix}\begin{pmatrix}m_{1}\\
m_{2}\\
m_{3}\\
m_{4}\\
m_{5}\\
m_{6}\\
m_{7}\\
m_{8}\end{pmatrix}=\begin{pmatrix}r_{b}\\
r_{c}\\
r_{d}\\
r_{bc}\\
r_{bd}\\
r_{cd}\\
r_{bcd}\\
r_{db0|d0c}-r_{b}-r_{c}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\begin{pmatrix}
1&1&1&1&1&0&0&0\\
1&1&0&1&0&1&1&0\\
1&0&1&1&0&1&0&1\\
1&1&1&2&1&1&1&0\\
1&1&1&2&1&1&0&1\\
1&1&1&2&0&1&1&1\\
1&1&1&2&1&1&1&1\\
1&0&1&1&0&1&0&1\end{pmatrix}\begin{pmatrix}n_{1}\\
n_{2}\\
n_{3}\\
n_{4}\\
n_{5}\\
n_{6}\\
n_{7}\\
n_{8}\end{pmatrix}=\begin{pmatrix}r_{e}\\
r_{f}\\
r_{g}\\
r_{e|f}\\
r_{e|g}\\
r_{f|g}\\
r_{e|f|g}\\
r_{gg|e0|0f}-r_{e}-r_{f}\end{pmatrix}.\end{aligned}$$ Solving for $m_{i},n_{i},(i=1,\ldots,8)$ gives (\[equh025\])-(\[equh0210\]).
Wang et. al. [@QWWangandyushaowen] did not give the values of $m_{i},~(i=1,2,\ldots,8)$ in Lemma \[lemma00\]. As a special case of Theorem \[theorem01\], we can derive all the dimensions of identity matrices in the equivalence canonical form of triple real quaternion matrices $(B,C,D)$, i.e., the values of $m_{i},~(i=1,2,\ldots,8)$ in Lemma \[lemma00\]: $$\begin{aligned}
m_{1}=r_{b}+r_{c}+r_{d}-r_{db0|d0c},
~
m_{2}=r_{db0|d0c}-r_{bc}-r_{d},\end{aligned}$$ $$\begin{aligned}
m_{3}=r_{db0|d0c}-r_{bd}-r_{c},
~
m_{4}=r_{bc}+r_{cd}+r_{bd}-r_{bcd}-r_{db0|d0c},\end{aligned}$$ $$\begin{aligned}
m_{5}=r_{bcd}-r_{cd},~
m_{6}=r_{db0|d0c}-r_{cd}-r_{b},~
~
m_{7}=r_{bcd}-r_{bd},~
m_{8}=r_{bcd}-r_{bc}.\end{aligned}$$On the other hand, the values of $m_{i},n_{i},(i=1,\ldots,8)$ play an important role in investigating the range of ranks of the general solutions to (\[system002\]) and (\[system001\]).
**Some applications of the simultaneous decomposition of (\[array1\])**
=======================================================================
The simultaneous decomposition of (\[array1\]) is useful in solving the following questions:
- §3.2. Give some solvability conditions and an expression of the general solution to the real quaternion matrix equation (\[system002\]).
- §3.3. Give the range of ranks of the general solution in the real quaternion matrix equation (\[system002\]).
- §3.4. Give some solvability conditions and an expression of the general solution to the real quaternion matrix equation (\[system001\]).
- §3.5. Give the range of ranks of the general solution in the real quaternion matrix equation (\[system001\]).
**Preliminaries**
-----------------
In this section, we give some lemmas which are used in the further development of this paper. The following Lemmas are due to [@CHU3], [@cohen1] and [@Woerdeman1]-[@Woerdeman3] which can be generalized to $\mathbb{H}.$
\[lemma04\]([@CHU3]-[@hezhuoheng], [@Woerdeman1]-[@Woerdeman3]) Let $$\begin{aligned}
H(X,Y)=\begin{pmatrix}A_{1}&B_{1}&C_{1}\\D_{1}&X&E_{1}\\
F_{1}&G_{1}&Y\end{pmatrix},\end{aligned}$$ where $A_{1}\in \mathbb{H}^{\tilde{n}\times n},B_{1}\in \mathbb{H}^{\tilde{n}\times m},C_{1}\in \mathbb{H}^{\tilde{n}\times p},
D_{1}\in \mathbb{H}^{\tilde{m}\times n},E_{1}\in \mathbb{H}^{\tilde{m}\times p},F_{1}\in \mathbb{H}^{\tilde{p}\times n}$ and $G_{1}\in \mathbb{H}^{\tilde{p}\times m}$ are given, and $X\in \mathbb{H}^{\tilde{m}\times m}$ and $Y\in \mathbb{H}^{\tilde{p}\times p}$ are variable matrices. Then, $$\begin{aligned}
\mathop {\max }\limits_{ X,Y } r\left[ {H\left( {X,Y} \right)} \right]=\min \left\{\tilde{m}+\tilde{p}+r_{a_{1}b_{1}c_{1}},
\tilde{m}+p+r_{a_{1}b_{1}|f_{1}g_{1}},m+\tilde{p}+r_{a_{1}c_{1}|d_{1}e_{1}},m+p+r_{a_{1}|d_{1}|f_{1}}\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ X,Y } r\left[ {H\left( {X,Y} \right)} \right]=r_{a_{1}b_{1}c_{1}}+r_{a_{1}|d_{1}|f_{1}}+
\max \left\{r_{a_{1}c_{1}|d_{1}e_{1}}-r_{a_{1}c_{1}}-r_{a_{1}|d_{1}},r_{a_{1}b_{1}|f_{1}g_{1}}-r_{a_{1}b_{1}}-r_{a_{1}|f_{1}} \right\}.\end{aligned}$$
\[lemma01\]([@CHU3]-[@hezhuoheng], [@Woerdeman1]-[@Woerdeman3]) Let $$\begin{aligned}
\label{equ031}
M(X,Y)=\begin{pmatrix}A_{1}&X\\Y&B_{1}\end{pmatrix}\end{aligned}$$ where $A_{1}\in \mathbb{H}^{m\times n}$ and $B_{1}\in \mathbb{H}^{p\times q}$ are given, and $X\in \mathbb{H}^{m\times q}$ and $Y\in \mathbb{H}^{p\times n}$ are variable matrices. Then, $$\begin{aligned}
\mathop {\max }\limits_{ X,Y } r\left[ {M\left( {X,Y} \right)} \right]=\min \left\{m+p,n+q,r_{a_{1}}+p+q,r_{b_{1}}+m+n \right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ X,Y } r\left[ {M\left( {X,Y} \right)} \right]=\max \left\{r_{a_{1}},~r_{b_{1}} \right\}.\end{aligned}$$
\[lemma03\]([@cohen1], [@hezhuoheng], [@Woerdeman1]-[@Woerdeman3]) Let $$\begin{aligned}
M_{2}=\begin{pmatrix}Y&D_{1}\\B_{1}&A_{1}\end{pmatrix},\end{aligned}$$where $A_{1},B_{1}$ and $D_{1}$ are given, and $Y\in \mathbb{H}^{n\times m}$ is a variable matrix. Then, $$\begin{aligned}
\mathop {\max }\limits_{Y\in \mathbb{H}^{n\times m} } r\left( {M_{2}} \right)=\mathop {\min }\left\{ n+r_{a_{1}b_{1}},m+r_{a_{1}|d_{1}} \right\},~
\mathop {\min }\limits_{Y\in \mathbb{H}^{n\times m} } r\left( {M_{2}} \right)=r_{a_{1}b_{1}}+r_{a_{1}|d_{1}}-r_{a_{1}}.\end{aligned}$$
**Some solvability conditions and the general solution to (\[system002\])**
---------------------------------------------------------------------------
In this section, the simultaneous decomposition of (\[array1\]) will be used to solve the real quaternion matrix equation (\[system002\]).
\[theorem04\] Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Then the equation (\[system002\]) is consistent if and only if $$\begin{aligned}
A_{94}=A_{96},~A_{49}=A_{69},~A_{64}=A_{46},\end{aligned}$$ $$\begin{aligned}
r_{abcd|e000|f000|g000}=r_{bcd}+r_{e|f|g},\quad \left(A_{1,10}^{*},~ \cdots, ~A_{9,10}^{*}\right)=0,\quad
\left(A_{10,1},~ \cdots, ~A_{10,9}\right)=0,\end{aligned}$$ $$\begin{aligned}
A_{29}=0,~A_{92}=0,~A_{38}=0,~A_{83}=0,~A_{48}=0,~A_{84}=0,~A_{56}=0,~A_{65}=0,\end{aligned}$$ $$\begin{aligned}
\label{equ00401}
A_{57}=0,~A_{75}=0,~A_{58}=0,~A_{85}=0,~A_{59}=0,~A_{95}=0,~A_{89}=0,~A_{98}=0.\end{aligned}$$
In this case, the general solution to (\[system002\]) can be expressed as $$\begin{aligned}
X=T_{1}^{-1}\widehat{X}V_{1}^{-1},\quad Y=T_{2}^{-1}\widehat{Y}V_{2}^{-1},\quad Z=T_{3}^{-1}\widehat{Z}V_{3}^{-1},\end{aligned}$$ where $$\begin{aligned}
\label{equ0041}
\widehat{X}=\bordermatrix{
~& n_{1}&n_{2} & n_{3}&n_{4}&n_{5}&q_{1}-r_{e} \cr
m_{1}&X_{11}&X_{12}&X_{13}&X_{14}&A_{15}&X_{16} \cr
m_{2}&X_{21}&X_{22}&A_{23}&A_{24}&A_{25}&X_{26} \cr
m_{3}&X_{31}&A_{32}&X_{33}&A_{34}-A_{36}&A_{35}&X_{36} \cr
m_{4}&X_{41}&A_{42}&A_{43}-A_{63}&A_{44}-A_{64}&A_{45}&X_{46} \cr
m_{5}&A_{51}&A_{52}&A_{53}&A_{54}&A_{55}&X_{56} \cr
p_{1}-r_{b}&X_{61}&X_{62}&X_{63}&X_{64}&X_{65}&X_{66}},\end{aligned}$$ $$\begin{aligned}
\label{equ0042}
\widehat{Y}=\bordermatrix{
~& n_{4}&n_{6} & n_{7}&n_{1}&n_{2}&q_{2}-r_{f} \cr
m_{4}&A_{66}-A_{64}&A_{67}-A_{47}&A_{68}&A_{61}-A_{41}+X_{41}&A_{62}&Y_{16} \cr
m_{6}&A_{76}-A_{74}&Y_{22}&A_{78}&Y_{24}&A_{72}&Y_{26} \cr
m_{7}&A_{86}&A_{87}&A_{88}&A_{81}&A_{82}&Y_{36} \cr
m_{1}&A_{16}-A_{14}+X_{14}&Y_{42}&A_{18}&Y_{44}&A_{12}-X_{12}&Y_{46} \cr
m_{2}&A_{26}&A_{27}&A_{28}&A_{21}-X_{21}&A_{22}-X_{22}&Y_{56} \cr
p_{2}-r_{c}&Y_{61}&Y_{62}&Y_{63}&Y_{64}&Y_{65}&Y_{66}},\end{aligned}$$ $$\begin{aligned}
\label{equ0043}
\widehat{Z}=\bordermatrix{
~& n_{8}&n_{4} & n_{6}&n_{3}&n_{1}&q_{3}-r_{g} \cr
m_{8}&A_{99}&A_{96}&A_{97}&A_{93}&A_{91}&Z_{16} \cr
m_{4}&A_{69}&A_{64}&A_{47}&A_{63}&A_{41}-X_{41}&Z_{26} \cr
m_{6}&A_{79}&A_{74}&A_{77}-Y_{22}&A_{73}&A_{71}-Y_{24}&Z_{36} \cr
m_{3}&A_{39}&A_{36}&A_{37}&A_{33}-X_{33}&A_{31}-X_{31}&Z_{46} \cr
m_{1}&A_{19}&A_{14}-X_{14}&A_{17}-Y_{42}&A_{13}-X_{13}&Z_{55}&Z_{56} \cr
p_{3}-r_{d}&Z_{61}&Z_{62}&Z_{63}&Z_{64}&Z_{65}&Z_{66}},\end{aligned}$$ $A_{ij},T_{i},V_{i}$ are defined in Theorem \[theorem01\], the remaining $X_{ij},Y_{ij},Z_{ij}$ in (\[equ0041\])-(\[equ0043\]) are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.
It follows from Theorem \[theorem01\] that the matrix equation (\[system002\]) is equivalent to the matrix equation $$\begin{aligned}
\label{equ0044}
S_{B}(T_{1}XV_{1})S_{E}+S_{C}(T_{2}YV_{2})S_{F}+S_{D}(T_{3}ZV_{3})S_{G}=S_{A}.\end{aligned}$$ Let the matrices $$\begin{aligned}
\label{equ0045}
\widehat{X}=T_{1}XV_{1}=\begin{pmatrix}X_{11}&\cdots&X_{16}\\
\vdots&\ddots&\vdots\\
X_{61}&\cdots&X_{66}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\widehat{Y}=T_{2}YV_{2}=\begin{pmatrix}Y_{11}&\cdots&Y_{16}\\
\vdots&\ddots&\vdots\\
Y_{61}&\cdots&Y_{66}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\label{equ0047}
\widehat{Z}=T_{3}ZV_{3}=\begin{pmatrix}Z_{11}&\cdots&Z_{16}\\
\vdots&\ddots&\vdots\\
Z_{61}&\cdots&Z_{66}\end{pmatrix},\end{aligned}$$ be partitioned in accordance with (\[equ0044\]). Then it follows from (\[equ0022\])-(\[equ0024\]) and (\[equ0044\])-(\[equ0047\]) that $$\begin{aligned}
\begin{pmatrix}
\scriptstyle X_{11}+Y_{44}+Z_{55}&\scriptstyle X_{12}+Y_{45}&\scriptstyle X_{13}+Z_{54}&\scriptstyle X_{14}+Z_{52}&\scriptstyle X_{15} &\scriptstyle Y_{41}+Z_{52} &\scriptstyle Y_{42}+Z_{53}&\scriptstyle Y_{43}&\scriptstyle Z_{51} &\scriptstyle 0 &\scriptstyle 0 \\
\scriptstyle X_{21}+Y_{54}&\scriptstyle X_{22}+Y_{55}&\scriptstyle X_{23}&\scriptstyle X_{24}&\scriptstyle X_{25} &\scriptstyle Y_{51} &\scriptstyle Y_{52}&\scriptstyle Y_{53} &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 \\
\scriptstyle X_{31}+Z_{45}&\scriptstyle X_{32}&\scriptstyle X_{33}+Z_{44}&\scriptstyle X_{34}+Z_{42}&\scriptstyle X_{35} &\scriptstyle Z_{42} &\scriptstyle Z_{43} &\scriptstyle 0 &\scriptstyle Z_{41} &\scriptstyle 0&\scriptstyle 0 \\
\scriptstyle X_{41}+Z_{25}&\scriptstyle X_{42}&\scriptstyle X_{43}+Z_{24}&\scriptstyle X_{44}+Z_{22}&\scriptstyle X_{45} &\scriptstyle Z_{22}&\scriptstyle Z_{23} &\scriptstyle 0&\scriptstyle Z_{21} &\scriptstyle 0 &\scriptstyle 0 \\
\scriptstyle X_{51}&\scriptstyle X_{52}&\scriptstyle X_{53}&\scriptstyle X_{54}&\scriptstyle X_{55} &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0\\
\scriptstyle Y_{14}+Z_{25} &\scriptstyle Y_{15} &\scriptstyle Z_{24} &\scriptstyle Z_{22} &\scriptstyle 0 &\scriptstyle Y_{11} +Z_{22}&\scriptstyle Y_{12} +Z_{23} &\scriptstyle Y_{13}&\scriptstyle Z_{21} &\scriptstyle 0 &\scriptstyle 0 \\
\scriptstyle Y_{24}+Z_{35} &\scriptstyle Y_{25} &\scriptstyle Z_{34} &\scriptstyle Z_{32} &\scriptstyle 0 &\scriptstyle Y_{21} +Z_{32}&\scriptstyle Y_{22}+Z_{33} &\scriptstyle Y_{23} &\scriptstyle Z_{31}&\scriptstyle 0 &\scriptstyle 0 \\
\scriptstyle Y_{34} &\scriptstyle Y_{35}&\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle Y_{31} &\scriptstyle Y_{32} &\scriptstyle Y_{33} &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0\\
\scriptstyle Z_{15} &\scriptstyle 0&\scriptstyle Z_{14} &\scriptstyle Z_{12} &\scriptstyle 0 &\scriptstyle Z_{12} &\scriptstyle Z_{13} &\scriptstyle 0 &\scriptstyle Z_{11} &\scriptstyle 0 &\scriptstyle 0 \\
\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0\\
\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0&\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0
\end{pmatrix}\end{aligned}$$ $$\begin{aligned}
\label{equ0048}
=\begin{pmatrix}
A_{11} & A_{12} & A_{13} &A_{14} & A_{15} & A_{16} &A_{17} & A_{18} & A_{19} & A_{1,10} & 0 \\
A_{21} & A_{22} & A_{23} &A_{24} & A_{25} & A_{26} &A_{27} & A_{28} & A_{29} & A_{2,10} & 0 \\
A_{31} & A_{32} & A_{33} &A_{34} & A_{35} & A_{36} &A_{37} & A_{38} & A_{39} & A_{3,10} & 0 \\
A_{41} & A_{42} & A_{43} &A_{44} & A_{45} & A_{46} &A_{47} & A_{48} & A_{49} & A_{4,10} & 0 \\
A_{51} & A_{52} & A_{53} &A_{54} & A_{55} & A_{56} &A_{57} & A_{58} & A_{59} & A_{5,10} & 0\\
A_{61} & A_{62} & A_{63} &A_{64} & A_{65} & A_{66} &A_{67} & A_{68} & A_{69} & A_{6,10} & 0 \\
A_{71} & A_{72} & A_{73} &A_{74} & A_{75} & A_{76} &A_{77} & A_{78} & A_{79} & A_{7,10} & 0 \\
A_{81} & A_{82} & A_{83} &A_{84} & A_{85} & A_{86} &A_{87} & A_{88} & A_{89} & A_{8,10} & 0\\
A_{91} & A_{92} & A_{93} &A_{94} & A_{95} & A_{96} &A_{97} & A_{98} & A_{99} & A_{9,10} & 0 \\
A_{10,1} & A_{10,2} & A_{10,3} &A_{10,4} & A_{10,5} & A_{10,6} &A_{10,7} & A_{10,8} & A_{10,9} & 0 &0\\
0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & I_{t}\end{pmatrix}.\end{aligned}$$
If the equation (\[system002\]) has a solution $(X,Y,Z)$, then by (\[equ0048\]), we have that the equalities in (\[equ00401\]) hold, and $$\begin{aligned}
&X_{11}+Y_{44}+Z_{55}=A_{11},~X_{12}+Y_{45}=A_{12},~X_{13}+Z_{54}=A_{13},~X_{14}+Z_{52}=A_{14},~X_{15}=A_{15}, \\
&Y_{41}+Z_{52}=A_{16},~Y_{42}+Z_{53}=A_{17},~ Y_{43}=A_{18},~ Z_{51}=A_{19},X_{21}+Y_{54}=A_{21},~X_{22}+Y_{55}=A_{22},\\
&X_{23}=A_{23},~X_{24}=A_{24},~X_{25}=A_{25},~Y_{51}=A_{26},~Y_{52}=A_{27},~Y_{53}=A_{28},~X_{31}+Z_{45}=A_{31}, \\
&X_{32}=A_{32}, ~X_{33}+Z_{44}=A_{33}, ~X_{34}+Z_{42}=A_{34}, ~X_{35}=A_{35}, ~ Z_{42}=A_{36}, ~Z_{43}=A_{37}, ~ Z_{41}=A_{39},\\
&X_{41}+Z_{25}=A_{41}, ~X_{42}=A_{42}, ~X_{43}+Z_{24}=A_{43}, ~X_{44}+Z_{22}=A_{44}, ~X_{45}=A_{45}, ~ Z_{22}=A_{46}, \\
&Z_{23}=A_{47}, ~ Z_{21}=A_{49}, ~X_{51}=A_{51}, ~X_{52}=A_{52}, ~X_{53}=A_{53}, ~X_{54}=A_{54}, ~X_{55}=A_{55}\\
&Y_{14}+Z_{25}=A_{61}, ~ Y_{15}=A_{62}, ~ Z_{24}=A_{63}, ~Z_{22}=A_{64}, ~ Y_{11}+Z_{22}=A_{66}, ~Y_{12}+Z_{23}=A_{67}, \\
& Y_{13}=A_{68}, ~Z_{21}=A_{69}, ~Y_{24}+Z_{35}=A_{71}, ~ Y_{25}=A_{72}, ~ Z_{34}=A_{73}, ~Z_{32}=A_{74}, ~ Y_{21} +Z_{32}=A_{76}, \\
& Y_{22}+Z_{33}=A_{77}, ~Y_{23}=A_{78} , ~ Z_{31}=A_{79}, ~Y_{34}=A_{81}, ~Y_{35}=A_{82}, ~ Y_{31}=A_{86}, ~Y_{32}=A_{87}, \\
& Y_{33}=A_{88},~Z_{15}=A_{91}, ~ Z_{14}=A_{93}, ~Z_{12}=A_{94}, ~ Z_{12}=A_{96}, ~Z_{13}=A_{97}, ~ Z_{11}=A_{99}.\end{aligned}$$ Hence, $(X,Y,Z)$ can be expressed as (\[equ0041\])-(\[equ0043\]) by (\[equ0045\])-(\[equ0047\]).
Conversely, assume that the equalities in (\[equ00401\]) hold, then by (\[equ0022\])-(\[equ0024\]) and (\[equ0044\])-(\[equ0048\]), it can be verified that the matrices have the forms of (\[equ0041\])-(\[equ0043\]) is a solution of (\[equ0044\]), i.e., (\[system002\]).
In our opinion the presented expression of the general solution is more useful than the expression found by Wang et al. [@QWWangandyushaowen], since the latter can not be used to consider the maximal and minimal ranks of the general solution to (\[system002\]).
**The range of ranks of the general solution to (\[system002\])**
-----------------------------------------------------------------
In this section, we consider the maximal and minimal ranks of the general solution to the real quaternion matrix equation (\[system002\]).
Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Assume that equation (\[system002\]) is consistent. Then, $$\begin{aligned}
\mathop {\max }\limits_{BXE+CYF+DZG=A } r \left({Z }\right)
=\min\left\{ p_{3},~q_{3},~p_{3}+q_{3}+r_{a|e|f}-r_{d}-r_{e|f|g},~p_{3}+q_{3}+r_{abc}-r_{g}-r_{bcd},\right.\end{aligned}$$ $$\begin{aligned}
p_{3}+q_{3}+r_{d0ab0|da00c|0e000|00f00}-r_{gg|e0|0f}-r_{bd}-r_{cd},~p_{3}+q_{3}+r_{ac|e0}-r_{cd}-r_{e|f},\end{aligned}$$ $$\begin{aligned}
\left. p_{3}+q_{3}+r_{ab|f0}-r_{f|g}-r_{bd},~p_{3}+q_{3}+r_{gg00|0ab0|a00c|e000|0f00}-r_{db0|d0c}-r_{e|g}-r_{f|g}\right\},\end{aligned}$$ $$\begin{aligned}
&\mathop {\min }\limits_{ BXE+CYF+DZG=A } r \left({Z }\right)
\\=&r_{d0ab0|da00c|0e000|00f00}+r_{gg00|0ab0|a00c|e000|0f00}+r_{abc}+r_{a|e|f}+r_{bd}+r_{e|g}+r_{bc}+r_{e|f}\\&-r_{d0ab00|da00cb|0e0000|00f000}
-r_{gg00|0ab0|a00c|e000|0f00|0e00}
\\&+ \max\big\{ r_{ac|e0}-r_{d0ab0|da00c|0e000|00f00|00e00}-r_{gg000|0ab0c|a00c0|e0000|0f000},\\&
\qquad \qquad r_{ab|f0}-r_{d0ab0|da00c|0e000|00f00|0f000}-r_{gg000|0ab00|a00cb|e0000|0f000}\big\}.\end{aligned}$$
It follows from Theorem \[theorem04\] that the expression of $Z$ in (\[system002\]) can be expressed as $Z=T_{3}^{-1}\widehat{Z}V_{3}^{-1},$ where $\widehat{Z}$ is given in (\[equ0043\]). Clearly, $r(Z)=r(T_{3}^{-1}\widehat{Z}V_{3}^{-1})=r(\widehat{Z}).$ Now we consider the maximal and minimal ranks of $\widehat{Z}$. Applying Lemma \[lemma01\] to the variable matrices $(Z_{61},~Z_{62},~Z_{63},~Z_{64},~Z_{65})$ and $\begin{pmatrix}\begin{smallmatrix}Z_{16}\\Z_{26}\\Z_{36}\\Z_{46}\\Z_{56}\end{smallmatrix}\end{pmatrix}$ of $\widehat{Z}$, we obtain $$\begin{aligned}
\mathop {\max }\limits_{ Z_{i6},~Z_{6i},~i=1,\ldots,5 } r ({\widehat{Z }})
=\min\left\{ p_{3},q_{3},p_{3}+q_{3}-r_{d}-r_{g}+r(\Theta),r(Z_{66})+r_{d}+r_{g}\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ Z_{i6},~Z_{6i},~i=1,\ldots,5 } r ({\widehat{Z} })
= \max\left\{ r(\Theta),r(Z_{66})\right\},\end{aligned}$$ where $$\begin{aligned}
\Theta=\bordermatrix{
~& n_{8}&n_{4} & n_{6}&n_{3}&n_{1} \cr
m_{8}&A_{99}&A_{96}&A_{97}&A_{93}&A_{91} \cr
m_{4}&A_{69}&A_{64}&A_{47}&A_{63}&A_{41}-X_{41} \cr
m_{6}&A_{79}&A_{74}&A_{77}-Y_{22}&A_{73}&A_{71}-Y_{24} \cr
m_{3}&A_{39}&A_{36}&A_{37}&A_{33}-X_{33}&A_{31}-X_{31} \cr
m_{1}&A_{19}&A_{14}-X_{14}&A_{17}-Y_{42}&A_{13}-X_{13}&Z_{55} }.\end{aligned}$$ Note that $$\mathop {\max } r\left( {Z_{66}} \right)=\mathop {\min } \left\{p_{3}-r_{d},q_{3}-r_{g}\right\},~
\mathop {\max } r\left( {Z_{66}} \right)=0.$$ Hence, we have $$\begin{aligned}
\mathop {\max }\limits_{ Z_{i6},~Z_{6i},~i=1,\ldots,6 } r ({\widehat{Z }})
=\min\left\{ p_{3},q_{3},p_{3}+q_{3}-r_{d}-r_{g}+r(\Theta)\right\},
~
\mathop {\min }\limits_{ Z_{i6},~Z_{6i},~i=1,\ldots,6 } r ({\widehat{Z} })
= r(\Theta).\end{aligned}$$ Applying Lemma \[lemma03\] to the variable matrices $\begin{pmatrix} A_{14}-X_{14},&A_{17}-Y_{42},&A_{13}-X_{13},&Z_{55} \end{pmatrix}$ of $\Theta$, we obtain $$\begin{aligned}
\mathop {\max }\limits_{ \begin{pmatrix}\begin{smallmatrix}A_{14}-X_{14},&A_{17}-Y_{42},&A_{13}-X_{13},&Z_{55}\end{smallmatrix}\end{pmatrix} } r \left({\Theta }\right)
=\min\left\{ m_{1}+r(\Theta_{1}),n_{1}+n_{3}+n_{4}+n_{6}+r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\\
A_{19}\end{pmatrix}\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ \begin{pmatrix}\begin{smallmatrix}A_{14}-X_{14},&A_{17}-Y_{42},&A_{13}-X_{13},&Z_{55}\end{smallmatrix}\end{pmatrix} } r \left({\Theta }\right)
= r(\Theta_{1})+r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\\
A_{19}\end{pmatrix}-r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\end{pmatrix},\end{aligned}$$ where $$\begin{aligned}
\Theta_{1}=\bordermatrix{
~& n_{8}&n_{4} & n_{6}&n_{3}&n_{1} \cr
m_{8}&A_{99}&A_{96}&A_{97}&A_{93}&A_{91} \cr
m_{4}&A_{69}&A_{64}&A_{47}&A_{63}&A_{41}-X_{41} \cr
m_{6}&A_{79}&A_{74}&A_{77}-Y_{22}&A_{73}&A_{71}-Y_{24} \cr
m_{3}&A_{39}&A_{36}&A_{37}&A_{33}-X_{33}&A_{31}-X_{31} }.\end{aligned}$$ Applying Lemma \[lemma03\] to the variable matrices $\begin{pmatrix}\begin{smallmatrix}A_{41}-X_{41}\\A_{71}-Y_{24}\\A_{31}-X_{31}\end{smallmatrix}\end{pmatrix}$ of $\Theta_{1}$, we obtain $$\begin{aligned}
\mathop {\max }\limits_{ \begin{pmatrix}\begin{smallmatrix}A_{41}-X_{41}\\A_{71}-Y_{24}\\A_{31}-X_{31}\end{smallmatrix}\end{pmatrix} } r \left({\Theta_{1} }\right)
=\min\left\{ m_{3}+m_{4}+m_{6}+r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93},&A_{91}\end{pmatrix},n_{1}+r(\Theta_{2})\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ \begin{pmatrix}\begin{smallmatrix}A_{41}-X_{41}\\A_{71}-Y_{24}\\A_{31}-X_{31}\end{smallmatrix}\end{pmatrix} } r \left({\Theta_{1} }\right)
= r(\Theta_{2})+r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93},&A_{91}\end{pmatrix}-r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93}\end{pmatrix},\end{aligned}$$ where $$\begin{aligned}
\Theta_{1}=\bordermatrix{
~& n_{8}&n_{4} & n_{6}&n_{3} \cr
m_{8}&A_{99}&A_{96}&A_{97}&A_{93} \cr
m_{4}&A_{69}&A_{64}&A_{47}&A_{63} \cr
m_{6}&A_{79}&A_{74}&A_{77}-Y_{22}&A_{73} \cr
m_{3}&A_{39}&A_{36}&A_{37}&A_{33}-X_{33} }.\end{aligned}$$ Applying Lemma \[lemma04\] to the variable matrices $A_{77}-Y_{22}$ and $A_{33}-X_{33}$ of $\Theta_{2}$, we obtain $$\begin{aligned}
&\mathop {\max }\limits_{ A_{77}-Y_{22},~A_{33}-X_{33} } r \left({\Theta_{2} }\right)
\nonumber\\=&\min\left\{ m_{3}+m_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{97}&A_{93}\\A_{69}&A_{64}&A_{47}&A_{63}\end{pmatrix},
n_{3}+m_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47}\\A_{39}&A_{36}&A_{37}\end{pmatrix},\right.\end{aligned}$$ $$\begin{aligned}
\left. m_{3}+n_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63}\\A_{79}&A_{74}&A_{73}\end{pmatrix},
n_{3}+n_{6}+r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74}\\A_{39}&A_{36}\end{pmatrix}
\right\}.\end{aligned}$$ $$\begin{aligned}
&\mathop {\min }\limits_{ A_{77}-Y_{22},~A_{33}-X_{33} } r \left({\Theta_{2} }\right)
=r\begin{pmatrix}A_{99}&A_{96}&A_{97}&A_{93}\\A_{69}&A_{64}&A_{47}&A_{63}\end{pmatrix}+r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74}\\A_{39}&A_{36}\end{pmatrix}\\&+\max\left\{ r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47}\\A_{39}&A_{36}&A_{37}\end{pmatrix}
-r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47} \end{pmatrix}-r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{39}&A_{36} \end{pmatrix},\right.\end{aligned}$$ $$\begin{aligned}
\left. r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63}\\A_{79}&A_{74}&A_{73}\end{pmatrix}-
r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63} \end{pmatrix}-
r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74} \end{pmatrix}
\right\}.\end{aligned}$$ Hence, $$\begin{aligned}
\mathop {\max }\limits_{BXE+CYF+DZG=A } r \left({Z }\right)
=\min\left\{ p_{3},q_{3},s_{1},s_{2},s_{3},s_{4},s_{5},s_{6}\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{BXE+CYF+DZG=A } r \left({Z }\right)
=\max\left\{ s_{7},s_{8}\right\},\end{aligned}$$ where $$\begin{aligned}
s_{1}=p_{3}+q_{3}-r_{d}-r_{g}+n_{1}+n_{3}+n_{4}+n_{6}+r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\\
A_{19}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
s_{2}=p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+m_{3}+m_{4}+m_{6}+r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93},&A_{91}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
s_{3}=p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+m_{3}+m_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{97}&A_{93}\\A_{69}&A_{64}&A_{47}&A_{63}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
s_{4}=p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+ n_{3}+m_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47}\\A_{39}&A_{36}&A_{37}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
s_{5}=p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+m_{3}+n_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63}\\A_{79}&A_{74}&A_{73}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
s_{6}=p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+n_{3}+n_{6}+r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74}\\A_{39}&A_{36}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
s_{7}= &r\begin{pmatrix}A_{99}&A_{96}&A_{97}&A_{93}\\A_{69}&A_{64}&A_{47}&A_{63}\end{pmatrix}+r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74}\\A_{39}&A_{36}\end{pmatrix}+r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\\
A_{19}\end{pmatrix}-r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\end{pmatrix}\\&+r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93},&A_{91}\end{pmatrix}-r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93}\end{pmatrix}\\&+ r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47}\\A_{39}&A_{36}&A_{37}\end{pmatrix}
-r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47} \end{pmatrix}-r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{39}&A_{36} \end{pmatrix},\end{aligned}$$ $$\begin{aligned}
s_{8}=& r\begin{pmatrix}A_{99}&A_{96}&A_{97}&A_{93}\\A_{69}&A_{64}&A_{47}&A_{63}\end{pmatrix}+r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74}\\A_{39}&A_{36}\end{pmatrix}+r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\\
A_{19}\end{pmatrix}-r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\end{pmatrix}\\&+r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93},&A_{91}\end{pmatrix}-r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93}\end{pmatrix}\\&+
r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63}\\A_{79}&A_{74}&A_{73}\end{pmatrix}-
r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63} \end{pmatrix}-
r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74} \end{pmatrix}.\end{aligned}$$ Now we pay attention to the ranks of the block matrices in $s_{i}$. Upon construction and computation, we obtain $$\begin{aligned}
\label{equh514}
r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74}\\A_{39}&A_{36}\end{pmatrix}=&r\begin{pmatrix}S_{G}&S_{G}&0&0\\0&S_{A}&S_{B}&0\\S_{A}&0&0&S_{C}\\S_{E}&0&0&0\\0&S_{F}&0&0\end{pmatrix}
-r(S_{E})-r(S_{F})-r(S_{B})-r(S_{C})\nonumber\\&-n_{3}-n_{4}-n_{6}-n_{8}\nonumber\\
=&r_{gg00|0ab0|a00c|e000|0f00}-r_{e}-r_{f}-r_{b}-r_{c}-n_{3}-n_{4}-n_{6}-n_{8},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99}&A_{96}&A_{97}&A_{93}\\A_{69}&A_{64}&A_{47}&A_{63}\end{pmatrix}
=&r\begin{pmatrix}S_{D}&0&S_{A}&S_{B}&0\\S_{D}&S_{A}&0&0&S_{C}\\0&S_{E}&0&0&0\\0&0&S_{F}&0&0\end{pmatrix}
-r(S_{E})-r(S_{F})-r(S_{B})-r(S_{C})\nonumber\\&-m_{3}-m_{4}-m_{6}-m_{8}\nonumber\\=&
r_{d0ab0|da00c|0e000|00f00}-r_{e}-r_{f}-r_{b}-r_{c}-m_{3}-m_{4}-m_{6}-m_{8},\end{aligned}$$ $$\begin{aligned}
r(A_{99},~A_{96},~A_{97},~A_{93},~A_{91})=
r(S_{A},~S_{B},~S_{C})-r(S_{B},~S_{C})=
r_{abc}-r_{bc},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\\
A_{19}\end{pmatrix}=r\begin{pmatrix}S_{A}\\ S_{E}\\ S_{F}\end{pmatrix}-r\begin{pmatrix}S_{E}\\ S_{F}\end{pmatrix}
=r_{a|e|f}-r_{e|f},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}
A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\end{pmatrix}
=&r\begin{pmatrix}S_{G}&S_{G}&0&0\\0&S_{A}&S_{B}&0\\S_{A}&0&0&S_{C}\\S_{E}&0&0&0\\0&S_{F}&0&0\\0&S_{E}&0&0\end{pmatrix}
-r\begin{pmatrix}S_{E}\\S_{F}\end{pmatrix}-r(S_{E})-r(S_{B})-r(S_{C})\nonumber\\&-n_{4}-n_{6}-n_{8}\nonumber\\
=&r_{gg00|0ab0|a00c|e000|0f00|0e00}-r_{e|f}-r_{e}-r_{b}-r_{c}-n_{4}-n_{6}-n_{8},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93}\end{pmatrix}=&
r\begin{pmatrix}S_{D}&0&S_{A}&S_{B}&0&0\\S_{D}&S_{A}&0&0&S_{C}&S_{B}\\0&S_{E}&0&0&0&0\\0&0&S_{F}&0&0&0\end{pmatrix}
-r(S_{E})-r(S_{F})-r(S_{B})\nonumber\\&-r(S_{B},~S_{C})-m_{4}-m_{6}-m_{8}\nonumber\\=&
r_{d0ab00|da00cb|0e0000|00f000}-r_{e}-r_{f}-r_{b}-r_{bc}-m_{4}-m_{6}-m_{8},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63}\\A_{79}&A_{74}&A_{73}\end{pmatrix}
=r\begin{pmatrix}S_{A}&S_{B}\\S_{F}&0\end{pmatrix}-r(S_{B})-r(S_{F})=r_{ab|f0}-r_{b}-r_{f},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47}\\A_{39}&A_{36}&A_{37}\end{pmatrix}
=r\begin{pmatrix}S_{A}&S_{C}\\S_{E}&0\end{pmatrix}-r(S_{E})-r(S_{C})=r_{ac|e0}-r_{e}-r_{c},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{39}&A_{36} \end{pmatrix}
=&r\begin{pmatrix}S_{G}&S_{G}&0&0&0\\0&S_{A}&S_{B}&0&S_{C}\\S_{A}&0&0&S_{C}&0\\S_{E}&0&0&0&0\\0&S_{F}&0&0&0\end{pmatrix}
-r(S_{E})-r(S_{F})-r(S_{C})-r(S_{B},~S_{C})\nonumber\\&-n_{3}-n_{4}-n_{6}-n_{8}\nonumber\\
=&r_{gg000|0ab0c|a00c0|e0000|0f000}-r_{e}-r_{f}-r_{c}-r_{bc}-n_{3}-n_{4}-n_{6}-n_{8},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74} \end{pmatrix}
=&r\begin{pmatrix}S_{G}&S_{G}&0&0&0\\0&S_{A}&S_{B}&0&0\\S_{A}&0&0&S_{C}&S_{B}\\S_{E}&0&0&0&0\\0&S_{F}&0&0&0\end{pmatrix}
-r(S_{E})-r(S_{F})-r(S_{B})-r(S_{B},~S_{C})\nonumber\\&-n_{3}-n_{4}-n_{6}-n_{8}\nonumber\\
=&r_{gg000|0ab00|a00cb|e0000|0f000}-r_{e}-r_{f}-r_{b}-r_{bc}-n_{3}-n_{4}-n_{6}-n_{8},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47} \end{pmatrix}
=&r\begin{pmatrix}S_{D}&0&S_{A}&S_{B}&0\\S_{D}&S_{A}&0&0&S_{C}\\0&S_{E}&0&0&0\\0&0&S_{F}&0&0\\0&0&S_{E}&0&0\end{pmatrix}
-r(S_{B})-r(S_{C})-r(S_{E})-r\begin{pmatrix}S_{E}\\S_{F}\end{pmatrix}\nonumber\\&
-m_{3}-m_{4}-m_{6}-m_{8}\nonumber\\
=&r_{d0ab0|da00c|0e000|00f00|00e00}-r_{b}-r_{c}-r_{e}-r_{e|f}-m_{3}-m_{4}-m_{6}-m_{8},\end{aligned}$$ $$\begin{aligned}
\label{equh523}
r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63} \end{pmatrix}
=&r\begin{pmatrix}S_{D}&0&S_{A}&S_{B}&0\\S_{D}&S_{A}&0&0&S_{C}\\0&S_{E}&0&0&0\\0&0&S_{F}&0&0\\0&S_{F}&0&0&0\end{pmatrix}
-r(S_{B})-r(S_{C})-r(S_{F})-r\begin{pmatrix}S_{E}\\S_{F}\end{pmatrix}\nonumber\\&-m_{3}-m_{4}-m_{6}-m_{8}\nonumber\\
=&r_{d0ab0|da00c|0e000|00f00|0f000}-r_{b}-r_{c}-r_{f}-r_{e|f}-m_{3}-m_{4}-m_{6}-m_{8}.\end{aligned}$$ Hence from (\[equh025\])-(\[equh0210\]) and (\[equh514\])-(\[equh523\]), we deduce that $$\begin{aligned}
s_{1}=p_{3}+q_{3}-r_{d}-r_{g}+n_{1}+n_{3}+n_{4}+n_{6}+r\begin{pmatrix}A_{99}\\
A_{69}\\
A_{79}\\
A_{39}\\
A_{19}\end{pmatrix}=p_{3}+q_{3}+r_{a|e|f}-r_{d}-r_{e|f|g},\end{aligned}$$ $$\begin{aligned}
s_{2}=&p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+m_{3}+m_{4}+m_{6}+r\begin{pmatrix}A_{99},&A_{96},&A_{97},&A_{93},&A_{91}\end{pmatrix}\\=&
p_{3}+q_{3}+r_{abc}-r_{g}-r_{bcd},\end{aligned}$$ $$\begin{aligned}
s_{3}=&p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+m_{3}+m_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{97}&A_{93}\\A_{69}&A_{64}&A_{47}&A_{63}\end{pmatrix}\\=&
p_{3}+q_{3}-r_{b}-r_{c}-r_{d}-r_{e}-r_{f}-r_{g}+r_{d0ab0|da00c|0e000|00f00}+n_{1}+m_{1}-m_{4}-m_{8}\\=&
p_{3}+q_{3}+r_{d0ab0|da00c|0e000|00f00}-r_{gg|e|f}-r_{bd}-r_{cd},\end{aligned}$$ $$\begin{aligned}
s_{4}=&p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+ n_{3}+m_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{97}\\A_{69}&A_{64}&A_{47}\\A_{39}&A_{36}&A_{37}\end{pmatrix}
\\=&p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+ n_{3}+m_{6}+r_{ac|e0}-r_{e}-r_{c}
=p_{3}+q_{3}+r_{ac|e0}-r_{cd}-r_{e|f},\end{aligned}$$ $$\begin{aligned}
s_{5}=&p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+m_{3}+n_{6}+r\begin{pmatrix}A_{99}&A_{96}&A_{93}\\A_{69}&A_{64}&A_{63}\\A_{79}&A_{74}&A_{73}\end{pmatrix}\\
=&p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+m_{3}+n_{6}+r_{ab|f0}-r_{b}-r_{f}=p_{3}+q_{3}+r_{ab|f0}-r_{f|g}-r_{bd},\end{aligned}$$ $$\begin{aligned}
s_{6}=&p_{3}+q_{3}-r_{d}-r_{g}+m_{1}+n_{1}+n_{3}+n_{6}+r\begin{pmatrix}A_{99}&A_{96} \\A_{69}&A_{64} \\A_{79}&A_{74}\\A_{39}&A_{36}\end{pmatrix}\\=&
p_{3}+q_{3}-r_{b}-r_{c}-r_{d}-r_{e}-r_{f}-r_{g}+r_{gg00|0ab0|a00c|e000|0f00}+m_{1}+n_{1}-n_{4}-n_{8}\\=&
p_{3}+q_{3}+r_{gg00|0ab0|a00c|e000|0f00}-r_{db0|d0c}-r_{e|g}-r_{f|g},\end{aligned}$$ $$\begin{aligned}
s_{7}=&r_{d0ab0|da00c|0e000|00f00}+r_{gg00|0ab0|a00c|e000|0f00}+r_{abc}+r_{a|e|f}+r_{bd}+r_{e|g}+r_{bc}+r_{e|f}\\&-r_{d0ab00|da00cb|0e0000|00f000}
-r_{gg00|0ab0|a00c|e000|0f00|0e00}
\\&+r_{ac|e0}-r_{d0ab0|da00c|0e000|00f00|00e00}-r_{gg000|0ab0c|a00c0|e0000|0f000},\end{aligned}$$ $$\begin{aligned}
s_{8}=&r_{d0ab0|da00c|0e000|00f00}+r_{gg00|0ab0|a00c|e000|0f00}+r_{abc}+r_{a|e|f}+r_{bd}+r_{e|g}+r_{bc}+r_{e|f}\\&-r_{d0ab00|da00cb|0e0000|00f000}
-r_{gg00|0ab0|a00c|e000|0f00|0e00}
\\&+ r_{ab|f0}-r_{d0ab0|da00c|0e000|00f00|0f000}-r_{gg000|0ab00|a00cb|e0000|0f000}.\end{aligned}$$
It is hard to derive the maximal and minimal ranks of the general solution to (\[system002\]) if we do not know the values of $m_{i},n_{i},(i=1,2,3,4,6)$.
Similarly, we can get the corresponding results on $X$ and $Y$. The proof is omitted.
Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Assume that equation (\[system002\]) is consistent. Then, $$\begin{aligned}
\mathop {\max }\limits_{BXE+CYF+DZG=A } r \left({X }\right)
=\min\left\{ p_{1},~q_{1},~p_{1}+q_{1}+r_{a|f|g}-r_{b}-r_{e|f|g},~p_{1}+q_{1}+r_{acd}-r_{e}-r_{bcd},\right.\end{aligned}$$ $$\begin{aligned}
p_{1}+q_{1}+r_{b0ad0|ba00c|0g000|00f00}-r_{gg|e0|0f}-r_{bd}-r_{bc},~p_{1}+q_{1}+r_{ac|g0}-r_{bc}-r_{f|g},\end{aligned}$$ $$\begin{aligned}
\left. p_{1}+q_{1}+r_{ad|f0}-r_{e|f}-r_{bd},~p_{1}+q_{1}+r_{ee00|0ad0|a00c|g000|0f00}-r_{db0|d0c}-r_{e|g}-r_{e|f}\right\},\end{aligned}$$ $$\begin{aligned}
&\mathop {\min }\limits_{ BXE+CYF+DZG=A } r \left({X }\right)
\\=&r_{b0ad0|ba00c|0g000|00f00}+r_{ee00|0ad0|a00c|g000|0f00}+r_{acd}+r_{a|f|g}+r_{bd}+r_{e|g}+r_{cd}+r_{f|g}\\&
-r_{b0ad00|ba00cd|0g0000|00f000}-r_{ee00|0ad0|a00c|g000|0f00|0g00}
\\&+ \max\big\{ r_{ac|g0}-r_{b0ad0|ba00c|0g000|00f00|00g00}-r_{ee000|0ad0c|a00c0|g0000|0f000},\\&\qquad \qquad
r_{ad|f0}-r_{b0ad0|ba00c|0g000|00f00|0f000}-r_{ee000|0ad00|a00cd|g0000|0f000}\big\}.\end{aligned}$$
Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Assume that equation (\[system002\]) is consistent. Then, $$\begin{aligned}
\mathop {\max }\limits_{BXE+CYF+DZG=A } r \left({Y }\right)
=\min\left\{ p_{2},~q_{2},~p_{2}+q_{2}+r_{a|e|g}-r_{c}-r_{e|f|g},~p_{2}+q_{2}+r_{abd}-r_{f}-r_{bcd},\right.\end{aligned}$$ $$\begin{aligned}
p_{2}+q_{2}+r_{c0ab0|ca00d|0e000|00g00}-r_{gg|e0|0f}-r_{bc}-r_{cd},~p_{2}+q_{2}+r_{ad|e0}-r_{cd}-r_{e|g},\end{aligned}$$ $$\begin{aligned}
\left. p_{2}+q_{2}+r_{ab|g0}-r_{f|g}-r_{bc},~p_{2}+q_{2}+r_{ff00|0ab0|a00d|e000|0g00}-r_{db0|d0c}-r_{e|f}-r_{f|g}\right\},\end{aligned}$$ $$\begin{aligned}
&\mathop {\min }\limits_{ BXE+CYF+DZG=A } r \left({Y }\right)
\\=&r_{c0ab0|ca00d|0e000|00g00}+r_{ff00|0ab0|a00d|e000|0g00}+r_{abd}+r_{a|e|g}+r_{bc}+r_{e|f}+r_{bd}+r_{e|g}\\&
-r_{c0ab00|ca00db|0e0000|00g000}-r_{ff00|0ab0|a00d|e000|0g00|0e00}
\\&+ \max\big\{ r_{ad|e0}-r_{c0ab0|ca00d|0e000|00g00|00e00}-r_{ff000|0ab0d|a00d0|e0000|0g000},\\&\qquad \qquad
r_{ab|g0}-r_{c0ab0|ca00d|0e000|00g00|0g000}-r_{ff000|0ab00|a00db|e0000|0g000}\big\}.\end{aligned}$$
**Some solvability conditions and the general solution to (\[system001\])**
---------------------------------------------------------------------------
In this section, the simultaneous decomposition of (\[array1\]) will be used to solve the real quaternion matrix equation (\[system001\]).
\[theorem05\] Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Then the equation (\[system001\]) is consistent if and only if $$\begin{aligned}
\label{solvab340}
r_{abcd|e000|f000|g000}=r_{bcd}+r_{e|f|g},~ A_{89}=0,~A_{98}=0,~A_{i,10}=0,~A_{10,i}=0,~ (i=6,7,8,9).\end{aligned}$$
In this case, the general solution to (\[system001\]) can be expressed as $$\begin{aligned}
X=T_{1}^{-1}\widehat{X}Q, \quad W=P\widehat{W}V_{1}^{-1},\quad Y=T_{2}^{-1}\widehat{Y}V_{2}^{-1},\quad Z=T_{3}^{-1}\widehat{Z}V_{3}^{-1},\end{aligned}$$ where $$\begin{aligned}
\label{equh340}
\widehat{X}=\bordermatrix{
~& n_{1}&n_{2} & n_{3}&n_{4}&n_{5}&n_{4}&n_{6}&n_{7}&n_{8}&n-r_{e|f|g} \cr
m_{1}&X_{11}&X_{12}&X_{13}&X_{14}&X_{15}&X_{16}&X_{17}&X_{18}&X_{19}&A_{1,10} \cr
m_{2}&X_{21}&X_{22}&X_{23}&X_{24}&X_{25}&X_{26}&X_{27}&X_{28}&A_{29}&A_{2,10} \cr
m_{3}&X_{31}&X_{32}&X_{33}&X_{34}&X_{35}&X_{36}&X_{37}&A_{38}&X_{39}&A_{3,10} \cr
m_{4}&X_{41}&X_{42}&X_{43}&X_{44}&X_{45}&X_{46}&X_{47}&A_{48}&A_{49}-A_{69}&A_{4,10} \cr
m_{5}&X_{51}&X_{52}&X_{53}&X_{54}&X_{55}&A_{56}&A_{57}&A_{58}&A_{59}&A_{5,10} \cr
p_{1}-r_{b}&X_{61}&X_{62}&X_{63}&X_{64}&X_{65}&X_{66}&X_{67}&X_{68}&X_{69}&X_{6,10} },\end{aligned}$$ $$\begin{aligned}
\widehat{W}=\bordermatrix{
~&\scriptstyle n_{1}&\scriptstyle n_{2} &\scriptstyle n_{3}&\scriptstyle n_{4}&\scriptstyle n_{5}&\scriptstyle q_{2}-r_{e} \cr
\scriptstyle m_{1}&\scriptstyle W_{11} &\scriptstyle W_{12}&\scriptstyle W_{13}&\scriptstyle W_{14}&\scriptstyle A_{15}-X_{15}&\scriptstyle W_{16} \cr
\scriptstyle m_{2}&\scriptstyle W_{21} &\scriptstyle W_{22} &\scriptstyle A_{23}-X_{23}&\scriptstyle A_{24}-X_{24}&\scriptstyle A_{25}-X_{25}&\scriptstyle W_{26} \cr
\scriptstyle m_{3}&\scriptstyle W_{31} &\scriptstyle A_{32}-X_{32}&\scriptstyle W_{33} &\scriptstyle W_{34} &\scriptstyle A_{35}-X_{35}&\scriptstyle W_{36} \cr
\scriptstyle m_{4}&\scriptstyle W_{41} &\scriptstyle A_{42}-X_{42}&\scriptstyle A_{43}-A_{63}+W_{63}-X_{43} &\scriptstyle A_{44}-A_{46}+X_{46}-X_{44} &\scriptstyle A_{45}-X_{45}&\scriptstyle W_{46} \cr
\scriptstyle m_{5}&\scriptstyle A_{51}-X_{51}&\scriptstyle A_{52}-X_{52}&\scriptstyle A_{53}-X_{53}&\scriptstyle A_{54}-X_{54}&\scriptstyle A_{55}-X_{55}&\scriptstyle W_{56} \cr
\scriptstyle m_{4}&\scriptstyle W_{61}&\scriptstyle W_{62}&\scriptstyle W_{63}&\scriptstyle A_{64}-A_{46}+X_{46}&\scriptstyle A_{65}&\scriptstyle W_{66} \cr
\scriptstyle m_{6}&\scriptstyle W_{71}&\scriptstyle W_{72}&\scriptstyle W_{73}&\scriptstyle W_{74}&\scriptstyle A_{75}&\scriptstyle W_{76} \cr
\scriptstyle m_{7}&\scriptstyle W_{81}&\scriptstyle W_{82}&\scriptstyle A_{83}&\scriptstyle A_{84}&\scriptstyle A_{85}&\scriptstyle W_{86} \cr
\scriptstyle m_{8}&\scriptstyle W_{91}&\scriptstyle A_{92}&\scriptstyle W_{93}&\scriptstyle A_{94}-A_{96}&\scriptstyle A_{95}&\scriptstyle W_{96} \cr
\scriptstyle m-r_{bcd}&\scriptstyle A_{10,1}&\scriptstyle A_{10,2}&\scriptstyle A_{10,3}&\scriptstyle A_{10,4}&\scriptstyle A_{10,5}&\scriptstyle W_{10,6} },\end{aligned}$$ $$\begin{aligned}
&\widehat{Y}=\nonumber\\&\bordermatrix{
~&\scriptscriptstyle n_{4}&\scriptscriptstyle n_{6} &\scriptscriptstyle n_{7}&\scriptscriptstyle n_{1}&\scriptscriptstyle n_{2}&\scriptscriptstyle q_{2}-r_{f} \cr
\scriptscriptstyle m_{4}&\scriptscriptstyle A_{66}-A_{64}+X_{46}&\scriptscriptstyle A_{67}-A_{47}+X_{47}&\scriptscriptstyle A_{68}&\scriptscriptstyle A_{61}-A_{41}+X_{41}+W_{41}-W_{61}&\scriptscriptstyle A_{62}-W_{62}&\scriptscriptstyle Y_{16} \cr
\scriptscriptstyle m_{6}&\scriptscriptstyle A_{76}-A_{74}+W_{74}&\scriptscriptstyle Y_{22}&\scriptscriptstyle A_{78}&\scriptscriptstyle Y_{24}&\scriptscriptstyle A_{72}-W_{72}&\scriptscriptstyle Y_{26} \cr
\scriptscriptstyle m_{7}&\scriptscriptstyle A_{86}&\scriptscriptstyle A_{87}&\scriptscriptstyle A_{88}&\scriptscriptstyle A_{81}-W_{81}&\scriptscriptstyle A_{82}-W_{82}&\scriptscriptstyle Y_{36} \cr
\scriptscriptstyle m_{1}&\scriptscriptstyle A_{16}-A_{14}+X_{14}-X_{16}-W_{14}&\scriptscriptstyle Y_{42}&\scriptscriptstyle A_{18}-X_{18}&\scriptscriptstyle Y_{44}&\scriptscriptstyle A_{12}-(X_{12}+W_{12})&\scriptscriptstyle Y_{46} \cr
\scriptscriptstyle m_{2}&\scriptscriptstyle A_{26}-X_{26}&\scriptscriptstyle A_{27}-X_{27}&\scriptscriptstyle A_{28}-X_{28}&\scriptscriptstyle A_{21}-(X_{21}+W_{21})&\scriptscriptstyle A_{22}-(X_{22}+W_{22})&\scriptscriptstyle Y_{56} \cr
\scriptscriptstyle p_{2}-r_{c}&\scriptscriptstyle Y_{61}&\scriptscriptstyle Y_{62}&\scriptscriptstyle Y_{63}&\scriptscriptstyle Y_{64}&\scriptscriptstyle Y_{65}&\scriptscriptstyle Y_{66} },\end{aligned}$$ $$\begin{aligned}
\label{equh343}
&\widehat{Z}=\nonumber\\&\bordermatrix{
~&\scriptstyle n_{8}&\scriptstyle n_{4} &\scriptstyle n_{6}&\scriptstyle n_{3}&\scriptstyle n_{1}&\scriptstyle q_{3}-r_{g} \cr
\scriptstyle m_{8}&\scriptstyle A_{99}&\scriptstyle A_{96}&\scriptstyle A_{97}&\scriptstyle A_{93}-W_{93}&\scriptstyle A_{91}-W_{91}&\scriptstyle Z_{16} \cr
\scriptstyle m_{4}&\scriptstyle A_{69}&\scriptstyle A_{46}-X_{46}&\scriptstyle A_{47}-X_{47}&\scriptstyle A_{63}-W_{63}&\scriptstyle A_{41}-(X_{41}+W_{41})&\scriptstyle Z_{26} \cr
\scriptstyle m_{6}&\scriptstyle A_{79}&\scriptstyle A_{74}-W_{74}&\scriptstyle A_{77}-Y_{22}&\scriptstyle A_{73}-W_{73}&\scriptstyle A_{71}-(Y_{24}+W_{71})&\scriptstyle Z_{36} \cr
\scriptstyle m_{3}&\scriptstyle A_{39}-X_{39}&\scriptstyle A_{34}-(X_{34}+W_{34})&\scriptstyle A_{37}-X_{37}&\scriptstyle A_{33}-(X_{33}+W_{33})&\scriptstyle A_{31}-(X_{31}+W_{31})&\scriptstyle Z_{46} \cr
\scriptstyle m_{1}&\scriptstyle A_{19}-X_{19}&\scriptstyle A_{14}-(X_{14}+W_{14})&\scriptstyle A_{17}-(X_{17}+Y_{42})&\scriptstyle A_{13}-(X_{13}+W_{13})&\scriptstyle Z_{55}&\scriptstyle Z_{56} \cr
\scriptstyle p_{3}-r_{d}&\scriptstyle Z_{61}&\scriptstyle Z_{62}&\scriptstyle Z_{63}&\scriptstyle Z_{64}&\scriptstyle Z_{65}&\scriptstyle Z_{66} },\end{aligned}$$ $P,Q,A_{ij},T_{i},V_{i}$ are defined in Theorem \[theorem01\], the remaining $X_{ij},Y_{ij},Z_{ij}$ in (\[equh340\])-(\[equh343\]) are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.
From Theorem \[theorem01\], we know that rewrite matrix equation (\[system001\]) is consistent is equivalent to that the following matrix equation $$\begin{aligned}
P(S_{B}T_{1}XQ^{-1}+P^{-1}WV_{1}S_{E}+S_{C}T_{2}YV_{2}S_{F}+S_{D}T_{3}ZV_{3}S_{G})Q=PS_{A}Q.\end{aligned}$$ Because $P,Q$ are nonsingular, the matrix equation (\[system001\]) can be written as $$\begin{aligned}
\label{equ70036}
S_{B}T_{1}XQ^{-1}+P^{-1}WV_{1}S_{E}+S_{C}T_{2}YV_{2}S_{F}+S_{D}T_{3}ZV_{3}S_{G}=S_{A}.\end{aligned}$$Let the matrices $$\begin{aligned}
\label{equ70037}
\widehat{X}=T_{1}XQ^{-1}=\begin{pmatrix}X_{11}&\cdots&X_{1,11}\\
\vdots&\ddots&\vdots\\
X_{61}&\cdots&X_{6,11}\end{pmatrix},~\widehat{W}=P^{-1}WV_{1}=\begin{pmatrix}W_{11}&\cdots&W_{1,6}\\
\vdots&\ddots&\vdots\\
W_{11,1}&\cdots&W_{11,6}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\label{equ70038}
\widehat{Y}=T_{2}YV_{2}=\begin{pmatrix}Y_{11}&\cdots&Y_{16}\\
\vdots&\ddots&\vdots\\
Y_{61}&\cdots&Y_{66}\end{pmatrix},~\widehat{Z}=T_{3}ZV_{3}=\begin{pmatrix}Z_{11}&\cdots&Z_{16}\\
\vdots&\ddots&\vdots\\
Z_{61}&\cdots&Z_{66}\end{pmatrix},\end{aligned}$$be partitioned in accordance with (\[equ70036\]). Substituting (\[equ70037\]) and (\[equ70038\]) into (\[equ70036\]) yields $$\begin{aligned}
\label{equh347}
S_{A}-S_{B}\widehat{X}-\widehat{W}S_{E}-S_{C}\widehat{Y}S_{F}-S_{D}\widehat{Z}S_{G}\triangleq\begin{pmatrix}\Omega_{11}&\Omega_{12}\\ \Omega_{21}&\Omega_{22}\end{pmatrix}=0,\end{aligned}$$ where $$\begin{aligned}
\label{equh349}
&\Omega_{11}=\nonumber\\&\begin{pmatrix}
\scriptscriptstyle A_{11}-(X_{11}+W_{11}+Y_{44}+Z_{55})&\scriptscriptstyle \scriptscriptstyle A_{12}-(X_{12}+W_{12}+Y_{45})&\scriptscriptstyle \scriptscriptstyle A_{13}-(X_{13}+W_{13}+Z_{54})&\scriptscriptstyle \scriptscriptstyle A_{14}-(X_{14}+W_{14}+Z_{52})&\scriptscriptstyle \scriptscriptstyle A_{15}-(X_{15}+W_{15})\\
\scriptscriptstyle A_{21}-(X_{21}+W_{21}+Y_{54})&\scriptscriptstyle A_{22}-(X_{22}+W_{22}+Y_{55})&\scriptscriptstyle A_{23}-(X_{23}+W_{23})&\scriptscriptstyle A_{24}-(X_{24}+W_{24})&\scriptscriptstyle A_{25}-(X_{25}+W_{25})\\
\scriptscriptstyle A_{31}-(X_{31}+W_{31}+Z_{45})&\scriptscriptstyle A_{32}-(X_{32}+W_{32})&\scriptscriptstyle A_{33}-(X_{33}+W_{33}+Z_{44})&\scriptscriptstyle A_{34}-(X_{34}+W_{34}+Z_{42})&\scriptscriptstyle A_{35}-(X_{35}+W_{35})\\
\scriptscriptstyle A_{41}-(X_{41}+W_{41}+Z_{25})&\scriptscriptstyle A_{42}-(X_{42}+W_{42})&\scriptscriptstyle A_{43}-(X_{43}+W_{43}+Z_{24})&\scriptscriptstyle A_{44}-(X_{44}+W_{44}+Z_{22})&\scriptscriptstyle A_{45}-(X_{45}+W_{45})\\
\scriptscriptstyle A_{51}-(X_{51}+W_{51})&\scriptscriptstyle A_{52}-(X_{52}+W_{52})&\scriptscriptstyle A_{53}-(X_{53}+W_{53})&\scriptscriptstyle A_{54}-(X_{54}+W_{54})&\scriptscriptstyle A_{55}-(X_{55}+W_{55})
\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
&\Omega_{12}=\nonumber\\&\begin{pmatrix}
\scriptstyle A_{16}-(Y_{41}+Z_{52}+X_{16}), &\scriptstyle A_{17}-(Y_{42}+Z_{53}+X_{17}), &\scriptstyle A_{18}-(Y_{43}+X_{18}), &\scriptstyle A_{19}-(Z_{51}+X_{19}), &\scriptstyle A_{1,10}-X_{1,10}, &\scriptstyle -X_{1,11} \\
\scriptstyle A_{26}-(Y_{51}+X_{26}), &\scriptstyle A_{27}-(Y_{52}+X_{27}),&\scriptstyle A_{28}-(Y_{53}+X_{28}), &\scriptstyle A_{29}-X_{29}, &\scriptstyle A_{2,10}- X_{2,10}, &\scriptstyle -X_{2,11} \\
\scriptstyle A_{36}-(Z_{42}+X_{36}), &\scriptstyle A_{37}-(Z_{43}+X_{37}),&\scriptstyle A_{38}-X_{38}, &\scriptstyle A_{39}-(Z_{41}+X_{39)}, &\scriptstyle A_{3,10}- X_{3,10},&\scriptstyle -X_{3,11} \\
\scriptstyle A_{46}-(Z_{22}+X_{46}), &\scriptstyle A_{47}-(Z_{23}+X_{47}), &\scriptstyle A_{48}-X_{48},&\scriptstyle A_{49}-(Z_{21}+X_{49}), &\scriptstyle A_{4,10}-X_{4,10}, &\scriptstyle -X_{4,11} \\
\scriptstyle A_{56}-X_{56}, &\scriptstyle A_{57}-X_{57} &\scriptstyle A_{58}- X_{58}, &\scriptstyle A_{59}- X_{59},&\scriptstyle A_{5,10}-X_{5,10},&\scriptstyle -X_{5,11}
\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\Omega_{21}=\begin{pmatrix}
\scriptstyle A_{61}-(Y_{14}+W_{61}+Z_{25}) &\scriptstyle A_{62}-(Y_{15}+W_{62}) &\scriptstyle A_{63}-(Z_{24}+W_{63}) &\scriptstyle A_{64}-(Z_{22}+W_{64}) &\scriptstyle A_{65}-W_{65}\\
\scriptstyle A_{71}-(Y_{24}+W_{71} +Z_{35})&\scriptstyle A_{72}-(Y_{25}+W_{72}) &\scriptstyle A_{73}-(Z_{34}+W_{73}) &\scriptstyle A_{74}-(Z_{32}+W_{74}) &\scriptstyle A_{75}-W_{75} \\
\scriptstyle A_{81}-(Y_{34} +W_{81})&\scriptstyle A_{82}-(Y_{35}+W_{82})&\scriptstyle A_{83}-W_{83} &\scriptstyle A_{84}-W_{84} &\scriptstyle A_{85}-W_{85}\\
\scriptstyle A_{91}-(Z_{15}+W_{91}) &\scriptstyle A_{92}-W_{92}&\scriptstyle A_{93}-(Z_{14}+W_{93}) &\scriptstyle A_{94}-(Z_{12}+W_{94}) &\scriptstyle A_{95}- W_{95}\\
\scriptstyle A_{10,1}-W_{10,1}&\scriptstyle A_{10,2}-W_{10,2} &\scriptstyle A_{10,3}-W_{10,3} &\scriptstyle A_{10,4}-W_{10,4}&\scriptstyle A_{10,5}-W_{10,5} \\
-W_{11,1} &\scriptstyle -W_{11,2}&\scriptstyle -W_{11,3} &\scriptstyle - W_{11,4} &\scriptstyle - W_{11,5}
\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
\label{equh352}
\Omega_{22}=\begin{pmatrix}
\scriptstyle A_{66}-(Y_{11} +Z_{22})&\scriptstyle A_{67}-(Y_{12} +Z_{23}) &\scriptstyle A_{68}-Y_{13}&\scriptstyle A_{69}-Z_{21} &\scriptstyle A_{6,10} &\scriptstyle 0 \\
\scriptstyle A_{76}-(Y_{21} +Z_{32})&\scriptstyle A_{77}-(Y_{22}+Z_{33}) &\scriptstyle A_{78}-Y_{23} &\scriptstyle A_{79}-Z_{31}&\scriptstyle A_{7,10} &\scriptstyle 0 \\
\scriptstyle A_{86}-Y_{31} &\scriptstyle A_{87}-Y_{32} &\scriptstyle A_{88}-Y_{33} &\scriptstyle A_{89} &\scriptstyle A_{8,10}&\scriptstyle 0\\
\scriptstyle A_{96}-Z_{12} &\scriptstyle A_{97}-Z_{13} &\scriptstyle A_{98} &\scriptstyle A_{99}-Z_{11} &\scriptstyle A_{9,10} &\scriptstyle 0 \\
\scriptstyle A_{10,6} &\scriptstyle A_{10,7} &\scriptstyle A_{10,8}&\scriptstyle A_{10,9} &\scriptstyle 0&\scriptstyle 0\\
0 &\scriptstyle 0&\scriptstyle 0 &\scriptstyle 0 &\scriptstyle 0 &\scriptstyle I_{t}
\end{pmatrix}.\end{aligned}$$
If the equation (\[system001\]) has a solution $(X,W,Y,Z)$, then by (\[equh347\]), we have that the equalities in (\[solvab340\]) hold, and $$\Omega_{11}=0,\Omega_{12}=0,\Omega_{21}=0,\Omega_{22}=0.$$
Conversely, assume that the equalities in (\[solvab340\]) hold, then by (\[equ0022\])-(\[equ0024\]) and (\[equh349\])-(\[equh352\]), it can be verified that the matrices have the forms of (\[equh340\])-(\[equh343\]) is a solution of (\[equ70036\]), i.e., (\[system002\]).
The presented expressions of $X$ and $W$ are more useful than the expressions found by Wang and He [@wanghe], since the latter can not be used to discuss the range of ranks of $X$ and $W$ to (\[system001\]).
**The range of ranks of the general solution to (\[system001\])**
-----------------------------------------------------------------
We in this section consider the range of ranks of the general solution to (\[system001\]).
Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Assume that equation (\[system001\]) is consistent. Then, $$\begin{aligned}
\mathop {\max }\limits_{BX+WE+CYF+DZG=A } r \left({X }\right)
=&\min\big\{ p_{1},~n,~p_{1}+r_{acd|e00}-r_{bcd},~p_{1}+r_{a|e|f|g}-r_{b},\\&
p_{1}+r_{b0ad0|ba00c|0g000|00f00|0e000|00e00}-r_{e|f|g}-r_{bc}-r_{bd},\\&\qquad p_{1}+r_{ac|e0|g0}-r_{bc},~p_{1}+r_{ad|e0|f0}-r_{bd}\big\},\end{aligned}$$ $$\begin{aligned}
&\mathop {\min }\limits_{ BX+WE+CYF+DZG=A } r \left({X }\right)
\\=&r_{a|e|f|g}+r_{acd|e00}-r_{e}-r_{b0ad00|ba00cd|0g0000|00f000|0e0000|00e000}\\&+r_{bd}+r_{cd}+r_{b0ad0|ba00c|0g000|00f00|0e000|00e00}+r_{e|f|g}\\&+
\min\big\{r_{ac|e0|g0}-r_{b0ad0|ba00c|0g000|00f00|00g00|0e000|00e00}-r_{ad0c|a0c0|e000|f000|g000},
\\&\qquad \qquad r_{ad|e0|f0}-r_{b0ad0|ba00c|0g000|00f00|0e000|00e00|0f000}-r_{ad00|a0cd|e000|f000|g000} \big\}.\end{aligned}$$
It follows from Theorem \[theorem05\] that the expression of $X$ in (\[system001\]) can be expressed as $X=T_{1}^{-1}\widehat{X}Q,$ where $\widehat{X}$ is given in (\[equh340\]). Clearly, $r(X)=r(T_{1}^{-1}\widehat{X}Q)=r(\widehat{X}).$ Now we consider the maximal and minimal ranks of $\widehat{X}$. Applying Lemma \[lemma02\] to the variable matrices $\begin{pmatrix}\begin{smallmatrix}X_{11}&\cdots&X_{15}\\ \vdots&\ddots&\vdots\\X_{51}&\cdots&X_{55}\end{smallmatrix}\end{pmatrix}$ and $(X_{66},~X_{67},~X_{68},~X_{69},~X_{6,10})$ of $\widehat{X}$, we obtain $$\begin{aligned}
&\mathop {\max }\limits_{ \begin{pmatrix}\begin{smallmatrix}X_{11}&\cdots&X_{15}\\ \vdots&\ddots&\vdots\\X_{51}&\cdots&X_{55}\end{smallmatrix}\end{pmatrix},
X_{6i},(i=6,\ldots,10)} r ({\widehat{X }})
\\&=\min\left\{ p_{1},n,p_{1}-r_{b}+r_{e}+r(\Phi_{1}),r(X_{61},X_{62},X_{63},X_{64},X_{65})+r_{b}+n_{4}+n_{6}+n_{7}+n_{8}+n-r_{e|f|g}\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ \begin{pmatrix}\begin{smallmatrix}X_{11}&\cdots&X_{15}\\ \vdots&\ddots&\vdots\\X_{51}&\cdots&X_{55}\end{smallmatrix}\end{pmatrix},
X_{6i},(i=6,\ldots,10)} r ({\widehat{X }})
= \mathop {\max }\left\{r(\Phi_{1}),~r(X_{61},X_{62},X_{63},X_{64},X_{65})\right\},\end{aligned}$$ where $$\begin{aligned}
\Phi_{1}=\bordermatrix{
~& n_{4}&n_{6}&n_{7}&n_{8}&n-r_{e|f|g} \cr
m_{1}& X_{16}&X_{17}&X_{18}&X_{19}&A_{1,10} \cr
m_{2}& X_{26}&X_{27}&X_{28}&A_{29}&A_{2,10} \cr
m_{3}& X_{36}&X_{37}&A_{38}&X_{39}&A_{3,10} \cr
m_{4}& X_{46}&X_{47}&A_{48}&A_{49}-A_{69}&A_{4,10} \cr
m_{5}& A_{56}&A_{57}&A_{58}&A_{59}&A_{5,10} }.\end{aligned}$$ Note that $$\mathop {\max }\left\{ r(X_{61},X_{62},X_{63},X_{64},X_{65})\right\}=\mathop {\min }\left\{p_{1}-r_{b},~r_{b}\right\},
~\mathop {\min }\left\{ r(X_{61},X_{62},X_{63},X_{64},X_{65})\right\}=0.$$ Hence, we obtain $$\begin{aligned}
\mathop {\max }\limits_{ \begin{pmatrix}\begin{smallmatrix}X_{11}&\cdots&X_{15}\\ \vdots&\ddots&\vdots\\X_{51}&\cdots&X_{55}\end{smallmatrix}\end{pmatrix},
X_{6,i},(i=1,\ldots,10)} r ({\widehat{X }})
=\min\left\{ p_{1},n,p_{1}-r_{b}+r_{e}+r(\Phi_{1})\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ \begin{pmatrix}\begin{smallmatrix}X_{11}&\cdots&X_{15}\\ \vdots&\ddots&\vdots\\X_{51}&\cdots&X_{55}\end{smallmatrix}\end{pmatrix},
X_{6,i},(i=1,\ldots,10)} r ({\widehat{X }})
= r(\Phi_{1}).\end{aligned}$$ Applying Lemma \[lemma03\] to the variable matrices $\begin{pmatrix}\begin{smallmatrix}X_{16}&X_{17}\\X_{26}&X_{27}\\X_{36}&X_{37}\\X_{46}&X_{47} \end{smallmatrix}\end{pmatrix}$ of $\Phi_{1}$, we obtain $$\begin{aligned}
&\mathop {\max }\limits_{ \begin{pmatrix}\begin{smallmatrix}X_{16}&X_{17}\\X_{26}&X_{27}\\X_{36}&X_{37}\\X_{46}&X_{47} \end{smallmatrix}\end{pmatrix}} r \left({\Phi_{1} }\right)
=\nonumber\\&\min\left\{ m_{1}+m_{2}+m_{3}+m_{4}+r\begin{pmatrix}A_{56},&A_{57},&A_{58},&A_{59},&A_{5,10}\end{pmatrix},n_{4}+n_{6}+r(\Phi_{2})\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ \begin{pmatrix}\begin{smallmatrix}X_{16}&X_{17}\\X_{26}&X_{27}\\X_{36}&X_{37}\\X_{46}&X_{47} \end{smallmatrix}\end{pmatrix}} r \left({\Phi_{1} }\right)
= r(\Phi_{2})+r\begin{pmatrix}A_{56},&A_{57},&A_{58},&A_{59},&A_{5,10}\end{pmatrix}-r\begin{pmatrix}A_{58},&A_{59},&A_{5,10}\end{pmatrix},\end{aligned}$$ where $$\begin{aligned}
\Phi_{2}=\bordermatrix{
~& n_{7}&n_{8}&n-r_{e|f|g} \cr
m_{1}& X_{18}&X_{19}&A_{1,10} \cr
m_{2}& X_{28}&A_{29}&A_{2,10} \cr
m_{3}& A_{38}&X_{39}&A_{3,10} \cr
m_{4}& A_{48}&A_{49}-A_{69}&A_{4,10} \cr
m_{5}& A_{58}&A_{59}&A_{5,10} }.\end{aligned}$$ Applying Lemma \[lemma03\] to the variable matrices $(X_{18},~X_{19})$ of $\Phi_{2}$, we obtain $$\begin{aligned}
\mathop {\max }\limits_{ (X_{18},~X_{19}) } r \left({\Phi_{2} }\right)
=\min\left\{ n_{7}+n_{8}+r\begin{pmatrix}A_{1,10}\\A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix},m_{1}+r(\Phi_{3})\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ (X_{18},~X_{19}) } r \left({\Phi_{2} }\right)
= r(\Phi_{3})+r\begin{pmatrix}A_{1,10}\\A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}
-r\begin{pmatrix}A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix},\end{aligned}$$ where $$\begin{aligned}
\Phi_{3}=\bordermatrix{
~& n_{7}&n_{8}&n-r_{e|f|g} \cr
m_{2}& X_{28}&A_{29}&A_{2,10} \cr
m_{3}& A_{38}&X_{39}&A_{3,10} \cr
m_{4}& A_{48}&A_{49}-A_{69}&A_{4,10} \cr
m_{5}& A_{58}&A_{59}&A_{5,10} }.\end{aligned}$$ Applying Lemma \[lemma04\] to the variable matrices $X_{28}$ and $X_{39}$ of $\Phi_{3}$, we obtain $$\begin{aligned}
&\mathop {\max }\limits_{ X_{28},X_{39}} r \left({\Phi_{3}}\right)
\nonumber\\=&\min\left\{ m_{2}+m_{3}+r\begin{pmatrix}A_{48}&A_{49}-A_{69}&A_{4,10}\\
A_{58}&A_{59}&A_{5,10}\end{pmatrix},
m_{2}+n_{8}+r\begin{pmatrix}A_{38}&A_{3,10}\\A_{48}&A_{4,10}\\A_{58}&A_{5,10}\end{pmatrix},\right.\end{aligned}$$ $$\begin{aligned}
\left. m_{3}+n_{7}+r\begin{pmatrix}A_{29}&A_{2,10}\\A_{49}-A_{69}&A_{4,10}\\A_{59}&A_{5,10}\end{pmatrix},
n_{7}+n_{8}+r\begin{pmatrix}A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}
\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{ X_{28},X_{39} } r \left({\Phi_{3}}\right)
=&r\begin{pmatrix}A_{48}&A_{49}-A_{69}&A_{4,10}\\
A_{58}&A_{59}&A_{5,10}\end{pmatrix}+r\begin{pmatrix}A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}\\&+\max\left\{ r\begin{pmatrix}A_{38}&A_{3,10}\\A_{48}&A_{4,10}\\A_{58}&A_{5,10}\end{pmatrix}
-r\begin{pmatrix}A_{48}&A_{4,10}\\A_{58}&A_{5,10}\end{pmatrix}
-r\begin{pmatrix}A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix},\right.\end{aligned}$$ $$\begin{aligned}
\left. r\begin{pmatrix}A_{29}&A_{2,10}\\A_{49}-A_{69}&A_{4,10}\\A_{59}&A_{5,10}\end{pmatrix}-r\begin{pmatrix}A_{49}-A_{69}&A_{4,10}\\A_{59}&A_{5,10}\end{pmatrix}-
r\begin{pmatrix}A_{2,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}
\right\}.\end{aligned}$$ Hence, $$\begin{aligned}
\mathop {\max }\limits_{BX+WE+CYF+DZG=A } r \left({X }\right)
=\min\left\{ p_{1},n,t_{1},t_{2},t_{3},t_{4},t_{5},t_{6}\right\},\end{aligned}$$ $$\begin{aligned}
\mathop {\min }\limits_{BX+WE+CYF+DZG=A } r \left({X }\right)
=\max\left\{ t_{7},t_{8}\right\},\end{aligned}$$ where $$\begin{aligned}
t_{1}=p_{1}-r_{b}+r_{e}+m_{1}+m_{2}+m_{3}+m_{4}+r\begin{pmatrix}A_{56},&A_{57},&A_{58},&A_{59},&A_{5,10}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
t_{2}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+n_{7}+n_{8}+r\begin{pmatrix}A_{1,10}\\A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
t_{3}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+m_{1}+m_{2}+m_{3}+r\begin{pmatrix}A_{48}&A_{49}-A_{69}&A_{4,10}\\
A_{58}&A_{59}&A_{5,10}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
t_{4}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+m_{1}+m_{2}+n_{8}+r\begin{pmatrix}A_{38}&A_{3,10}\\A_{48}&A_{4,10}\\A_{58}&A_{5,10}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
t_{5}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+m_{1}+m_{3}+n_{7}+r\begin{pmatrix}A_{29}&A_{2,10}\\A_{49}-A_{69}&A_{4,10}\\A_{59}&A_{5,10}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
t_{6}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+m_{1}+n_{7}+n_{8}+r\begin{pmatrix}A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}\geq t_{2},\end{aligned}$$ $$\begin{aligned}
t_{7}=&r\begin{pmatrix}A_{1,10}\\A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}+r\begin{pmatrix}A_{56},&A_{57},&A_{58},&A_{59},&A_{5,10}\end{pmatrix}
+r\begin{pmatrix}A_{48}&A_{49}-A_{69}&A_{4,10}\\
A_{58}&A_{59}&A_{5,10}\end{pmatrix}
\\&-r(A_{58},~A_{59},~A_{5,10})+r\begin{pmatrix}A_{38}&A_{3,10}\\A_{48}&A_{4,10}\\A_{58}&A_{5,10}\end{pmatrix}
-r\begin{pmatrix}A_{48}&A_{4,10}\\A_{58}&A_{5,10}\end{pmatrix}
-r\begin{pmatrix}A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix},\end{aligned}$$ $$\begin{aligned}
t_{8}=&r\begin{pmatrix}A_{1,10}\\A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}+r\begin{pmatrix}A_{56},&A_{57},&A_{58},&A_{59},&A_{5,10}\end{pmatrix}
+r\begin{pmatrix}A_{48}&A_{49}-A_{69}&A_{4,10}\\
A_{58}&A_{59}&A_{5,10}\end{pmatrix}\\&-r(A_{58},~A_{59},~A_{5,10})+r\begin{pmatrix}A_{29}&A_{2,10}\\A_{49}-A_{69}&A_{4,10}\\A_{59}&A_{5,10}\end{pmatrix}-r\begin{pmatrix}A_{49}-A_{69}&A_{4,10}\\A_{59}&A_{5,10}\end{pmatrix}-
r\begin{pmatrix}A_{2,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}.\end{aligned}$$ Now we pay attention to the ranks of the block matrices in $t_{i}$. Upon construction and computation, we obtain $$\begin{aligned}
\label{equh366}
r\begin{pmatrix}A_{56},&A_{57},&A_{58},&A_{59},&A_{5,10}\end{pmatrix}=&r\begin{pmatrix}S_{A}&S_{C}&S_{D}\\S_{E}&0&0\end{pmatrix}-r(S_{C},~S_{D})-r(S_{E})
\nonumber\\=&r_{acd|e00}-r_{cd}-r_{e},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{1,10}\\A_{2,10}\\A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}=r\begin{pmatrix}S_{A}\\S_{E}\\S_{F}\\S_{G}\end{pmatrix}
-r\begin{pmatrix}S_{E}\\S_{F}\\S_{G}\end{pmatrix}
=r_{a|e|f|g}-r_{e|f|g},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{48}&A_{49}-A_{69}&A_{4,10}\\
A_{58}&A_{59}&A_{5,10}\end{pmatrix}=
&r\begin{pmatrix}S_{B}&0&S_{A}&S_{D}&0\\S_{B}&S_{A}&0&0&S_{C}\\0&S_{G}&0&0&0\\0&0&S_{F}&0&0\\0&S_{E}&0&0&0\\0&0&S_{E}&0&0\end{pmatrix}
-r(S_{C})-r(S_{D})\nonumber\\&-r\begin{pmatrix}S_{E}\\S_{F}\end{pmatrix}-r\begin{pmatrix}S_{E}\\S_{G}\end{pmatrix}-m_{2}-m_{3}-m_{4}-m_{5}
\nonumber\\
=&r_{b0ad0|ba00c|0g000|00f00|0e000|00e00}\nonumber\\&-r_{c}-r_{d}-r_{e|f}-r_{e|g}-m_{2}-m_{3}-m_{4}-m_{5},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{38}&A_{3,10}\\A_{48}&A_{4,10}\\A_{58}&A_{5,10}\end{pmatrix}=r\begin{pmatrix}S_{A}&S_{C}\\S_{E}&0\\S_{G}&0\end{pmatrix}
-r\begin{pmatrix}S_{E}\\S_{G}\end{pmatrix}-r(S_{C})=r_{ac|e0|g0}-r_{e|g}-r_{c},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{29}&A_{2,10}\\A_{49}-A_{69}&A_{4,10}\\A_{59}&A_{5,10}\end{pmatrix}=r\begin{pmatrix}S_{A}&S_{D}\\S_{E}&0\\S_{F}&0\end{pmatrix}
-r(S_{D})-r\begin{pmatrix}S_{E}\\S_{F}\end{pmatrix}=r_{ad|e0|f0}-r_{d}-r_{e|f},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{3,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}=&r\begin{pmatrix}S_{A}&S_{D}&0&S_{C}\\S_{A}&0&S_{C}&0\\S_{E}&0&0&0\\S_{F}&0&0&0\\S_{G}&0&0&0\end{pmatrix}
-r(S_{C},~S_{D})-r(S_{C})-r\begin{pmatrix}S_{E}\\S_{F}\\S_{G}\end{pmatrix}\nonumber\\
=&r_{ad0c|a0c0|e000|f000|g000}-r_{cd}-r_{c}-r_{e|f|g},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix}A_{2,10}\\A_{4,10}\\A_{5,10}\end{pmatrix}
=&r\begin{pmatrix}S_{A}&S_{D}&0&0\\S_{A}&0&S_{C}&S_{D}\\S_{E}&0&0&0\\S_{F}&0&0&0\\S_{G}&0&0&0\end{pmatrix}-r(S_{C},~S_{D})-r(S_{D})
-r\begin{pmatrix}S_{E}\\S_{F}\\S_{G}\end{pmatrix}\nonumber\\
=&r_{ad00|a0cd|e000|f000|g000}-r_{cd}-r_{d}-r_{e|f|g},\end{aligned}$$ $$\begin{aligned}
r(A_{58},~A_{59},~A_{5,10})=
&r\begin{pmatrix}S_{B}&0&S_{A}&S_{D}&0&0\\S_{B}&S_{A}&0&0&S_{C}&S_{D}\\0&S_{G}&0&0&0&0\\0&0&S_{F}&0&0&0\\0&S_{E}&0&0&0&0\\0&0&S_{E}&0&0&0\end{pmatrix}
\nonumber\\&-r(S_{B},~S_{D})-r(S_{C},~S_{D})-r\begin{pmatrix}S_{E}\\S_{F}\end{pmatrix}-r\begin{pmatrix}S_{E}\\S_{G}\end{pmatrix}\nonumber\\=
&r_{b0ad00|ba00cd|0g0000|00f000|0e0000|00e000}\nonumber\\&-r_{bd}-r_{cd}-r_{e|f}-r_{e|g},\end{aligned}$$ $$\begin{aligned}
r\begin{pmatrix} A_{49}-A_{69}&A_{4,10}\\
A_{59}&A_{5,10}\end{pmatrix}
=&r\begin{pmatrix}S_{B}&0&S_{A}&S_{D}&0\\S_{B}&S_{A}&0&0&S_{C}\\0&S_{G}&0&0&0\\0&0&S_{F}&0&0\\0&S_{E}&0&0&0\\0&0&S_{E}&0&0\\0&S_{F}&0&0&0\end{pmatrix}
-r(S_{C})-r(S_{D})-r\begin{pmatrix}S_{E}\\S_{F}\\S_{G}\end{pmatrix}-r\begin{pmatrix}S_{E}\\S_{F}\end{pmatrix} \nonumber\\&-m_{2}-m_{3}-m_{4}-m_{5} \nonumber\\
=&r_{b0ad0|ba00c|0g000|00f00|0e000|00e00|0f000}\nonumber\\&-r_{c}-r_{d}-r_{e|f|g}-r_{e|f}-m_{2}-m_{3}-m_{4}-m_{5},\end{aligned}$$ $$\begin{aligned}
\label{equh373}
r\begin{pmatrix} A_{48}&A_{4,10}\\
A_{58}&A_{5,10}\end{pmatrix}
=&r\begin{pmatrix}S_{B}&0&S_{A}&S_{D}&0\\S_{B}&S_{A}&0&0&S_{C}\\0&S_{G}&0&0&0\\0&0&S_{F}&0&0\\0&0&S_{G}&0&0\\0&S_{E}&0&0&0\\0&0&S_{E}&0&0\end{pmatrix}
-r(S_{C})-r(S_{D})-r\begin{pmatrix}S_{E}\\S_{F}\\S_{G}\end{pmatrix}-r\begin{pmatrix}S_{E}\\S_{G}\end{pmatrix}\nonumber\\&-m_{3}-m_{4}-m_{5}\nonumber\\
=&r_{b0ad0|ba00c|0g000|00f00|00g00|0e000|00e00}\nonumber\\&-r_{c}-r_{d}-r_{e|f|g}-r_{e|g}-m_{2}-m_{3}-m_{4}-m_{5}.\end{aligned}$$ Hence from (\[equh025\])-(\[equh0210\]) and (\[equh366\])-(\[equh373\]), we deduce that $$\begin{aligned}
t_{1}=p_{1}-r_{b}+r_{e}+m_{1}+m_{2}+m_{3}+m_{4}+r_{acd|e00}-r_{cd}-r_{e}=p_{1}+r_{acd|e00}-r_{bcd},\end{aligned}$$ $$\begin{aligned}
t_{2}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+n_{7}+n_{8}+r_{a|e|f|g}-r_{e|f|g}=p_{1}+r_{a|e|f|g}-r_{b},\end{aligned}$$ $$\begin{aligned}
t_{3}=p_{1}+r_{b0ad0|ba00c|0g000|00f00|0e000|00e00}-r_{e|f|g}-r_{bc}-r_{bd},\end{aligned}$$ $$\begin{aligned}
t_{4}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+m_{1}+m_{2}+n_{8}+r_{ac|e0|g0}-r_{e|g}-r_{c}=p_{1}+r_{ac|e0|g0}-r_{bc},\end{aligned}$$ $$\begin{aligned}
t_{5}=p_{1}-r_{b}+r_{e}+n_{4}+n_{6}+m_{1}+m_{3}+n_{7}+r_{ad|e0|f0}-r_{d}-r_{e|f}=p_{1}+r_{ad|e0|f0}-r_{bd},\end{aligned}$$ $$\begin{aligned}
t_{7}=&r_{a|e|f|g}+r_{acd|e00}-r_{e}-r_{b0ad00|ba00cd|0g0000|00f000|0e0000|00e000}
\\&+r_{bd}+r_{cd}+r_{b0ad0|ba00c|0g000|00f00|0e000|00e00}+r_{e|f|g}
\\&+r_{ac|e0|g0}-r_{b0ad0|ba00c|0g000|00f00|00g00|0e000|00e00}-r_{ad0c|a0c0|e000|f000|g000},\end{aligned}$$ $$\begin{aligned}
t_{8}=& r_{a|e|f|g}+r_{acd|e00}-r_{e}-r_{b0ad00|ba00cd|0g0000|00f000|0e0000|00e000}
\\&+r_{bd}+r_{cd}+r_{b0ad0|ba00c|0g000|00f00|0e000|00e00}+r_{e|f|g} \\&+
r_{ad|e0|f0}-r_{b0ad0|ba00c|0g000|00f00|0e000|00e00|0f000}-r_{ad00|a0cd|e000|f000|g000}.\end{aligned}$$
Similarly, we can get the corresponding results on $W,Y,$ and $Z$.
Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Assume that equation (\[system001\]) is consistent. Then, $$\begin{aligned}
\mathop {\max }\limits_{BX+WE+CYF+DZG=A } r \left({W }\right)
=&\min\big\{ q_{1},~m,~q_{1}+r_{ab|f0|g0}-r_{e|f|g},~q_{1}+r_{abcd}-r_{e},\\&
q_{1}+r_{ee0000|0ad0b0|a00c0b|g00000|0f0000}-r_{bcd}-r_{e|f}-r_{e|g},\\&\qquad \qquad q_{1}+r_{abd|f00}-r_{e|f},~q_{1}+r_{abc|g00}-r_{e|g}\big\},\end{aligned}$$ $$\begin{aligned}
&\mathop {\min }\limits_{ BX+WE+CYF+DZG=A } r \left({W }\right)\nonumber
\\=&r_{abcd}+r_{ab|f0|g0}-r_{b}-r_{ee0000|0ad0b0|a00c0b|g00000|0f0000|0g0000}+r_{e|g}+r_{f|g}\\&+r_{ee0000|0ad0b0|a00c0b|g00000|0f0000}+r_{bcd}\\&+
\min\big\{r_{abd|f00}-r_{ee00000|0ad00b0|a00cd0b|g000000|0f00000}-r_{aadcb|g0000|0f000|f0000},\\&\qquad \qquad
r_{abc|g00}-r_{ee00000|0ad0b0c|a00c0b0|g000000|0f00000}-r_{aabcd|g0000|0f000|0g000} \big\}.\end{aligned}$$
Let $A\in \mathbb{H}^{m\times n},
B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n}$ and $G\in \mathbb{H}^{q_{3}\times n}$ be given. Assume that equation (\[system001\]) is consistent. Then, $$\begin{aligned}
&\mathop {\max }\limits_{BX+WE+CYF+DZG=A } r \left({Y }\right)
=\\&\min\big\{ p_{2},~q_{2},~p_{2}+q_{2}+r_{ab|e0|g0}-r_{e|f|g}-r_{bc},~p_{2}+q_{2}+r_{abd|e00}-r_{e|f}-r_{bcd}\big\},\\
&
\mathop {\min }\limits_{ BX+WE+CYF+DZG=A } r \left({Y }\right)
=r_{ab|e0|g0}+r_{abd|e00}-r_{abd|e00|g00}-r_{b}-r_{e},
\\&\mathop {\max }\limits_{BX+WE+CYF+DZG=A } r \left({Z }\right)
=\\&\min\big\{ p_{3},~q_{3},~p_{3}+q_{3}+r_{ab|e0|f0}-r_{e|f|g}-r_{bd},~p_{3}+q_{3}+r_{abc|e00}-r_{e|g}-r_{bcd}\big\},
\\&
\mathop {\min }\limits_{ BX+WE+CYF+DZG=A } r \left({Z }\right)
=r_{ab|e0|f0}+r_{abc|e00}-r_{abc|e00|f00}-r_{b}-r_{e}.\end{aligned}$$
All the results are true over octonion algebra.
**Conclusion**
==============
We have established the simultaneous decomposition of the general real quaternion matrix array (\[array1\]). We have derived all the dimensions of identity matrices in the equivalence canonical form of the matrix array (\[array1\]). Using the simultaneous decomposition of the general matrix array (\[array1\]), we have presented necessary and sufficient conditions for the existence and the general solutions to the real matrix equations (\[system002\]) and (\[system001\]), respectively. Moreover, we have given the range of ranks of the general solutions to (\[system002\]) and (\[system001\]), respectively.
As a special case of the matrix array (\[array1\]), we have derived all the dimensions of identity matrices in the equivalence canonical form of triple matrices with the same row or column numbers, which perfect the results in [@QWWangandyushaowen]. The presented expression of the general solution is more useful than the expression found in [@QWWangandyushaowen], since the latter can not be used to consider the maximal and minimal ranks of the general solution to (\[system002\]). On the other hand, Wang and He [@wanghe] gave the range of ranks of $Y$ and $Z$ to (\[system001\]), but the two present authors did not derive the range of ranks of $X$ and $W$. We in this paper have solved this problem.
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[^1]: This research was supported by the grants from the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080).
\* Corresponding author
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Michel Boileau[^1] and Steve Boyer[^2]'
title: 'On character varieties, sets of discrete characters, and non-zero degree maps'
---
Introduction
============
General introduction {#gen}
--------------------
Character variety methods have proven an essential tool for the investigation of problems in low-dimensional topology and have been instrumental in the resolution of many well-known problems. In this paper we use them to study homomorphisms between the fundamental groups of $3$-manifolds, in particular those induced by non-zero degree maps. We assume throughout that our manifolds are compact, connected, orientable, and $3$-dimensional. A [*knot manifold*]{} is a compact, connected, irreducible, orientable $3$-manifold whose boundary is an incompressible torus. We shall restrict our attention, for the most part, to [*small*]{} knot manifolds, that is, those which contain no closed essential surfaces. This is a simplifying hypothesis and though many of the results discussed in the paper extend to the general setting, we will not discuss them. A small knot manifold is atoroidal and Haken, hence it is either hyperbolic or admits a Seifert fibred structure with base orbifold of the form $D^2(p, q)$ for some integers $p, q \geq 2$.
We call a homomorphism $\varphi: \Gamma_1 \to \Gamma_2$ between two groups a [*virtual epimorphism*]{} if its image is of finite index in $\Gamma_2$. For instance, a non-zero degree map between manifolds induces a virtual epimorphism on the level of fundamental groups. The first part of the paper investigates virtual epimorphisms between the fundamental groups of small knot manifolds. We will see that the existence of such homomorphisms places constraints on the algebraic decomposition of a knot manifold’s $PSL_2(\mathbb C)$-character variety and, as a consequence, we will determine a priori bounds on the number of virtual epimorphisms between the fundamental groups of small knot manifolds with a fixed domain. This work yields minimality results which will be applied to illustrate the results of the second part of the paper. There we fix a small knot manifold $M$ and investigate various sets of characters of representations $\rho: \pi_1(M) \to PSL_2(\mathbb C)$ whose images are discrete. It turns out that the topology of these sets is intimately related to the algebraic structure of the $PSL_2(\mathbb C)$-character variety of $M$ as well as dominations of manifolds by $M$ and its Dehn fillings. In particular, we apply our results to study families of non-zero degree maps $f_n: M(\alpha_n) \to V_n$ where $M(\alpha_n)$ is the $\alpha_n$-Dehn filling $M$ and $V_n$ is either a hyperbolic manifold or $\widetilde{SL_2}$ manifold. Using this, the existence of infinite families of small, closed, connected, orientable manifolds which do not admit non-zero degree maps, other than homeomorphisms, to any hyperbolic manifold, or even manifolds which are either reducible, Haken, or admit a geometric structure, is determined.
In the remainder of the introduction we give a more detailed description of our results and the organization of the paper. Here is some notation and terminology we shall use.
Throughout, $\Gamma$ will denote a finitely generated group. We call a homomorphism $\rho: \Gamma \to PSL_2(\mathbb C)$ discrete, non-elementary, torsion free, abelian, etc. if its image has this property. If $\chi_\rho \in X_{PSL_2}(M)$ is the character of $\rho$ we will call it discrete, non-elementary, torsion free, abelian, etc. if each representation $\rho': \pi_1(M) \to PSL_2(\mathbb C)$ with $\chi_{\rho'} = \chi_\rho$ has this property. For instance we can unambiguously refer to a character as being either irreducible, non-elementary, or torsion free.
A [*slope*]{} on the boundary of a knot manifold $M$ is a $\partial M$-isotopy class of essential simple closed curves. Slopes correspond bijectively with $\pm$ pairs of primitive elements of $H_1(\partial M)$ in the obvious way. The [*longitudinal slope*]{} on $\partial M$ is the unique slope $\lambda_M$ having the property that it represents a torsion element of $H_1(M)$. When $M$ is the exterior of a knot $K$ in a closed $3$-manifold $W$, there is a unique slope $\mu_K$ on $\partial M$, called the [*meridinal slope*]{}, which is homologically trivial in a tubular neighbouhood of the knot. If $W$ is a $\mathbb Z$-homology $3$-sphere, then $\mu_K$ and $\lambda_M$ are dual in the sense that the homology classes they carry form a basis for $H_1(\partial M)$.
Each slope $\alpha$ on $\partial M$ determines an element of $\pi_1(M)$ well-defined up to conjugation and taking inverse. We will sometimes use this connection to evaluate a representation on a slope, but only in a context where the statement being made is independent of the choice of element of $\pi_1(M)$. For instance we may say that $\rho(\alpha) \in PSL_2(\mathbb C)$ is parabolic, or loxodromic, or trivial.
A representation $\rho: \pi_1(M) \to PSL_2(\mathbb C)$ is [*peripherally nontrivial*]{} if $\rho(\pi_1(\partial M))$ does not equal $\{\pm I\}$. When $M$ is small, there are only finitely many characters of representations which are not peripherally non-trivial. Indeed, there are only finitely many characters $\chi_\rho$ for which $\rho(\pi_1(\partial M))$ is trivial or a parabolic subgroup of $PSL_2(\mathbb C)$ (cf. Corollary \[smallcharactervariety\]). Thus, apart from finitely many exceptions, a discrete, torsion-free character is the character of a representation $\rho$ for which there is a unique slope $\alpha$ on $\partial M$ such that $\rho(\alpha) = \pm I$. In this case we call $\alpha$ the *slope of $\rho$*.
A *hyperbolic manifold* is one whose interior admits a complete, finite volume, hyperbolic structure. A closed manifold which admits an $\widetilde{SL_2}$ structure is called an $\widetilde{SL_2}$ manifold. Similarly we will refer to closed $Nil$ manifolds, $\mathbb E^3$ manifolds, etc. Two families of manifolds we will focus on are the family ${\cal H}$ of hyperbolic $3$-manifolds and the family ${\cal M}$ of $3$-manifolds which are either reducible, Haken, or admit a geometric structure. According to Thurston’s Geometrization Conjecture, which has been claimed by Perelman, ${\cal M}$ is the set of all compact, connected, orientable $3$-manifolds. We shall assume this holds below.
We say that $M$ *dominates* $N$, written $M \geq N$, if there is a continuous, proper map from $M$ to $N$ of non-zero degree. Moreover, if $N$ is not homeomorphic to $M$ we say that $M$ *strictly dominates* $N$. The relation $\geq$ is a partial order when restricted to manifolds in ${\cal M}$ which are aspherical but are neither torus (semi) bundles or Seifert manifolds with zero Euler number [@Wan1], [@Wan2]. This partial order is far from well-understood, even when restricted to hyperbolic 3-manifolds.
A knot manifold is [*minimal*]{} if the only knot manifold it dominates is itself. (Note that each knot manifold dominates $S^1 \times D^2$, but also that the latter is not a knot manifold.) For example, using elementary $3$-manifold topology we can see that the total space of a punctured torus bundle is minimal if and only if its monodromy is not a proper power (see \[BWa1, Prop. 2.6\]). A closed, connected, orientable $3$-manifold is [*minimal*]{} if the only manifold it dominates is one with finite fundamental group. (It is easy to see that every closed, connected, orientable $3$-manifold dominates each such manifold with finite fundamental group.) A manifold $V$ is ${\cal H}$-[*minimal*]{} if the only manifold in ${\cal H}$ it dominates is itself. Note that we do not require that $V \in {\cal H}$. Similarly we can define the notion of an ${\cal M}$-[*minimal*]{} manifold. If the reader prefers not to assume the geometrization theorem of Perelman, then in what follows, a closed, minimal $3$-manifold should be taken to mean a closed, ${\cal M}$-minimal manifold.
Virtual epimorphisms of knot manifold groups and domination
-----------------------------------------------------------
Let $X_0$ be an algebraic component of $X_{PSL_2}(\Gamma)$, the $PSL_2(\mathbb C)$-character variety of $\Gamma$ (see §\[varieties\]). In §\[subvarieties\] we note that $X_0$ determines a normal subgroup $\hbox{Ker}(X_0) \subset \Gamma$ with the property that for the generic character $\chi_\rho \in X_0$, $\hbox{Ker}(X_0) = \hbox{kernel}(\rho)$. For a small knot manifold $M$ there is a closely related normal subgroup $K_M(X_0)$ (§\[seminorm\]). Define ${\cal I}_M$ to be the set of isomorphism classes of groups $\pi_1(M) / K_M(X_0)$ where $X_0$ ranges over the set of algebraic components which contain an irreducible character. In Theorem \[epibound\] we show that the cardinal $|{\cal I}_M|$ gives an upper bound for number of isomorphism classes of groups $\pi_1(N)$ where $N$ is a small knot manifold for which there is an epimorphism $\varphi: \pi_1(M) \to \pi_1(N)$. For epimorphisms induced by non-zero degree maps we obtain an interesting refinement.
The set of algebraic components of $X_{PSL_2}(M)$, the $PSL_2(\mathbb C)$-character variety of the fundamental group of a small knot manifold $M$, are partitioned into two types - those whose Culler-Shalen seminorms are norms and those which are not (see §\[culler-shalen\]). Let ${\cal N}_M$ be the set of isomorphism classes of groups $\pi_1(M) / K_M(X_0)$ where $X_0$ ranges over the set of algebraic components whose associated Culler-Shalen seminorm is a norm.
We will consider two such maps $f_j: M \to N_j$ ($j = 1,2$) between knot manifolds to be [*equivalent*]{} if there is a homeomorphism $h: N_1 \to N_2$ such that the following diagram commutes up to homotopy: $$\xymatrix@R=20pt@C=70pt
{& (N_1, \partial N_1)\ar[dd]^{h} \\
(M, \partial M)\ar[ur]^{f_1}\ar[dr]_{f_2} & \\
& (N_2, \partial N_2)}$$ The following theorem is proven in §\[seminorm\] (see Theorem \[dombound\]).
\[domboundintro\] Let $M$ be a small knot manifold.\
$(1)$ The number of equivalence classes of $\pi_1$-surjective non-zero degree maps $M \to N$ is bounded above by $|{\cal I}_M|$. More precisely,\
$(a)$ The number of equivalence classes of $\pi_1$-surjective non-zero degree maps $M \to N$\
where $N$ is hyperbolic is bounded above by $|{\cal N}_M|$.\
$(b)$ The number of equivalence classes of $\pi_1$-surjective non-zero degree maps $M \to N$\
where $N$ is Seifert is bounded above by $|{\cal I}_M| - |{\cal N}_M|$.\
$(2)(a)$ The number of equivalence classes of non-zero degree maps from $M$ to a hyperbolic\
manifold is bounded above by a constant depending only on $X_{PSL_2}(M)$.\
$(b)$ The number of homeomorphism classes of Seifert fibred manifolds dominated by $M$\
is bounded above by a constant depending only on $X_{PSL_2}(M)$.
This result can be used to give many examples of minimal and ${\cal H}$-minimal knot exteriors. For instance it is shown in Example \[twistpretzel\] that the exterior of the $(-2,3,n)$ pretzel knot is ${\cal H}$-minimal while those of twist knots or $(-2,3,n)$ pretzel knots ($n \not \equiv 0$ (mod $3$)) are minimal. On the other hand, the theorem’s usefulness is tempered by the difficulty, in general, of determining the algebraic decomposition of $X_{PSL_2}(M)$. Indeed, there only handful families of knot manifolds whose character varieties have been explicitly determined. For instance, little definite information is known on the number of algebraic components of the character varieties of two-bridge knot exteriors. We can, nevertheless, use character variety methods to study dominations by such manifolds. In §\[rigidity\] we associate to each small knot exterior $M$ a function $d_M: \pi_1(M) \to \mathbb Z$ with the property that if $\varphi: \pi_1(M) \to \pi_1(N)$ is a virtual epimorphism, then $d_N(\varphi(\gamma)) \leq d_M(\gamma)$ for all $\gamma \in \pi_1(M)$. Further, if $\gamma$ is not rigid (§\[rigidity\]), then $d_N(\varphi(\gamma)) = d_M(\gamma)$ implies that $\varphi$ is 1-1. This leads to our next result (see Theorem \[homomorphismsequence\]).
\[intro:domseqbound\] Let $M$ be a small knot manifold and consider a sequence of virtual epimorphisms $$\pi_1(M) \stackrel{\varphi_1}{\longrightarrow} \pi_1(N_1) \stackrel{\varphi_2}{\longrightarrow} \cdots \stackrel{\varphi_n}{\longrightarrow} \pi_1(N_n)$$ none of which is injective. If $N_i$ is small for each $i$, then $n \leq d_M(\gamma)$ for each totally non-rigid element $\gamma \in \pi_1(M)$. Moreover, if $n = d_M(\gamma)$, then $N_n$ is a twisted $I$-bundle over the Klein bottle.
The non-injectivity of $\varphi_i$ is a necessary condition since small knot manifolds which are Seifert fibred admit self-coverings of arbitrarily large degree.
Precise calculations of $d_M(\gamma)$ can be made for various families of knot manifolds. In the case where $M$ is the exterior of a two-bridge knot we obtain the following explicit bounds. (See Theorems \[2bridgebound\] and \[2bridgedegree1bound\].)
\[twobridgedomboundintro\] Let $M_{p/q}$ denote the exterior of the $(\frac{p}{q})$ two-bridge knot and consider a sequence of virtual epimorphisms none of which is injective: $$\pi_1(M_{p/q}) \stackrel{\varphi_1}{\longrightarrow} \pi_1(N_1) \stackrel{\varphi_2}{\longrightarrow} \cdots \stackrel{\varphi_n}{\longrightarrow} \pi_1(N_n)$$ $(1)$ If $N_i$ is small for each $i$, then $n < \frac{p-1}{2}$.\
$(2)$ If each $\varphi_i$ is induced by a non-zero degree map, then $n+1$ is bounded above by the number of distinct divisors of $p$.\
$(3)$ If each $\varphi_i$ is induced by a degree one map, then $n+1$ is bounded above by the number of distinct prime divisors of $p$.
Theorem \[twobridgedomboundintro\] immediately yields an infinite family of minimal two-bridge knot exteriors.
$\;$\
$(1)$ If $p$ is prime, then $M_{p/q}$ is minimal if and only if it is hyperbolic $($i.e. $q \not \equiv \pm 1 \hbox{ $($mod $p$ $)$}$.$)$\
$(2)$ If $p$ is a prime power, any degree one map $M_{p/q} \to N$, $N$ a knot manifold, is homotopic to a homeomorphism.
Two-bridge knot exteriors are not minimal in general. For instance, T. Ohtsuki, R. Riley, and M. Sakuma have given a systematic construction of degree one maps between two-bridge knot exteriors.
Families of discrete characters and domination
----------------------------------------------
The fundamental group of a small knot manifold $M$ admits many discrete, non-elementary representations with values in $PSL_2(\mathbb C)$. For instance when $M$ is hyperbolic, its holonomy representation is discrete and non-elementary, as are the holonomy representations of the hyperbolic Dehn fillings of $M$. Similarly, when $M$ is Seifert fibred but not a twisted $I$-bundle over the Klein bottle, a holonomy representation of its base orbifold is discrete and non-elementary, as are those of the base orbifolds of the generic Dehn filling of $M$. One of the problems we investigate in the second part of the paper is to determine to what extent these are the only discrete non-elementary representations of $\pi_1(M)$.
Set $$D(M) = \{ \chi_\rho \in X_{PSL_2}(M): \rho \hbox{ is discrete and non-elementary}\}.$$ Classic work of Jørgensen and Marden shows that $D(M)$ is closed in $X_{PSL_2}(M)$ (see §\[convkleinhyp\]). Their results combine with the work of Culler and Shalen on ideal points of curves of $PSL_2(\mathbb C)$-characters to show that if $D(X_0) = D(M) \cap X_0$ is not compact for some component $X_0$ of $X_{PSL_2}(M)$, there is a connected essential surface $S$ in $M$ such that the restriction of each character in $X_0$ to $\pi_1(S)$ is elementary. It follows that $S$ is an annulus if $X_0$ contains the character of a faithful representation. In particular, $M$ is not hyperbolic. (Morgan and Shalen used this approach to prove that if the set of discrete faithful characters of the fundamental group of a compact $3$-manifold is not compact, then the group splits non-trivially along a virtually abelian subgroup [@MS3].) We use these ideas to construct various infinite families of small hyperbolic knot exteriors $M$ for which $D(M)$ is compact (see §\[unbounded\]).
To each representation $\rho: \pi_1(M) \to PSL_2(\C)$ is associated a volume *vol($\rho$)* $\in \mathbb R$ defined by taking any pseudo-developing map from the universal cover $\tilde M$ into $\H^3$ and integrating the pull-back of the hyperbolic volume form on a fundamental domain of $M$ (see [@Dun], [@Fra] for more details). This value depends only on the character of $\rho$ so it makes sense to talk of the volume of a character. Moreover, the associated volume function $vol: X_{PSL_2}(M) \to \mathbb R$ is continuous (indeed analytic). Here is a natural way to construct representations with non-zero volume.
Let $V$ be a compact, connected, orientable, hyperbolic manifold with holonomy representation $\rho_V: \pi_1(V) \to PSL_2(\mathbb C)$. If $V$ is a knot manifold, $f: M \to V$ a non-zero degree map, and $\rho = \rho_V \circ f_\#$, then $vol(\rho) = \hbox{degree}(f) vol(V) \ne 0$. Similarly, if $V$ is closed, $M(\alpha)$ is a Dehn filling of $M$, $f: M(\alpha) \to V$ a non-zero degree map, and $\rho$ the composition $\pi_1(M) \to \pi_1(M(\alpha))\stackrel{f_\#}{\longrightarrow} \pi_1(V) \stackrel{\rho_V}{\longrightarrow} PSL_2(\mathbb C)$, then $vol(\rho) = vol(\rho_V \circ f_\#) = \hbox{degree}(f) vol(V) \ne 0$. Note that each of these non-zero volume representations is discrete and torsion free.
There is a converse to this construction. The image of a discrete, torsion free, non-zero volume representation $\rho$ is the fundamental group of an element $V$ of ${\cal H}$ which is either a knot manifold or closed depending on whether or not $\rho|\pi_1(\partial M)$ is injective (Lemma \[standardimage\]) In the former case it is easy to see that $\rho$ is induced, as above, by a map $M \to V$. Since $0 \ne vol(\rho) = \hbox{degree}(f) vol(V)$ so $\hbox{degree}(f) \ne 0$. In the latter case, we use the fact that the abelian subgroups of $\pi_1(V)$ are cyclic to see that there is at least one slope $\alpha$ on $\partial M$ such that $\rho(\alpha) = \pm I$. It follows that $\rho$ is obtained, as above, from a map $f: M(\alpha) \to V$ with $\hbox{degree}(f) \ne 0$. Thus discrete, non-elementary, torsion free, non-zero volume representations of the fundamental group of a knot exterior $M$ correspond to non-zero degree maps of $M$ or its Dehn fillings to a hyperbolic manifold.
A [*principal component*]{} $X_0$ of the $PSL_2(\mathbb C)$-character variety of a finitely generated group $\Gamma$ is a component which contains the character of a discrete, faithful, irreducible representation of $\Gamma / Z(\Gamma)$, where $Z(\Gamma)$ denotes the centre of $\Gamma$. Our next result is a combination of Theorem \[convord\] and Lemma \[projectiveconvergence\].
\[thm:characters\] Let $M$ be a small hyperbolic knot manifold, $X_0$ a component of $X_{PSL_2}(M)$, and suppose that $\{\chi_n\} \subset X_0$ is a sequence of distinct characters of non-zero volume representations $\rho_n: \pi_1(M) \to PSL_2(\mathbb C)$ with image a torsion-free, cocompact, discrete group $\Gamma_n$. For $n \gg 0$, let $\alpha_n$ be the slope of $\rho_n$. Then up to taking a subsequence, one of the following two possibilities arises:\
$(a)$ the slopes $\alpha_n$ converge projectively to the class of a boundary slope of $M$; or\
$(b)$ $\lim \chi_n$ exists and is the character of a discrete, non-elementary, torsion free, non-zero volume representation $\rho_0$ such that:\
$\rho_0|\pi_1(\partial M)$ is 1-1 and $\rho_0(\pi_1(M))$ is a finite index subgroup of the fundamental group\
of a $1$-cusped hyperbolic manifold $V$.\
there are slopes $\beta_n$ on $\partial V$ such that for each $n$ the fundamental group of the Dehn\
filled manifold $V(\beta_n)$ is isomorphic to $\Gamma_n$ and the character $\chi_n$ is induced by the\
composition $\pi_1(M) \to \rho_0(\pi_1(M)) \hookrightarrow \pi_1(V) \to \pi_1(V(\beta_n)) \cong \Gamma_n$.\
$X_0 = \rho_0^*(Y_0)$ for a principal component $Y_0$ of $X_{PSL_2}(V)$
In certain circumstances we can guarantee that conclusion (b) of the theorem holds. For instance, this is the case when $M$ is hyperbolic and the characters $\chi_n$ lie on a principal component $X_0$ of the $PSL_2(\mathbb C)$-character variety of $\pi_1(M)$ (Corollary \[principalcompact\]). More generally, it is ruled out if we suppose that the characters $\chi_n$ lie on a curve component $X_0$ of $X_{PSL_2}(M)$ such that one of the following two conditions holds:\
(a) for each ideal point $x_0$ of $X_0$ there are a component $S_0$ of an essential surface assoc-\
iated to $x_0$ and a character $\chi \in X_0$ such that $\chi|\pi_1(S_0)$ is non-elementary; or\
(b) the Culler-Shalen seminorm of $X_0$ is a norm and each ideal point of $X_0$ has an assoc-\
iated essential surface $S_0$ with $|\partial S_0| \leq 2$.\
See Corollary \[nonelementarycompact\] for the justification of case (a) and Corollary \[lessthan2\] for that of case (b). The following is a consequence of Theorem \[thm:characters\] (see Corollary \[discetecharsprincipal\].)
\[cor:finite volume\] Let $M$ be a small hyperbolic knot manifold. Then all but finitely many of the discrete, non-zero volume characters on a principal curve $X_0$ of the $PSL_2(\mathbb C)$-character variety of $\pi_1(M)$ are induced by the complete hyperbolic structure on the interior of $M$ or by Dehn fillings of manifolds finitely covered by $M$.
One of our principal motives for investigating families of discrete characters was to address the following question posed by Shicheng Wang: If there are non-zero degree maps between infinitely many distinct Dehn fillings of two knot manifolds $M$ and $N$, are they induced by a non-zero degree map $M \to N$? Here is a version of Theorem \[thm:characters\] for non-zero degree maps which provides a partial answer (see §\[dominationhyperbolic\]).
\[thm:domination\] Let $M$ be a small hyperbolic knot manifold and suppose that there is a slope $\alpha_0$ on $\partial M$ such that the Dehn filled manifold $M(\alpha_0)$ does not dominate a hyperbolic manifold. Let $\{\alpha_n\}_{n \geq 1}$ be a sequence of distinct slopes on $\partial M$ which do not subconverge projectively to a boundary slope. If there are dominations $f_n: M(\alpha_n) \geq V_n$ where $V_n$ is a hyperbolic manifold, then there exist a compact hyperbolic manifold $V_0$ with a domination $f: M \geq V_0$, a subsequence $\{j\}$ of $\{n\}$, and slopes $\beta_j$ on $\partial V_0$ such that:\
For each $j$, $V_0(\beta_j) \cong V_j$.\
The following diagrams are commutative up to homotopy : $$\begin{array}{ccc}
M & \stackrel{f}{\longrightarrow} & V_0 \\
\downarrow & & \downarrow \\
M(\alpha_j) & \stackrel{f_j}{\longrightarrow} & V_j \cong V_0(\beta_j)
\end{array}$$ Thus infinitely many of the dominations $f_n: M(\alpha_n) \geq V_n$ are induced by $f$. If we assume further that the dominations $f_n: M(\alpha_n) \geq V_n$ are strict, then $f: M \geq V_0$ is strict as well.
The only known example of closed hyperbolic ${\cal H}$-minimal $3$-manifold is $\frac12$ surgery on the figure eight knot [@RWZ]. The following consequences of Theorem \[thm:domination\] show that closed ${\cal H}$-minimal manifolds are actually quite plentiful.
We will denote the projective space of $H_1(M; \mathbb R)$ by $\mathbb P(H_1(\partial M; \mathbb R))$ and the class of non-zero element $\beta \in H_1(\partial M; \mathbb R)$ by $[\beta]$. The following is part (1) of Theorem \[minimal hyperbolic\].
\[thm:h-minimal\] Let $M$ be a small, hyperbolic ${\cal H}$-minimal knot manifold and suppose that there is a slope $\alpha_0$ on $\partial M$ such that the Dehn filled manifold $M(\alpha_0)$ does not dominate any closed hyperbolic manifold. If $U \subset \mathbb P(H_1(\partial M; \mathbb R))$ is the union of disjoint closed arc neighbourhoods of the finite set of boundary slopes of $M$, then $\mathbb P(H_1(\partial M; \mathbb R)) \setminus U$ contains only finitely many projective classes of slopes $\alpha$ such that $M(\alpha)$ is not ${\cal H}$-minimal. In particular, $M$ admits infinitely many ${\cal H}$-minimal Dehn fillings.
This theorem applies to many hyperbolic knot manifolds. For instance, it applies to punctured torus bundles whose monodromies are pseudo-Anosov and not proper powers, or the exterior of the $(-2,3,n)$ pretzel ($n \ne 1,3,5$).
The meridinal slope of a knot in the $3$-sphere whose exterior is small is never a boundary slope (see Theorem 2.0.3 of [@CGLS]). Thus Theorem \[thm:h-minimal\] implies:
\[cor:integer surgery\] Let $M$ be the exterior of a small hyperbolic knot in $S^3$ and let $\mu,\lambda \in H_1(\partial M)$ represent the meridinal and longuitudinal slope respectively. If $M$ is ${\cal H}$-minimal, then for all but finitely many $n \in \Z$, the Dehn filled manifold $M(n\mu + \lambda)$ is ${\cal H}$-minimal.
For certain two-bridge knot exteriors we can say more (see §\[sec:h-minimal\]).
\[cor:2-bridge\] Let $M$ be the exterior of a hyperbolic $\frac{p}{q}$ two-bridge knot with $p$ prime. Then all but finitely many Dehn fillings of $M$ yield ${\cal H}$-minimal manifolds.
In order to construct families of closed minimal manifolds it is necessary to prove a version of Theorem \[thm:characters\] for discrete representations to $PSL_2(\mathbb R)$. Set $$D(M; \mathbb R) = \{ \chi_\rho \in D(M) : \rho \hbox{ has image in } PSL_2(\mathbb R)\}$$ and $D(X_0; \mathbb R) = D(M; \mathbb R) \cap X_0$ where $X_0$ is a subvariety of $X_{PSL_2}(M)$. In section §\[realconvergent\] we prove a result on convergent sequences of characters in $D(M; \mathbb R)$ whose topological interpretation is investigated in §\[psl2domination\]. An example of the sort of result we obtain is the next theorem (see Corollary \[cor:s-domination\]).
\[cor:s-dominationintro\] Let $M$ will be a small hyperbolic knot manifold with $H_1(M) \cong \mathbb Z$, $\{\alpha_n\}$ a sequence of distinct slopes on $\partial M$, and $\{\chi_n\} \subset D(M; \mathbb R)$ a sequence of characters of representations $\rho_n$ such that $\rho_n(\alpha_n) = \pm I$ for all $n$. If there are infinitely many distinct characters $\chi_n$ and the sequence $\{\chi_n\}$ subconverges to a character $\chi_{\rho_0}$ such that $\rho_0(\lambda_M) \ne \pm I$, then $M$ strictly dominates a Seifert manifold with incompressible boundary.
If we remove the condition that $H_1(M) \cong \mathbb Z$ from the hypotheses of the theorem, we can still construct a non-zero degree map from $M$ to a Seifert orbifold with incompressible boundary. We cannot rule out, though, the possibility that the manifold underlying the orbifold is not a solid torus.
This last theorem can be used to construct infinite families of closed minimal manifolds. For instance, we have the following theorem (see Theorem \[finitesl2domination\]).
\[finitesl2dominationintro\] Suppose that $M$ is a small ${\cal H}$-minimal hyperbolic knot manifold which has the following properties:
: There is a slope $\alpha_0$ on $\partial M$ such that $M(\alpha_0)$ is ${\cal H}$-minimal.
: For each norm curve $X_0 \subset X_{PSL_2}(M)$ and for each essential surface $S$ associated to an ideal point of $X_0$ there is a character $\chi_\rho \in X_0$ which restricts to a strictly irreducible character on $\pi_1(S)$.
: There is no surjective homomorphism from $\pi_1(M)$ onto a Euclidean triangle group.
: There is no epimorphism $\rho: \pi_1(M) \to \Delta(p,q,r) \subset PSL_2(\mathbb R)$ such that $\rho(\pi_{1} (\partial M))$ is elliptic or trivial.
Then all but finitely many Dehn fillings $M(\alpha)$ yield a minimal manifold.
As a consequence, we will prove our next result in Corollary \[cor:s-minimal2\].
\[prop:g-minimalintro\] If $M$ is the exterior of a hyperbolic twist knot, then all but finitely many Dehn filling $M(\alpha)$ yield a minimal manifold.
Our final results show that quite general hypotheses on a minimal knot exterior imply that it admits infinitely many minimal Dehn fillings. (See Theorems \[uinfinite\] and \[minginfinite\].)
\[uinfiniteintro\] Let $M$ be an ${\cal H}$-minimal, small, hyperbolic knot manifold and suppose that $H_1(M) \cong \mathbb Z \oplus T$ where $T$ is torsion prime to $6$ and $H_1(\partial M) \to H_1(M)/T \cong \mathbb Z$ is surjective. Suppose as well that\
$(a)$ there is a slope $\alpha_0$ on $\partial M$ such that $\pi_1(M(\alpha_0))$ admits no homomorphism onto a\
non-elementary Kleinian group or a Euclidean triangle group, and\
$(b)$ either\
$(i)$ there is no discrete, non-elementary representation $\rho \in R_{PSL_2(\mathbb R)}(M)$ such that\
$\rho(\pi_1(M))$ is isomorphic to a free product of two non-trivial cyclic groups and\
$\rho(\pi_1(\partial M))$ is parabolic, or\
$(ii)$ $T = \{0\}$, $M$ is minimal and there is no representation $\rho: \pi_1(M(\lambda_M)) \to$\
$PSL_2(\mathbb R)$ such that $\rho(\pi_1(M(\lambda_M)))$ is a free product of two non-trivial\
cyclic groups and $\rho(\pi_1(\partial M))$ is parabolic.\
Then there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal.
When $T \neq \{0\}$, this corollary applies to the exterior of many knots in lens spaces. For instance, to the knot manifold obtained by $m$-Dehn surgery on one component of the right-hand Whitehead link, $m$ odd. See Example \[lensspace\]. When $T = \{0\}$ it applies to the exterior of many knots in the $3$-sphere. For instance it is remarked in Example \[s3ginfiniteegs\] that if the Alexander polynomial of a knot manifold $M$ with $H_1(M) \cong \mathbb Z$ is not divisible by the Alexander polynomial of a non-trivial torus knot, there is no homomorphism of $\pi_1(M)$ onto the free product of two non-trivial finite cyclic groups. Thus if $M$ is minimal, small, and hyperbolic, there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal. The proof of the following corollary is also discussed in this example.
\[intro:2bridgefillingminimal\] Let $M$ be the exterior of a $\frac{p}{q}$ two-bridge knot with $p$ is prime and $q \not \equiv \pm 1$ $($mod $p$$)$, or of a $(-2, 3, n)$ pretzel with $n \not \equiv 0$ $($mod $3$$)$. Then there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal.
Organization of the paper and acknowledgements
----------------------------------------------
The basic properties of $PSL_2(\mathbb C)$-character varieties and Culler-Shalen theory are described in §\[varieties\]. The main result of this section states that the morphism $\varphi^*: X_{PSL_2}(\Gamma_2) \to X_{PSL_2}(\Gamma_1)$ induced by a virtual epimorphism $\varphi: \Gamma_1 \to \Gamma_2$ is generically 1-1 when restricted to the union of the positive dimensional algebraic components of $X_{PSL_2}(\Gamma_2)$ which contain a strictly irreducible character (Corollary \[1-1\]). Section 3 deals with virtual epimorphisms of the fundamental groups of knot manifolds and contains the proofs of Theorems \[domboundintro\], \[intro:domseqbound\], and \[twobridgedomboundintro\]. In §\[sec:characters\] we begin our study of families of discrete characters. In particular, unbounded sequences of such characters are studied in §\[unbounded\] and convergent ones in §\[convergent\]. This leads to the proofs of Theorems \[thm:characters\] and \[thm:domination\]. We discuss ${\cal H}$-minimal Dehn filling in §\[sec:h-minimal\], including the proof of Theorem \[thm:h-minimal\]. The last two sections of the paper deal with families of discrete $PSL_2(\mathbb R)$-characters (§\[psl2rcharacters\]) and constructing minmal manifolds (§\[gminimal\]). The proofs of Theorems \[cor:s-dominationintro\], \[finitesl2dominationintro\], \[uinfiniteintro\] and Corollaries \[prop:g-minimalintro\], \[intro:2bridgefillingminimal\] are found here. Finally, three appendices are included which deal, respectively, with the following topics: a smoothness criterion for dihedral characters; peripheral values of representations of the fundamental groups of twist knot exteriors; the bending construction.
The authors wish to thank Shicheng Wang for many stimulating conversations and for bringing to their attention his question about Dehn fillings and non-zero degree maps.
Varieties of $PSL_2(\mathbb C)$-characters {#varieties}
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Generalities
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In what follows we shall refer to the elements of $PSL_2(\mathbb C)$ as matrices. Denote by ${\cal D}$ the abelian subgroup of $PSL_2(\mathbb C)$ consisting of diagonal matrices and by ${\cal N}$ the subgroup consisting of those matrices which are either diagonal or have diagonal coefficients $0$. Note that ${\cal D}$ has index $2$ in ${\cal N}$ and any element in ${\cal N} \setminus {\cal D}$ has order $2$. Further, the centre of ${\cal N}$, which we will denote by $Z({\cal N})$, is isomorphic to $\mathbb Z/2$ and is generated by $\pm \left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)$.
The action of $SL_2(\mathbb C)$ on $\mathbb C^2$ descends to one of $PSL_2(\mathbb C)$ on $\mathbb CP^1$. We call a representation $\rho$ with values in $PSL_2(\mathbb C)$ [*irreducible*]{} if the associated action on $\mathbb CP^1$ is fixed point free, otherwise we call it [*reducible*]{}. We call it [*strictly irreducible*]{} if the action has no invariant subset in $\mathbb CP^1$ with fewer than three points. Note that
- $\rho$ is reducible if and only if it is conjugate to a representation whose image consists of upper-triangular matrices.
- $\rho$ is conjugate to a representation with image in ${\cal D}$ if and only if the action on $\mathbb
CP^1$ has at least two fixed points. It is conjugate into ${\cal
N}$ if and only if it leaves a two point subset of $\mathbb CP^1$ invariant.
- $\rho$ is is strictly irreducible if and only if it is irreducible but not conjugate into ${\cal N}$.
- if $\rho$ is irreducible and $A \in PSL_2(\mathbb C)$ satisfies $A \rho A^{-1}
= \rho$, then either $A = \pm I$ or up to conjugation, $A = \pm \left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)$ and $\rho$ conjugates into ${\cal N}$. Thus if $\rho$ is strictly irreducible, then $A = \pm I$.
The action of $PSL_2(\mathbb C)$ on $\mathbb CP^1 = S^2_\infty$ extends over $\mathbb H^3$ yielding an identification $PSL_2(\mathbb C) = \hbox{Isom}_+(\mathbb H^3)$. A representation is called [*elementary*]{} if the associated action on $\overline{\mathbb H}^3$ has a finite orbit. Equivalently, the representation is reducible or conjugates to one with image in either $SO(3) = PSU(2)$ or ${\cal N}$.
Let $\Gamma$ be a finitely generated group. The set $R_{PSL_2}(\Gamma)$ of representations of $\Gamma$ with values in $PSL_2(\mathbb C)$ admits the structure of a $\mathbb C$-affine algebraic set \[LM\] called the [*$PSL_2(\mathbb C)$-representation variety*]{} of $\Gamma$. The action of $PSL_2(\mathbb C)$ on $R_{PSL_2}(\Gamma)$ determines an algebro-geometric quotient $X_{PSL_2}(\Gamma)$ whose coordinate ring is $\mathbb C[R_{PSL_2}(\Gamma)]^{PSL_2(\mathbb C)}$ and a regular map $t: R_{PSL_2}(\Gamma) \to X_{PSL_2}(\Gamma)$ \[LM\]. This quotient is called the [*$PSL_2(\mathbb C)$-character variety*]{} of $\Gamma$. For $\rho \in R_{PSL_2}(\Gamma)$, we denote $t(\rho)$ by $\chi_\rho$ and refer to it as the [*character*]{} of $\rho$. If $\chi_{\rho_1} = \chi_{\rho_2}$ and $\rho_1$ is irreducible, then $\rho_1$ and $\rho_2$ are conjugate representations. We can therefore call a character $\chi_\rho$ reducible, irreducible, or strictly irreducible if $\rho$ has that property. Each reducible character is the character of a diagonal representation, that is, one with image in ${\cal D}$. The property of an irreducible representation being conjugate into $SO(3)$ is also determined by its character (see Proposition III.1.1 of [@MS1] for instance) and so if $\chi_{\rho_1} = \chi_{\rho_2}$ and $\rho_1$ is elementary, then so is $\rho_2$. In this case we call the character elementary.
When $\Gamma$ is the fundamental group of a path-connected space $Y$, we write $R_{PSL_2}(Y)$ for $R_{PSL_2}(\pi_1(Y))$, $X_{PSL_2}(Y)$ for $X_{PSL_2}(\pi_1(Y))$, and refer to them respectively as the $PSL_2(\mathbb C)$-representation variety of $Y$ and $PSL_2(\mathbb C)$-character variety of $Y$.
Each $\gamma \in \Gamma$ determines an element $f_\gamma$ of the coordinate ring $\mathbb C [X_{PSL_2}(\Gamma)]$ according to the formula $$f_\gamma(\chi_\rho) = \mbox{trace}(\rho(\gamma))^2 - 4.$$
A homomorphism $\varphi: \Gamma_1 \to \Gamma_2$ determines morphisms $\varphi^*: R_{PSL_2}(\Gamma_2) \to R_{PSL_2}(\Gamma_1), \; \rho \mapsto \rho \circ \varphi$ and $\varphi^*: X_{PSL_2}(\Gamma_2) \to X_{PSL_2}(\Gamma_1), \; \chi_\rho \mapsto \chi_{\rho \circ \varphi}$. For $\gamma \in \Gamma_1$ and $\chi_\rho \in X_{PSL_2}(\Gamma_2)$ we have $$f_\gamma(\varphi^*(\chi_\rho)) = f_{\varphi(\gamma)}(\chi_\rho). \eqno{(2.1.1)}$$
We end this section with a useful observation
\[closed\] If the image of $\varphi: \Gamma_1 \to \Gamma_2$ is of finite index $n$ in $\Gamma_2$, then $\varphi^*: X_{PSL_2}(\Gamma_2) \to X_{PSL_2}(\Gamma_1)$ is a closed map with respect to the Zariski topology.
Let $X_0$ be a Zariski closed subset of $X_{PSL_2}(\Gamma_2)$ and let $Y_0 = \overline{\varphi^*(X_0)}$. If $\bar X_0, \bar Y_0$ are projective closures of $X_0, Y_0$, then $\varphi^*$ determines a surjective projective morphism $\bar \varphi^*: \bar X_0 \to \bar Y_0$. Let $y_0 \in Y_0$ and choose $x_0 \in \bar X_0$ such that $\bar \varphi^*(x_0) = y_0$, and a projective curve $C \subseteq \bar X_0$ which contains $x_0$. Set $C_0 = C \cap C_0$ and note that if $x_0 \not \in C_0$, there is some $\gamma \in \pi_1(M)$ such that $f_\gamma(x_0) = \infty$ (cf. Theorem 2.1.1 of [@CS]). For $A, B \in SL_2(\mathbb C)$ we have $\hbox{trace}(AB) + \hbox{trace}(A^{-1} B) = \hbox{trace}(A) \hbox{trace}(B)$, and this identity can be used inductively to show that $f_{\gamma^n}$ is a degree $|n|$ polynomial in $f_{\gamma}$. In particular, $f_{\gamma^n}(x_0) = \infty$. On the other hand there is some $\delta \in \Gamma_1$ such that $\varphi(\delta) = \gamma^n$. Then $f_\delta(y_0) = f_\delta(\bar \varphi^*(x_0)) = f_{\varphi(\delta)}(x_0) = f_{\gamma^n}(x_0) = \infty$. But this contradicts the fact that $y_0 \in Y_0$. Thus $x_0 \in C_0 \subseteq X_0$ and so $\varphi^*$ is onto $Y_0$.
Subvarieties of $X_{PSL_2}(\Gamma)$ {#subvarieties}
-----------------------------------
The set of reducible representations $R_{PSL_2}^{red}(\Gamma) \subseteq R_{PSL_2}(\Gamma)$ ($\Gamma$ a finitely generated group) is a closed algebraic subset (cf. the proof of Corollary 1.4.5 of [@CS]). The sets $R_{SO(3)}(\Gamma), R_{{\cal D}}(\Gamma)$, and $R_{{\cal N}}(\Gamma)$ of representations of $\Gamma$ with values in $SO(3), {\cal D}$, and ${\cal N}$ are also closed algebraic subsets of $R_{PSL_2}(\Gamma)$. A similar statement holds for their images $X_{SO(3)}(\Gamma), X_{PSL_2}^{red}(\Gamma)$, and $X_{{\cal N}}(\Gamma)$ in $X_{PSL_2}(\Gamma)$. In particular, the set $X_{Elem}(\Gamma) = X_{SO(3)}(\Gamma) \cup X_{{\cal N}}(\Gamma)$ of characters is Zariski closed in $X_{PSL_2}(\Gamma)$.
A subvariety $X_0$ of $X_{PSL_2}(\Gamma)$ is called [*non-trivial*]{} if it contains the character of an irreducible representation. It is called [*strictly non-trivial*]{} if it contains the character of a strictly irreducible representation. The property of being (strictly) irreducible is open so that the generic character of a (strictly) non-trivial subvariety of $X_{PSL_2}(\Gamma)$ is (strictly) irreducible. Let $X_+^{irr}(\Gamma)$ denote the union of the positive dimensional non-trivial components of $X_{PSL_2}(\Gamma)$ and $X_+^{str}(\Gamma)$ the union of its positive dimensional strictly non-trivial components.
For each non-trivial subvariety $X_0$ of $X_{PSL_2}(\Gamma)$ there is a subvariety $R_{X_0}$ of $R_{PSL_2}(\Gamma)$ uniquely determined by the condition that it is conjugation invariant and $t(R_{X_0}) = X_0$ (cf. Lemma 4.1 of \[BZ1\]). We define the *kernel* of $X_0$ to be the normal subgroup of $\Gamma$ given by $$\hbox{Ker}(X_0) = \bigcap \limits_{\rho \in R_{X_0}} \hbox{kernel}(\rho).$$ For instance $\hbox{Ker}(X_0) = \{1\}$ if $R_{X_0}$ contains an injective representation.
\[ker\] Let $X_0$ be a non-trivial subvariety of $X_{PSL_2}(\Gamma)$.\
$(1)$ There is a subset $V$ of $R_{X_0}$ which is a countable union of proper, closed, conjugation invariant algebraic subsets of $R_{X_0}$ such that for $\rho \in R_{X_0} \setminus V$, $\hbox{kernel}(\rho) = \hbox{Ker}(X_0)$.\
$(2)$ If $\varphi: \Gamma_1 \to \Gamma_2$ is a homomorphism and $X_0$ is a subvariety of $X_{PSL_2}(\Gamma_2)$ such that $Y_0 = \overline{\varphi^*(X_0)}$ is non-trivial, then $\hbox{Ker}(Y_0) = \varphi^{-1}(\hbox{Ker}(X_0))$. In particular, $\hbox{kernel}(\varphi) \subseteq \hbox{Ker}(Y_0)$.
\(1) For each $\gamma \in \pi_1(M)$ set $V_\gamma = \{\rho \in R_{X_0}
\; | \; \rho(\gamma) = \pm I\}$. Then $V_\gamma$ is a closed, conjugation invariant algebraic subset of $R_{X_0}$. It is clear that $\gamma \in \hbox{Ker}(X_0)$ if and only if $V_\gamma = R_{X_0}$. Set $V = \bigcup_{\gamma \not \in \hbox{Ker}(X_0)} V_\gamma$ and observe that $\rho \in R_{X_0} \setminus V$ if and only if $\rho(\gamma) \ne \pm I$ for each $\gamma \not \in \hbox{Ker}(X_0)$. In particular, $\hbox{kernel}(\rho) =
\hbox{Ker}(X_0)$ for such $\rho$. This proves (1).
\(2) Now $\varphi^*(X_0) = t(\varphi^*(R_{X_0})) \subseteq t(\overline{\varphi^*(R_{X_0})}) \subseteq \overline{t(\varphi^*(R_{X_0}))} = \overline{\varphi^*(X_0)} = Y_0$ and since $\overline{\varphi^*(R_{X_0})}$ is closed and conjugation invariant in $R_{PSL_2}(\Gamma_1)$, Theorem 3.3.5(iv) of [@Ne] implies that $t(\overline{\varphi^*(R_{X_0})})$ is Zariski closed in $X_{PSL_2}(\Gamma_1)$. It follows that $R_{Y_0} = \overline{\varphi^*(R_{X_0})}$. Hence noting that $\varphi^*(\rho)(\gamma) = \rho(\varphi(\gamma)) = \pm I$ whenever $\gamma \in \varphi^{-1}(\hbox{Ker}(X_0))$ and $\rho \in R_{X_0}$, it follows that $\rho'(\gamma) = \pm I$ for all $\rho' \in R_{Y_0}$. In other words, $\gamma \in \hbox{Ker}(Y_0)$. Conversely if $\gamma \in \hbox{Ker}(Y_0)$ and $\rho \in R_{X_0}$, then $\rho(\varphi(\gamma)) = \varphi^*(\rho)(\gamma) = \pm I$. Thus $\gamma \in \varphi^{-1}(\hbox{Ker}(X_0))$. This proves (2).
We call a component $X_0$ of $X_+^{irr}(\Gamma)$ [*principal*]{} if it contains the character of a discrete, faithful, irreducible representation of $\Gamma/Z(\Gamma)$ where $Z(\Gamma)$ denotes the centre of $\Gamma$. It is clear that $\hbox{Ker}(X_0) \subseteq Z(\Gamma)$.
\[kerprincipal\] $\;$\
$(1)$ If $X_0$ is a principal component of $X_+^{irr}(\Gamma)$, then $\hbox{Ker}(X_0) = Z(\Gamma)$.\
$(2)$ If $\varphi: \Gamma_1 \to \Gamma_2$ is a homomorphism and $X_0$ is a subvariety of $X_{PSL_2}(\Gamma_2)$ such that $\overline{\varphi^*(X_0)}$ is principal, then $\hbox{kernel}(\varphi) \subseteq Z(\Gamma_1)$.
\(1) It suffices to show that $Z(\Gamma) \subseteq \hbox{Ker}(X_0)$. To that end we note that if $\rho \in R_{PSL_2}(\Gamma)$ is irreducible, then every element in $\rho(Z(\Gamma))$ has order $1$ or $2$. In particular for $\gamma \in Z(\Gamma)$ and $\rho \in R_{X_0}$ we have $f_\gamma(\chi_\rho) \in \{0, -4\}$. Hence $f_\gamma|X_0$ is constant and since it vanishes at a discrete faithful character, it is identically zero. Thus $\rho(\gamma) = \pm I$ for all $\rho \in R_{X_0}$ and therefore $Z(\Gamma) \subseteq \hbox{Ker}(X_0)$.
Part (2) follows from part (1) and part (2) of the previous lemma.
Restriction
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If $\varphi: \Gamma_1 \to \Gamma$ is surjective, it is easy to see that $\varphi^*: X_{PSL_2}(\Gamma) \to X_{PSL_2}(\Gamma_1)$ is injective. The goal of this section is to show that a similar conclusion is true for virtual epimorphisms $\varphi: \Gamma_1 \to \Gamma$ as long as we restrict $\varphi^*$ to $X_{+}^{str}(\Gamma)$.
Let $D_n, T_{12}, O_{24}$ denote, respectively, the dihedral group of order $2n$, the tetrahedral group of order $12$, and the octahedral group of order $24$. Set $${\cal K} = \{ \pm I, \pm \left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right), \pm \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right), \pm \left(\begin{array}{cc} 0 & i \\ i & 0 \end{array} \right)\} \subset {\cal N}$$ and observe that ${\cal K} \cong D_2$. It is well-known and easy to verify that the centraliser $Z_{PSL_2}({\cal K})$ of ${\cal K}$ in $PSL_2(\mathbb C)$ is ${\cal K}$ and its normaliser $N_{PSL_2}({\cal K})$ is isomorphic to $O_{24}$. The only other subgroups of $PSL_2(\mathbb C)$ which contain ${\cal K}$ as a normal subgroup are the (unique) subgroup of ${\cal N}$ isomorphic to $D_4$ and the (unique) index $2$ subgroup of $N_{PSL_2}({\cal K})$, which is isomorphic to $T_{12}$.
Given any homomorphisms $\rho \in R_{\cal N}(\Gamma)$ and $\varphi: \Gamma \to Z({\cal N})$, the equation $$\rho'(\gamma) = \varphi(\gamma) \rho(\gamma)$$ defines an element $\rho' \in R_{{\cal N}}(\Gamma)$ which will denote by $\varphi \cdot \rho$.
\[determined\] Let $\Gamma_0$ be a normal subgroup of a finitely generated group $\Gamma$ and suppose that $\rho_1, \rho_2 \in R_{PSL_2}(\Gamma)$ restrict to the same irreducible representation $\rho_0 \in R_{PSL_2}(\Gamma_0)$. Then either\
$(a)$ $\rho_1 = \rho_2$, or\
$(b)$ after a similtaneous conjugation of $\rho_1$ and $\rho_2$ we have $\rho_j(\Gamma) \subset {\cal N}$ for $j = 1, 2$ and there is a homomorphism $\varphi: \Gamma/\Gamma_0 \to Z({\cal N})$ such that $\rho_2 = \varphi \cdot \rho_1$, or\
$(c)$ $\rho_0(\Gamma_0) \cong D_2$ and $\rho_1(\Gamma) = \rho_2(\Gamma)$ is isomorphic to one of $D_2, D_4, T_{12},$ or $O_{24}$. There are only finitely many orbits in $R_{PSL_2}(\Gamma)$ for which this case arises.
Fix $\gamma \in \Gamma$ and set $\rho_j(\gamma) = A_j$ for $j = 1, 2$. Then for $\gamma_0
\in \Gamma_0$ we have $A_1
\rho_0(\gamma_0) A_1^{-1} = \rho_0(\gamma \gamma_0 \gamma^{-1}) = A_2
\rho_0(\gamma_0) A_2^{-1}$. Thus $A_2^{-1} A_1 \in Z_{PSL_2}(\rho_0(\Gamma_0))$, the centraliser of $\rho_0(\Gamma_0)$ in $PSL_2(\mathbb C)$. The irreducibility of $\rho_0$ then implies that either $A_2 = A_1$ or, after conjugation, $\rho_0(\Gamma_0) \subset {\cal N}$ and $A_2^{-1} A_1 = \pm \left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)$. If the former occurs for each $\gamma \in \Gamma$ then $\rho_1 = \rho_2$ and so we are in case (a). Suppose then that the latter arises for some $\gamma \in \Gamma$.
Similtaneously conjugate $\rho_1$ and $\rho_2$ so that $\rho_0(\Gamma_0) \subset {\cal N}$. If some $B \in \rho_0(\Gamma_0)$ has order larger than $2$, then $B \in {\cal D}$ and for any matrix $A \in \rho_j(\Gamma)$ (either $j$) we have $A B A^{-1} \in {\cal D}$. Thus $A B A^{-1} = B^\epsilon$ for some $\epsilon = \pm 1$ and therefore $A \in {\cal N}$. It follows that $\rho_j(\Gamma) \subset {\cal N}$ for both $j$. From the previous paragraph we know that if $\gamma \in \Gamma$ and $A_j = \rho_j(\gamma)$, then $\pm (A_2^{-1} A_1)^2 = \pm I$. Moreover, $\pm (A_2^{-1} A_1) B (A_2^{-1} A_1)^{-1} = B$ and our restrictions on $B$ then imply that $A_2^{-1} A_1 \in {\cal D}$. Thus $\rho_2(\gamma)^{-1}\rho_1(\gamma) \in \{\pm I, \pm \left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right)\} = Z_{PSL_2}({\cal N})$. It follows that $\gamma \mapsto \rho_1(\gamma)^{-1}\rho_2(\gamma)$ induces a homomorphism $\varphi: \Gamma/\Gamma_0 \to Z_{PSL_2}({\cal N})$ such that $\rho_2 = \varphi \cdot \rho_1$. This is case (b).
Suppose then that each $\pm I \ne B \in \rho_0(\Gamma_0) \subset {\cal N}$ has order $2$. Then $\rho_0(\Gamma_0) = {\cal K}$ and is a normal subgroup of $\rho_j(\Gamma)$ ($j = 1, 2$). Hence by our remarks above we know that $\rho_j(\Gamma)$ is a subgroup of $N_{PSL_2}({\cal K}) \cong O_{24}$ and is isomorphic to one of $D_2, D_4, T_{12}$, and $O_{24}$. Conjugation induces an exact sequence $1 \to {\cal K} \to \rho_j(\Gamma) \to Aut({\cal K}) \cong S_3$ and we saw above that for each $\gamma \in \Gamma$, $\rho_1(\gamma)^{-1} \rho_2(\gamma) \in Z_{PSL_2}({\cal N}) \subset {\cal K}$. Therefore the images of $\rho_1(\Gamma)$ and $\rho_2(\Gamma)$ in $Aut({\cal K})$ coincide and so $|\rho_1(\Gamma)| = |\rho_2(\Gamma)|$. Since ${\cal K} \subseteq \rho_j(\Gamma)$, we actually have $\rho_1(\Gamma) = \rho_2(\Gamma)$. Finally, since two finite subgroups of $PSL_2(\mathbb C)$ which are abstractly isomorphic are conjugate in $PSL_2(\mathbb C)$ and $\Gamma$ is finitely generated, there are only finitely many orbits in $R_{PSL_2}(\Gamma)$ of representations with image either $D_2, D_4, T_{12},$ or $O_{24}$. This is case (c).
\[str11\] Let $\Gamma_0$ be a finitely generated normal subgroup of a finitely generated group $\Gamma$ and suppose that $\rho_1, \rho_2 \in R_{PSL_2}(\Gamma)$ are strictly irreducible with images different from $T_{12}$ and $O_{24}$. Suppose further that $\chi_{\rho_1|\Gamma_0} = \chi_{\rho_2|\Gamma_0}$ and is irreducible. Then $\chi_{\rho_1} = \chi_{\rho_2}$.
\[inj\] Let $X_0$ be a positive dimensional non-trivial subvariety of $X_{PSL_2}(\Gamma)$ and $\Gamma_0$ a finitely generated normal subgroup of $\Gamma$. Then one of the following three situations arises.\
$(a)$ $R_{X_0} \subset R_{PSL_2}(\Gamma/\Gamma_0)$. That is, $\rho(\Gamma_0) = \{\pm I\}$ for each $\rho \in R_{X_0}$.\
$(b)$ The restriction to $\Gamma_0$ of each $\rho \in R_{X_0}$ is conjugate into ${\cal N}$ and $X_0 \subset X_{{\cal N}}(\Gamma)$.\
$(c)$ The restriction to $\Gamma_0$ of the generic $\rho \in R_{X_0}$ is strictly irreducible. Moreover, there is a Zariski open, conjugation invariant, connected subset $U$ of $R_{X_0}$ such that if $\rho_1, \rho_2 \in U$ and their restrictions to $\Gamma_0$ have the same characters, then $\chi_{\rho_1} = \chi_{\rho_2}$.
We shall suppose that conclusion (a) of the lemma does not occur. In particular, there is some $\rho_0 \in R_{X_0}$ such that $\rho_0(\Gamma_0) \ne \{\pm I\}$. Then there is a Zariski open, conjugation invariant, connected subset $U_0 \subset R_{X_0}$ consisting of irreducible representations such that $\rho(\Gamma_0) \ne \{\pm I\}$ for all $\rho \in U_0$. As there are only finitely many conjugacy classes of representations with image isomorphic to $D_2$, we also suppose that $\rho(\Gamma) \not \cong D_2$ for $\rho \in U_0$.
As a first case, assume that the restriction to $\Gamma_0$ of each $\rho \in U_0$ is reducible. If $\rho(\Gamma_0)$ is not diagonalisable for some $\rho \in U_0$, it has a unique fixed point in $\mathbb CP^1$, and so the fact that $\rho(\Gamma_0)$ is normal in $\rho(\Gamma)$ implies that $\rho$ is reducible, which is not the case. Thus $\rho(\Gamma_0)$ is diagonalisable for $\rho \in
U_0$. It follows that for such $\rho$, $\rho(\Gamma)$ leaves invariant a two element subset of $\mathbb C P^1$ and therefore conjugates to a representation with image in ${\cal N}$. Since we have assumed that $\rho(\Gamma) \not \cong D_2$, there is a unique index $2$ subgroup $\Gamma_\rho$ of $\Gamma$ such that $\rho|\Gamma_\rho$ is diagonalisable while $\rho(\Gamma_\rho) \not \subset \mathbb Z/2$. Fix generators $\gamma_0, \gamma_1, \ldots , \gamma_n$ of $\Gamma$ so that $\rho(\gamma_j)^2 = \pm I$ if and only if $j = 0$. Then $\Gamma_\rho$ is generated by $\gamma_0^2, \gamma_1, \ldots , \gamma_n, \gamma_1', \ldots , \gamma_n'$ where $\gamma_j' = \gamma_0 \gamma_j \gamma_0^{-1}$. For all $\rho' \in U_0$ close enough to $\rho$ and $j \geq 1$ we have $\rho'(\gamma_j)^2 \ne \pm I$ and therefore $\rho'(\gamma_j) \in {\cal D}$. Hence $\Gamma_\rho \subseteq \Gamma_{\rho'} $ for $\rho'$ in an open neighbourhood of $\rho$. It follows that $\Gamma_\rho$ is independent of $\rho \in U_0$. Denote this common subgroup by $\Gamma_1$. Each $\rho_0 \in R_{X_0}$ is a limit of representations $\rho_n \in U_0$. It follows that $\rho_0|\Gamma_1$ is reducible and as above we see that $\rho_0$ is either reducible or conjugates into ${\cal N}$. In either case, $\chi_{\rho_0} \in X_{{\cal N}}(\Gamma)$. Thus conclusion (b) holds.
Next assume that the restriction to $\Gamma_0$ of the generic $\rho \in R_{X_0}$ is irreducible but not strictly irreducible. For such $\rho$, after a conjugation we may suppose that $\rho(\Gamma_0) \subset {\cal N}$. If $\rho(\Gamma_0) \ne {\cal K}$ then $\rho(\Gamma) \subset {\cal N}$, while if $\rho(\Gamma_0) = {\cal K}$ then $\rho(\Gamma)$ is isomorphic to one of ${\cal K}, D_4, T_{12}$ or $O_{24}$ (c.f. the proof of Lemma \[determined\]). There are only finitely many characters of representations $\rho \in R_{PSL_2}(\Gamma)$ with image $T_{12}$ or $O_{24}$, so the generic character in $X_0$ lies in $X_{\cal N}(\Gamma)$. It follows that conclusion (b) holds.
Finally suppose that the restriction to $\Gamma_0$ of some $\rho \in R_{X_0}$ is strictly irreducible. Then there is a Zariski open subset $U \subseteq U_0$ such that for each $\rho \in U$, $\rho|\Gamma_0$ is strictly irreducible and the image of $\rho$ is neither $T_{12}$ nor $O_{24}$. If $\rho_1, \rho_2 \in U$ restrict to representations of $\Gamma_0$ with the same character, then Lemma \[determined\] shows that $\chi_{\rho_1} = \chi_{\rho_2}$. This is conclusion (c).
\[1-1\] Let $\varphi: \Gamma_1 \to \Gamma$ be a virtual epimorphism. Then $Y := \varphi^*(X_+^{str}(\Gamma))$ is a Zariski closed subset of $X_+^{str}(\Gamma_1)$. Further, $\varphi^*$ sends distinct components of $X_+^{str}(\Gamma)$ to Zariski dense subsets of distinct components of $Y$ and is generically one-to-one on each of these components.
It suffices to prove the result when $\Gamma_1$ is a finite index subgroup of $\Gamma$ and $\varphi$ is the inclusion.
Fix a finite index subgroup $\Gamma_0 \subseteq \Gamma_1$ which is normal in $\Gamma$ and a component $X_0$ of $X_+^{str}(\Gamma)$. Since $X_0$ is positive dimensional, it cannot be contained in $X_{PSL_2}(\Gamma/ \Gamma_0)$. Thus the third option in Proposition \[inj\] must arise with respect to the restriction map $R_{PSL_2}(\Gamma) \to R_{PSL_2}(\Gamma_0)$. In particular, the induced map $X_0 \to X_{PSL_2}(\Gamma_0)$ is generically $1-1$ and so its image is a positive dimensional subvariety of $X_{PSL_2}^{str}(\Gamma_0)$. This image is also closed by Lemma \[closed\]. Corollary \[str11\] implies that distinct components of $X_+^{str}(\Gamma)$ are sent to distinct subvarieties of $X_{PSL_2}^{str}(\Gamma_0)$. Finally, since the restriction $X_+^{str}(\Gamma) \to X_{PSL_2}(\Gamma_0)$ factors through the map $X_+^{str}(\Gamma) \to X_{PSL_2}(\Gamma_1)$, the conclusions of the corollary hold.
Culler-Shalen theory {#culler-shalen}
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In this section, $M$ will denote compact, connected, orientable, irreducible $3$-manifold whose boundary is a torus.
Any complex affine curve $C$ admits an affine desingularisation $C^\nu \stackrel{\nu}{\longrightarrow} C$ where $\nu$ is surjective and regular. Moreover, the smooth projective model $\tilde C$ of $C$ is obtained by adding a finite number of ideal points to $C^\nu$. Thus $\tilde C = C^\nu \cup {\cal I}(C)$ where ${\cal I}(C)$ is the set of ideal points of $C$. There are natural identifications between the function fields of $C, C^\nu,$ and $\tilde C$. Thus to each $f \in \mathbb C(C)$ we have corresponding $f^\nu \in \mathbb C(C^\nu) = \mathbb
C(C)$ and $\tilde f \in \mathbb C(\tilde
C) = \mathbb C(C)$ where $f^\nu = f \circ \nu = \tilde f|C^\nu$.
The proof of the following basic result can be found in Proposition 3.2.1 of [@CS].
[[(Thurston)]{}]{} \[posdim\] Let $V$ be a compact orientable $3$-manifold and $\rho \in R_{PSL_2}(V)$ an irreducible representation such that $\rho(\pi_1(T)) \ne \{\pm I\}$ for each toral boundary component $T$ of $V$. Then the dimension of any component $X_0$ of $X_{PSL_2}(V)$ which contains $\chi_{\rho}$ is at least $3t - \frac{3}{2}\chi(\partial V)$ where $t$ is the number of toral boundary components of $V$.
Recall that each $\gamma \in \pi_1(M)$ determines an element $f_\gamma$ of the coordinate ring $\mathbb C [X_{PSL_2}(M)]$ satisfying $$f_\gamma(\chi_\rho) = \mbox{trace}(\rho(\gamma))^2 - 4$$ where $\rho \in R_{PSL_2}(M)$. Each $\alpha \in H_1(\partial M) = \pi_1(\partial M)$ defines an element of $\pi_1(M)$ well-defined up to conjugation and therefore determines an element $f_\alpha \in \mathbb C [X_{PSL_2}(M)]$. Similarly each slope $\alpha$ on $\partial M$ determines an element of $\pi_1(M)$ well-defined up to conjugation and taking inverse, and so defines $f_\alpha \in \mathbb C[X_{PSL_2}(M)]$.
To each curve $X_0$ in $X_{PSL_2}(M)$ we associate the function $$d_{X_0}: \pi_1(M) \to \mathbb Z, \; d_{X_0}(\gamma) = \hbox{degree}(f_\gamma: X_0 \to \mathbb C).$$ Standard trace identities imply that for $n \in \mathbb Z$, $$d_{X_0}(\gamma^n) = |n| d_{X_0}(\gamma).$$ More generally, it was shown in[@CGLS] that $d_{X_0}$ has nice properties when restricted to abelian subgroups of $\pi_1(M)$. For instance, when restricted to $\pi_1(\partial M)$ it gives rise to a [*Culler-Shalen seminorm*]{} $$\| \cdot \|_{X_0}: H_1(\partial M; \mathbb R) \to [0, \infty)$$ where for each $\alpha \in H_1(\partial M) = \pi_1(\partial M)$ we have $\|\alpha\|_{X_0} = d_{X_0}(\alpha)$.
We say that a curve $X_0 \subset X_{PSL_2}(M)$ is a [*norm curve*]{} if $\|\cdot\|_{X_0}$ is a norm. If $\|\cdot\|_{X_0} \ne 0$, though it is not a norm, there is a primitive element $\beta \in H_1(\partial M)$ well-defined up to sign such that $\|\beta\|_{X_0} = 0$. In this case we say that $X_0$ is a [*$\beta$-curve*]{}.
For $x \in \tilde X_0$ and $\gamma \in \pi_1(M)$, we denote by $Z_x(\tilde f_\gamma), \Pi_x(\tilde f_\gamma)$ the multiplicity of $x$ as a zero, respectively pole, of $\tilde
f_\gamma$. From the definition of $\|\cdot\|_{X_0}$ we see that for each $\alpha \in H_1(\partial M)$ we have $$\|\alpha\|_{X_0} = \sum_{x \in \tilde X_0} Z_x(\tilde f_\alpha) =
\sum_{x \in {\cal I}(X_0)} \Pi_x(\tilde f_\alpha). \eqno{(2.4.1)}$$ When $M$ is hyperbolic, there is an essentially canonical choice of curve $X_0 \subset X_{PSL_2}(M)$ characterized by the fact that it contains the character of a discrete, faithful representation. In this case, it is known [@CGLS] that $\|\cdot\|_{X_0}$ is a norm.
Consider a curve $X_0$ in $X_{PSL_2}(M)$. We say that a sequence of characters $\chi_n \in X_0$ [*converges to an ideal point*]{} $x_0 \in \tilde X_0$ if there are a sequence $\{x_n\}$ in $X_0^\nu \subset \tilde X_0$ and an ideal point $x_0 \in {\cal I}(X_0)$ such that $\nu(x_n) = \chi_n$ for all $n$ and $\lim_n x_n = x_0$.
For a path-connected space $X$, a representation $\rho \in R_{PSL_2}(X)$, a path-connected subspace $Q$ of a space $X$ with inclusion map $i: Q \to X$, set $$\rho^Q := \rho\circ i_\#: \pi_1(Q) \to PSL_2(\mathbb C).$$ Since $\rho_Q$ is determined up to conjugation, there is a well-defined $$\chi_\rho^Q = \chi_{\rho^Q}.$$
\[limitsreps\] [([@CS])]{} Suppose that $X_0$ is a curve in $X_{PSL_2}(M)$ and $\rho_n \in R_{X_0}
\subset R_{PSL_2}(M)$ is a sequence of representations whose characters $\chi_n$ converge to an ideal point $x_0$ of $\tilde X_0$. Then there is an essential surface $S \subset M$ whose complementary components $A_1, A_2, \ldots , A_n$ satisfy the following properties.\
$(a)$ For each $i$, the characters $\chi_n^{A_i}$ converge to a character $\chi_0^{A_i}$. Thus if $S_j$ is a component of $S$, then $\chi_0^{S_j} := \lim_n
\chi_n^{S_j} \in X_{PSL_2}(S_j)$ exists. Further, $\chi_0^{S_j}$ is reducible.\
$(b)$ For each $i$, there are conjugates $\sigma_n^{A_i}$ of $\rho_n^{A_i}$ which converge to a representation $\sigma_0^{A_i} \in R_{PSL_2}(A_i)$ for which $\chi_{\sigma_0^{A_i}} = \chi_0^{A_i}$.
A representation $\sigma_0^{A_i} \in R_{PSL_2}(A_i)$ obtained as a limit of some conjugates of $\rho_n^{A_i}$ is said to be a [*limiting representation associated to the sequence $\{\rho_n\}$*]{}.
Any essential surface $S \subset M$ as described in Proposition \[limitsreps\] is said to be [*associated*]{} to the ideal point $x_0$.
\[idealvalue\] [([@CS], [@CGLS], [@CCGLS])]{} Let $x_0$ be an ideal point of a curve $X_0$ in $X_{PSL_2}(M)$. There is at least one primitive class $\alpha \in H_1(\partial M)$ such that $\tilde f_\alpha(x_0) \in \mathbb C$. Further,\
$(1)$ if there is exactly one such class (up to sign), then it is a boundary class and any surface $S$ associated to $x_0$ has non-empty boundary of slope $\alpha$. Further, $\tilde f_\alpha(x_0) = (\lambda - \lambda^{-1})^2$ where $\lambda$ is a root of unity,\
$(2)$ if there are rationally independent classes $\alpha, \beta \in
H_1(\partial M)$ such that $\tilde f_\alpha(x_0), \tilde f_\beta(x_0) \in \mathbb C$, then $\tilde f_\gamma(x_0) \in \mathbb C$ for each $\gamma \in H_1(\partial M)$ and the surface $S$ can be chosen to be closed.
\[smallcharactervariety\] Suppose that $M$ is a small knot exterior.\
$(1)$ If $X_0$ is a non-trivial component of $X_{PSL_2}(M)$ and $x_0$ an ideal point of $X_0$, there is a primitive class $\alpha \in H_1(\partial M)$ such that $\Pi_{x_0}(\tilde f_{\alpha}) > 0$. Thus $\|\cdot \|_{X_0}$ is either a norm curve or a $\beta$-curve for some primitive $\beta \in H_1(\partial M)$.\
$(2)$ If $\alpha \in H_1(\partial M)$ is a slope such that $X_{PSL_2}(M(\alpha))$ is infinite, then $\alpha$ is a boundary slope.
The first statement follows immediately from the previous proposition. For the second, assume that $X_{PSL_2}(M(\alpha))$ is infinite and choose a curve $X_0 \subset X_{PSL_2}(M(\alpha)) \subset X_{PSL_2}(M)$. Since $M$ is small, any essential surface in $M$ associated to an ideal point $x_0$ has boundary. Moreover, since $X_0 \subset X_{PSL_2}(M(\alpha))$, $\tilde f_\alpha(x_0) = 0$. Part (1) of the corollary shows that $\alpha$ is the unique slope with this property and therefore part (1) of the previous proposition shows that it is a boundary slope.
\[normcondition\] Let $X_0$ be a curve in $X_{PSL_2}(M)$ containing the characters of two discrete representations $\rho_1, \rho_2$ such that $\rho_j(\pi_1(\partial M))$ contains a non-trivial loxodromic element of $PSL_2(\mathbb C)$. If there are rationally independent classes $\alpha_1, \alpha_2 \in H_1(\partial M)$ such that $\rho_j(\alpha_j) = \pm I$ for $j = 1, 2$, then $\|\cdot\|_{X_0}$ is a norm.
Our hypotheses imply that $\rho_j(\pi_1(\partial M)) \cong \mathbb Z \oplus \mathbb Z/c_j$ for some $c_j \geq 1$ and that any element of infinite order in this group is loxodromic. In particular this is the case for any element of $H_1(\partial M)$ which is rationally independent of $\alpha_j$ and therefore $f_{\alpha_1}(\chi_{\rho_2}) = |\mbox{trace}(\rho_2(\alpha_1))|^2 - 4 \ne 0$. Since $f_{\alpha_1}(\chi_{\rho_1}) = 0$, $f_{\alpha_1}|X_0$ is not constant. If there is a primitive class $\beta \in H_1(\partial M)$ such that $f_\beta|X_0$ is constant, then for some $j$, $\alpha_j$ and $\beta$ are rationally independent and so $\rho_j(\beta)$ is loxodromic. It follows that $f_\beta \equiv f_\beta(\chi_j) = (\lambda - \lambda^{-1})^2$ where $\lambda$ is not a root of unity. In particular $\tilde f_\beta$ takes on this value at each ideal point of $X_0$. Proposition \[idealvalue\] now shows that $\tilde f_{\alpha_1}(x_0) \in \mathbb C$ for each ideal point $x_0$ of $X_0$. But this impossible as it would imply that $f_{\alpha_1}|X_0$ is constant. Thus $\|\cdot\|_{X_0}$ is a norm.
Dominations between small knot manifolds {#sec:knot exterior}
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Bounds on dominations between small knot manifolds {#seminorm}
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Let $M$ be a small knot manifold and denote by $T_1(M)$ the torsion subgroup of $H_1(M)$ and $F_1(M) = H_1(M)/T_1(M) \cong \mathbb Z$ its free part. If $K_M = \hbox{kernel}(\pi_1(M) \to H_1(M) \to F_1(M))$, then $$Z(\pi_1(M)) \cap K_M = \{1\}$$ where $Z(\pi_1(M))$ denotes the centre of $\pi_1(M)$. This is obvious when $M$ is hyperbolic. When it is Seifert fibred, it admits a Seifert structure with base orbifold $D^2(p,q)$ and so $Z(\pi_1(M))$ is generated by the class of the fibre. Since $\partial M \ne \emptyset$, $M$ admits a horizontal surface, and as $D^2(p,q)$ is orientable, the surface is non-separating. Thus the fibre class has infinite order in $H_1(M)$ which yields the desired conclusion. (See [@Wal1] for more details.)
Given a component $X_0$ of $X_+^{irr}(M)$, set $$K_M(X_0) = \hbox{Ker}(X_0) \cap K_M.$$
Recall that a component $X_0 \subseteq X_+^{irr}(M)$ is called [*principal*]{} if it contains the character of an irreducible, discrete faithful representation of $\pi_1(M)/Z(\pi_1(M))$. The character varieties of small knot manifolds have dimension $1$ \[CCGLS\] and since they are either hyperbolic or Seifert fibred, they contain at least one principal component. Moreover, such a component is contained in $X_+^{str}(M)$ unless $M$ is a twisted $I$-bundle over the Klein bottle.
For a small knot manifold define $${\cal I}_M = \{\pi_1(M) / K_M(Y_0) : Y_0 \hbox{ an algebraic component of } X_{+}^{irr}(M)\} / \hbox{isomorphism}$$ and note $$|{\cal I}_M| \leq \# \hbox{ algebraic components of } X_{+}^{irr}(M).$$
\[epibound\] Let $M$ and $N$ be small knot manifolds and $X_0$ a principal component of $X_{PSL_2}(N)$. If $\varphi: \pi_1(M) \to \pi_1(N)$ is an epimorphism and $Y_0 = \varphi^*(X_0) \subseteq X_+^{irr}(M)$, then $\hbox{kernel}(\varphi) = K_M(Y_0)$. Thus $|{\cal I}_M|$ is an upper bound for the number of isomorphism classes of groups $\pi_1(N)$ where $N$ is a small knot manifold for which there is an epimorphism $\varphi: \pi_1(M) \to \pi_1(N)$.
By Lemmas \[ker\] and \[kerprincipal\] we have $\hbox{Ker}(Y_0) = \varphi^{-1}(\hbox{Ker}(X_0)) = \varphi^{-1}(Z(\pi_1(N)))$. Since $\varphi$ is surjective, it induces an isomorphism $F_1(M) \to F_1(N)$ and therefore $\varphi^{-1}(K_N) = K_M$. It follows that $K_N(Y_0) = \varphi^{-1}(Z(\pi_1(N))) \cap \varphi^{-1}(K_N) = \varphi^{-1}(Z(\pi_1(N)) \cap K_N) = \varphi^{-1}(1) = \hbox{kernel}(\varphi)$. Hence $\pi_1(N) \cong \pi_1(M) / K_M(Y_0)$ represents an element of ${\cal I}_M$. This completes the proof.
\[cor:canonical curve\] Suppose that $M$ is a small knot manifolds such that $X_+^{irr}(M)$ contains only principal components. Then any non-zero degree map $f: M \to N$ is homotopic to a cover. In particular, if $M$ covers no orientable manifold other than itself, it is minimal.
Since $\hbox{degree}(f) \ne 0$, $f_\#(\pi_1(M))$ has finite index in $\pi_1(N)$ and so consideration of the cover $\tilde N \to N$ corresponding to $\hbox{image}(f_\#)$ and $\tilde f: M \to N$, the lift of $f$, we can suppose, without loss of generality, that $f_\#$ is surjective. Theorem \[epibound\] implies that $\hbox{kernel}(f_\#) = K_M(Y_0)$ for a principal component $Y_0$ of $X_{PSL_2}(M)$. Thus by Lemma \[kerprincipal\] $\hbox{kernel}(f_\#) = Z(\pi_1(M)) \cap K_M = \{1\}$ and therefore $f_\#$ is an isomorphism which preserves the peripheral structure. Hence $f$ is homotopic to a homeomorphism [@Wal2]. This completes the proof.
\[twistpretzel\] [The $PSL_2(\mathbb C)$-character variety of the exterior of a non-trivial twist knot or a $(-2, 3, n)$ pretzel knot, $n \not \equiv 0$ (mod $3$), has a unique non-trivial component (([@Bu], [@Mat]). Further, any such manifold covers no orientable manifold but itself as otherwise the cover would be regular [@GW] and so the knot would admit a free symmetry. But by [@GLM] and [@BolZ] (see also [@Ha]), this eventuality only occurs in the case of the trefoil knot exterior where the result is readily verified. Thus we have two infinite families of minimal manifolds. It is interesting to note that when $n \equiv 0$ (mod $3$), the character variety of the exterior of the $(-2, 3, n)$ pretzel knot has precisely two non-trivial components, one principal and the other corresponding to a strict domination of the trefoil knot exterior. (This follows from the analysis in the section “$r$-curves" of [@Mat].) Thus $M$ is ${\cal H}$-minimal in this case.]{}
We say that non-zero degree maps $f_j: M \to N_j$ ($j=1,2$) to be [*equivalent*]{} if there is a homeomorphism $g: N_1 \to N_2$ such that $f_2 \simeq g \circ f_1$.
Set $${\cal N}_M = \{\pi_1(M) / K_M(Y_0) : Y_0 \hbox{ a norm curve in } X_+^{irr}(M) \} / \hbox{isomorphism} \subseteq {\cal I}_M.$$
\[dombound\] Let $M$ be a small knot manifold.\
$(1)$ The number of equivalence classes of $\pi_1$-surjective non-zero degree maps $M \to N$ is bounded above by $|{\cal I}_M|$. More precisely,\
$(a)$ The number of equivalence classes of $\pi_1$-surjective non-zero degree maps $M \to N$\
where $N$ is hyperbolic is bounded above by $|{\cal N}_M|$.\
$(b)$ The number of equivalence classes of $\pi_1$-surjective non-zero degree maps $M \to N$\
where $N$ is Seifert is bounded above by $|{\cal I}_M| - |{\cal N}_M|$.\
$(2)(a)$ The number of equivalence classes of non-zero degree maps from $M$ to a hyperbolic\
manifold is bounded above by a constant depending only on $X_{PSL_2}(M)$.\
$(b)$ The number of homeomorphism classes of Seifert fibred manifolds dominated by $M$\
is bounded above by a constant depending only on $X_{PSL_2}(M)$.
[Small Seifert knot manifolds have base orbifolds of the form $D^2(p,q)$ where $p, q \geq 2$. They are also surface bundles over the circle with periodic monodromies. If $F$ is the fibre and $h: F \to F$ the monodromy, then $D^2(p, q) = F/h$. It is clear that such manifolds admit self-covering maps of arbitrarily high degree. On the other hand, if $M$ is a hyperbolic knot manifold, it is well-known that the degree of a proper map $f: M \to N$ is bounded above by $vol(M)$. These contrasting facts are the root of the difference in the statements of parts (2)(a) and (b) of the theorem. ]{}
[**Proof of Theorem \[dombound\].**]{} A standard transversality argument shows that if $M \to N$ is a non-zero degree map, then $N$ is small.
\(1) Let $f: M \to N$ be a $\pi_1$-surjective non-zero degree map. According to Theorem \[epibound\], if $X_0$ is a principal component of $X_+^{irr}(N)$ and $Y_0 = f_\#^*(X_0)$, then $\hbox{kernel}(f_\#) = K_M(Y_0)$. Suppose that $f': M \to N'$ is another $\pi_1$-surjective non-zero degree map and $X_0'$ is a principal component of $X_+^{irr}(N')$ such that $Y_0 = (f')_\#^*(X'_0)$ and therefore $\hbox{kernel}((f')_\#) = K_M(Y_0)$. We claim that $f$ and $f'$ are equivalent. To see this, observe that by construction, there is an isomorphism $\varphi: \pi_1(N) \to \pi_1(N')$ such that $f'_\# = \varphi \circ f_\#$. Since $N$ and $N'$ are Haken, it suffices to prove that $\varphi(\pi_1(\partial N)) \subseteq \pi_1(\partial N')$ [@Wal2]. This is clear when $N$, and therefore $N'$, is hyperbolic. Suppose then that they are Seifert fibred manifolds with base orbifolds ${\cal B}, {\cal B}'$. Since $\varphi(Z(\pi_1(N))) = Z(\pi_1(N'))$, $\varphi$ induces an isomorphism $\bar \varphi: \pi_1({\cal B}) \to \pi_1({\cal B}')$. By hypothesis, the finite index subgroup $f_\#(\pi_1(\partial M))$ of $\pi_1(\partial N)$ is sent into $\pi_1(\partial N')$ by $\varphi$ and therefore some positive power of a generator of $\pi_1(\partial {\cal B}) \cong \mathbb Z$ is sent by $\bar \varphi$ into $\pi_1(\partial {\cal B}')$. As $\pi_1({\cal B})$ is isomorphic to a free product of two finite cyclic groups ($\mathbb Z/p * \mathbb Z/q$ if ${\cal B} = D^2(p,q)$), the centralizer of any element is cyclic and so we conclude that $\bar \varphi(\pi_1(\partial {\cal B})) \subseteq \pi_1(\partial {\cal B}')$. This fact together with the identity $\varphi(Z(\pi_1(N))) = Z(\pi_1(N'))$ imply that $\varphi$ preserves peripheral subgroups and so is induced by a homeomorphism $g: N \to N'$. Thus the number of equivalence classes of $\pi_1$-surjective non-zero degree maps is bounded above by $|{\cal I}_M|$.
The proof of (1)(a) follows from the fact that if $N$ is hyperbolic, then a principal curve $X_0 \subset X_{PSL_2}(N)$ is a norm curve (cf. §\[culler-shalen\]). Thus if $f: M \to N$ has non-zero degree, then $f_\#|\pi_1(\partial M)$ is injective and so $Y_0 = f_\#^*(X_0)$ is also a norm curve. Part (1)(b) follows similarly. Simply note that a principal curve $X_0$ for a small Seifert knot manifold is never a norm curve since the class $\gamma \in \pi_1(N)$ of a regular fibre is sent to $\pm I$ by each $\rho \in R_{X_0}$ (Proposition \[kerprincipal\](1)).
(2)(a) Consider a non-zero degree map $f: M \to N$ where $N$ is hyperbolic. Since $M$ is small and $f_\#(\pi_1(M))$ has finite index in $\pi_1(N)$, $M$ must also be hyperbolic. Now $f$ factors $M \stackrel{\tilde f}{\longrightarrow} \tilde N \stackrel{g}{\longrightarrow} N$ where $\tilde f$ is $\pi_1$-surjective and $g$ is a cover of degree at most $vol(M)$, a constant determined by $X_{PSL_2}(M)$. By part (1), there are only finitely many possibilities for $M \stackrel{\tilde f}{\longrightarrow} \tilde N$ up to equivalence. Hence as $vol(\tilde N) \leq vol(M)$, we are reduced to proving the following claim.
\[hypbound\] Given a hyperbolic $3$-manifold $W$ with $vol(W) \leq vol(M)$, the number of equivalence classes of covers $p: W \to V$ is bounded above by a constant depending only on $vol(M)$.
A $d$-fold covering $p: W \to V$ is determined up to equivalence by a homomorphism $\rho: \pi_1(V) \to {\cal S}_d$, where ${\cal S}_d$ is the symmetric group on a set of $d$ elements. Further, $\rho$ induces a transitive action of $\pi_1(V)$ on the set for which $\pi_1(W)$ is the stabilizer of an element. Thus $K_\rho := \hbox{kernel}(\rho) \subseteq \pi_1(W)$ and has index at most $(d-1)!$. It follows that $K_\rho$ is the fundamental group of a finite cover $\hat W \to W$ whose volume is bounded above by $(d-1)! vol(M)$. Therefore, the isometry group $Out(K_\rho)$ of $\hat W$ is a finite group of cardinality at most $24 (d-1)! vol(M)$ since by [@Mey] the volume of a cusped hyperbolic $3$-orbifold is $\geq 1/6$. Since $d \leq vol(W) \leq vol(M)$, $|Out(K_\rho)|$ is bounded by a constant depending only on $vol(M)$.
Let $H_V = \rho(\pi_1(V)) \subseteq {\cal S}_d$ and consider the exact sequence $$1 \to K_\rho \to \pi_1(V) \stackrel{\rho}{\longrightarrow} H_V \to 1$$ This extension is determined by the associated homomorphism $H_V \to \hbox{Out}(K_\rho)$ since $K_\rho$ has trivial centre (see Chap. IV.6 of [@Brn]). The number of such homomorphisms is bounded above by $|Out(K_\rho)|^{d!}$, which in turn is bounded by a constant depending only on $vol(M)$. Since the number of possible groups $H_V$ is bounded above by $2^{d!}$, it remains to show that the number of possibilities for $K_\rho$ is bounded above by a constant depending only on $vol(M)$. But there is a universal constant C such that the rank of $\pi_1(W)$ is no more than $C vol(W) \leq C vol(M)$ [@Ad]. On the other hand, the number of normal subgroups of $\pi_1(W)$ of index at most $(d-1)!$ is bounded above by $((d-1)!)!)^{rank(\pi_1(W))}$. This completes the proof of the Claim and therefore of (2)(a). (Claim \[hypbound\])
(2)(b) Part (1)(b) shows that the number of $\pi_1$-surjective dominations of $M$ to a Seifert manifold is bounded above by $|{\cal I}_M| - |{\cal N}_M|$. Given such a domination $M \to \tilde N$, let $D^2(p,q)$ be the base orbifold of $\tilde N$ and note that $p, q$ are determined by the associated curve in $X_+^{irr}(M)$. To complete the proof, it suffices to show that the number of homeomorphism types of knot manifolds $N$ finitely covered by $\tilde N$ is bounded above by a constant depending only on $p, q$.
Let $\tilde N \to N$ be a cover and suppose $D^2(a,b)$ is the base orbifold of $N$. There is an induced orbifold cover $D^2(p,q) \to D^2(a,b)$ of some degree $d \geq 1$. An elementary calculation based on Euler characteristics shows that if $d > 1$, then up to permutation of $a, b$ either\
(a) $a = d = 2$ and $p = q = b$, or\
(b) $a = b = p = q = 2$.\
A small Seifert knot manifold $N$ with base orbifold $D^2(a,b)$ is the union of two vertical solid tori along a vertical annulus where the solid tori are of fibred type $(a,r)$ and $(b,s)$ where $1 = \gcd(a,r) = \gcd(b,s)$. The homeomorphism type of $N$ is unchanged if we alter the gluing map by a homeomorphism which extends over either solid torus. Hence the number of homeomorphism types of small Seifert knot manifolds with base $D^2(a,b)$ is at most $\frac{ab}{4} \leq \frac{pq}{4}$. Thus we are done.
Rigidity in $\pi_1(M)$ and bounds on sequences of dominations {#rigidity}
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We assume that $M$ is a small knot manifold in this section.
Call $\gamma \in \pi_1(M)$ [*rigid*]{} if $f_\gamma|X_0$ is constant for some principal curve $X_0$ of $X_{PSL_2}(M)$. Equivalently, $d_{X_0}(\gamma) = 0$ (cf. §\[culler-shalen\]). (This condition is independent of the choice of principal curve.) For instance, if a positive power of $\gamma \in \pi_1(M)$ is central, then $\gamma$ is rigid. We call $\gamma$ [*non-rigid*]{} otherwise. Finally we call $\gamma \in \pi_1(M)$ [*totally non-rigid*]{} if $f_\gamma|X_0$ is non-constant for all curves $X_0 \subseteq X_{+}^{irr}(M)$.
\[non-rigid\] Let $M$ be a small knot manifold.\
$(1)$ If $M$ is hyperbolic, every non-trivial element of $\pi_1(\partial M)$ is non-rigid.\
$(2)$ If $M$ is Seifert, an element of $\pi_1(M)$ is rigid if and only if some non-zero power of it is central.\
$(3)$ If $\alpha \in H_1(\partial M) = \pi_1(\partial M) \subset \pi_1(M)$ is a slope which is not a boundary slope, then $\alpha$ is totally non-rigid.
Part (1) is proved in Proposition 1.1.1 of [@CGLS].
\(2) Suppose that $M$ is Seifert. Since it is small, its base orbifold is of the form ${\cal B} = D^2(p, q)$ for some integers $p, q \geq 2$. If no positive power of $\gamma \in \pi_1(M)$ is central, then $\gamma$ projects to an element $\bar \gamma \in \pi_1({\cal B}) \cong \mathbb Z/p * \mathbb Z/q$ of reduced length at least $2$ with respect to any generators $x$ of $\mathbb Z/p$ and $y$ of $\mathbb Z / q$. It follows as in the proof of Theorem 1 of [@BMS] that $f_{\bar \gamma}$ is non-constant on each non-trivial curve of $X_{PSL_2}(\mathbb Z/p * \mathbb Z/q)$. Thus $\gamma$ is non-rigid.
\(3) Let $X_0$ be a non-trivial curve in $X_{PSL_2}(M)$ and suppose that $f_\alpha|X_0$ is constant. Then for any ideal point $x$ of $X_0$, $f_\alpha (x) \in \mathbb C$. But this impossible as otherwise Proposition \[idealvalue\] implies that either $M$ is large or $\alpha$ is a boundary slope. Thus $f_\alpha|X_0$ is not constant.
\[samedegree\] Suppose that $M$ and $N$ are small knot manifolds and $\varphi: \pi_1(M) \to
\pi_1(N)$ is a virtual epimorphism.\
$(1)$ $\varphi^*$ induces a birational isomorphism between $X_+^{str}(N)$ and a union of algebraic components of $X_+^{str}(M)$. In particular if $Y_0$ is a component of $X_{+}^{str}(N)$, then $X_0 = \varphi^*(Y_0)$ is a component of $X_{+}^{str}(M)$ and for each $\gamma \in \pi_1(M)$ we have $$d_{X_0}(\gamma) = d_{Y_0}(\varphi(\gamma)).$$ $(2)$ If there is a principal component $X_0$ of $X_{PSL_2}(M)$ contained in $\varphi^*(X_{+}^{str}(N))$, then $\varphi$ is injective.
Parts (1) follows from the remark in the opening paragraph of this section and Corollary \[1-1\]. We consider part (2) then.
Suppose that $\varphi^*(X_{+}^{str}(Y_0)) = X_0$ for some component $Y_0$ of $X_{+}^{str}(N)$ and principal component $X_0$ of $X_{PSL_2}(M)$. Lemma \[ker\] (2) shows that $\hbox{kernel}(\varphi) \subseteq \hbox{Ker}(X_0) \subseteq Z(\pi_1(M))$. Thus if $\hbox{kernel}(\varphi) \ne \{1\}$, $M$ is Seifert fibred, and as $\pi_1(M)$ and $\pi_1(N)$ are torsion free, $\hbox{kernel}(\varphi) = Z(\pi_1(M)) \cong \mathbb Z$. But this is impossible as it would imply that $\pi_1(N)$ contains a subgroup isomorphic to $\pi_1(M)/Z(\pi_1(M))$, which is the free product of two finite cyclic groups. Thus $\hbox{kernel}(\varphi) = \{1\}$.
We define the [*strict degree*]{} of an element $\gamma$ of the fundamental group of a small knot manifold $M$ to be the sum $$d_M (\gamma)= \sum_{\stackrel{components \; X_0 \; of}{X_+^{str}(M)}} d_{X_0}(\gamma).$$ Note that $d_M (\gamma) > 0$ if $\gamma$ is non-rigid as long as $M$ is not a twisted $I$-bundle over the Klein bottle. The following lemma is of use in this case.
\[kleinbottle\] Let $M, N$ be small manifolds and $\varphi: \pi_1(M) \to \pi_1(N)$ a virtual epimorphism.\
$(1)$ If $N$ is a twisted $I$-bundle over the Klein bottle and $X_0$ the principal curve in $X_+^{irr}(N)$, then $\varphi^*(X_0)$ is a non-trivial curve in $X_{PSL_2}(M)$.\
$(2)$ If $M$ is a twisted $I$-bundle over the Klein bottle then so is $N$ and $\varphi$ is injective.
\(1) Let $\tilde N \stackrel{g}{\longrightarrow} N$ be the cover corresponding to $\varphi(\pi_1(M))$. A finite cover of $N$ is either homeomorphic to $N$ or $S^1 \times S^1 \times I$ and so as $M$ has first Betti number $1$, $\tilde N$ is also a twisted $I$-bundle over the Klein bottle. The reader will then verify that $Y_0 = (g_\#)^*(X_0)$ is a principal curve for $\tilde N_i$. But if $\tilde \varphi: \pi_1(M) \to \pi_1(\tilde N)$ is the surjection induced by $\varphi$, $\tilde \varphi^*(Y_0) = \varphi^*(X_0)$ is a non-trivial curve in $X_{PSL_2}(M)$.
\(2) If $M$ is a twisted $I$-bundle over the Klein bottle, then $\pi_1(N)$ has a finite index abelian subgroup and therefore is also a twisted $I$-bundle over the Klein bottle. As in the proof of (1), if $\tilde N \to N$ is the cover corresponding to $\varphi(\pi_1(M))$, then $\tilde N$ is also a twisted $I$-bundle over the Klein bottle. But the fundamental group of such a manifold is Hopfian so the induced epimorphism $\pi_1(M) \to \pi_1(\tilde N)$ is an isomorphism. Thus $\varphi$ is injective.
\[degreeinject\] Let $\varphi: \pi_1(M) \to \pi_1(N)$ be a virtual epimorphism.\
$(1)$ $d_N(\varphi(\gamma)) \leq d_M(\gamma)$ for all $\gamma \in \pi_1(M)$.\
$(2)$ If $\gamma \in \pi_1(M)$ is not rigid and $d_N(\varphi(\gamma)) = d_M(\gamma)$, then $\varphi$ is injective.
The first assertion is a consequence of part (1) of Lemma \[samedegree\]. To prove the second, note that Lemma \[samedegree\](2) shows that we can suppose there is a principal component $X_0$ of $X_{PSL_2}(M)$ which is not contained in the Zariski closure of the image of $\varphi^*$. Lemma \[kleinbottle\](2) shows that we can also suppose that $M$ is not a twisted $I$-bundle over the Klein bottle. Thus as $\gamma$ is not rigid, $d_{X_0}(\gamma) > 0$ and therefore $d_M (\gamma) \geq d_N(\varphi(\gamma)) + d_{X_0}(\gamma)> d_N(\varphi(\gamma)) = d_M (\gamma)$, which is impossible.
[Note that under the hypotheses of part (2) of the theorem, work of Waldhausen [@Wal2] implies that $\varphi$ is induced by a covering map $M \to N$ as long as it preserves the peripheral subgroups of $\pi_1(M)$ and $\pi_1(N)$. This is automatically satisfied if $N$ is hyperbolic. ]{}
Our next result gives an a priori bound on the length of certain sequences of homomorphisms between the fundamental groups of small knot manifolds.
\[homomorphismsequence\] Let $M$ be a small knot manifold and consider a sequence of homomorphisms $$\pi_1(M) \stackrel{\varphi_1}{\longrightarrow} \pi_1(N_1) \stackrel{\varphi_2}{\longrightarrow} \cdots \stackrel{\varphi_n}{\longrightarrow} \pi_1(N_n)$$ none of which is injective. If $N_i$ is small and $\varphi_i$ is a virtual epimorphism for each $i$, then $n \leq d_M(\gamma)$ for each totally non-rigid element $\gamma \in \pi_1(M)$. Moreover, if $n = d_M(\gamma)$ for some such $\gamma$, then $N_n$ is a twisted $I$-bundle over the Klein bottle.
Set $\psi_i = \varphi_i \circ \cdots \circ \varphi_1$ and let $\gamma \in \pi_1(M)$ be totally non-rigid. If $\psi_i(\gamma)$ is rigid for some $1 \leq i \leq n$ and $X_0 \subset X_{+}^{irr}(N_i)$ is a principal curve, then $f_{\psi_i(\gamma)}|X_0$ is constant, or equivalently, $f_\gamma|\psi_i^*(X_0)$ is constant (Identity (2.1.1)). Since $\gamma$ is totally non-rigid, $\psi_i^*(X_0)$ cannot be a non-trivial curve and therefore $X_0 \not \subset X_+^{str}(N_i)$ (Corollary \[1-1\]). Thus $N_i$ is a twisted $I$-bundle over the Klein bottle. But Lemma \[kleinbottle\](1) shows that this case does not arise under our assumptions. It follows that $\psi_i(\gamma)$ is non-rigid for $1 \leq i \leq n$. Moreover, Lemma \[kleinbottle\](2) shows that if $N_i$ is a twisted $I$-bundle over the Klein bottle for some $i$, then $i = n$. Theorem \[degreeinject\] now implies that $$d_{M}(\gamma) > d_{N_{1}}(\psi_1(\gamma)) > \cdots > d_{N_{i}}(\psi_i(\gamma)) \cdots > d_{N_{n}}(\psi_{n}(\gamma)) \geq 0$$ with $d_{N_{n}}(\psi_{n}(\gamma)) = 0$ if and only if $N_n$ is a twisted $I$-bundle over the Klein bottle. This completes the proof.
Dominations by two-bridge knot exteriors {#twobridgedomination}
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Consider relatively prime integers $p,q$ where $p \geq 1$ is odd and let $k_{p/q}$ denote the two-bridge knot corresponding to the rational number $p/q$. Thus the $2$-fold cover of $S^3$ branched over $k_{p/q}$ is the lens space $L(p,q)$. It is a theorem of Schubert \[Sch\] that $k_{p/q}$ is equivalent to $k_{p'/q'}$ if and only if $L(p,q)$ is homeomorphic to $L(p',q')$. The exterior $M_{p/q}$ of $k_{p/q}$ is known to be small \[HT\]. Moreover it is hyperbolic unless $q \equiv \pm 1$ (mod $p$), in which case it is a $(p,2)$ torus knot.
The proof of the following unpublished result of Tanguay is contained in Appendix A.
\[meridiandegree\] [[([@Tan])]{}]{} Let $M$ be the exterior of the two-bridge knot of type $p/q$. If $\mu \in \pi_1(M)$ is a meridinal class, then $d_M(\mu)= \frac{p-1}{2}.$
As a consequence we deduce:
Consider a sequence of homomorphisms $$\pi_1(M_{p/q}) \stackrel{\varphi_1}{\longrightarrow} \pi_1(N_1) \stackrel{\varphi_2}{\longrightarrow} \cdots \stackrel{\varphi_n}{\longrightarrow} \pi_1(N_n)$$ none of which is injective. If $N_i$ is small and $\varphi_i$ is a virtual epimorphism for each $i$, then $n < \frac{p-1}{2}$.
The meridinal slope $\mu$ of a two-bridge knot is not a boundary slope [@HT] so Lemma \[non-rigid\](3) shows that it is totally non-rigid in $\pi_1(M_{p/q})$. Theorem \[homomorphismsequence\] then yields the inequality $n \leq d_{M_{p/q}}(\mu) = \frac{p-1}{2}$ with equality only if $N_n$ is a twisted $I$-bundle over the Klein bottle. We saw in the proof of Lemma \[kleinbottle\](1) that if $\tilde N_n \to N_n$ is the cover corresponding to the image of $\varphi_n \circ \varphi_{n-1} \circ \ldots \circ \varphi_1$, then $\tilde N_n$ is also a twisted $I$-bundle over the Klein bottle. But this is impossible since $H_1(M_{p/q})$ is cyclic while $H_1(\tilde N_n)$ is not. Thus $n < \frac{p-1}{2}$.
This result can be significantly strengthened if the homomorphisms are induced by non-zero degree maps. This is the goal of the remainder of this section.
\[2bridgemini\] Let $N$ be a knot manifold and $\varphi: \pi_1(M_{p/q}) \to \pi_1(N)$ a homomorphism such that the image $\varphi(\mu)$ of a meridian $\mu$ is peripheral.\
$(1)$ If $\varphi$ is an epimorphism, then $N$ is homeomorphic to the exterior $M_{p'/q'}$ of a 2-bridge knot in $S^3$. Moreover either $M_{p'/q'} = M_{p/q}$ or $p = kp'$ with $k > 1$.\
$(2)$ If $\varphi(\pi_1(M))$ is of finite index $d$ in $\pi_1(N)$, then either $d = 1$ and the conclusions of part $(1)$ hold, or $N$ is Seifert fibred and $\varphi$ factors through an epimorphism $\tilde \varphi: \pi_1(M_{p/q}) \to \pi_1(M_{p'})$ to which the conclusions of part $(1)$ apply. Further, $\gcd(2p',d) = 1$.
$(1)$ Since $\mu$ normally generates $\pi_1(M_{p/q})$, $ \varphi(\mu)$ does the same for $\pi_1(N)$. In particular $\varphi(\mu) \ne 1$ so if $\mu'$ is the slope on $\partial N$ corresponding to the projective class of $\varphi_*(\mu) \in H_1(\partial N)$, then the manifold $ W = N( \mu')$ obtained by Dehn filling $\partial N$ along the slope $\mu'$ is a homotopy $3$-sphere.
Let $k'$ be the core of the surgery in $W = N(\mu')$ and let $\widehat{W}_2(k')$ be the $2$-fold cover of $W$ branched over $k'$. There is an induced surjective homomorphism $\mathbb Z/p \cong \pi_1(L(p,q)) \to \pi_1(\widehat{W}_2(k'))$ and so the latter group is finite cyclic $\mathbb Z/p'$ with $p'$ dividing $p$. Since $\pi_1(M_{p/q})$ is generated by two elements, the same holds for $\pi_1(N)$, hence $k'$ is a $2$-generator knot in the homotopy sphere $W$. It follows as in [@Wed] that $k'$ is prime and thus $N$ cannot contain an essential annulus with slope $\mu'$. Thus the $2$-fold branched covering $\widehat{W}_2(k')$ of $k'$ is irreducible and by the orbifold theorem (\[BP\], \[BLP\], \[CHK\]) $\widehat{W}_2(k')$ it is itself a lens space and the covering involution conjugates to an orthogonal involution. Therefore $W = N(\varphi(\mu)) \cong S^3$ and $k'$ is a two-bridge knot. In other words, $(N(\varphi(\mu)), k') \cong (S^3, k_{p'/q'})$ for some integers $p' \geq 1, q'$ with $p'$ dividing $p$ and $q'$ coprime with $p'$. Property P for two-bridge knots [@Tak] implies that $\varphi(\mu) = \mu'$ is a meridian of $k_{p'/q'}$.
According to Theorem \[degreeinject\] and Proposition \[meridiandegree\], either $\varphi$ is an isomorphism or $\frac{p-1}{2} = d_{M_{p/q}}(\mu) > d_{M_{p'/q'}}(\mu') =
\frac{p'-1}{2}$. In the first case $k_{p/q} = k_{p'/q'}$ while in the second $p = kp'$ with $k > 1$. This is the conclusion of (1).
$(2)$ Let $\tilde N \to N$ be the cover corresponding to the image of $\varphi$ and define $\tilde \varphi: \pi_1(M) \to \pi_1(\tilde N)$ and $\psi: \pi_1(\tilde N) \to
\pi_1(N)$ in the obvious way. Part $(1)$ implies that $\tilde N$ is homeomorphic to the exterior $M_{p'/q'}$ of a 2-bridge knot in $S^3$. Hence by [@GW], the cover $\tilde N \to N$ is regular and cyclic. If $N$ is hyperbolic, $\tilde N = N$ since hyperbolic 2-bridge knot exteriors admit no free symmetries by [@GLM] (see also [@Ha]), and therefore we are in case (1). Otherwise $N$ is Seifert and so $\tilde N$ is the exterior $M_p'$ of $k_{p'}$, which is the $(p',2)$ torus knot. Thus the Seifert structure on $M_{p'}$ has base orbifold $D^2(2, p')$. Since $p'$ is odd, the proof of part (2)(b) of Theorem \[dombound\] shows that the cover $\tilde N \to N$ induces a homeomorphism of the underlying orbifolds. Thus the cover is a degree $d$ unwinding of a regular fibre of $N$ and so if $F$ is the fibre and $h: F \to F$ the monodromy of the realization of $N$ as a surface bundle over the circle, then $h^d$ is the monodromy of the realization of $\tilde N$ as a surface bundle. The induced homeomorphism on the level of orbifolds is $F/h \to F/h^d$ and so $d$ must be coprime with the order of $h$. But since $F/h \cong D^2(2,p')$, this order is a multiple of $2p'$.
The following two results are immediate consequences of the previous theorem:
\[2bridgebound\] Consider a sequence of non-zero degree maps $$M_{p_0/q_0} = N_0 \stackrel{f_1}{\longrightarrow} N_1 \stackrel{f_2}{\longrightarrow} \cdots \stackrel{f_n}{\longrightarrow} N_n$$ between knot manifolds, none of which is homotopic to a homeomorphism. If $N_{n-1}$ is hyperbolic, there are coprime pairs $p_j, q_j$ $(1 \leq j \leq n)$ such that $N_j = M_{p_j / q_j}$ $(1 \leq j \leq n-1)$, $N_n$ is finitely covered by some $M_{p_n / q_n}$, and $p_{j-1} = k_j p_j$ for some integer $k_j > 1$ $(1 \leq j \leq n)$. Hence, $n + 1$ is bounded above by the number of distinct multiplicative factors of $p$.
\[2bridgemini2\] If $p$ is an odd prime, then $M_{p/q}$ is minimal if and only if it is hyperbolic $($i.e. $q \not \equiv \pm 1 \hbox{ $($mod p$)$}$$)$.
If we consider domination via degree-one maps (i.e. 1-domination), we obtain stronger results:
Let $N$ be a knot manifold and $f: M_{p/q} \to N$ a degree-one map. Then either $f$ is homotopic to a homeomorphism or $N$ is a two-bridge knot exterior $M_{p'/q'}$ where $p = kp', k > 1,$ and $\gcd(k, p') = 1$.
Since a degree-one map induces an epimorphism on the level of fundamemtal groups, case $(1)$ of the previous theorem shows that if $f$ is not homotopic to a homeomorphism, then $N = M_{p'/q'}$ where $p = kp', k > 1$. Moreover, $f$ induces a degree-one map $L(p,q) \to L(p',q')$ between the $2$-fold branched covers. By Corollary 6 of \[RoW\], there is an integer c such that $q' \equiv (\frac{p}{p'}) c^2 q$ (mod $p'$). In particular this implies that $\gcd(\frac{p}{p'}, p') = 1$.
\[2bridgedegree1bound\] Consider a sequence of degree-one maps $$M_{p/q} \stackrel{f_1}{\longrightarrow} N_1 \stackrel{f_2}{\longrightarrow} \cdots \stackrel{f_n}{\longrightarrow} N_n$$ between knot manifolds, none of which is homotopic to a homeomorphism. Then $n + 1$ is bounded above by the number of distinct prime factors of $p$.
\[1minimal\] If $p$ is a prime power, the two-bridge knot exterior $M_{p/q}$ does not stictly $1$-dominate any knot manifold.
Sets of discrete $PSL_2(\mathbb C)$-characters {#sec:characters}
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We investigate sequences of $PSL_2(\mathbb C)$ characters of representations of the fundamental groups of small knot manifolds whose images are discrete. This leads us in particular to proofs of Theorems \[thm:characters\] and \[thm:domination\]. Our analysis relies fundamentally on the convergence theory of Kleinian groups and hyperbolic $3$-manifolds.
Convergence of Kleinian groups and hyperbolic $3$-manifolds {#convkleinhyp}
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A metric space is [*proper*]{} if all of its closed and bounded subsets are compact. A sequence of proper pointed metric spaces $(X_n, x_n)$ is said to converge *geometrically* to a metric space $(X_0, x_0)$ if for every $r > 0$, the sequence of compact metric balls $\{B_{X_n}(x_n; r)\}$ converges in the Gromov bilipschitz topology to $B_{X_0}(x_0; r)$. (See chapter 3 of [@Gro] and also chapter E [@BeP], chapter 7 [@MT].)
We recall the thick/thin decomposition of a complete, finite volume hyperbolic $3$-manifold $V$ (chapter D of [@BeP]): given a positive constant $0< \mu \leq \mu_0$, where $\mu_0$ is the Margulis constant, $V$ decomposes as $V_{[\mu, \infty)} \cup V_{(0,\mu]}$ such that:
$V_{[\mu, \infty)} = \{ x \in V : \hbox{inj}(x) \geq \mu \}$ is the $\mu$-thick part of $V$
$V_{(0,\mu]} = \{ x \in V : \hbox{inj}(x) \leq \mu \}$ is the $\mu$-thin part of $V$.
For $\mu \leq \mu_0,$ each component of $\mu$-thin part of $V$ is either empty, or a geodesic neighborhood of a closed geodesic (a Margulis tube, homeomorphic to $S^1 \times D^2$) or a cusp with torus cross sections (homeomorphic to $T^2 \times [0, \infty)$).
Let $\{V_n\}$ be a sequence of pointed, closed, connected, orientable, hyperbolic $3$-manifolds whose volumes are bounded above. There is a sequence of base points $x_n \in (V_{n})_{[\mu_0, \infty)}$ such that some subsequence $\{(V_j, x_j)\}$ converges to a pointed, complete, finite volume, hyperbolic $3$-manifold $(V, x)$. In particular this implies that given $\varepsilon > 0$ and $0< \mu \leq \mu_0$, for $j \geq n_{0}(\varepsilon,\mu)$ the $\mu$-thick parts of $V_j$ and $V$ are $(1+ \varepsilon)$-bilipschitz homeomorphic. Moreover $$vol(V) = \lim_j vol(V_j),$$ (see chapter E of [@BeP], theorem 7.9 of [@MT]). Further, if $V$ is closed, then $V = V_{j}$ for $j \gg 0$ and if $V$ is not closed, $V_j$ is obtained by Dehn filling $V$ for $j \gg 0$ (see chapter 5 of [@Thu], chapter E of [@BeP]). By a Dehn filling of a complete, non-compact, finite volume hyperbolic $3$-manifold $V$ we mean a Dehn filling of some compact core $V_0$ of $V$.
In order to simplify the presentation, base points for fundamental groups and pointed metric spaces will often be supressed from the notation. In particular we will say that a sequence $\{V_n\}$ of hyperbolic manifolds converges geometrically to a hyperbolic manifold $V$ if it does so under a suitable choice of base points.
We come now to the algebraic counterpart of this notion of geometric convergence. A good source on this topic is the paper [@JM] of Jørgensen and Marden. The torsion-free case is dealt with in chapter 7 of [@MT].
The [*envelope*]{} of a sequence $\{\Gamma_n\}$ of subgroups of $PSL_2(\mathbb C)$ is defined as $$Env(\{\Gamma_n\}) := \{\gamma = \lim_n \gamma_n : \gamma_n \in \Gamma_n \mbox{ for all } n\} \subset PSL_2(\mathbb C).$$ Clearly, $Env(\{\Gamma_n\})$ is a subgroup of $PSL_2(\mathbb C)$.
A [*Kleinian*]{} group is a discrete subgroup of $PSL_2(\mathbb C)$. A [*Fuchsian*]{} group is a discrete subgroup of $PSL_2(\mathbb R)$.
\[elemordiscr\] [(cf. Lemmas 3.2 and 3.6, [@JM] ) ]{}$\;$\
$(1)$ If each $\Gamma_n$ is a non-elementary Kleinian group, then $Env(\{\Gamma_n\})$ is either elementary or discrete.\
$(2)$ If $Env(\{\Gamma_n\})$ is non-elementary, then each $\gamma \in \overline{\cup (\Gamma_n \setminus \{\pm I\})}$ is either loxodromic, or parabolic, or elliptic of finite order $m \geq 2$. In the latter case, for any subsequence $\{\gamma_{n'}\}$ which converges to $\gamma$, $\gamma_{n'}$ has order $m$ for $n' \gg 0$.
\[torsionfree\] If each $\Gamma_n$ is a torsion-free non-elementary Kleinian group, then $Env(\{\Gamma_n\})$ is either abelian or discrete, and if it is non-abelian, each $\gamma \in \overline{\cup (\Gamma_n \setminus \{1\})}$ is either loxodromic or parabolic.
A sequence $\{\Gamma_n\}$ of subgroups of $PSL_2(\mathbb C)$ is said to [*converge geometrically*]{} to a subgroup $\Gamma_0$ of $PSL_2(\mathbb C)$ if $\Gamma_0 = Env(\{\Gamma_{j}\})$ for every subsequence $\{ j \}$ of $\{ n \}$. The sequence $\{\Gamma_n\}$ is said to [*converge algebraically*]{} to $\Gamma_0$ if there is a finitely generated group $\pi$ and representations $\rho_n \in R_{PSL_2}(\pi)$ ($n \geq 0$) such that $\Gamma_n = \rho_n(\pi)$ and $\lim_n \rho_n = \rho_0$. Note that if $\{\Gamma_n\}$ converges algebraically to $\Gamma_0$ and geometrically to $\Gamma$, then $\Gamma_0 \subseteq \Gamma \subseteq Env(\{\Gamma_{n}\})$.
We record the following result for later use. Proofs in the torsion-free case can be found in Theorems 7.6, 7.7, 7.12, 7.13, and 7.14 of [@MT]. The general case can be dealt with using the results of [@JM].
\[convergence\] Suppose that $\pi$ is a finitely generated group and $\rho_n: \pi \to PSL_2(\mathbb C)$ is a sequence which converges to $\rho_0 \in R_{PSL_2}(\pi)$. For $n \geq 0$ set $\Gamma_n = \rho_n(\pi)$ and suppose that for $n \geq 1$, $\Gamma_n$ is a non-elementary Kleinian group. Then\
$(1)$ $\Gamma_0$ is a non-elementary Kleinian group.\
$(2)$ for $n \gg 0$ there is a homomorphism $\theta_n: \Gamma_0 \to
\Gamma_{n}$ such that $\rho_{n} = \theta_n \circ \rho$. Further,\
$\lim \theta_n =
1_{\Gamma_0}$.\
$(3)$ there are a non-elementary Kleinian group $\Gamma$ containing $\Gamma_0$ and a subsequence $\{j\}$ of\
$\{n\}$ such that $\{\Gamma_{j}\}$ converges geometrically to $\Gamma$. Moreover, the homomorphisms $\theta_{j}$ of\
part $(2)$ extend to homomorphisms $\Gamma \to \Gamma_{j}$, which we continue to denote $\theta_{j}$, in such a\
way that $\lim_j \theta_j =
1_{\Gamma}$.\
$(4)$ the quotient spaces $\mathbb H^3/\Gamma_{j}$ converge geometrically to $\mathbb
H^3/\Gamma$
In the remainder of the paper we investigate sets of discrete characters and apply our results to study sequences of non-zero degree maps between closed manifolds. Let $M$ be a knot manifold and $X_0$ a subvariety of $X_{PSL_2}(M)$. Set $$D(X_0) = \{ \chi_\rho \in X_0 : \rho \hbox{ is discrete and non-elementary}\}$$ $$D^*(X_0) = \{ \chi_\rho \in D(X_0) : \rho \hbox{ is torsion free}\}$$ $$D_{0}^*(X_0) = \{ \chi_\rho \in D^*(X_0) : \rho \hbox{ has non-zero volume}\}.$$ Note that the image of any $\rho \in R_{PSL_2}(M)$ whose character is contained in $D_0^*(X_0)$ is the fundamental group of a complete hyperbolic $3$-manifold.
\[standardimage\] Let $\chi_\rho \in D^*_0(X_0)$, $\Gamma = \rho(\pi_1(M))$, and $V = \mathbb H^3 / \Gamma$.\
$(1)$ If $\rho|\pi_1(\partial M)$ is injective, then a compact core $V_0$ of $V$ is a hyperbolic knot manifold and there is a proper non-zero degree map $f: M \to V_0$ such that $f_\#: \pi_1(M) \to \pi_1(V_0) = \Gamma$ is conjugate to $\rho$.\
$(2)$ If $\rho|\pi_1(\partial M)$ is not injective, $V$ is closed and there is a slope $\alpha$ on $\partial M$ and a non-zero degree map $f: M(\alpha) \to V$ such that the composition $\pi_1(M) \to \pi_1(M(\alpha)) \stackrel{f_\#}{\longrightarrow} \pi_1(V) = \Gamma$ is conjugate to $\rho$.\
$(3)$ If $v_0 > 0$ is the minimal volume for complete, connected, orientable, hyperbolic $3$-manifolds, then $|vol(\chi_\rho)| \geq v_0$.
If $\rho|\pi_1(\partial M)$ is injective, there is a compact core $V_0$ of $V$ and a torus $T$ in $\partial V_0$ such that $\rho(\pi_1(\partial M)) \subset \pi_1(T)$. Thus there is a proper map $f: (M, \partial M) \to (V_0, T)$ realizing $\rho$. By the definition of the volume of a representation ([@Dun]), $|\hbox{degree}(f)| vol(V) = |vol(\rho)| \ne 0$. In particular $|\hbox{degree}(f)| \ne 0$, which implies that $\partial V_0 = T$ and $|vol(\rho)| \geq vol(V) \geq v_0$. On the other hand, if $\hbox{kernel}(\rho| \pi_1(\partial M)) \ne \{\pm I\}$, $\rho$ factors $\pi_1(M) \to \pi_1(M(\alpha)) \stackrel{\bar \rho}{\longrightarrow} \pi_1(V) = \Gamma$ for some slope $\alpha$ since the image of $\rho$ is torsion free. There is a map $f: M(\alpha) \to V$ associated to the homomorphism $\bar \rho$ and Lemma 2.5.4 of [@Dun] implies that $vol(\bar \rho) = vol(\rho)$. Then $|\hbox{degree}(f)| vol(V) = |vol(\bar \rho)| = |vol(\rho)| \ne 0$ and again we see that $|\hbox{degree}(f)| \ne 0$ so that $V$ must be closed and $|vol(\rho)| \geq vol(V) \geq v_0$. This completes the proof.
Here is a simple application of the results of this section.
\[closedinX\_0\] $D(X_0), D^*(X_0),$ and $D^*_0(X_0)$ are closed in $X_0$.
Suppose that $\lim_n \chi_n = \chi_0 \in X_0$ where $\chi_n \in D(X_0)$ for all $n$. Proposition 1.4.4 of [@CS] (or Corollary 2.1 of [@CL]) shows that there are a subsequence $\{j\}$ of $\{n\}$ and a convergent sequence of representations $\{\rho_j\} \subset R_{X_0}$ such that $\chi_j = \chi_{\rho_j}$. Set $\rho_0 = \lim_j \rho_j$ and note that $\chi_0 = \chi_{\rho_0}$. Proposition \[convergence\] implies that $\chi_{\rho_0} \in D(X_0)$. Moreover, if we assume that each $\chi_n \in D^*(X_0)$, then part (2) of Proposition \[convergence\] implies that $\chi_{\rho_0} \in D^*(X_0)$. Thus $D(X_0)$ and $D^*(X_0)$ are closed in $X_0$. In particular, if $\chi_n \in D^*_0(X_0)$ for all $n$, then $\chi_0 \in D^*(X_0)$. From the previous lemma and the continuity of the volume function we have $|vol(\chi_0)| = \lim_n |vol(\chi_n)| \geq v_0 > 0$. Thus $\chi_0 \in D_0^*(X_0)$, which completes the proof.
Unbounded sequences of discrete $PSL_2(\mathbb C)$-characters {#unbounded}
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In this section $M$ will be a small knot manifold and $X_0$ a non-trivial component of $X_{PSL_2}(M)$. We are interested in the asymptotic behaviour of the sets $D(X_0)$ and $D^*(X_0)$. Consider a sequence $\{\chi_n\} \subset D(X_0)$ which converges to an ideal point $x_0$ of $X_0$. Fix $\rho_n \in R_{X_0}$ such that $\chi_n = \chi_{\rho_n}$ and let $\alpha_0$ be the $\partial$-slope associated to $x_0$.
\[projectiveconvergence\] Let $M$ be a small knot manifold, $X_0$ a curve component of $X_{PSL_2}(M)$, and $\{\chi_n\} \subset D(X_0)$ a sequence which converges to an ideal point $x_0$ of $X_0$. Fix $\rho_n \in R_{X_0}$ such that $\chi_n = \chi_{\rho_n}$ and let $\alpha_0$ be the $\partial$-slope associated to $x_0$.\
$(1)$ For $n \gg 0$, $\hbox{kernel}(\rho_n|\pi_1(\partial M)) \cong \mathbb Z$ and $\rho_n(\pi_1(\partial M)) \cong \mathbb Z \oplus \mathbb Z/c_n$ where the $\mathbb Z$ factor is generated by a loxodromic element and $c_n \geq 1$.\
$(2)$ Let $\alpha_n \in H_1(\partial M)$ be the element, unique up to sign, which generates the kernel of $\rho_n|\pi_1(\partial M)$ $($ $n \gg0$ $)$. Then $\lim_n [\alpha_n] = [\alpha_0]$.\
$(3)$ If $[\alpha_n] \ne [\alpha_m]$ for some $m, n \gg 0$, then $X_0$ is a norm curve.
\(1) Since $M$ is small, there is a slope $\beta$ on $\partial M$ such that $\tilde f_\beta$ has a pole at $x_0$. Thus $\rho_n(\beta)$ is loxodromic for large $n$ and so for such $n$, $\rho_n|\pi_1(\partial M))$ contains no parabolics. On the other hand, a discrete subgroup of $PSL_2(\mathbb C)$ isomorphic to $\mathbb Z^2$ contains parabolic matrices. Thus $\hbox{kernel}(\rho_n|\pi_1(\partial M)) \ne \{\pm I\}$, which implies (1).
\(2) Since $M$ is small, Proposition \[idealvalue\] shows that there is a unique slope $\alpha_0 \in H_1(\partial M)$ such that $\tilde f_{\alpha_{0}}(x_0) \in \mathbb C$. Further, $\alpha_0$ is a boundary slope, any surface $S$ associated to $x_0$ has non-empty boundary of slope $\alpha_0$, and if $\alpha_{0}^{*} \in H_1(\partial M)$ is a slope dual to $\alpha_0$ (i.e. $\alpha_0 \cdot \alpha_0^* = 1$), then $\lim_n f_{\alpha_{0}^{*}}(\chi_n) = \infty$. We must show that $\lim_n [\alpha_n] = [\alpha_0]$.
To that end set $\alpha_n = p_n \alpha_0 + q_n \alpha_{0}^{*}$. By construction, $(p_n, q_n) \ne (0,0)$ and for $n$ large, $\rho_n(\alpha_{0}^{*})$ is loxodromic. By choice of $\alpha_n$ we have $(\rho_n(\alpha_0))^{p_n} = (\rho_n(\alpha_{0}^{*}))^{-q_n}$ and therefore the minimal translation lengths $\ell(\rho_n(\alpha_0))$ and $\ell((\rho_n(\alpha_{0}^{*}))$ of $\rho_n(\alpha_0)$ and $\rho_n(\alpha_{0}^{*})$ satisfy: $$\vert p_n \vert \ell(\rho_n(\alpha_0)) = \vert q_n \vert \ell((\rho_n(\alpha_{0}^{*})) > 0.$$ If $\pm A \in PSL_2(\mathbb C)$, then $\ell(\pm A) = |\log(|\frac{\hbox{trace}(A)}{2} + \sqrt{(\frac{\hbox{trace}(A)}{2})^2 - 4}|)|$ and so our hypotheses imply that $\lim_n \ell((\rho_n(\alpha_{0}^{*})) = \infty$ while $\lim_n \ell(\rho_n(\alpha_0))$ is bounded. Thus $\lim_n \frac{q_n}{p_n} = 0$ or equivalently, $\alpha_n$ converge projectively to $[\alpha_0]$.
\(3) follows from (1) and Corollary \[normcondition\].
\[elementaryimage\] Let $M$ be a small knot manifold, $X_0$ a non-trivial component of $X_{PSL_2}(M)$, and $\{\chi_n\} \subset D(X_0)$ a sequence which converges to an ideal point $x_0$ of $X_0$. If $S_0$ is a component of an essential surface associated to $x_0$ and $i_\#: \pi_1(S_0) \to \pi_1(M)$ is the inclusion induced homomorphism, then either\
$(a)$ $\overline{i_\#^*(X_0)} \subset X_{{\cal N}}(S_0)$, or\
$(b)$ $i_\#^*(X_0) = \{\chi_\rho\}$ where $\rho(\pi_1(S_0))$ is either the tetrahedral group, the octahedral group,\
or the icosahedral group.
Fix $\rho_n \in R_{X_0}$ such that $\chi_n = \chi_{\rho_n}$ and let $S_0$ be a component of an essential surface $S$ in $M$ associated to $x_0$. Since $\chi_n|\pi_1(S_0)$ converges to a character $\chi_\sigma \in X_{PSL_2}(S_0)$ (Proposition \[limitsreps\]), we can replace the $\rho_n$ by conjugate representations so that after passing to a subsequence $\{j\}$, we have $\lim \rho_j|\pi_1(S_0) = \sigma$ where $\chi = \chi_\sigma$ (see Proposition 1.4.4 of [@CS] or Corollary 2.1 of [@CL]). We also know that $\sigma$ is reducible (Proposition \[limitsreps\]) and so by taking $j \gg 0$, $\rho_j^{S_0}$ is discrete and elementary (Proposition \[convergence\]). A discrete elementary subgroup of $PSL_2(\mathbb C)$ is either reducible, conjugates into ${\cal N}$, or is isomorphic to a [*polyhedral group*]{} (i.e. the tetrahedral group, the octahedral group, or the icosahedral group). Thus $i_\#^*(\chi_j)$ is contained in $X_{{\cal N}}(S_0)$ or $\rho_j$ has polyhedral image. This proves the lemma when $i_\#^*(X_0)$ is a single character. Suppose, on the other hand, that $\overline{i_\#^*(X_0)}$ is a curve $Y_0 \subset X_{PSL_2}(S_0)$. Then $i_\#^*: X_0 \to Y_0$ is finite-to-one and since there only finitely many characters of representations in $R_{PSL_2}(S_0)$ with image a polyhedral group, $Y_0 \cap X_{{\cal N}}(S_0)$ is infinite. But $X_{{\cal N}}(S_0)$ is Zariski closed in $X_{PSL_2}(S_0)$, and so it contains $Y_0$.
\[nonelementarycompact\] Let $M$ be a small knot manifold, $X_0$ a non-trivial component of $X_{PSL_2}(M)$. Suppose that for each ideal point $x_0$ of $X_0$ there are a component $S_0$ of an essential surface associated to $x_0$ and a character $\chi \in X_0$ such that $\chi|\pi_1(S_0)$ is non-elementary. Then $D(X_0), D^*(X_0)$, and $D_0^*(X_0)$ are compact.
Theorem \[elementaryimage\] shows that $D(X_0)$ does not accumulate to an ideal point of $X_0$. The result then follows from Proposition \[closedinX\_0\].
\[principalcompact\] Let $M, N$ be small hyperbolic knot manifolds and suppose that $\varphi: \pi_1(M) \to \pi_1(N)$ is a virtual epimorphism. Fix a principal component $Y_0 \subset X_{PSL_2}(N)$ and set $X_0 = \varphi^*(Y_0)$. Then $D(X_0), D^*(X_0)$, and $D_0^*(X_0)$ are compact. In particular this is true for a principal component of $X_{PSL_2}(M)$.
First suppose that $M = N$ and $\varphi$ is the identity. By Corollary \[nonelementarycompact\] it suffices to show that for each connected essential surface $S_0$ in $M$, there is a character $\chi_\rho \in X_0$ such that $\rho(\pi_1(S_0))$ is non-elementary. Fix such a surface and note that $\pi_1(S_0)$ is a non-abelian free group since $M$ is small and hyperbolic. Moreover, since $X_0$ is principal, it contains the character of a discrete faithful representation $\rho_0$ of $\pi_1(M)$. Thus $\rho_0(\pi_1(S_0))$ is a discrete and free of rank at least $2$ and as such is non-elementary. Thus $D(X_0), D^*(X_0)$, and $D_0^*(X_0)$ are compact.
Now consider the general case and let $\chi_\rho \in D(X_0)$. By Lemma \[closed\] there is a $\chi_{\rho'} \in Y_0$ such that $\varphi^*(\chi_{\rho'}) = \chi_\rho$. Since $\chi_\rho$ is irreducible we can suppose that $\rho = \rho' \circ \varphi$. Then $\rho'$ is non-elementary and since the image of $\varphi$ has finite index in $\pi_1(N)$, it is also discrete. In other words, $\chi_{\rho'} \in D(Y_0)$ and so $\chi_\rho = \varphi^*(\chi_{\rho'}) \in \varphi^*(D(Y_0))$. Hence $D(X_0)$ is contained in the compact subset $\varphi^*(D(Y_0))$ of $X_0$.
[Corollary \[principalcompact\] implies that the set of discrete, non-elementary characters in the $PSL_2(\mathbb C)$ character variety of the exterior of either a hyperbolic twist knot or the $(-2, 3, n)$ pretzel knot, $n \not \equiv 0$ (mod $3$) is compact (cf. Example \[twistpretzel\]).]{}
\[pqnoncompact\] [The corollary is false if we assume that $N$ is Seifert fibred but not a twisted $I$-bundle over the Klein bottle. Indeed, suppose that $N$ has base orbifold $D(p,q)$ where $p, q \geq 2, (p,q) \ne (2,2)$. Each pair $\pm I \ne A_0, B_0 \in {\cal D}$ such that $A_0^p = B_0^q = \pm I$ determines a curve $Y_0 \subset X_{PSL_2}(\mathbb Z/p * \mathbb Z/q)$ consisting of the characters of homomorphisms sending a generator of $\mathbb Z/p$ to $A_0$ and one of $\mathbb Z/q$ to a conjugate $B$ of $B_0$ (see Example 3.2, [@BZ1]). Further, if a sequence $\{B_n\}$ of such conjugates is chosen so that $\lim_n |\hbox{trace}(A_0 B_n)| = \infty$, the associated characters tend to the unique ideal point of $Y_0$ (Example 3.2, [@BZ1]). On the other hand, if $A_0, B_0 \in PSL_2(\mathbb R)$ are chosen to have extreme negative trace (page 293, [@Kn]), they generate a discrete group isomorphic to $\mathbb Z/p * \mathbb Z/q \cong \pi_1(D(p,q))$ as long as $|\hbox{trace}(A_0B)| \geq 2$ (Theorem 2.3 [@Kn]). In particular, they determine a principal component $Y_0 \subset X_{PSL_2}(\mathbb Z/p * \mathbb Z/q) = X_{PSL_2}(\pi_1(D(p,q)) \subset X_{PSL_2}(N)$ for which $D(Y_0)$ is non-compact. By hypothesis, $Y_0 \subset X_+^{str}(N)$, so $X_0 := \varphi^*(Y_0) \subset X_+^{str}(M)$ (Corollary \[1-1\]) and by construction, $D(X_0)$ is non-compact.]{}
\[torsionfreeconvideal\] Let $M$ be a small knot manifold, $X_0$ a norm curve component of $X_{PSL_2}(M)$, and $\{\chi_n\} \subset D^*(X_0)$ a sequence which converges to an ideal point $x_0$ of $X_0$. If $S$ is an essential surface associated to $x_0$ and $S_0$ a component of $S$, then $S_0$ is separating and there is a complementary component $A$ of $S_0$ such that $\rho(\pi_1(A)))$ is abelian for each $\rho \in R_{X_0}$. There is a subsequence $\{j\}$ of $\{n\}$ such that $\rho_j(\pi_1(A)))$ is cyclic for all $j$.
Choose $\rho_n \in R_{X_0}$ such that $\chi_n = \chi_{\rho_n}$. By Lemma \[projectiveconvergence\] we may suppose that $\rho_n(\pi_1(\partial M))$ is loxodromic and since $\rho_n$ is torsion free, the lemma imlies that there is a unique slope $\alpha_n$ on $\partial M$ satisfying $\rho_n(\alpha_n) = \pm I$. We may suppose that the $\alpha_n$ are distinct, since $X_0$ is a norm curve, and that none of them are boundary slopes [@Hat].
Since $M$ is small, $S_0$ has non-empty boundary of slope $\alpha_0$, say. Then by construction, $\rho_n(\alpha_0) \ne \pm I$ is loxodromic for $n \gg 0$. According to Theorem \[elementaryimage\], $\rho_n(\pi_1(S_0))$ is elementary and since it is discrete, torsion free, and contains a loxodromic ($n \gg 0$), it is a cyclic subgroup of $PSL_2(\mathbb C)$. In this case we can apply the bending construction to $\chi_n$ along $\pi_1(S_0)$ (Appendix \[bending\]). We claim that for $n \gg 0$, the bending of $\chi_n$ along $\pi_1(S_0)$ is trivial. For such $n$, $\chi_{n} \in X_0$ is contained in a unique component of $X_{PSL_2}(M)$, and so if the claim is false $X_0$ is obtained by bending $\chi_{n}$ along $\pi_1(S_0)$. In particular $\rho_m(\alpha_0)$ is independent of $m$, at least up to conjugation. But then $f_{\alpha_0}|X_0$ is constant and so $X_0$ cannot be a norm curve, contrary to our hypotheses.
Suppose that $S_0$ is non-separating and write $M = A/ \{S_0^+ = S_0^-\}$ where $A$ is the complementary component of $S_0$ in $M$ and $S_0^+ \sqcup S_0^- \subseteq \partial A$ are parallel copies of $S_0$. Note that $\pi_1(M)$ is generated by $\pi_1(A)$ and $\gamma$, a homotopy class represented by a loop which intersects $S_0$ once transversely. Fix $n \gg 0$ and note that since $\rho_n$ cannot be bent non-trivially along $\pi_1(S_0)$, either $\rho_n(\pi_1(M)) \subset {\cal N}$ or $\rho_n$ is reducible (Lemma \[constbendnonsep\]). As neither of these possibilities is satisfied in our situation, $S_0$ must be separating. Hence if $M = A \cup_{S_0} B$ where $A$ and $B$ are the complementary components of $S_0$ in $M$, the fact that for large $n$ the bending of $\chi_n$ along $\pi_1(S_0)$ is trivial, at least one of $\rho_n^A, \rho_n^B$ has cyclic image. Further, this image is trivial if the image of $\rho_n^{S_0}$ is trivial (Lemma \[constconj\]). By passing to a subsequence and possibly exchanging $A$ and $B$, we can assume that for $n \gg 0$, $\rho_n^A$ has cyclic image and $\rho_n(\pi_1(A)) = \{\pm I\}$ if $\rho_n(\pi_1(S_0)) = \{\pm I\}$.
Let ${\cal O}(\rho_n)$ denote the $PSL_2(\mathbb C)$ orbit of $\rho_n$. Since $\cup_{m \geq n} {\cal O}(\rho_n)$ is Zariski dense in $R_{X_0}$, $n \geq 1$, the previous paragraph shows that $\rho(\pi_1(S_0))$ is abelian for each $\rho \in R_{X_0}$.
\[lessthan2\] Let $M$ be a small knot manifold and $X_0$ a norm curve of $X_{PSL_2}(M)$. Then $D^*(X_0)$ is a compact subset of $X_0$ as long as the following condition holds: Any ideal point of a norm curve in $X_{PSL_2}(M)$ has an associated essential surface with a component $S_0$ having no more than two boundary components.
By Proposition \[closedinX\_0\], it suffices to show that $D^*(X_0)$ is contained in a compact subset of $X_0$. Suppose then that $\{\chi_n \} \subset D^*(X_0)$ is a sequence which converges to an ideal point $x_0$ of $X_0$ and choose $\rho_n \in R_{X_0}$ whose character is $\chi_n$. Fix a component $S_0$ of an essential surface $S$ associated to $x_0$ with $|\partial S_0| \leq 2$. Theorem \[torsionfreeconvideal\] implies that $\rho(\pi_1(A)))$ is abelian for each $\rho \in R_{X_0}$. Since $X_0$ is a norm curve, $f_{\alpha_0}|X_0$ is non-constant and so there is a Zariski dense subset in $R_{X_0}$ of representations $\rho \in R_{X_0}$ such that $\rho(\alpha_0)$ is loxodromic. Fix such a representation $\rho_0$ and conjugate it so that $\rho_0(\alpha_0)$ is diagonal. Then both $\rho_0(\pi_1(A))$ and $\rho_0(\pi_1(\partial M))$ are contained in ${\cal D}$.
Let $B$ be the other complementary component of $S_0$. A maximal compression of $\partial B$ in $B$ must yield a family of $2$-spheres as $M$ is small. Thus $B$ is a handlebody and therefore $\pi_1(\partial B) \to \pi_1(B)$ is surjective. Consider a class $\sigma \in \pi_1(M)$ represented by a product of a path in $S_0$ followed by one in $\partial M \cap B$. By hypothesis $|\partial S_0| = 2$ and so $\sigma$ is the product of an element of $\pi_1(A)$ and one in $\pi_1(\partial M)$. It follows that $\rho_0(\sigma) \in {\cal D}$. Since $\pi_1(B)$ is generated by such classes and $\pi_1(S_0)$, we see that $\rho_n(\pi_1(B)) \subset {\cal D}$. But then the image of $\rho_0$ is abelian, which is impossible as $R_{X_0}$ contains a Zariski dense subset of such representations. Thus $D^*(X_0)$ contained in a compact subset of $X_0$.
Ohtsuki [@Oht] has shown that two-bridge knot exteriors satisfy the condition of the previous corollary.
\[2bridge compact\] Let $X_0$ be a norm curve in the character variety of a two-bridge knot exterior. Then $D^*(X_0)$ and $D_0^*(X_0)$ are compact subsets of $X_0$.
Convergent sequences of discrete, co-compact $PSL_2(\mathbb C)$-characters with non-zero volume {#convergent}
-----------------------------------------------------------------------------------------------
Let $M$ will be a small hyperbolic knot manifold and $X_0$ a non-trivial component of $X_{PSL_2}(M)$. Consider a sequence $\{\chi_{\rho_n}\} \subset D_0^*(X_0)$ of distinct characters which converge to some $\chi_{\rho_0} \in X_0$. Fix $\rho_n \in R_{X_0}$ whose character is $\chi_n$, set $\Gamma_n = \rho_n(\pi_{1}(M))$, and let $V_n = \mathbb H^3/\Gamma_n$.
Since $M$ is small, $\|\cdot \|_{X_0} \ne 0$ and so there are only finitely many $n$ such that $\rho_n(\pi_1(\partial M))$ is either $\{\pm I\}$ or contains a paraboic element of $PSL_2(\mathbb C)$. (Otherwise $\rho_n(\pi_1(\partial M))$ would be contained in a paraboic subgroup for infinitely many $n$ and therefore $f_\gamma|X_0 \equiv 0$ for every peripheral $\gamma$). We suppose then that $\rho_n(\pi_1(\partial M))$ contains a loxodromic for each $n$. Since $\Gamma_n$ is torsion free, this implies that $\rho_n(\pi_1(\partial M)) \cong \mathbb Z$ and so there is a unique slope $\alpha_n$ on $\partial M$ which generates $\hbox{kernel}(\rho_n|\pi_{1}(\partial M))$. It follows from Lemma \[standardimage\] that $V_n$ is a closed hyperbolic $3$-manifold. If $\bar \rho_n \in R_{PSL_2}(M(\alpha_n))$ is the homomorphism induced by $\rho_n$, the proof of part (3) of this lemma shows that $$vol(V_n) \leq |vol(\bar \rho_n)| \leq |vol(\rho_n)| \leq vol(M(\alpha_n)) < vol(M).$$
\[convord\] Assume that the sequence $\{\chi_n\} \subset D_0^*(X_0)$, as above, converges to a character $\chi_{\rho_0}$ and that the slopes $\alpha_n$ associated to $\rho_n$ are distinct. Then there are
[$\;$ (a)]{}
: a subsequence $\{j\}$ of $\{n\}$ such that $\{V_{j}\}$ converges geometrically to a $1$-cusped hyperbolic $3$-manifold $V$ whose fundamental group containes $\rho_0(\pi_{1}(M))$ as a finite index subgroup.
[$\;$ (b)]{}
: a proper non-zero degree map $f_0: M \to V_0$ such that $V_0$ is a compact core of $V$ and if $k_0 : V_0 \to V$ is the inclusion, then $\rho_0 = (k_0)_\# \circ (f_0)_\#$.
[$\;$ (c)]{}
: slopes $\beta_j$ on $\partial V_0$ and identifications $V_j \cong V_0(\beta_j)$, such that $(f_0|\partial M)_*(\alpha_j)$ is a multiple of $\beta_j \in H_1(\partial V)$ and if $k_j: V_0 \to V_0(\beta_j)$ is the inclusion, then $\chi_j$ is the character of the composition $(k_j)_\# \circ (f_0)_\#$.
[$\;$ (d)]{}
: non-zero degree maps $f_j: M(\alpha_n) \to V_0(\beta_j)$ such that the following diagrams are commutative up to homotopy: $$\begin{array}{ccc}
M & \stackrel{f_0}{\longrightarrow} & V_0 \\
\downarrow & & \downarrow \\
M(\alpha_j) & \stackrel{f_j}{\longrightarrow} & V_j \cong V_0(\beta_j)
\end{array}$$
Moreover, $X_0 = (f_0)_\#^*(Y_0)$ where $Y_0$ is a principal curve for $V_0$. In particular, $X_0$ is a norm curve.
After replacing the $\rho_n$ by conjugate representations and passing to a subsequence, we may suppose that $\lim \rho_n = \rho_0$ (see Corollary 2.1 of [@CL] for example). Then $\{\Gamma_n\}$ converges algebraically to $\Gamma_0 = \rho_0(\pi_1(M))$. Proposition \[convergence\], $\Gamma_0$ is a non-elementary Kleinian group and there are homomorphisms $\theta_n: \Gamma_0 \to \Gamma_{n}$ such that $\rho_{n} = \theta_n \circ \rho_0$ for $n \gg 0$. By passing to a subsequence we may suppose that this is true for $n \geq 1$.
Since ${\rm kernel}(\rho_n|\pi_1(\partial M)) = \langle \alpha_n \rangle$, $\rho_{n} = \theta_n \circ \rho_0$, and the slopes $\alpha_n$ are distinct, it follows that $\rho_0|\pi_1(\partial M)$ is injective. By Proposition \[convergence\] implies that after passing to a subsequence we can suppose that
\(i) $\{\Gamma_n\}$ converges geometrically to a non-elementary Kleinian group $\Gamma$ containing $\Gamma_0$\
and $\theta_n$ extends to homomorphisms $\Gamma \to \Gamma_n$ which we still denote by $\theta_n$.
\(ii) $\lim V_n = V := \mathbb H^3/\Gamma$ in the sense of Gromov bilipschitz topology.
As we noted above, $vol(V_n) < vol(M)$ and therefore $vol(V) = \lim vol(V_n) \leq vol(M)$. It follows that $V$ is a complete, connected, orientable, finite volume hyperbolic $3$-manifold. Further, $V$ has at least one cusp since $\Gamma$ contains $\rho_0(\pi_1(\partial M)) \cong \mathbb Z \oplus \mathbb Z$. Thus $V_n$ is obtained from $V$ by hyperbolic Dehn filling for large $n$ (cf. §\[convkleinhyp\]).
Let $k_0: V_0 \to V$ be the inclusion of a compact core of $V$. Since $M$ and $V_0$ are $K(\pi,1)$-spaces, there is a map $f_0: M \to V_0$ such that $\rho_0 = (k_0)_\# \circ (f_0)_\#$. (We have fixed an identification of $\pi_1(V)$ with $\Gamma$ here.) Since $V_0$ is atoroidal, there is a torus $T \subseteq \partial V_0$ such that $\rho_0(\pi_1(\partial M)) \subseteq \pi_1(T)$, at least up to conjugation. Homotope $f_0$ so that $f_0(\partial M) \subseteq T$. Then $(f_0)\#: \pi_1(\partial M) \to \pi_{1}(T)$ and is injective (by construction), which shows that ${\rm degree}(f_{0}|: \partial M \to T) \ne 0$. On the other hand, if $[\partial M] \in H_2(M)$ and $[T] \in H_2(V_0)$ are fundamental classes for $\partial M$ and $T$, then $0 = [\partial M]$ so that $0 = (f_0)_*([\partial M]) = {\rm degree}(f_{0}|) [T]$ and therefore $\partial V_0 = T$.
Recall that $V_n$ is obtained by hyperbolic Dehn filling on $V$. There is some slope $\beta_n$ on $T$ such that $V_n = V_0(\beta_n)$. If $k_n: V_0 \to V_n$ denotes the inclusion, then $\theta_n \circ (k_0)_\# = (k_n)_\#$ (cf. Theorem 7.17 of \[MT\]) and so $\mbox{kernel} (\theta_n) = (k_0)_\#(\langle \langle \beta_n \rangle \rangle_{\pi_1(V_0)})$ where $\langle \langle \beta_n \rangle \rangle_{\pi_1(V_0)}$ is the normal closure in $\pi_1(V_0)$ of the element corresponding to the slope $\beta_n$. Thurston’s hyperbolic Dehn filing theorem (see chapter 5 of [@Thu] or the appendix to [@BoP]) implies that $\pi_1(\partial V_0) \cap \mbox{kernel} (\theta_n) = \langle \beta_n \rangle \cong \mathbb Z$ for large $n$. By passing to a subsequence we can arrange for it to hold for all $n$.
Since $\rho_n = \theta_n \circ \rho_0 = \theta_n \circ (k_0)_\# \circ (f_0)_\# = (k_n)_\# \circ (f_0)_\#$, we have $1 = \rho_n(\alpha_n) = (k_n)_\# ((f_0)_\#(\alpha_n))$ and therefore $(f_0)_\#(\alpha_n) \in \pi_1(\partial V_0) \cap \mbox{kernel} (\theta_n) =\langle \beta_n \rangle$. Thus $f_0$ induces a map $f_n: M(\alpha_n) \to V(\beta_n)$ with ${\rm degree}(f_n) = {\rm degree}(f_0)$. If $i_n: M \to M(\alpha_n) $ denotes the inclusion, we have $(f_n)_\# \circ (i_n)_\# = \theta_n \circ (k_0)_\#\circ (f_{0})\# = (k_n)_\# \circ (f_0)\# = \rho_n$. Hence for large $n$ the following diagrams are commutative up to homotopy $$\begin{array}{ccc}
M & \stackrel{f_0}{\longrightarrow} & V_0 \\
\downarrow & & \downarrow \\
M(\alpha_j) & \stackrel{f_j}{\longrightarrow} & V_j \cong V_0(\beta_j)
\end{array}$$
To complete the proof, we must show that $X_0 = (f_0)_\#^*(Y_0)$ where $Y_0$ is a principal curve for $V_0$. To that end we note that Thurston’s hyperbolic Dehn filling theorem proves that if $Y_0$ is the principal component of $X_{PSL_2}(V_0)$ which contains the character $\chi_0'$ of $\pi_1(V_0) = \Gamma$, then $\chi_0' = \lim_n \chi_n'$ where $\chi_n' \in Y_0$ is the character of our identification $\pi_1(V_0(\beta_n) = \Gamma_n$. By construction $(f_0)_\#^*(\chi_n') = \chi_n$ so that $X_0 \cap (f_0)_\#^*(Y_0)$ is infinite. Lemma \[closed\] then shows that $X_0 = (f_0)_\#^*(Y_0)$.
\[discetecharsprincipal\] Let $M$ be a small hyperbolic knot manifold and $X_0$ a principal component of $X_{PSL_2}(M)$. Then all but finitely many of the elements of $D_0^*(X_0)$ are induced by the complete hyperbolic structure on the interior of $M$ or by Dehn fillings of manifolds finitely covered by $M$.
We know from Corollary \[principalcompact\] that $D_0^*(X_0)$ is contained in a compact subset of $X_0$, Thus if the result is false, we could find a convergent sequence $\{\chi_n\} \subset D_0^*(X_0)$ of distinct characters no one of which is induced by a holonomy character of $M$ or one of the Dehn fillings of an oriented manifold it finitely covers. As above we can assume that $\chi_n$ is the character of a representation $\rho_n$ which is peripherally non-trivial. Let $\alpha_n$ be its slope and note that since $\|\cdot\|$ is a norm curve, the function $n \mapsto \alpha_n$ is finite-to one. Thus we can take a subsequence $\{j\}$ of $\{n\}$ for which the $\alpha_j$ are distinct and apply Theorem \[convord\] to see that there are a non-zero degree map $f_0: M \to N$ where $N$ is hyperbolic, a principal component $Y_0$ of $X_{PSL_2}(N)$ such that $X_0 = (f_0)_\#^*(Y_0)$, and, for infinitely many $j$, $\chi_j$ is the image under $(f_0)_\#^*$ of the holonomy character of some Dehn filling of $N$. Lemma \[kerprincipal\](2) shows that $(f_0)_\#$ is injective and so we can take $f_0: M \to N$ to be a covering map ([@Wal2]). But then infinitely many $\chi_j$ are induced from a Dehn filling of an orientable manifold covered by $M$ contrary to our hypotheses. This completes the proof.
\[characterizingdiscretecharacters\] Let $M$ be a small knot manifold and suppose that there is a norm curve $X_0$ in $X_{PSL_2}(M)$ for which $D_0^*(X_0)$ has an accumulation point in $X_0$. Then there are a hyperbolic manifold $N$, a non-zero degree map $f_0: M \to N$, and a principal component $Z_0$ of $X_{PSL_2}(N)$ such that $X_0 = (f_0)_\#^*(Z_0)$. Further, all but finitely many characters in $D_0^*(X_0)$ are the images under $(f_0)_\#^*$ of the holonomy character of $N$ or one of the Dehn fillings of an oriented manifold finitely covered by $N$.
The hypotheses can be used with Theorem \[convord\] to see that there are a hyperbolic manifold $N_0$, a non-zero degree map $f_0: M \to N_0$, and a principal component $Y_0$ of $X_{PSL_2}(N_0)$ such that $X_0 = f_\#^*(Y_0)$. Let $N \to N_0$ be the cover corresponding to the image of $(f_0)_\#$, $\tilde f_0: M \to N$ a lift of $f_0$, and $Z_0$ the principal curve in $X_{PSL_2}(N)$ obtained by restriction from $Y_0$. Clearly $X_0 = (\tilde f_0)_\#^*(Z_0)$ and final claim of the corollary is a consequence of the previous result applied to $Z_0$. The details are left to the reader.
The final results of this section follow immediately from Theorem \[torsionfreeconvideal\] and Corollaries \[lessthan2\], \[2bridge compact\], and \[characterizingdiscretecharacters\].
Let $M$ be a small knot manifold and suppose that there is a norm curve $X_0$ in $X_{PSL_2}(M)$ for which $D_0^*(X_0)$ has an accumulation point in $X_0$. Then $D_0^*(X_0)$ is compact and has a unique accumulation point corresponding the holonomy character of a hyperbolic knot manifold under a non-zero degree map $M \to N$.
\[compactconditions\] Let $M$ be a small knot manifold and $X_0 \subset X_{PSL_2}(M)$ a norm curve. Then $D_0^*(X_0)$ is compact in $X_0$ with at most one accumulation point if for each ideal point $x_0$ of $X_0$ there is a component $S_0$ of an essential surface associated to $x_0$ such that at least one of the following two conditions holds:\
$(i)$ $\chi|\pi_1(S_0)$ is non-elementary for some $\chi \in X_0$; or\
$(ii)$ $|\partial S_0| \leq 2$.
\[twobridgecompact\] Let $X_0$ be a norm curve in the character variety of a two-bridge knot exterior $M$. Then $D_0^*(X_0)$ is either finite or is a compact subset of $X_0$ with a unique accumulation point. In the latter case there are a two-bridge knot exterior $N$, a non-zero degree map $f: M \to N$, and a principal component $Y_0$ of $X_{PSL_2}(N)$ such that $X_0 = f_\#^*(Y_0)$.
The theorem follows from the results cited above and Theorem \[2bridgemini\] once we note that $M$ must be hyperbolic if $X_{PSL_2}(M)$ is to contain a norm curve.
Domination and hyperbolic Dehn filling {#dominationhyperbolic}
--------------------------------------
In this section we prove Theorem \[thm:domination\].
Let $M$ be a small knot manifold and $\{\alpha_n\}_{n \geq 1}$ a sequence of distinct slopes on $\partial M$ such that for each $n$ we have a map $f_n: M(\alpha_n) \to V_n$ of degree $d_n \geq 1$ where $V_n$ is hyperbolic. We suppose as well that $\{\alpha_n\}$ does not subconverge projectively to a boundary slope and that there is a slope $\alpha_0$ on $\partial M$ such that $M(\alpha_0)$ does not dominate any hyperbolic $3$-manifold.
Let $p_n: \tilde V_n \to V_n$ be the finite cover corresponding to $(f_n)_\#(\pi_1(M))$. We can suppose that $p_n$ is a local isometry. Fix a lift $\tilde f_n: M \to \tilde V_n$ of $f_n$ of degree $\tilde d_n \geq 1$ say. If $v_0 > 0$ is the minimal volume for closed, connected, orientable, hyperbolic $3$-manifolds, then for each $n$ we have $vol(M) > vol(M(\alpha_n)) \geq \tilde d_n vol(\tilde V_n) = d_n vol(V_n) \geq d_n v_0 \geq \tilde d_n v_0.$ Thus the $d_n$ and $\tilde d_n$ are bounded so we can assume, after passing to a subsequence, that they are constant, say $\hbox{degree}(f_n) = d \geq 1, \; \hbox{degree}(\tilde f_n) = \tilde d.$ The degree of each $p_n$ is $d/ \tilde d$.
We identify $\pi_1(V_n)$ with a subgroup $\Gamma_n$ of $PSL_2(\mathbb C)$ and set $(p_n)_\#(\pi_1(\tilde V_n)) = \tilde \Gamma_n \subseteq \Gamma_n$. Let $i_n: M \to M(\alpha_n)$ be the inclusion and define $\rho_n \in R_{PSL_2}(M)$ to be the composition $\pi_1(M) \stackrel{(i_n)_\#}{\longrightarrow} \pi_1(M(\alpha_n)) \stackrel{(f_n)_\#}{\longrightarrow} \tilde \Gamma_n \subseteq \Gamma_n \subset PSL_2(\mathbb C)$. The character of $\rho_n$ will be denoted by $\chi_n$. These objects combine in the following commutative diagram.
& & \_1(V\_n)\
& (2,2)[ (f\_n)\_\#]{} & \_[(p\_n)\_\#]{}\
\_1(M(\_n)) & \^[ (f\_n)\_\#]{} & \_n =\_1(V\_n)\
\^[(i\_n)\_\#]{} & &\
\_1(M) & \^[ \_n]{} & PSL\_2(C)
We claim that each $\rho_n$ is peripherally non-trivial. Otherwise the composition $M \to M(\alpha_n) \stackrel{f_n}{\longrightarrow} V_n$ extends to a map $M(\alpha_0) \to V_n$ of degree $d$, contrary to our hypothesis. Since $\alpha_n \ne \alpha_m$ for $n \ne m$, the characters $\chi_n$ are distinct. After passing to a subsequence we can suppose that they are contained in a non-trivial curve $X_0 \subset X_{PSL_2}(M)$. Proposition \[normcondition\] shows that $X_0$ is a norm curve. Finally, noting that $vol(\chi_n) = d vol(V_n) \ne 0$ we see that $\chi_n \in D_0^*(X_0)$. Since the slopes $\{\alpha_n\}_{n \geq 1}$ do not projectivly subconverge to a $\partial$-slope, Lemma \[projectiveconvergence\] shows that there is a subsequence of characters $\{\chi_k\}$ which converge to a character $\chi_{\rho_0} \in D_0^*(X_0)$, and so the conditions of Theorem \[convord\] are satisfied.
[**Proof of Theorem \[thm:domination\]**]{}. By Theorem \[convord\], the sequence $\tilde V_k$ converges geometrically to a $1$-cusped hyperbolic $3$-manifold $\tilde V$ for which there are:
a proper non-zero degree map $\tilde f_0: M \to \tilde V_0$ such that $\tilde V_0$ is a compact core of $\tilde V$ and if $j_0 : \tilde V_0 \to \tilde V$ is the inclusion map, then $\rho_0 = (j_0)_\# \circ (\tilde f)_\#$;
slopes $\tilde \beta_k$ on $\partial \tilde V_0$ and identifications $\tilde V_k = \tilde V_{0}(\tilde \beta_k)$, such that $(\tilde f_0|\partial M)_*(\alpha_k)$ is a multiple of $\tilde \beta_k \in H_1(\partial \tilde V_0)$ and if $\tilde j_k: \tilde V_0 \to \tilde V_0(\tilde \beta_k)$ is the inclusion, then $\chi_k$ is induced by the composition $(p_k)_\# \circ (\tilde j_k)_\# \circ (\tilde f_0)_\#$.
non-zero degree maps $\tilde f'_k: M(\alpha_k) \to \tilde V(\tilde \beta_k)$ such that the following diagrams are commutative up to homotopy: $$\begin{array}{ccc}
M & \stackrel{\tilde f_0}{\longrightarrow} & \tilde V_0 \\
\downarrow & & \downarrow \\
M(\alpha_k) & \stackrel{\tilde f'_k}{\longrightarrow} & \tilde V_k \cong \tilde V_0(\tilde \beta_k)
\end{array} \eqno{(4.2.1)}$$
Since $(p_k)_\# \circ (\tilde f_k)_\# \circ (i_k)_\# = \rho_k = (p_k)_\# \circ (\tilde j_k)_\# \circ (\tilde f_0)_\# = (p_k)_\# \circ (\tilde f'_k)_\# \circ (i_k)_\# $, it follows that $f_k = p_k \circ \tilde f_k$ is homotopic to $f'_k = p_k \circ \tilde f'_k$. In particular $\hbox{degree}(\tilde f_0) = \hbox{degree}(\tilde f'_k) = \hbox{degree}(\tilde f_k) = \tilde d$ and $\hbox{degree}(f'_k) = \hbox{degree}(f_k) = d$.
Now $\lim_k vol(V_k) = \lim_k (\frac{\tilde d}{d}) vol(\tilde V_k) = (\frac{\tilde d}{d}) vol(\tilde V)$, and so after passing to a subsequence we may assume that $\{V_k\}$ converges geometrically to a complete hyperbolic $3$-manifold $V$ with finite volume $vol( V) = (\frac{\tilde d}{d}) vol(\tilde V)$. For $k \gg 0$, $vol(\tilde V) > vol(\tilde V_k)$ and therefore $vol(V) > vol(V_k)$. Thus $V$ has at least one cusp. On the other hand, $p_k$ is a local isometry so for $\tilde x \in \tilde V_k$ we have $\hbox{inj}(\tilde x) \leq (\frac{d}{\tilde d}) \hbox{inj}(p_k(\tilde x))$. Thus if $\mu_0$ is the Margulis constant and $\mu \leq \frac{\tilde d \mu_0}{d}$, we have $p_{k}^{-1}((V_{k})_{(0,\mu]}) \subseteq (\tilde V_k)_{(0,\frac{d \mu}{\tilde d}]}$ ($k \gg 0$). Since there is a sequence $\mu_k \to 0$ such that $(\tilde V_k)_{(0,\frac{d
\mu_k}{\tilde d}]}$ is a Margulis tube about a geodesic $\tilde \gamma_k$, $(V_{k})_{(0,\mu_k]}$ is a Margulis tube about a geodesic $\gamma_k$, because a geodesic is unique in its homotopy class. Thus $V$ has only one cusp. We note, moreover, that $p_k^{-1}(\gamma_k) = \tilde \gamma_k$ and therefore $\overline{(\tilde V_k)_{(0,\frac{d \mu}{\tilde d}]} \setminus p_{k}^{-1}((V_{k})_{(0,\mu]})} \cong \partial (\tilde V_k)_{(0,\frac{d \mu}{\tilde d}]} \times I$. Thus for large $k$ we can identify $(V_k)_{[\mu, \infty)}$ with a compact core $V_0$ of $V$ and $p_{k}^{-1}((V_k)_{[\mu, \infty)})$ with a compact core $\tilde V_0$ of $\tilde V$. In this way $p_k$ induces a covering map $p_k^0: \tilde V_0 \to V_0$ of degree $d/ \tilde d$. Since $V_0$ and $\tilde V_0$ admit complete finite volume hyperbolic structures on their interiors, after pre-composition by an isotopy of $\tilde V_0$, we can take $p_n^0$ to be a local isometry on the interior of $\tilde V_0$. Now $V_0$ has only finitely many (pointed) covers of degree $d/ \tilde d$ up to equivalence and the isometry group of $\hbox{int}(\tilde V_0)$ is finite, therefore we can restrict to a subsequence and suppose that for all $n, m$, we have $p_n^0 = p_m^0 = p$, say.
The geometric convergence of $V_k$ to $V$ implies that for large $k$ there are slopes $\beta_k$ on $\partial V_0$ such that $V_k = V_0(\beta_k)$. From the previous paragraph we see that any component of $p^{-1}(\beta_k)$ is isotopic to $\tilde \beta_k$ on $\partial \tilde V_0$. Therefore the following diagrams are commutative up to homotopy:
$$\begin{array}{ccc}
\tilde V_0 & \stackrel{p}{\longrightarrow} & V_0 \\
\downarrow & & \downarrow \\
\tilde V_k \cong \tilde V_0(\tilde\beta_k) & \stackrel{p_k}{\longrightarrow} & V_k \cong V_0(\beta_k)
\end{array} \eqno{(4.2.2)}$$
Since $f_k = p_k \circ \tilde f_k$ is homotopic to $f'_k = p_k \circ \tilde f'_k$, by putting together diagrams (4.2.1) and (4.2.2) one deduces that the proper map $f = p \circ \tilde f_0: M \to V_0$ of degree $d \geq 1$ makes the following diagrams commute up to homotopy:
$$\begin{array}{ccc}
M & \stackrel{f}{\longrightarrow} & V_0 \\
\downarrow & & \downarrow \\
M(\alpha_k) & \stackrel{f_k}{\longrightarrow} & V_k \cong V_0(\beta_k)
\end{array}$$
If we assume further that the dominations $f_k : M(\alpha_k) \to V_k \cong V(\beta_k)$ are strict, then the domination $f_0: M \to V$ must also be strict. Otherwise $V_0$ is homeomorphic to $M$ and $d = \hbox{degree}(f_0) = 1$ (since $vol(M) = vol(V_0)$). Then $(f_0)_\#: \pi_{1} (M) \to \pi_{1} (V_0)$ is surjective and therefore an isomorphism since $\pi_{1} (M)$ is Hopfian. By Mostow rigidity theorem we can suppose that $f_0$ is a homeomorphism. But then the induced maps $f'_k : M(\alpha_k) \to V(\beta_k)$ are homeomorphisms homotopic to $f_k$, in contradiction with our assumption that $f_k$ is a strict domination.
${\cal H}$-minimal Dehn filling {#sec:h-minimal}
===============================
The goal of this section is to construct collections of infinitely many ${\cal H}$-minimal closed hyperbolic $3$-manifolds by proving Theorem \[thm:h-minimal\] and Corollaries \[cor:integer surgery\] and \[cor:2-bridge\]. The proofs of these results rely on the following theorem:
\[minimal hyperbolic\] Let $M$ be an ${\cal H}$-minimal, small hyperbolic knot manifold and suppose that there is a slope $\alpha_0$ on $\partial M$ such that the Dehn filled manifold $M(\alpha_0)$ does not dominate any closed hyperbolic manifold.\
$(1)$ If $U \subset \mathbb P(H_1(\partial M; \mathbb R))$ is the union of disjoint closed arc neighbourhoods of the finite set of boundary slopes of $M$, then $\mathbb P(H_1(\partial M; \mathbb R)) \setminus U$ contains only finitely many projective classes of slopes $\alpha$ such that $M(\alpha)$ is not ${\cal H}$-minimal. In particular, $M$ admits infinitely many ${\cal H}$-minimal Dehn fillings.\
$(2)$ If $D_0^*(X_0)$ is a compact subset of $X_0$ for each norm curve in $X_{PSL_2}(M)$, then there are only finitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is not ${\cal H}$-minimal. In particular, this conclusion holds if for each ideal point $x_0$ of a norm curve $X_0$, there is a component $S_0$ of an essential surface associated to $x_0$ such that at least one of the following two conditions holds:\
$(i)$ $\chi|\pi_1(S_0)$ is non-elementary for some $\chi \in X_0$; or\
$(ii)$ $|\partial S_0| \leq 2$.
\(1) Suppose that there are infinitely many projective classes of slopes $\alpha$ in $\mathbb P(H_1(\partial M; \mathbb R)) \setminus U$ such that $M(\alpha)$ is not $\cal H$-minimal. Then there are an infinite sequence of distinct slopes $\alpha_n$ on $\partial M$ which does not subconverge to a $\partial$-slope and strict finite dominations $f_n: M(\alpha_n) \to V_n$, where $V_n$ are closed hyperbolic $3$-manifolds. The sequence $\{\alpha_n\}$ verifies the hypotheses of Theorem \[thm:domination\], hence there is a strict domination $f_0: M \to V$, where $V$ is a $1$-cusped, complete, hyperbolic $3$-manifold, contrary to the $\cal H$-minimality of $M$.
\(2) The first assertion follows from the argument in the proof of part (1) while the second follows from Corollary \[compactconditions\].
[**Proofs of Theorem \[thm:h-minimal\] and Corollary \[cor:2-bridge\]**]{}. Theorem \[thm:h-minimal\] is the first assertion of Theorem \[minimal hyperbolic\] while Corollary \[cor:integer surgery\] follows from second assertion and Corollary \[2bridge compact\].
Sets of discrete $PSL_2(\mathbb R)$-characters {#psl2rcharacters}
==============================================
Discrete $PSL_2(\mathbb R)$-representations of the fundamental groups of small knot manifolds {#realdiscrete}
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In this section we specialize our study to sets of discrete $PSL_2(\mathbb R)$-characters and apply our conclusions to obtain results on $\widetilde{SL_2}$-minimality. This will lead us, for instance, to a proof of Corollary \[prop:g-minimalintro\] and thus the construction of infinitely many closed minimal manifolds.
Let $M$ be a small knot manifold and set $$D(M; {\mathbb R}) = \{ \chi_\rho \in X_{PSL_2}(M) : \rho \hbox{ is a discrete, non-elementary $PSL_2(\mathbb R)$ representation}\}.$$ If $X_0$ is a component of $X_{PSL_2}(M)$ let $$D(X_0; {\mathbb R}) = \{ \chi_\rho \in X_0 : \rho \hbox{ is a discrete, non-elementary $PSL_2(\mathbb R)$ representation}\}.$$ Thus $D(X_0; \mathbb R) = D(X_0) \cap X_{PSL_2(\mathbb R)}(M)$ and so is closed in $X_0$ (cf. Proposition \[closedinX\_0\]).
Fix $\rho \in R_{PSL_2}(M)$ such that $\chi_\rho \in D(M; {\mathbb R})$, set $\Delta = \rho(\pi_{1}(M))$, and let ${\cal B} = \mathbb H^2/\Delta$. The underlying surface $|{\cal B}|$ of ${\cal B}$ is orientable and therefore has only cone singularities.
\[smallfuchsian\] $\;$\
$(1)$ $\Delta$ is either a hyperbolic triangle group or a free product of two finite cyclic groups.\
$(2)$ $\rho(\pi_1(\partial M)) \cong \mathbb Z/c$ for some $c \geq 0$ and if $c > 0$, $\Delta$ is a hyperbolic triangle group.\
$(3)$ Suppose that $\chi_{\rho_n} \in D(X_0; \mathbb R), (n \geq 1)$ are distinct and $\rho_n(\pi_1(\partial M)) \cong \mathbb Z / c_n$ with $c_n \geq 1$. Then $\lim_n c_n = \infty$.
\(1) If $|{\cal B}|$ is non-compact, then $\Delta = \pi_1({\cal B}) \cong \pi_1(|{\cal B}|) * \mathbb Z/a_1 * \ldots * \mathbb Z/ a_k$ where $a_1, a_2, \ldots , a_k \geq 2$ are the orders of the cone points. On the other hand, we can identify $X_{PSL_2}(\Delta)$ with a closed algebraic subset of $X_{PSL_2}(M)$. Since the latter has complex dimension $1$, either $\pi_1(|{\cal B}|) \cong \{1\}$ and $k \leq 2$ or $\pi_1(|{\cal B}|) \cong \mathbb Z$ and $k = 0$. The latter is impossible since it implies that $\Delta \cong \mathbb Z$. Thus $\Delta$ is a free product of two finite cyclic groups. In this case, if $\alpha \in \hbox{kernel}(\rho|\pi_1(\partial M))$, $X_{PSL_2}(M(\alpha))$ has dimension $1$ and since $M$ is small, $\alpha$ is a boundary slope.
Next suppose that $|{\cal B}|$ is closed. The relation which associates a holonomy representation to a hyperbolic structure determines an embedding of the Teichmüller space ${\cal T}({\cal B})$ of ${\cal B}$ in $X_{PSL_2(\mathbb R)}(\pi_1({\cal B})) \subset X_{PSL_2}(\pi_1({\cal B})) \subset X_{PSL_2}(M)$. Thus ${\cal T}({\cal B})$ has real dimension at most $1$. But this dimension is given by $-3\chi(|{\cal B}|) + 2k$ where $k$ is the number of cone points in ${\cal B}$ (Corollary 13.3.7 [@Thu]). Since $|{\cal B}|$ is orientable, the only possibility is for it to be of the form $S^2(a,b,c)$ so that $\Delta$ is a hyperbolic triangle group.
\(2) The first assertion of (2) follows from the elementary observation that an abelian subgroup of $PSL_2(\mathbb R)$ is cyclic. For the second, suppose that $c > 0$ and note that there are infinitely many slopes in $\hbox{kernel}(\rho|\pi_1(\partial M))$. Fix one such slope $\alpha$ and suppose that $\Delta$ is a free product of two finite cyclic groups. There is a principal curve $Y_0 \subset X_{PSL_2}(\Delta) \subset X_{PSL_2}(M(\alpha))$ and so $M(\alpha)$ admits a closed essential surface. Since $M$ is small, $\alpha$ is a boundary slope, and as there are only finitely many such slopes [@Hat], we obtain a contradiction. Thus $\Delta$ is a hyperbolic triangle group.
\(3) Otherwise there is a subsequence $\{j\}$ and $c \geq 1$ such that $c_n = c$ for all $j$. Then for any peripheral class $\gamma$ there are only finitely many possibilities for $f_\gamma(\chi_j)$. Since the $\chi_j$ are distinct this implies that each $f_\gamma$ is constant, which contradicts the smallness of $M$.
\[triangleisolated\] If the image of $\rho \in R_{PSL_2}(M)$ is a discrete hyperbolic triangle group, then $\chi_\rho$ is an isolated point of $D(M; \mathbb R)$.
Suppose that there is a sequence $\{\chi_{\rho_n}\}$ in $D(M; \mathbb R) \setminus \{\chi_\rho\}$ which converges to $\chi_\rho$. By passing to a subsequence and replacing the $\rho_n$ by conjugate representations we may suppose that $\lim_n \rho_n = \rho$ (Lemma 2.1 [@CL]) and find homomorphisms $\theta_n: \rho(\pi_1(M)) \to PSL_2(\mathbb C)$ such that $\rho_n = \theta_n \circ \rho$ (Proposition \[convergence\]). We claim that the $PSL_2(\mathbb C)$ character varieties of triangle groups are finite. Assuming this for the moment, by again passing to a subsequence we may find $A_n \in PSL_2(\mathbb C)$ such that $\theta_n = A_n \theta_1 A_n^{-1}$. Then $\rho_n = \theta_n \circ \rho= A_n (\theta_1 \circ \rho) A_n^{-1}$. Hence $\chi_\rho = \lim_n \chi_{\rho_n} = \lim_n \chi_{\rho_1} = \chi_{\rho_1} \in D(M; \mathbb R) \setminus \{\chi_\rho\}$, which is impossible. Thus $\chi_\rho$ is an isolated point of $D(M; \mathbb R)$.
To see that the character variety of the $(p,q,r)$-triangle group $\Delta(p,q,r) = \langle x,y : x^p = y^q = (xy)^r = 1 \rangle$ is finite, note that there is a natural embedding $X_{PSL_2}(\Delta(p,q,r)) \subset X_{PSL_2}(\mathbb Z/p * \mathbb Z/q)$. Indeed, $X_{PSL_2}(\Delta(p,q,r))$ is contained in the set of points where the regular function $f: X_{PSL_2}(\mathbb Z/p * \mathbb Z/q) \to \mathbb C, \chi_\rho \mapsto \hbox{trace}(\rho(xy))^2$ takes on the value $4 \cos^2(\frac{\pi j}{r})$ for some integer $j$. Now $X_{PSL_2}(\mathbb Z/p * \mathbb Z/q)$ consists of a finite union of curves and isolated points (Example 3.2 [@BZ1]) and it is simple to see from the parameterizations given in that example that the restriction of $f$ to any of the curves is non-constant. Thus it takes on the value $4 \cos^2(\frac{\pi j}{r})$ at only finitely many points and therefore $X_{PSL_2}(\Delta(p,q,r))$ is finite.
\[productnotisolated\] Suppose that the image $\Delta$ of $\rho \in R_{PSL_2}(M)$ is isomorphic to $\mathbb Z / p * \mathbb Z / q$. Then $\chi_\rho$ is an accumulation point of $D(X_0; \mathbb R)$ where $X_0 = \rho^*(Y_0)$ for some principal component $Y_0$ of $X_{PSL_2}(\Delta)$. Further, $D(X_0; \mathbb R)$ is non-compact in $X_0$ and there is a compact subset $K \subset X_0$ such that\
$($a$)$ $\hbox{int}(K)$ contains all characters in $D(X_0; \mathbb R)$ of representations whose images are\
hyperbolic triangle groups, and\
$($b$)$ $\overline{(X_0 \setminus K)} \cap D(X_0; \mathbb R)$ contains all characters in $D(X_0; \mathbb R)$ of representations whose\
images are $\mathbb Z / p * \mathbb Z / q$.
The inclusion $\Delta \to PSL_2(\mathbb C)$ is contained in a unique curve $Y_0 \subset X_{PSL_2}(\Delta)$ (cf. Example 3.2 [@BZ1]). Set $X_0 = \rho^*(Y_0)$. The remaining assertions of the lemma are a consequence of the discussion in Remark \[pqnoncompact\] and Theorem 2.3 of [@Kn].
\[smallproduct\] Let $\{\chi_n\} \subset D(X_0; \mathbb R)$ be a sequence of distinct characters of representations $\rho_n$ with image a free product of two finite cyclic groups. Then there are an epimorphism $\pi_1(M) \to \mathbb Z/p * \mathbb Z / q$ $(2 \leq p, q)$ and a principal curve $Y_0 \subset X_{PSL_2}(\mathbb Z/p * \mathbb Z/q)$ which maps bijectively to $X_0$ under the inclusion $X_{PSL_2}(\mathbb Z/p * \mathbb Z/q) \subset X_{PSL_2}(M)$. In particular, $X_0$ is an $\alpha_0$-curve for some slope $\alpha_0$ on $\partial M$ and $\rho_n(\pi_1(M)) \cong \mathbb Z/p * \mathbb Z / q$ for all $n$.
Choose $n \gg 0$ such that $\chi_n$ is a simple point of $X_{PSL_2}(M)$. By hypothesis, the image $\Delta$ of $\rho_n$ is isomorphic to $\mathbb Z/p * \mathbb Z / q$ for some $2 \leq p, q$. There is a principal curve $Y_0 \subset X_{PSL_2}(\Delta)$ containing the inclusion $\Delta \to PSL_2(\mathbb C)$ and since $\chi_n$ is a simple point, its image in $X_{PSL_2}(M)$ is $X_0$. Lemma \[smallfuchsian\](2) implies that $X_0$ is an $\alpha_0$-curve for some slope $\alpha_0$. Finally, for each $n$, there is an epimorphism $\mathbb Z / p * \mathbb Z / q \cong \Delta \to \rho_n(\pi_1(M)) \cong \mathbb Z / r * \mathbb Z / s$ for some $r, s \geq 2$. It follows from Example 3.2 of [@BZ1] that the induced homomorphisms $\mathbb Z / p, \mathbb Z / q \to \mathbb Z / r * \mathbb Z / s$ are injective. Further, since these images conjugate into one of $\mathbb Z / r, \mathbb Z / s$ and generate $\mathbb Z / r * \mathbb Z / s$, we must have $\mathbb Z / r * \mathbb Z / s \cong \mathbb Z / p * \mathbb Z / q$.
Unbounded sequences of discrete $PSL_2(\mathbb R)$-characters {#realunbounded}
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Let $M$ be a small knot manifold and $X_0$ a component of $X_{PSL_2}(M)$. Consider a sequence $\{\chi_n\}$ in $D(X_0; \mathbb R)$ which converges to an ideal point $x_0$ of $X_0$. The following lemma is a consequence of Lemmas \[projectiveconvergence\](1) and \[smallfuchsian\].
\[projectiveconvergencereal\] Let $M$ be a small knot manifold, $X_0$ a curve component of $X_{PSL_2}(M)$, and $\{\chi_n\} \subset D(X_0; \mathbb R)$ a sequence which converges to an ideal point $x_0$ of $X_0$. Fix $\rho_n \in R_{X_0}$ such that $\chi_n = \chi_{\rho_n}$ and let $\alpha_0$ be the $\partial$-slope associated to $x_0$. For $n \gg 0$, $\hbox{kernel}(\rho_n|\pi_1(\partial M)) \cong \mathbb Z$ and $\rho_n(\pi_1(\partial M)) \cong \mathbb Z$ where the $\mathbb Z$ factor is generated by a loxodromic.
There is no subgroup of $PSL_2(\mathbb R)$ isomorphic to the tetrahedral, octahedral, or icosahedral group. Thus the next result follows directly from Theorem \[elementaryimage\].
\[convidealreal\] Let $M$ be a small knot manifold, $X_0$ a component of $X_{PSL_2}(M)$, and $\{\chi_n\} \subset D(X_0; \mathbb R)$ a sequence which converges to an ideal point $x_0$ of $X_0$. If $S_0$ is a component of an essential surface associated to $x_0$, then for $n \gg 0$, the image of $X_0$ in $X_{PSL_2}(S_0)$ is contained in $X_{{\cal N}}(S_0)$.
Convergent sequences of discrete $PSL_2(\mathbb R)$-characters {#realconvergent}
--------------------------------------------------------------
Let $M$ be a small knot manifold and $X_0$ a non-trivial component of $X_{PSL_2}(M)$. We are interested in the accumulation points of $D(X_0; \mathbb R)$ in $X_0$.
\[convordreal\] Let $M$ be a small knot manifold, $X_0$ a non-trivial component of $X_{PSL_2}(M)$, and $\{\chi_n\} \subset D(X_0; \mathbb R)$ a sequence of distinct characters which converge to some $\chi_{\rho_0} \in X_0$. Then\
$(1)$ $\rho_0(\pi_1(M)) = \Delta_0$ is discrete, non-elementary, and isomorphic to $\mathbb Z / p * \mathbb Z /q$ for some integers $2 \leq p, q$;\
$(2)$ there is a principal component $Y_0 \subset X_{PSL_2}( \Delta_0)$ such that $X_0 = \rho_0^*(Y_0)$;\
$(3)$ there is a unique slope $\alpha_0$ on $\partial M$ such that $\rho_0(\alpha_0) = \pm I$ and $X_0$ is an $\alpha_0$-curve;\
$(4)$ if $\rho_n(\pi_1(\partial M))$ is finite for infinitely many $n$, $\rho_0(\pi_1(\partial M)) \cong \mathbb Z$ is generated by a parabolic.
Fix $\rho_n \in R_{X_0}$ whose character is $\chi_n$. After replacing the $\rho_n$ by conjugate representations (over $PSL_2(\mathbb R)$) and passing to a subsequence, we may suppose that $\lim \rho_n = \rho_0$. Let $\Delta_n$ the image of $\rho_n$ ($n \geq 0$) and ${\cal B}_n = \mathbb H^2/ \Delta_n$ ($n \geq 1$). Lemma \[smallfuchsian\] shows that for $n \geq 1$, $\Delta_n$ is either a free product of two finite cyclic groups or a hyperbolic triangle group. Thus the topological orbifold type of ${\cal B}_n$ is either $\mathbb R^2(p,q)$ or $S^2(p,q,r)$. Since $\{\Delta_n\}$ converges algebraically to $\Delta_0$, Proposition \[convergence\] implies that $\Delta_0$ is a non-elementary Kleinian group and after passing to a subsequence we may suppose that $\{\Delta_n\}$ converges geometrically to a Fuchsian group $\Delta$ containing $\Delta_0$. Further, there are homomorphisms $\theta_n: \Delta \to \Delta_n$ such that $\rho_n = \theta_n \circ \rho_0$ ($n \geq 1$) and $\lim_n \theta_n$ is the inclusion $\Delta \to PSL_2(\mathbb C)$.
Assume first that the image of $\rho_n$ is a free product of finite cyclic groups for infinitely many $n$. By Lemma \[smallproduct\] there are an integer $n \gg 0$, integers $p, q \geq 2$, and a principal component $Z_0$ of $X_{PSL_2}(\Delta_n)$ such that $\Delta_n \cong \mathbb Z / p * \mathbb Z /q$ and $X_0 = \rho_n^*(Z_0) \subset \rho_n^*(X_{PSL_2}(\Delta_n))$. Hence as $\rho_0$ is irreducible, we have $\rho_0 = \psi \circ \rho_n$ for some $\psi \in R_{PSL_2}(\Delta_n)$. It follows that we have surjective homomorphisms $\Delta_n \stackrel{\psi}{\longrightarrow} \Delta_0$ and $\Delta_0 \stackrel{\theta_n}{\longrightarrow} \Delta_n$. Since $\Delta_0$ and $\Delta_n$ are Hopfian, $\theta_n$ is an isomorphism. Thus $Y_0 = \theta_n^*(Z_0)$ is a principal component of $X_{PSL_2}(\Delta_0)$ and $X_0 = \rho_n^*(Z_0) = \rho_0^*(Y_0)$. Lemma \[smallfuchsian\] shows that the remaining conclusions (3) and (4) of the proposition hold.
Next assume that $\Delta_n$ is isomorphic to the $(a_n, b_n, c_n)$ triangle group $\Delta(a_n, b_n, c_n)$ where $2 \leq a_n \leq b_n \leq c_n$. Then ${\cal B}_n = S^2(a_n, b_n, c_n)$. We know that $\rho_0(\pi_1(\partial M)) \cong \mathbb Z / d$ for some $d \geq 0$. If $d > 0$, then $\rho_n(\pi_1(\partial M)) = \theta_n(\rho_0(\pi_1(\partial M)))$ is a quotient of the finite group $\mathbb Z / d$ for all $n$, which contradicts Lemma \[smallfuchsian\]. Thus $\rho_0(\pi_1(\partial M)) \cong \mathbb Z$. Let $\alpha_0$ be the unique slope such that $\rho_0(\alpha_0) = \pm I$.
\[limitpq\] The sequence $\{c_n\}$ tends to infinity and after passing to a subsequence we can find integers $2 \leq p \leq q$ such that $a_n = p, b_n = q$ for all $n$. Further, $\Delta \cong \mathbb Z/p * \mathbb Z/q$ and $\Delta_0$ has index at most $2$ in $\Delta$. If it has index $2$, then $\Delta \cong \mathbb Z/2 * \mathbb Z/q, \Delta_0 \cong \mathbb Z/q * \mathbb Z/q$, and $c_n$ is odd.
If $\{c_n\}$ is a bounded sequence, then so are $\{a_n\}, \{b_n\}$ and so after passing to a subsequence we can suppose that they are constants $a, b, c$. We know that $\mathbb H^2/ \Delta = \lim_n \mathbb H^2/ \Delta_n = \lim_n {\cal B}_n= S^2(a,b,c)$. Thus $\Delta \cong \Delta(a,b,c)$ and as this group is Hopfian, it follows that $\theta_n: \Delta \to \Delta_n$ is an isomorphism for all $n$. Since the groups $\Delta_n$ are conjugate in $PSL_2(\mathbb R)$ and the outer automorphism group of $\Delta(a,b,c)$ is finite, it follows that there are only finitely many conjugacy classes among the representations $\rho_n = \theta_n \circ \rho_0$, which contradicts our assumptions. Thus after passing to a subsequence we may suppose that $\lim_n c_n = \infty$.
If $\{a_n\}$ is not bounded, then up to passing to a subsequence we may suppose that $\lim_n a_n = \infty$. It follows that $\lim_n b_n = \infty$ and therefore $\mathbb H^2/ \Delta = \lim_n {\cal B}_n$ is a thrice-punctured sphere. But then $\Delta$ is a free group on two generators, and therefore the non-abelian group $\Delta_0$ is free. Thus the dimension of $X_{PSL_2}(\Delta_0)$ is at least $3$. But this is impossible as $\rho_0^*: X_{PSL_2}(\Delta_0) \to X_{PSL_2}(M)$ is injective. Thus $\{a_n\}$ is bounded so that after passing to a subsequence we may suppose that $a_n = p \geq 2$ for all $n$.
A similar argument shows that if $\{b_n\}$ is unbounded, then $\Delta_0$ is a non-abelian subgroup of $\Delta \cong \mathbb Z/p * \mathbb Z$. Then $\Delta_0$ is a free product of at least two cyclic groups, each of which is either free or has order dividing $p$. If there are either three such factors or two with one of them free, a contradiction is obtained as in the previous paragraph. On the other hand if $\Delta_0 \cong \mathbb Z/r * \mathbb Z/s$ where $r$ and $s$ divide $p$, then $\Delta(p, b_n, c_n) = \theta_n(\Delta_0)$ is generated by two elements of order dividing $p$. Knapp [@Kn] studied when two elliptics can generate a triangle group and determined necessary and sufficient conditions on their orders and the coefficients of the triangle group for this to occur. It follows from Theorem 2.3 of [@Kn] (and its proof) that if $\Delta(p, b_n, c_n)$ is generated by elements of bounded order, then $\{b_n\}$ is a bounded sequence, contrary to our assumptions. Thus by passing to a subsequence we may suppose that $b_n = q \geq p$ for all $n$.
The work above shows that $\mathbb H^2/ \Delta = \lim_n S^2(p,q, c_n) = \mathbb R^2(p,q)$ so that $\Delta \cong \mathbb Z / p * \mathbb Z / q$. Hence $\Delta_0 \subset \Delta$ is a free product of cyclic groups. and the smallness of $M$ implies that it must be of the form $\mathbb Z/r * \mathbb Z/s$ where each of $r, s$ divides at least one of $p, q$. It follows that $\Delta(p, q, c_n)$ is generated by two elements whose orders divide $r, s$ respectively. Given our constraints on $c_n$ and $r, s$, Theorem 2.3 of [@Kn] shows that the conclusion of the claim holds. (Claim \[limitpq\])
There is a principal component $Y_0$ of $X_{PSL_2}(\Delta_0)$ which contains the character of the inclusion $\Delta_0 \to PSL_2(\mathbb C)$. Since $\lim_n \theta_n$ is this inclusion and the algebraic components of $X_{PSL_2}(\Delta_0)$ are topological components (see Example 3.2 of [@BZ1]), if $n \gg 0$, $\chi_{\theta_n} \in Y_0$. On the other hand, $\chi_n$ is a simple point of $X_{PSL_2}(M)$ for $n \gg 0$. Since $\chi_{\rho_0} = \lim_n \chi_n = \lim_n \rho_0^*(\chi_{\theta_n})$ it follows that $X_0 = \rho_0^*(Y_0)$. This proves (1) and (2) while (3) is a consequence of the (1), (2), and Lemma \[smallfuchsian\]. Finally, to prove (4), note that if $\alpha_1 \ne \alpha_0$ is a slope, then $|\hbox{trace}(\rho_0)(\alpha_1)| = \lim_n |\hbox{trace}(\rho_n)(\alpha_1)| \leq 2$. On the other hand if $\rho_0(\alpha_1)$ is elliptic, then $\rho_n(\alpha_1)$ is elliptic of the same order for $n \gg 0$. This contradicts Lemma \[smallfuchsian\]. Thus $\rho_0(\alpha_1)$ is parabolic.
\[drintersectionfinite\] Suppose that $M$ is a small knot manifold.\
$(1)$ If $X_0$ is a norm curve component of $X_{PSL_2}(M)$, then the intersection of $D(X_0; \mathbb R)$ with any compact subset of $X_0$ is finite.\
$(2)$ If $\pi_1(M)$ does not surject onto a free product of non-trivial cyclic groups. Then the intersection of $D(M; \mathbb R)$ with any compact subset of $X_{PSL_2}(M)$ is finite.
\[propertyq\] [The character variety of a knot manifold $M$ whose fundamental group admits a discrete epimorphism onto a free product of finite cyclic groups contains an $\alpha_0$-curve for some slope $\alpha_0$. Hence Example \[twistpretzel\] gives many examples for which this does not occur. In particular Corollaries \[principalcompact\] and \[dmrfinite\] show that if $M$ is the exterior of a hyperbolic twist knot or a $(-2,3,n)$ pretzel knot with $n \not \equiv 0$ (mod $3$), then $D(M; \mathbb R)$ is finite. Gonzàlez-Acu$\tilde{n}$a and Ramirez [@GR1], [@GR2] have studied the problem of when the fundamental group of the exterior $M$ of a knot in the $3$-sphere admits an epimorphism onto a free product $\mathbb Z / p * \mathbb Z /q$ for some integers $p, q \geq 2$. It is simple to see that in this case $p, q$ are relatively prime. Hartley and Murasugi showed [@HM] that the epimorphism factors through a homomorphism $\pi_1(M) \to \pi_1(M_{p,q})$ whose image is normal with cokernel finite cyclict. This implies that the Alexander polynomial of $M$ is divisible by that of $M_{p,q}$. These conclusions hold more generally for manifolds $M$ with $H_1(M) \cong \mathbb Z$ (cf. the proof of Theorem \[pqdomination\]). Gonzàlez-Acuna and Ramirez [@GR1] have given an algorithm which determines which two-bridge knot exteriors have fundamental groups which admit such a representation. This work easily shows that the fundamental group of the exterior of the $\frac{p}{q}$ two-bridge knot, $p$ prime, admits no such representation. ]{}
\[dmrfinite\] $\;$\
$(1)$ If $M$ is a small knot manifold and $X_0$ is a non-trivial component of $X_{PSL_2}(M)$ such that for each connected, essential surface $S_0$ in $M$ there is a character $\chi \in X_0$ such that $\chi|\pi_1(S_0)$ is strictly irreducible, then $D(X_0; \mathbb R)$ is finite.\
$(2)$ Let $M$ and $N$ be small hyperbolic knot manifolds and suppose that $\varphi: \pi_1(M) \to \pi_1(N)$ is a virtual epimorphism. Then if $Y_0$ is a principal component of $X_{PSL_2}(N)$ and $X_0 = \varphi^*(Y_0)$, then $D(X_0; \mathbb R)$ is finite.
\(1) By Theorem \[convidealreal\] we deduce that $D(X_0; \mathbb R)$ is compact. If it has an accumulation point then Theorem \[convordreal\] implies that there is a surjection $\rho: \pi_1(M) \to \mathbb Z / p * \mathbb Z / q$ and a principal component $Y_0$ of $X_{PSL_2}(\mathbb Z / p * \mathbb Z / q)$ such that $X_0 = \rho^*(Y_0)$. But then Lemma \[productnotisolated\] shows that $D(X_0; \mathbb R)$ is not compact. Thus $D(X_0; \mathbb R)$ is finite.
\(2) Corollary \[principalcompact\] implies that $D(X_0; \mathbb R)$ is compact. If it has an accumulation point then Theorem \[convordreal\] implies that for each irreducible $\chi_\rho \in X_0$, $\rho(\pi_1(M))$ is generated by two torsion elements (cf. Remark \[pqnoncompact\]). But this is clearly not the case for the image by $\rho^*$ of the discrete faithful character of $\pi_1(N)$. Thus $D(X_0; \mathbb R)$ is finite.
Discrete $PSL_2(\mathbb R)$-representations and domination {#psl2domination}
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\[pqdomination\] Let $M$ be a knot manifold with $H_1(M) \cong \mathbb Z$ and suppose that there is a homomorphism $\rho_0 \in R_{PSL_2}(M)$ with discrete, non-elementary image $\Delta_0 \cong \mathbb Z / p * \mathbb Z / q$. Suppose further that $\rho_0(\lambda_M)$ is parabolic for any longitudinal class $\lambda_M \in \pi_1(\partial M)$. Then there are a Seifert fibred manifold $N$ whose interior has base orbifold $\mathbb H^2 / \Delta_0 \cong \mathbb R^2(p,q)$ and a domination $f: (M, \partial M) \to (N, \partial N)$ such that the composition $\pi_1(M) \stackrel{f_\#}{\longrightarrow} \pi_1(N) \to \Delta_0$ is conjugate to $\rho_0$.
Consider the central extension $$1 \to K \to \mbox{Isom}_0(\widetilde{SL_2}) \stackrel{\psi}{\longrightarrow} PSL_2(\mathbb R) \to 1$$ where $\mbox{Isom}_0(\widetilde{SL_2})$ is the component of the identity in $\mbox{Isom}(\widetilde{SL_2})$ and $K \cong \mathbb R$ (cf. pp. 464-465 of [@Sc]). It is simple to see that for each torsion element $x \in PSL_2(\mathbb R)$, there is a unique torsion element $\tilde A \in \psi^{-1}(A) \subset \mbox{Isom}_0(\widetilde{SL_2})$. Thus $\rho_0$ lifts to a representation $\tilde \rho_0: \pi_1(M) \to \mbox{Isom}_0(\widetilde{SL_2})$ whose image is isomorphic to $\Delta_0$. Fix a non-zero homomorphism $\phi: \pi_1(M) \to K$ and note that $$\tilde \rho: \pi_1(M) \to \mbox{Isom}_0(\widetilde{SL_2}), \gamma \mapsto \phi(\gamma) \tilde \rho_0(\gamma)$$ is another homomorphism which lifts $\rho_0$. Set $\tilde \Delta_\phi = \tilde \rho(\pi_1(M))$.
\[disretelift\] $\tilde \Delta_\phi$ is discrete, torsion free, and is the fundamental group of a Seifert manifold $N$ with base orbifold $D^2(p,q)$.
Since $\Delta_0$ is discrete in $PSL_2(\mathbb R)$, $\tilde \Delta_\phi$ is discrete in $\mbox{Isom}_0(\widetilde{SL_2})$ if and only if it intersects the central subgroup $K$ of $\mbox{Isom}_0(\widetilde{SL_2})$ in a discrete subgroup. This intersection is precisely $\tilde \rho(\mbox{kernel}(\rho_0)) = \phi(\mbox{kernel}(\rho_0)) \subset \phi(\pi_1(M)) \subset K$. The latter group is isomorphic to $\mathbb Z$ by construction, and so is discrete. Thus $\tilde \Delta_\phi$ is discrete.
Suppose that $\gamma \in \pi_1(M)$ and $\tilde \rho(\gamma)^n = 1$ for some positive $n$. Then up to conjugation, $\rho_0(\gamma)^n = \pm I$ is also torsion and therefore $\tilde \rho_0(\gamma)^n = 1$ as well. But then $1 = \tilde \rho(\gamma)^n = \phi(\gamma)^n \tilde \rho_0(\gamma)^n = \phi(\gamma)^n$. Since $K$ is torsion free we conclude that $\gamma \in \hbox{kernel}(\phi)$, and since $H_1(M) \cong \mathbb Z$ and $\phi \ne 0$, $\hbox{kernel}(\phi) = [\pi_1(M), \pi_1(M)]$. Hence $\gamma \in [\pi_1(M), \pi_1(M)]$ and therefore the image of $\rho_0(\gamma)$ in $H_1(\Delta_0)$ is zero. But $\Delta_0 \cong \mathbb Z / p * \mathbb Z / q$ so that $\Delta_0 \to H_1(\Delta_0)$ is injective on torsion elements. Thus $\rho_0(\gamma) = 1$ and therefore $\tilde \rho(\gamma) = \phi(\gamma) \tilde \rho_0(\gamma) = 1$. This proves that $\tilde \Delta_\phi$ is torsion free.
The conclusions of the two previous paragraphs imply that $\tilde \Delta_\phi$ acts freely and properly discontinuously on $\widetilde{SL_2}$. Let $W = \widetilde{SL_2}/\tilde \Delta_0^\phi$ be the quotient manifold. Now $\tilde \Delta_\phi \cap K \ne \{0\}$ as otherwise $\psi | \tilde \Delta_\phi \to \Delta_0$ would be an isomorphism, which contradicts the result of the last paragraph. Thus $\tilde \Delta_\phi \cap K \cong \mathbb Z$ and so $K/(\tilde \Delta_\phi \cap K) \cong S^1$. On the other hand, $\mathbb H^2/ \Delta_0 \cong \mathbb R^2(p,q)$. Thus there is an orbifold bundle $S^1 \to W \to \mathbb R^2(p,q)$ so that $W$ admits a compactification $N$ with boundary a torus. Further, $N$ admits a Seifert fibering with base orbifold $D^2(p,q)$. This completes the proof of the claim. (Claim \[disretelift\])
To complete the proof of the proposition we must show that there is a domination $M \geq N$. To that end, fix a map $f: M \to N$ which realizes $\tilde \rho: \pi_1(M) \to \tilde \Delta_\phi = \pi_1(N)$. We must show that $f_\#|\pi_1(\partial M)$ is injective and has image contained in a peripheral subgroup of $\pi_1(N)$.
By hypothesis $\rho_0(\lambda_M)$ is parabolic. In particular it has infinite order and is distinct from a primitive element $\alpha_0 \in \hbox{kernel}(\rho_0|\pi_1(\partial M))$ (cf. Theorem \[convordreal\](3)). It follows that $1 \ne \phi(\alpha_0) \in K$. It is easy to see that the restriction of $\tilde \rho$ to $\langle \lambda_M, \alpha_0 \rangle \cong \mathbb Z^2$ is injective and since $\tilde \Delta_\phi$ is torsion free, the same holds for its restriction to $\pi_1(\partial M)$.
Finally, to show that $\tilde \rho(\pi_1(\partial M))$ is peripheral, it suffices to see that $\rho_0(\pi_1(\partial M))$ is peripheral in $\Delta_0 = \pi_1(D^2(p,q))$. But this is clear since it is a parabolic subgroup of $\Delta_0$. This completes the proof.
[The condition that $H_1(M) \cong \mathbb Z$ was used to guarantee that $\tilde \Delta$ is torsion free. Without this condition we can still construct a proper non-zero degree map from $M$ to a $3$-dimensional Seifert orbifold, but the underlying space of the orbifold might be $S^1 \times D^2$. ]{}
\[seifertdomination\] Let $M$ be a small hyperbolic knot manifold with $H_1(M) \cong \mathbb Z$, $X_0$ a non-trivial component of $X_{PSL_2}(M)$, and $\{\chi_{\rho_n}\} \subset D(X_0; \mathbb R)$ a sequence of distinct characters which converges to $\chi_{\rho_0} \in X_0$. Suppose further that for each $n$, $\rho_n(\pi_1(\partial M))$ is finite. Then $\rho_0$ has discrete, non-elementary image isomorphic to a free product of two finite cyclic groups and $\rho_0(\pi_1(\partial M))$ is parabolic. If $\rho_n(\lambda_M) \ne \pm I$ for infinitely many $n$, there is a strict domination $M \geq N$ for some Seifert manifold $N$ with incompressible boundary.
Suppose that $\lim_n \chi_{\rho_n} = \chi_{\rho_0}$ and set $\Delta_0 = \rho_0(\pi_1(M))$. By Theorem \[convordreal\], $\Delta_0 \subset PSL_2(\mathbb R)$ is discrete and isomorphic to $\mathbb Z / p * \mathbb Z / q$ for some integers $2 \leq p, q$. Our hypotheses imply that $\rho_0(\lambda_M)$ is parabolic (cf. Proposition \[elemordiscr\](2) and Theorem \[convordreal\](4)). Thus Theorem \[pqdomination\] implies the desired conclusion.
\[cor:s-domination\] Let $M$ will be a small hyperbolic knot manifold, $\{\alpha_n\}$ a sequence of distinct slopes on $\partial M$, and $\{\chi_n\} \subset D(M; \mathbb R)$ a sequence of characters of representations $\rho_n$ such that $\rho_n(\alpha_n) = \pm I$ for all $n$. If there are infinitely many distinct $\chi_n$ and the sequence $\{\chi_n\}$ subconverges to a character $\chi_{\rho_0}$, then\
$(1)$ the image of $\rho_0$ is isomorphic to a discrete, non-elementary free product of two finite cyclic groups;\
$(2)$ $\rho_n(\pi_1(\partial M))$ is finite for infinitely many $n$ and $\rho_0(\pi_1(\partial M))$ is parabolic.\
$(3)$ if $H_1(M) \cong \mathbb Z$ and $\rho_0(\lambda_M) \ne \pm I$, $M$ strictly dominates a Seifert manifold with incompressible boundary.
After passing to a subsequence we can assume that the $\chi_n$ are distinct. Part (3) of Theorem \[convordreal\] shows that there is a unique slope $\alpha_0$ on $\partial M$ such that $\rho_0(\alpha_0) = \pm I$. Then for $n \gg 0$ we have $\rho_n(\alpha_0) = \pm I$ (Proposition \[elemordiscr\]). Since the $\alpha_n$ are distinct, this implies that for large $n$, $\rho_n(\pi_1(\partial M))$ is a finite cyclic group of order dividing $\Delta(\alpha_0, \alpha_n)$. Corollary \[seifertdomination\] then yields a strict domination $f: M \to N$ where $N$ is a Seifert manifold with incompressible boundary.
Minimal Dehn fillings {#gminimal}
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In the section we use the results of the paper to construct various infinite families of minimal closed $3$-manifold.
\[finiteredhaken\] $\;$\
$(1)$ If $M$ is a small knot manifold, there are only finitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is either reducible or Haken.\
$(2)$ A closed, connected, orientable manifold with infinite fundamental group is either reducible, Haken, or admits a geometric structure modelled on $Nil, \mathbb H^3$, or $\widetilde{SL_2}$.
\(1) If $M(\alpha)$ contains an essential surface $S$ and we isotope $S$ so as to minimize $|S \cap \partial M|$, then $S_0 := S \cap M$ is an essential surface in $M$. Since $M$ is small, $\partial S_0 \ne \emptyset$ and has slope $\alpha$. Thus $\alpha$ is a boundary slope. By [@Hat], there are at most finitely many such $\alpha$. Thus (1) holds.
\(2) By the geometrization theorem of Perelman we see that a closed, connected, orientable manifold $W$ which is irreducible though not Haken admits a geometric structure. If the structure is $Sol$, $W$ is Haken since it is irreducible and contains an essential torus ([@Sc]). If it is $\mathbb S^2 \times \mathbb R, \mathbb E^3$ or $\mathbb H^2 \times \mathbb R$, then $W$ amits a Seifert fibre structure with zero Euler number and therefore it is either reducible or Haken ([@Sc]). If it is $\mathbb S^3$, $\pi_1(W)$ is finite. This proves (2).
\[finitesl2domination\] Suppose that $M$ is a small ${\cal H}$-minimal hyperbolic knot manifold which has the following properties:
: There is a slope $\alpha_0$ on $\partial M$ such that $M(\alpha_0)$ is ${\cal H}$-minimal.
: For each norm curve $X_0 \subset X_{PSL_2}(M)$ and for each essential surface $S$ associated to an ideal point of $X_0$ there is a character $\chi_\rho \in X_0$ which restricts to a strictly irreducible character on $\pi_1(S)$.
: There is no surjective homomorphism from $\pi_1(M)$ onto a Euclidean triangle group.
: There is no epimorphism $\rho: \pi_1(M) \to \Delta(p,q,r) \subset PSL_2(\mathbb R)$ such that $\rho(\pi_{1} (\partial M))$ is elliptic or trivial.
Then all but finitely many Dehn fillings $M(\alpha)$ yield a minimal manifold.
By Theorem \[minimal hyperbolic\](2) and Lemma \[finiteredhaken\], we need only show that $M(\alpha)$ is $Nil$-minimal and $\widetilde{SL_2}$-minimal for all but finitely many slopes $\alpha$ on $\partial M$.
Suppose that there is a slope $\alpha$ and a domination $f$ from $M(\alpha)$ to a closed $Nil$-manifold $V$ with base orbifold ${\cal B}_n$. By passing to a cover of $V$ we may suppose that $f_\#$ is surjective. We can also suppose that $\alpha$ is not a boundary slope so that $M(\alpha)$ is not Haken. Since ${\cal B}$ is Euclidean, the only possibility is that ${\cal B} \cong S^2(a,b,c)$ for some Euclidean triple $(a,b,c)$. But then we would have an epimorphism $\pi_1(M) \to \pi_1(M(\alpha)) \to \pi_1(V) \to \pi_1(S^2(a,b,c)) \cong \Delta(a,b,c)$, which contradicts (c). Thus $M(\alpha)$ is $Nil$-minimal for all but finitely many $\alpha$.
Suppose next that there are a sequence of distinct slopes $\alpha_n$ and dominations $f_n$ from $M(\alpha_n)$ to a closed $\widetilde{SL_2}$-manifold $V_n$ with base orbifold ${\cal B}_n$. By passing to a cover of $V_n$, we may suppose that $(f_n)_\#$ is surjective for all $n$. Let $\rho_n$ be the composition $\pi_1(M) \to \pi_1(M(\alpha_n)) \stackrel{(f_n)_\#}{\longrightarrow} \pi_1(V_n) \to \pi_1({\cal B}_n) \subset PSL_2(\mathbb R) \subset PSL_2(\mathbb C)$. By passing to a subsequence we may suppose that $\chi_n \in X_0$ for some non-trivial curve $X_0$. Hypothesis (d) implies that $\hbox{kernel}(\rho_n|\pi_1(\partial M)) = \langle \alpha_n \rangle$ so that there are infinitely many distinct $\chi_n$ and $\rho_n(\pi_1(\partial M))$ is infinite. It follows that $\rho_n(\pi_1(\partial M))$ contains loxodromics. Thus Corollary \[normcondition\] implies that $X_0$ is a norm curve. But then hypothesis (b) and Corollary \[dmrfinite\](1) imply that there are only finitely many $\chi_n$, contrary to the construction. Thus there is no sequence $\{\alpha_n\}$ as above and so $M(\alpha)$ is $\widetilde{SL_2}$-minimal for all but finitely many $\alpha$.
\[cor:s-minimal2\] If $M$ is the exterior of a hyperbolic twist knot, then all but finitely many Dehn fillings $M(\alpha)$ yield a minimal manifold.
Hypothesis (a) of Theorem \[finitesl2domination\] clearly holds for $M$, and since the only non-trivial curve in $X_{PSL_2}(M)$ is a principal curve ([@Bu]), hypothesis (b) holds as well (cf. the proof of Corollary \[principalcompact\]). Finally, hypotheses (c) and (d) are true for $M$ by Proposition \[nil\] and Proposition \[prop:epi\].
Condition (d) of Theorem \[finitesl2domination\] is difficult to verify in general. Nevertheless, the following results show that we can still construct infinite families of minimal Dehn fillings in quite general situations. First we need to prove an elementary lemma.
\[infinitelymanyslopes\] Let $\alpha_0, \alpha_1, \ldots, \alpha_n$ be projectively distinct primitive elements of $\mathbb Z^2$ and suppose that $L_1, L_2, \ldots , L_m$ are subgroups of $\mathbb Z^2$ none of which contains $\alpha_0$. For each $i = 1, 2, \ldots , n$, let $U_i$ be an arc neighbourhood of $[\alpha_i] \in \mathbb P(\mathbb R^2)$ and suppose that $\overline{U_i} \cap \overline{U_j} = \emptyset$ for $i \ne j$. Then there are infinitely many primitive $\alpha \in \mathbb Z^2$ such that $\alpha \not \in L_1 \cup \ldots \cup L_m$ and $[\alpha] \not \in U_1 \cup \ldots \cup U_n$.
Since each $L_j$ is contained in a rank $2$ subgroup of $\mathbb Z^2$ in the complement of $\alpha_0$, we can assume, without loss of generality, that each $L_j$ has rank $2$. Let $\beta_0 \in \mathbb Z^2$ be dual to $\alpha_0$ and fix coprime integers $a, b$ such that $\delta_0 := a \alpha_0 + b \beta_0 \ne \alpha_i$, $0 \leq i \leq n$. Set $L_0 = \{\alpha_0 + n \delta_0 : n \in \mathbb Z\}$ and note that from the definition of $U = U_1 \cup \ldots \cup U_n$ and choice of $\delta_0$, there is some $k_0 > 0$ such that if $|k| \geq k_0$, $[\alpha_0 + k \delta_0] \not \in U$. Define $d \geq 1$ to be the index of $L_1 \cap L_2 \cap \ldots \cap L_m$ in $\mathbb Z^2$ and note that for each $k \in \mathbb Z$, the class $\alpha_k = \alpha_0 + dkb \delta_0 \not \in (L_1 \cup L_2 \cup \ldots \cup L_m)$. The proof is completed by observing that $\alpha_k = (1 + abkd) \alpha_0 + b^2kd \beta_0$ is primitive and $[\alpha_k] \not \in U$ for $|k| \geq k_0$.
\[uinfinite\] Let $M$ be an ${\cal H}$-minimal, small, hyperbolic knot manifold and suppose that $H_1(M) \cong \mathbb Z \oplus T$ where\
$(a)$ $H_1(\partial M) \to H_1(M)/T \cong \mathbb Z$ is surjective, and\
$(b)$ $\mathbb Z / a \oplus \mathbb Z / b$ is not a quotient of $T$ for $(a,b) = (2,3), (2,4), (3,3)$.\
Suppose as well that\
$(c)$ there is no discrete, non-elementary representation $\rho \in R_{PSL_2(\mathbb R)}(M)$ such that\
$\rho(\pi_1(M))$ is isomorphic to a free product of two non-trivial cyclic groups and $\rho(\pi_1(\partial M))$\
is parabolic;\
$(d)$ there is a slope $\alpha_0$ on $\partial M$ such that $\pi_1(M(\alpha_0))$ admits no homomorphism onto a\
non-elementary Kleinian group or a Euclidean triangle group.\
If $U \subset \mathbb P(H_1(\partial M; \mathbb R))$ is the union of disjoint closed arc neighbourhoods of the finite set of boundary slopes of $M$, then there are infinitely many slopes $\alpha$ such that $[\alpha] \in \mathbb P(H_1(\partial M; \mathbb R)) \setminus U$ and $M(\alpha)$ is minimal.
By Lemma \[finiteredhaken\] and Theorem \[minimal hyperbolic\](1), it suffices to show that there are infinitely many slopes $\alpha$ such that $[\alpha] \in \mathbb P(H_1(\partial M; \mathbb R)) \setminus U$ and $M(\alpha)$ is both $Nil$-minimal and $\widetilde{SL_2}$-minimal. As we argued in the proof of Theorem \[finitesl2domination\], if $\alpha$ is not a boundary slope and there is a domination $M(\alpha) \to V$ where $V$ is a $Nil$ or $\widetilde{SL_2}$ manifold, there is an epimorphism $\rho: \pi_1(M) \to \Delta(a,b,c)$ which can suppose lies in $D(M; \mathbb R)$ if $(a,b,c)$ is a hyperbolic triple.
Suppose first of all that $\rho: \pi_1(M) \to \Delta(a,b,c)$ is surjective and $(a,b,c)$ is a Euclidean triple with $a \leq b \leq c$. There is an epimorphism $\mathbb Z \oplus T = H_1(M) \to H_1(\Delta(a,b,c)) \cong \mathbb Z / a \oplus \mathbb Z / b$ where $(a,b)$ is one of the pairs $(2,3), (2,4), (3,3)$. Hence our hypotheses imply that there is some $\gamma \in \pi_1(\partial M)$ which is sent to a non-zero element of $H_1(\Delta(a,b,c))$ under the composition $\pi_1(M) \stackrel{\rho}{\longrightarrow} \Delta(a,b,c)) \to H_1(\Delta(a,b,c))$. It is a simple exercise to then show that $\rho(\gamma)$ has non-trivial finite order in $\Delta(a,b,c)$. (For instance, use the fact that $\Delta(a,b,c)$ can be considered a subgroup of the upper-triangular matrices in $PSL_2(\mathbb C)$.) Since an abelian subgroup of an infinite triangle group is cyclic, $\rho(\pi_1(\partial M)) \cong \mathbb Z / d$ where $d \in \{2,3,4, 6\}$. Thus there are only finitely many possibilities for $\hbox{kernel}(\rho|\pi_1(\partial M))$, say $L_1, \ldots , L_k$. By hypothesis none of them contain $\alpha_0$. Further, if $\alpha \not \in L_1 \cup \ldots \cup L_k$, then $\pi_1(M(\alpha))$ admits no homomorphism onto a Euclidean triangle group.
Next set $D_U(M; \mathbb R) := \{\chi_\rho \in D(M; \mathbb R) : \rho(\alpha) = \pm I \hbox{ for some slope } \alpha \hbox{ such that } [\alpha] \not \in U\}$ and suppose it is infinite. If $\alpha$ is a slope such that $[\alpha] \not \in U$, then $\alpha$ is not a boundary slope and so there are only finitely many $\chi_\rho \in D_U(M; \mathbb R)$ such that $\rho(\alpha) = \pm I$ (Corollary \[smallcharactervariety\]). Hence we can find a sequence of distinct slopes $\alpha_n$, a sequence of distinct characters $\chi_{\rho_n} \in D_U(M; \mathbb R)$, and a component $X_0$ of $X_{PSL_2}(M)$ such that $[\alpha_n] \not \in U$, $\rho_n(\alpha_n) = \pm I$, and $\chi_{\rho_n} \in X_0$. Lemma \[projectiveconvergence\] shows that $\{\chi_{\rho_n}\}$ does not accumulate to an ideal point of $X_0$. Thus we can suppose that it converges to some $\chi_{\rho_0} \in X_0$. Theorem \[convordreal\] implies that $\rho_0(\pi_1(M))$ is a free product of two finite cyclic groups and $\rho_0(\pi_1(\partial M))$ is parabolic. But this contradicts our hypotheses. Thus $D_U(M; \mathbb R)$ is finite, say $D_U(M; \mathbb R) = \{\chi_{\rho_1}, \chi_{\rho_2} , \ldots , \chi_{\rho_l}\}$. Set $L_j' = \hbox{kernel}(\rho_j|\pi_1(\partial M))$ ($1 \leq j \leq l$). Then $\alpha_0 \not \in (L'_1 \cup L'_2 \cup \ldots \cup L'_m)$ and if $\alpha \not \in (L'_1 \cup L'_2 \cup \ldots \cup L'_m)$ is a slope such that $[\alpha] \not \in U$, $\pi_1(M(\alpha))$ admits no homomorphism onto a hyperbolic triangle group. The proof is completed by applying Lemma \[infinitelymanyslopes\] to the subgroups $L_1, \ldots , L_k, L'_1, \ldots , L'_l$.
\[lensspace\] [The theorem applies to the exterior of many knots in lens spaces. For instance, it follows from work of Indurskis [@In] that if $M_m$ is the manifold obtained by $m$-Dehn filling on one component of the right-hand Whitehead link, then $X_{PSL_2}(M_m)$ has exactly one non-trivial component and is therefore minimal. For $|m| > 4$, $M_m$ is hyperbolic. If $\mu$ is the slope on $\partial M_m$ corresponding to a meridian of the Whitehead link, $M_m(\mu) \cong L(m, 1)$. Since $H_1(M_m) \cong \mathbb Z \oplus \mathbb Z / m$ and $H_1(\partial M_m) \to H_1(M_m) / T_1(M_m)$ is onto, the hypotheses of Theorem \[uinfinite\] are satisfied as long as $m \not \equiv 0$ [(mod $6$)]{}. For such $m$, $M_m(\alpha)$ is minimal for infinitely many slopes $\alpha$.]{}
\[minginfinite\] Suppose that $M$ is a minimal small hyperbolic knot manifold such that $H_1(M)$ $\cong \mathbb Z$ and that\
$(a)$ there is no homomorphism $\rho: \pi_1(M(\lambda_M)) \to PSL_2(\mathbb R)$ such that $\rho(\pi_1(M(\lambda_M)))$ is\
a free product of two non-trivial cyclic groups and $\rho(\pi_1(\partial M))$ parabolic.\
$(b)$ there is a slope $\alpha_0$ on $\partial M$ such that $\pi_1(M(\alpha_0))$ admits no homomorphism onto a\
non-elementary Kleinian group or a Euclidean triangle group.\
Then there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal.
The proof is similar to that of Theorem \[uinfinite\]. As before it suffices to show that there are subgroups $L_1, L_2, \ldots , L_m$ of $H_1(\partial M)$, none of which contain $\alpha_0$, such that if $\alpha \not \in L_1 \cup \ldots \cup L_m$ is a slope, though not a boundary slope, then $\pi_1(M(\alpha))$ admits no surjective homomorphism onto an infinite triangle group. Since $H_1(M) \cong \mathbb Z$, the homological conditions (a) and (b) from the statement of Theorem \[uinfinite\] hold and so there are subgroups $L_1, \ldots , L_k$ of $H_1(\partial M)$, none of which contain $\alpha_0$, such that if $\alpha \not \in L_1 \cup \ldots \cup L_k$, then $\pi_1(M(\alpha))$ admits no homomorphism onto a Euclidean triangle group.
To derive a similar conclusion for hyperbolic triangle groups, the proof of Theorem \[uinfinite\] shows that it suffices to fix a disjoint union $U \subset \mathbb P(H_1(\partial M; \mathbb R))$ of closed arc neighbourhoods of the finite set of boundary slopes of $M$ and prove that $D_U(M; \mathbb R) := \{\chi_\rho \in D(M; \mathbb R) : \rho(\alpha) = \pm I \hbox{ for some slope } \alpha \hbox{ such that } [\alpha] \not \in U\}$ is finite. Suppose otherwise and note that as in the proof of Theorem \[uinfinite\], we can find a representation $\rho_0: \pi_1(M) \to PSL_2(\mathbb R)$ with discrete image isomorphic to a free product of non-trivial cyclic groups such that $\rho_0(\pi_1(\partial M))$ is parabolic. Hypothesis (a) implies that $\rho_0(\lambda_M) \ne \pm I$ and so Theorem \[pqdomination\] implies that $M$ strictly dominates some Seifert manifold $N$ with incompressible boundary. This contradicts the minimality of $M$. Thus $D_U(M; \mathbb R)$ is finite and the proof proceeds as in that of Theorem \[uinfinite\].
\[s3ginfinite\] Let $M$ be a minimal, small, hyperbolic $3$-manifold which is the exterior of a knot $K$ in the $3$-sphere. If there is no homomorphism $\rho: \pi_1(M(\lambda_M)) \to PSL_2(\mathbb R)$ such that $\rho(\pi_1(M(\lambda_M)))$ is a free product of two non-trivial cyclic groups and $\rho(\pi_1(\partial M))$ parabolic, then there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal.
\[s3ginfiniteegs\] [If the Alexander polynomial of a knot $K \subset S^3$ with exterior $M$ is not divisible by the Alexander polynomial of a non-trivial torus knot, there is no homomorphism of $\pi_1(M)$ onto the free product of two non-trivial finite cyclic groups (cf. Remark \[propertyq\]). Thus if its exterior is minimal, small, and hyperbolic, there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal. This provides many examples. For others, let $K$ be the $(-2, 3, n)$ pretzel where $n \not \equiv 0$ (mod $3$). We noted in Example \[twistpretzel\] that there is a unique non-trivial component of $X_{PSL_2}(M)$ and it is principal and used this to deduce that $M$ is minimal. It also implies that there is no homomorphism $\rho: \pi_1(M(\lambda_M)) \to PSL_2(\mathbb R)$ such that $\rho(\pi_1(M(\lambda_M)))$ is a free product of two non-trivial cyclic groups (Lemma \[productnotisolated\]). Thus Corollary \[s3ginfinite\] may be applied to see that there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal. As a final example, Riley has shown that if $K$ is a two-bridge knot and $\rho \in R_{PSL_2}(M)$ is irreducible with $\rho(\pi_1(\partial M))$ parabolic, then $\rho(\lambda_M) \ne \pm I$ (Lemma 1 [@Ri]). In particular, if $M$ is the exterior of a $\frac{p}{q}$ two-bridge knot, it is minimal (Corollary \[2bridgemini2\]), small, and hyperbolic if $p$ is prime and $q \not \equiv \pm 1$ (mod $p$). Thus the corollary implies that there are infinitely many slopes $\alpha$ on $\partial M$ such that $M(\alpha)$ is minimal.]{}
On the smoothness of dihedral characters {#2-bridge}
========================================
One goal of this appendix is to prove that if $\mu$ is a meridinal class of the $p/q$ two-bridge knot, then $d_{M_{p/q}}(\mu) = \frac{p-1}{2}$. In order to do this, we determine a useful criterion for the smoothness of dihedral characters.
A cohomological calculation {#sec:cohocalcul}
---------------------------
Let $\Gamma$ be a finitely generated group, $V$ is a complex vector space, and $\theta: \Gamma \to GL(V)$ a homomorphism. We use $b_1(\Gamma; \theta)$ to denote the complex dimension of $H^1(\Gamma; V_\theta)$. For instance if $\rho \in R_{PSL_2}(\Gamma)$, the induced action of $\Gamma$ on $sl_2(\mathbb C)$ given by the composition $\Gamma \stackrel{\rho}{\longrightarrow} PSL_2(\mathbb C)
\stackrel{Ad}{\longrightarrow} Aut(sl_2(\mathbb C))$ gives rise to the cohomology group $H^1(\Gamma; sl_2(\mathbb C)_{Ad\rho})$ whose dimension is $b_1(\Gamma; Ad \rho)$.
Identify the dihedral group of $2n$ elements $D_n$ with the subgroup of ${\cal N}$ generated by the matrices $\pm \left(
\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}
\right)$ and $\pm \left( \begin{array}{cc} \zeta & 0 \\ 0 & \zeta^{-1}
\end{array} \right)$ where $\zeta =
\exp(\frac{2 \pi i}{2n})$. Any subgroup of $PSL_2(\mathbb C)$ abstractly isomorphic to $D_n$ is conjugate in $PSL_2(\mathbb C)$ to $D_n$.
For each divisor $d \geq 1$ of $n$ we have surjections $\theta_{n,d}: D_n
\to D_d$ given by $$\theta_{n,d}(\pm \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}
\right)) =
\pm \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \mbox{ and
} \theta_{n,d}(\pm \left(
\begin{array}{cc} \zeta & 0 \\ 0 & \zeta^{-1} \end{array} \right)) = \pm
\left( \begin{array}{cc}
\zeta^{\frac{n}{d}} & 0 \\ 0 & \zeta^{-\frac{n}{d}}\end{array} \right).$$
\[b1Adrho\] Let $\rho: \Gamma \to PSL_2(\mathbb C)$ be a representation whose image is $D_n$, $n > 1$. For each divisor $d \geq 1$ of $n$ let $\rho_d$ be the composition of $\rho$ with $\theta_{n,d}$ and set $\Gamma_{2d} = ker(\rho_d)$. Then $$b_1(\Gamma; Ad \rho) = b_1(\Gamma_2) - b_1(\Gamma) + \frac{1}{\phi(n)}
\sum_{d|n}
\mu(\frac{n}{d}) b_1(\Gamma_{2d})$$ where $\phi$ is Euler’s $\phi$-function and $\mu$ is the Möbius function.
Consider the real basis $$e_1 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right),
\;\;\;\;\;
e_2 = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right), \;\;\;\;\;
e_3 = \left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right)$$ of $sl_2(\mathbb C)$. Let $\Theta = Ad|: {\cal N} \to Aut(sl_2(\mathbb
C))$. The reader will verify that the span $\langle e_1 \rangle \cong \mathbb C$ of $e_1$ is invariant under $\Theta({\cal N})$ as is $\langle e_2, e_3 \rangle \cong \mathbb C^2$. Thus $$sl_2(\mathbb C)_{\Theta} = \mathbb C_{\Theta_1} \oplus \mathbb
C^2_{\Theta_2} \eqno{({\rm A}.1.1)}$$ where $\Theta_1: {\cal N} \to GL_1(\mathbb C)$ is given by $$\Theta_1(A) = \left\{\begin{array}{rl} 1_{\mathbb C} & \mbox{if } A \in
{\cal D} \\ -1_{\mathbb C} &
\mbox{if } A \in {\cal N} \setminus{\cal D} \end{array} \right.$$ and, in terms of the ordered basis $\{e_2, e_3\}$, $\Theta_2: {\cal N} \to
GL_2(\mathbb C)$ is given by $$\Theta_2(\pm \left(\begin{array}{cc} u & 0 \\ 0 & u^{-1} \end{array}
\right)) =
\left(\begin{array}{cc} u^2 & 0 \\ 0 & u^{-2, } \end{array} \right),
\Theta_2(\pm \left(\begin{array}{cc} 0 & v
\\ -v^{-1} & 0 \end{array} \right)) = \left(\begin{array}{cc} 0 & -v^2 \\
-v^{-2} & 0 \end{array} \right).$$
Without loss of generality we may suppose that the image of $\rho$ lies in ${\cal N}$. Then (A.1.1) yields the decomposition $sl_2(\mathbb C)_{Ad\rho} = \mathbb
C_{\theta_1} \oplus \mathbb
C^2_{\theta_2}$ where $\theta_j = \Theta_j \circ \rho$. Hence $$b_1(\Gamma; Ad\rho) = b_1(\Gamma; \theta_1) +
b_1(\Gamma; \theta_2).$$ The proof of the lemma now follows from Claims (1) and (2) below.
[**Claim 1**]{}. $\;$ $b_1(\Gamma; \theta_1) =
b_1(\Gamma_2) - b_1(\Gamma)$.
The $\Gamma$-module $\mathbb C[\Gamma/\Gamma_2] = \mathbb C[\mathbb
Z/2]_\Gamma$ splits into two $1$-dimensional modules $$\mathbb C[\mathbb Z/2]_\Gamma =
\mathbb C_1 \oplus \mathbb
C_{\theta_1}$$ where $\mathbb C_1$ is the trivial $\Gamma$-module. Then $H^1(\Gamma_2; \mathbb C) \cong
H^1(\Gamma;\mathbb C[\Gamma/\Gamma_2]) = H^1(\Gamma; \mathbb C[\mathbb
Z/2]_\Gamma) = H^1(\Gamma; \mathbb C_1)
\oplus H^1(\Gamma; \mathbb C_{\theta_1})$. Thus $b_1(\Gamma_2) =
b_1(\Gamma) + b_1(\Gamma; \theta_1)$. (of Claim 1)
[**Claim 2**]{}. $\;$ $b_1(\Gamma; \theta_2) =
\frac{1}{\phi(n)} \sum_{d|n}
\mu(\frac{n}{d}) b_1(\Gamma_{2d})$ [*where $\phi$ is Euler’s $\phi$-function and $\mu$ is the Möbius function.*]{}
Fix a divisor $d \geq 1$ of $n$ and observe that the $\Gamma$-module $\mathbb C[\Gamma/\Gamma_{2d}] \cong \mathbb C[D_d]$ splits as a sum $$\mathbb C[\Gamma/\Gamma_{2d}] = \mathbb C_1 \oplus \mathbb C_{\theta_1}
\oplus \bigoplus_{r=1}^d \mathbb C^2_{\delta_r} \eqno{({\rm A}.1.2)}$$ where $\delta_r = \Theta \circ \delta_r^0 \circ \rho_d: \Gamma \to
GL_2(\mathbb C)$ with $\delta_r^0: D_d \to D_d$ given by $$\delta_r^0(\pm \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}
\right)) =
\pm \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \mbox{ and
} \delta_r^0(\pm \left(
\begin{array}{cc} \zeta & 0 \\ 0 & \zeta^{-1} \end{array} \right)) = \pm
\left( \begin{array}{cc}
\zeta^{\frac{rn}{d}} & 0 \\ 0 & \zeta^{-\frac{rn}{d}}\end{array} \right).$$ (see §5.3 of [@Se] for example). If $r$ divides $d$, then $\delta_r^0 \circ \rho_d =
\rho_{\frac{d}{r}}$. Moreover, if $r_1$ and $r_2$ have the same order in $\mathbb Z/d$, there is a $\Gamma$-module isomorphism between $\mathbb
C^2_{\delta_{r_1}}$ and $\mathbb C^2_{\delta_{r_2}}$. (For under this condition there is an automorphism $\psi$ of the group $\theta_{r_1}(\Gamma)$ such that $\delta_{r_2} = \psi \circ
\delta_{r_1}$.) Combining these observations with (A.1.2) and Claim 1 shows that $$b_1(\Gamma_{2d}) - b_1(\Gamma_2) = \sum_{e|d} \phi(e) b_1(\Gamma; \Theta \circ \rho_e).$$ This formula holds for each $d$ which divides $n$, so the Möbius inversion formula (see §16.4 of [@HW] for example) yields $$b_1(\Gamma; \theta_2) = b_1(\Gamma; \Theta \circ
\rho_n)=
\frac{1}{n} \sum_{d|n} \mu(\frac{n}{d}) (b_1(\Gamma_{2d}) -
b_1(\Gamma_2)) = \frac{1}{\phi(n)} \sum_{d|n} \mu(\frac{n}{d})
b_1(\Gamma_{2d}),$$ as $n > 1$. (of Claim 2)
This completes the proof of Lemma \[b1Adrho\].
A criterion for the smoothness of dihedral characters of knot groups {#criterion}
--------------------------------------------------------------------
For a knot $K$ in a $\mathbb Z$-homology $3$-sphere $W$ we use $\widehat{W}_2(K) \to W$ to denote the $2$-fold cover of $W$ branched over $K$. It is well-known that any irreducible representation of the fundamental group of the exterior of $K$ with values in ${\cal N}$ has image $D_n$ for some odd $n$. Moreover, Klassen observed that if $\Delta_K$ is the Alexander polynomial of $K$, there are exactly $\frac{|\Delta_K(-1)| - 1}{2}$ characters of such representations (compare Theorem 10 of [@Kl]).
A [*simple point*]{} of a complex affine algebraic set $V$ is a point of $V$ which is contained in a unique algebraic component of $V$ and is a smooth point of that component.
\[b11\] Let $M$ be the exterior of a knot $K$ in a $\mathbb Z$-homology $3$-sphere $W$. Suppose that $\rho: \pi_1(M) \to PSL_2(\mathbb C)$ has image $D_n$ where $n > 1$. Then the associated $2n$-fold cover $\widetilde{M}_\rho \to M$ extends to a branched cover $p: \widehat{W}_\rho(K) \to W$, branched over $K$. Moreover\
$(1)$ $p$ factors through an $n$-fold cyclic (unbranched) cover $\widehat{W}_\rho(K) \to \widehat{W}_2(K)$ and the $2$-fold branched cyclic cover $\widehat{W}_2(K) \to W$.\
$(2)$ if $b_1(\widehat{W}_\rho(K)) = 0$, then $H^1(M; Ad\rho) \cong \mathbb C$ and $\chi_\rho$ is a simple point of $X_{PSL_2}(M)$.
\(1) Fix meridinal and longitudinal classes $\mu$ and $\lambda$ in $\pi_1(\partial M) \subset \pi_1(M)$. Denote by $\widetilde{M}_2 \to M$ the $2$-fold cover of $M$. Since $W$ is a $\mathbb Z$-homology $3$-sphere we have $b_1(M) = b_1(\widetilde{M}_2) = 1$.
The subgroup $\rho^{-1}({\cal D})$ has index $2$ in $\pi_1(M)$ and so equals $\pi_1(\widetilde{M}_2)$. Hence $\rho|\pi_1(\widetilde{M}_2)$ has image $D_n \cap {\cal D} \cong \mathbb Z/n$. Since $\pi_1(M)$ is generated by $\mu$ and $\pi_1(\widetilde{M}_2)$ we see that $\rho(\mu) \in {\cal N} \setminus {\cal D}$ and therefore has order $2$. Further, since $\lambda$ is a double commutator, $\pi_1(\partial \widetilde{M}_2) \subset \hbox{kernel}(\rho)$. In particular if $\widetilde{M}_\rho \to \widetilde{M}_2$ is the regular cover associated to $\rho|\pi_1(\widetilde{M}_2)$, then $|\partial \widetilde{M}_\rho| = n$.
Since $\rho(\mu^2) = \pm I$, $\rho|\pi_1(\widetilde{M}_2)$ factors through $\pi_1(\widehat{W}_2(K))$ and defines an $n$-fold cyclic cover $\widehat{W}_\rho(K) \to \widehat{W}_2(K)$ which composes with $\widehat{W}_2(K) \to W$ to produce the desired cover of $W$ branched over $K$. It is clear that $\widehat{W}_\rho(K) \to \widehat{W}_2(K)$ is obtained from $\widetilde{M}_\rho \to \widetilde{M}_2$ by equivariant Dehn filling.
\(2) Suppose now that $b_1(\widehat{W}_\rho(K)) = 0$. For each $d \geq 1$ which divides $n$, let $\rho_d = \theta_{n,d} \circ \rho$ and $\widetilde{M}_{2d} \to M$ the associated cover. The second paragraph of the proof of (1) shows that $|\partial \widetilde{M}_{2d}| = d$ and since each boundary component is a torus, $b_1(\widetilde{M}_{2d}) \geq d$. On the other hand, the third paragraph shows that there is a Dehn filling of $\widetilde{M}_{2d}$ which yields $\widehat{W}_{\rho_d}(K)$. Now by construction, $\widehat{W}_{\rho_d}(K)$ is covered by $\widehat{W}_{\rho}(K)$ and therefore $b_1(\widehat{W}_{\rho_d}(K)) \leq b_1(\widehat{W}_{\rho}(K)) = 0$. Thus $d \leq
b_1(\widetilde{M}_{2d}) \leq b_1(\widehat{W}_{\rho_d}(K)) + |\partial \widetilde{M}_{2d}| \leq d$. Plugging $b_1(\widetilde{M}_{2d}) = d$ into the conclusion of Lemma \[b1Adrho\] shows that $$b_1(M; Ad \rho) = b_1(\widetilde{M}_2) - b_1(M) + \frac{1}{\phi(n)}
\sum_{d|n} \mu(\frac{n}{d}) d = \frac{1}{\phi(n)}
\sum_{d|n} \mu(\frac{n}{d}) d.$$ It is well-known that for $n > 1$ we have $\frac{1}{\phi(n)} \sum_{d|n} \mu(\frac{n}{d}) d = 1$ (see Identity 16.3.1 of [@HW] for example) and thus, $b_1(M; Ad \rho) = 1$. Theorem 3 of \[BZ3\] now shows that $\chi_\rho$ is a simple point of $X_{PSL_2}(M)$. This completes the proof.
\[2br1\] If $K$ is a two-bridge knot with exterior $M$ and $\rho \in R_{PSL_2}(M)$ has image $D_n$ where $n > 1$, then $H^1(M; Ad\rho) \cong \mathbb C$ and $\chi_\rho$ is a simple point of $X_{PSL_2}(M)$.
Since the $2$-fold branched cyclic cover of $W = S^3$ over $K$ is a lens space, Proposition \[b11\](1) implies that $b_1(\widehat{W}_\rho(K)) = 0$. The desired conclusion now follows from conclusion (2) of Proposition \[b11\].
Proof of Proposition \[meridiandegree\] {#pf510}
---------------------------------------
Let $p \geq 1,q$ be relatively prime integers where $p$ is odd. We observed in §\[criterion\] that given a knot $K \subset S^3$ with exterior $M$, the image of any homomorphism $\rho: \pi_1(M) \to {\cal N}$ with non-abelian image is $D_n$ for some odd $n \geq 3$. Moreover, the number of characters of such representations is exactly $\frac{|\Delta_K(-1)| - 1}{2} = \frac{|H_1(\hat S^3(K))| - 1}{2} < \infty$. For $K = k_{p/q}$ we have $|\Delta_K(-1)| = |H_1(L(p,q))| = p$. This discussion yields our next lemma.
\[strnontriv\] Every non-trivial curve in the $PSL_2(\mathbb C)$ character variety of the exterior of a knot in $S^3$ is strictly non-trivial.
Consider a non-trivial curve $X_0 \subset X_{PSL_2}(M_{p/q})$. It is shown in [@HT] that the meridinal slope $\mu$ of $k_{p/q}$ is not a boundary slope. Since $M_{p/q}$ is small, Propositions \[idealvalue\] shows that for each ideal point $x$ of $\tilde X_0$, $\Pi_x(\tilde f_{\mu}) > 0$. Thus $\Pi_x(\tilde f_{\mu^2}) = \Pi_x(\tilde f_{\mu}(\tilde f_{\mu} + 4)) > 0$ as well. Then $Z_x(\tilde f_{\mu}) = Z_x(\tilde f_{\mu^2}) = 0$ and so by Identity (2.4.1) we have $$0 < d_M(\mu) = d_M(\mu^2) - d_M(\mu) =
\sum_{\stackrel{non-trivial}{X_0}} \; \sum_{x \in X_0^\nu} (Z_x(\tilde f_{\mu^2}) - Z_x(\tilde
f_{\mu})). \eqno{({\rm A}.3.1)}$$ It follows from Proposition 1.1.3 of [@CGLS] that $Z_x(\tilde f_{\mu}) \leq Z_x(\tilde f_{\mu^2})$ for each $x \in X_0^\nu$. Moreover, Proposition 1.5.4 of that paper shows that if $Z_x(\tilde
f_{\mu}) < Z_x(\tilde f_{\mu^2})$ for some $x \in X_0^\nu$ and $\nu(x) = \chi_\rho$, then $\rho(\mu^2) = \pm I$. In particular, the restriction of $\rho$ to the fundamental group of the $2$-fold cover of $M_{p/q}$ factors through the fundamental group of $L(p,q)$, the $2$-fold cover of $S^3$ branched over $k_{p/q}$. Thus the image of $\rho$ is finite and as we can suppose that it is not cyclic (Proposition 1.5.5 of [@CGLS]), it must be a non-abelian dihedral group.
Now $\rho(\mu)$ is neither parabolic nor $\pm I$, so that $Z_x(\tilde f_\mu(x)) = 0$. We know that $\chi_\rho$ is a simple point of $X_{PSL_2}(M_{p/q})$ by Corollary \[2br1\] and we claim (see Lemma \[simple\] below) that $Z_x(\tilde f_{\mu^2}) = 1$. Note that these two facts and Identity (A.3.1) show that $d_M(\mu)$ equals the number of irreducible, dihedral characters which lie on some non-trivial curve in $X_{PSL_2}(M_{p/q})$. But by Proposition \[posdim\], every such character lies on such a curve, and since there are $\frac{p-1}{2}$ irreducible, dihedral characters of $\pi_1(M_{p/q})$, we have $d_M(\mu) = \frac{p-1}{2}$, which is what we set out to prove.
\[simple\] Let $\chi_\rho$ be an irreducible, dihedral character of $\pi_1(M_{p/q})$, $X_0$ the unique curve in $X_{PSL_2}(M_{p/q})$ which contains it, and $x$ the unique point of $X_0^\nu$ such that $\nu(x) = \chi_\rho$. Then $Z_x(\tilde f_{\mu^2}) = 1$.
The proof that $Z_x(\tilde f_{\mu^2}) = 1$ is essentially identical to the proof of Theorem 2.1 (2) of [@BB], though with with some slight modifications as $2\mu$ is not a primitive class in $H_1(\partial M)$. These modifications are simple and we describe them next.
Let $M_{p/q}(2\mu)$ be the space obtained by attaching a solid torus to $M_{p/q}$ by a covering map which maps $S^1 \times \{1\}$ homeomorphically to $\lambda_{M_{p/q}}$ and is a $2$-fold cover of $\{*\} \times \partial D^2$ to $\mu$. Note that there is a unique $2$-fold cover of $M_{p/q}(2\mu)$ and its total space is $L(p,q)$, the $2$-fold cover of $S^3$ branched over $k_{p/q}$. Note as well that $\rho$ factors through $\pi_1(M_{p/q}(2\mu))$. Lemma \[b1Adrho\] applied to this situation shows that $H^1(M_{p/q}(2\mu)) = 0$. The proof of Lemma 1.8 of \[BB\] shows that if $u \in Z^1(\pi_1(M_{p/q}); Ad\rho)$ represents a non-zero class in $H^1(\pi_1(M_{p/q});
Ad\rho)$ then $u(\mu^2) \ne 0$. The calculation $Z_x(\tilde f_{\mu^2}) = 1$ now follows in a similar fashion to the calculation in the proof of Theorem 2.1 (2) of \[BB\].
Peripheral values of homomorphisms of twist knot groups {#twistknotperipheral}
=======================================================
In this appendix we show that the trefoil knot is the only twist knot whose group admits a homomorphism onto an infinite triangle group such that the image of the peripheral subgroup is finite.
After Hoste and Shanahan, we identify the $n$-twist knot $K_n$ with the knot $J(2, 2n)$ of [@HS1]. When $n = -1, 0, 1$, $K_n$ is the figure $8$ knot, the trivial knot, and the trefoil knot respectively.
The fundamental group of the exterior $M_n$ of $K_n$ admits a presentation $$\pi_1(M_n) = \langle a, b : a (ab^{-1}a^{-1}b)^n = (ab^{-1}a^{-1}b)^n b \rangle$$ where $a$ and $b$ are meridinal classes (cf. Proposition 1 of [@HS1]). Set $w = ab^{-1}a^{-1}b$ so that the relation becomes $a w^n = w^n b$.
\[nil\] If there is a surjective homomorphism $\rho: \pi_1(M_n) \to \Delta(p,q,r)$ where $(p,q,r)$ is a Euclidean triple, then $n = 1$ and hence $K_n$ is the trefoil knot. Further, $(p,q,r) = (2,3,6)$ up to permutation.
First we observe that any two elements of $\Delta(p,q,r)$ which are of infinite order commute since they correspond to translations under the natural embedding $\Delta(p,q,r) \to \hbox{Isom}_+(\mathbb E^2)$. Thus $\rho(a)$ and $\rho(b) = \rho(w)^{-n}\rho(a) \rho(w)^n$ must be elliptic. Since $H_1(\Delta(p,q,r))$ is necessarily cyclic, we have $(p,q,r) = (2,3,6)$ up to permutation. Further, $\rho(a)$ generates $H_1(\Delta(2,3,6)) \cong \mathbb Z/6$ so that $\rho(a)$ and $\rho(b)$ have order $6$. Then up to replacing $\rho$ by a conjugate representation we may suppose that $$\rho(a) = xy \in \langle x, y : x^2 = y^ 3 = (xy)^6 = 1 \rangle = \Delta(2,3,6).$$ We claim that up to conjugating $\rho$ by a power of $xy$, we can suppose that $\rho(b) = yx$. To see this, fix a tessellation ${\cal T}$ of $\mathbb E^2$ by triangles with angles $\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{6}$, and identify $\Delta(2,3,6) \subset \hbox{Isom}(\mathbb E^2)$ with the set of orientation preserving symmetries of ${\cal T}$. the elements $xy, yx$ are conjugate elements of order $6$ in $\Delta(2,3,6)$ and form a generating set. Moreover, the tessellation ${\cal T}$ can be described as follows. Let $A$ be the fixed point of $xy$, $B \ne A$ that of $yx$, and let $C$ be the midpoint of $[A, B]$. Denote by $L$ the line through $C$ which is orthogonal to $[A, B]$ and let $T(A, D, E)$ be the triangle with vertices $A, D, E$ where $D, E \in L$ are equidistant to $C$ and the angles at $A, D, E$ are $\frac{\pi}{3}, \frac{\pi}{3}, \frac{\pi}{6}$ respectively. The triangle $T(A,C,D)$ is a face of ${\cal T}$ and so the tessellation is its orbit under the action of $\Delta(2,3,6)$. Moreover, $T(A, D, E)$ is the union of the two adjacent faces $T(A,C,D)$ and $T(A,C,E)$ and so is a fundamental domain for $\Delta(2,3,6)$. Since $\mathbb E^2$ admits self-similarities of arbitrary scale factor, it is clear that any two elements of order $6$ in $\hbox{Isom}(\mathbb E^2)$ with distinct fixed points generate a subgroup isomorphic to $\Delta(2,3,6)$ with invariant tessellation and fundamental domain constructed as above. In particular this is the case for $\rho(a), \rho(b)$. Let ${\cal T}'$ and $T(A, C', E')$ be the associated tessellation and fundamental domain. Since $\rho(a), \rho(b)$ generate $\Delta(2,3,6)$, ${\cal T} = {\cal T}'$ and so $T(A, C, E)$ can be obtained from $T(A, C', E')$ by a rotation about $A$ of angle $\frac{2 \pi j}{3}$ for some integer $j$. This rotation is given by $(xy)^{\epsilon j}, \epsilon \in \{\pm 1\}$, so if we replace $\rho$ by $(xy)^{-\epsilon j} \rho (xy)^{\epsilon j}$, the new fixed point of $\rho(b)$ is $B$. Since $\rho(a)$ and $\rho(b)$ are conjugate, it follows that $\rho(b) = yx$.
With these calculations in hand, we see that $v := [a, b^{-1}] = x(yx)^3$ is a product of two elements of order $2$ with distinct fixed points. Thus $v$ is a translation which leaves ${\cal T}$ invariant. Since $\rho(b) = v^{-n} \rho(a) v^n$, $B$, the fixed point of $\rho(b)$, equals $v^{-n}(A)$. But examination of ${\cal T}$ shows the only way this is possible is for $n = 1$.
\[prop:epi\] If there is a surjective homomorphism $\rho: \pi_1(M_n) \to \Delta(p,q,r) \subset PSL_2(\mathbb R)$ such that $(p,q,r)$ is a hyperbolic triple and $\rho(a)$ is elliptic, then $n = 1$, so that $K_n$ is the trefoil knot. Further, $(p,q,r) = (2,3,r), \, r \geq 7$ up to permutation.
Suppose that there is a surjective homomorphism $\rho: \pi_1(M_n) \to \Delta(p,q,r) \subset PSL_2(\mathbb R)$ such that $(p,q,r)$ is a hyperbolic triple and $\rho(a)$ is elliptic. Clearly $n \ne 0$ and $\Delta(p,q,r)$ is generated by two conjugate elliptics. Theorem 2.3 of [@Kn] implies that up to permuting $p,q,r$, one of the following two scenarios arises.
[(a)]{}
: $(p,q,r) = (2,q, r)$ where $\rho(a)$ has order $q$ and $r \geq 3$ is odd. Further, there is an integer $s$ relatively prime to $q$ such that in the standard presentation $\Delta(2, q,r) = \langle x, y, z : x^2, y^q, z^r \rangle$ we have $\rho(a) = y^s, \rho(b) = xy^sx^{-1}$.
[(b)]{}
: $(p,q,r) = (2, 3, r)$ where where $\rho(a)$ has odd order $r \geq 7$. Further, there is an integer $s$ relatively prime to $r$ such that in the standard presentation $\Delta(2, 3, r) = \langle x, y, z : x^2, y^3, z^r \rangle$ we have $\rho(a) = z^s, \rho(b) = yxy^{-1}z^syxy^{-1}$.
Set $v = \rho(w)$ so that $$v = [\rho(a), \rho(b^{-1})] = \left\{ \begin{array}{ll} (y^sxy^{-s})((xy^{-s})x(xy^{-s})^{-1}) & \hbox{in scenario (a)} \\ ((z^sy)x(z^sy)^{-1})((yxy^{-1}z^{-s}y)x(yxy^{-1}z^{-s}y)^{-1}) & \hbox{in scenario (b)}. \end{array} \right.$$ In either case $v = uu'$ where $u, u'$ are of order $2$. Denote by $R, R'$ the fixed points of $u, u'$ and observe that if $R = R'$, then $u = u'$ and therefore $v = uu' = 1$. Then $\rho(b) = \rho(w^{-n}aw^n) = v^{-n} \rho(a) v^n = \rho(a)$, which is impossible. Thus $R \ne R'$ and it is easy to see that if $\gamma$ denotes the geodesic in $\mathbb H^2$ which contains both $R$ and $R'$, then $v$ is a hyperbolic element of $PSL_2(\mathbb R)$ with invariant geodesic $\gamma$ and translation length $2d_{\mathbb H^2}(R, R')$. The proof of the proposition is similar in the two possible scenarios. We analyse them separately.
Assume first that we are in scenario (a). There is a fundamental domain for $\Delta(2,q,r)$ in $\mathbb H^2$ which is a geodesic triangle $T = T(A,B,C)$ having vertices $A = \hbox{Fix}(y), B = \hbox{Fix}(xyx^{-1}), C = \hbox{Fix}(z)$ and the midpoint $P$ of $[A, B]$ is $\hbox{Fix}(x)$ (so $x(A) = B$). The angles of $T$ at $A, B, C$ are $\frac{\pi}{q}, \frac{\pi}{q}, \frac{2 \pi}{r}$ respectively. The hyperbolicity of $v$ implies that for $l \ne 0$, $v^l(C), v^l(P) \not \in T$ and so if $T \cap v^l(T) \ne \emptyset$, then up to replacing $l$ by its negative we have $T \cap v^l(T) = \{A\}$ and $v^l(B) = A$.
Since $xy^sx = \rho(b) = \rho(w^{-n}aw^n) = v^{-n} y^s v^n$, there is an integer $m$ such that $v^n = y^mx$. Then $v^n(B) = y^mx(B) = A$ so that $v^n(T) \cap T= \{A\}$. It now follows from the previous paragraph that if for some $l \ne 0$ we have $T \cap v^l(T) \ne \emptyset$, then up to replacing $l$ by its negative we have $v^l(B) = A = v^n(B)$. Thus $l = \pm n$ and so for $d \ne e$, $$v^{dn}(T) \cap v^{en}(T) = \left\{\begin{array}{cl}
v^{dn}(B) & e = d - 1 \\
v^{(d+1)n}(B) & e = d + 1 \\
\emptyset & e \ne d \pm 1 \end{array} \right.$$ and the reader will verify that $\Gamma_0 = \cup_d v^{dn}(T)$ is an infinite chain of geodesic triangles which is closed, properly embedded, and separating in $\mathbb H^2$. It follows that $n = \pm 1$ as otherwise $v(\Gamma_0) \cap \Gamma_0 = \emptyset$ and so the side of $\Gamma_0$ in $\mathbb H^2$ containing $v(\Gamma_0)$ is invariant under $v$. Thus $v^l(\Gamma_0) \cap \Gamma_0 = \emptyset$ for all $l > 0$, contrary to the fact that $v^{|n|}(\Gamma_0) = \Gamma_0$.
If $n = -1$, then $y^{m}x = v^{-1}$ so that $xy^{m}x^{-1} = (y^{-s}xy^{s})x(y^{s}xy^{-s})$. In particular, $(y^{-s}xy^{s})x(y^{s}xy^{-s})$ fixes $x(A) = B$. But $(y^{-s}xy^{s})x(y^{s}xy^{-s})$ is a product of three conjugates of $x$ with fixed points $P, y^s(P), y^{-s}(P)$ and the reader will verify that this is impossible because such a configuration of order 2 elliptics cannot fix $B$. (Alternately, we refer the reader to the proof of Theorem 11.5.2 of \[Beardon\] where the fixed points of a product of three order 2 elliptics are analysed. The analysis implies that if $(y^{-s}xy^{s})x(y^{s}xy^{-s})$ has a fixed point, then this fixed point and $B$ lie on opposite sides of the geodesic through $y^s(P)$ and $y^{-s}(P)$.)
Finally, if $n = 1$, $K_n$ is the trefoil knot and we have $y^{m}x = v = y^sxy^{-s}xy^{-s}xy^{s}x$ so that $y^{m-3s} = (xy^{-s})^3$. If $(xy^{-s})^3 \ne 1$ then $xy^{-s}$ fixes $A$, which is impossible. Thus $y^{m-3s} = (xy^{-s})^3 = 1$. We will show that $r = 3$ to complete this part of the proof. Let $D$ be the fixed point of $xy^{-s}$. Since $xy^{-s}(A) = B$, $D$ lies on the perpendicular $L$ to $[A,B]$ through $P$. Now $D \ne P$ as otherwise $y^s = 1$. On the other hand, $C$ and $x(C)$ are the closest points to $P$ of the given tessellation of $\mathbb H^2$ which lie on $L$. Further, since $z(B) = (xz^{-1})(B) = A$ we see that $\frac{2\pi}{3}$, the absolute value of the angle of rotation of $xy^{-s}$ is bounded above by that of $z$ at $C$ or $xz^{-1}x$ at $x(C)$, which is $\frac{2\pi}{r}$, with equality if and only if $D \in \{C, x(C)\}$. Thus $\frac{2\pi}{3} \leq \frac{2\pi}{r}$ which implies that $r=3$ and we are done.
Now suppose that we are in scenario (b) and consider the geodesic triangle $T_0$ in $\mathbb H^2$ with vertices $A, B, C$ such that $A = \hbox{Fix}(x), B = \hbox{Fix}(y), C = \hbox{Fix}(z)$. The angles of $T_0$ at $A, B, C$ are $\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{r}$ respectively where $r \geq 7$ is odd. Recall that $$v = [(z^sy)x(z^sy)^{-1}][(yxy^{-1}z^{-s}y)x(yxy^{-1}z^{-s}y)^{-1}]$$ is a product of two conjugates of $x$ and observe that that form of $\rho(b)$ given in scenario (b) implies that $$v^n = z^m yxy^{-1}$$ for some integer $m$.
Consider the geodesic triangle $T = T(C,D, E)$ containing $T_0$ with vertices $C, D = (yxy^{-1})(C) = \hbox{Fix}((yxy^{-1})z(yxy^{-1})), E = y(C) = \hbox{Fix}(yzy^{-1})$ of angles $\frac{\pi}{r}, \frac{\pi}{r}, \frac{4\pi}{r}$ respectively.
\[intersection\] For any integer $l \ne 0$, $v^l(T) \cap T$ is either empty or one of the vertices $C, D$.
(of Claim \[intersection\]) First we show that $v^l(\hbox{int}(T)) \cap \hbox{int}(T) = \emptyset$. Decompose $T$ as $T_1 \cup T_2 \cup T_3$ where $T_1 = T(B,C,E), T_2 = T(B, E, B')$ where $B' = (yxy^{-1})(B)$, and $T_3 = T(B', D, E)$ and note that each of $T_1, T_2, T_3$ is a fundamental domain for the action of $\Delta(2,3,r)$ on $\mathbb H^2$. Since $T_1$ is sent to the geodesic triangle $T_3$ by the elliptic element $(yxy^{-1})(xzx^{-1})(yxy^{-1})^{-1}$, $v^l(\hbox{int}(T_1)) \cap \hbox{int}(T_3) = \emptyset$. Similarly $v^l(\hbox{int}(T_1)) \cap \hbox{int}(T_2) = \emptyset$ so that $v^l(\hbox{int}(T_1)) \cap \hbox{int}(T) = \emptyset$. In the same way we see that $v^l(\hbox{int}(T_2)) \cap \hbox{int}(T) = v^l(\hbox{int}(T_3)) \cap \hbox{int}(T) = \emptyset$, which is what we needed to prove.
Second we claim that $v^l(E) \not \in T$. If this is false we have $v^l(E) \in \{C, D\}$ (i.e. the only valency $2r$ vertices in $T$ are $C,D,E$ and $v^l(E) \ne E$ as it is hyperbolic), say $v^l(E) = C$. Then the axis of $v$ is perpendicular to the perpendicular bisector $L$ of $[C, E]$. On the other hand, $v^n(D) = z^m yxy^{-1}(D) = C$, so the axis of $v$ is also perpendicular to the perpendicular bisector $L'$ of either $[C, D]$. But this is impossible since $L \cap L' = \{y(E)\} \ne \emptyset$. Hence $v^l(E) \ne C$ and a similar argument shows it does not equal $D$.
Third we observe that $v^l(T) \cap T$ contains no edge of the tessellation. By the first paragraph, such an edge would have to lie in $\partial T$ and by the second it could not contain $E$. Since $v^l$ preserves the combinatorial type of the vertices it is now easy to use the method of the first paragraph to obtain a contradiction.
These observations imply that if $v^l(T) \cap T$ is non-empty, then it is a vertex of $T$. This proves Claim \[intersection\]
Since $v^n(D) = C$, the claim implies that if $v^l(T) \cap T \ne \emptyset$, then after possibly replacing $l$ by its negative we have $v^l(D) = C$. Thus $l = \pm n$ and the intersection is $C$ if $l = n$ and $D$ otherwise. Arguing as in scenario (a) we have $|n| = 1$. If $n = -1$, we have $z^myxy^{-1} = v^{-1}$ and therefore $$(yxy^{-1})z^m(yxy^{-1}) = [(z^{-s}y)x(z^{-s}y)^{-1}](yxy^{-1})[(z^sy)x(z^sy)^{-1}].$$ Hence $D$ is fixed by the product $[(z^{-s}y)x(z^{-s}y)^{-1}](yxy^{-1})[(z^sy)x(z^sy)^{-1}]$ of three conjugates of $x$ with fixed points $y(P), (z^{s}y)(P), (z^{-s}y)(P)$. As in the analysis of the case $n = -1$ in scenario (a), it can be verified that this cannot occur. Thus $n = 1$ and $K_n$ is the trefoil knot.
Bending
=======
Let $\Gamma$ be a finitely generated group which splits over a subgroup $\Gamma_0$ and $\rho: \Gamma \to PSL_2(\mathbb C)$ a homomorphism such that $\rho(\Gamma_0)$ is abelian but not isomorphic to $\mathbb Z/2 \oplus \mathbb Z/2$. Under this condition we can perform a deformation operation on $\chi_\rho$ known as bending. The details of the construction depend on whether the splitting is a free product with amalgamation or an HNN extension and are dealt with in §\[fpa\] and §\[hnn\] respectively.
Recall the subgroups ${\cal D, N}$ of $PSL_2(\mathbb C)$ defined in §\[generalities\] and set $${\cal P}_+ = \{\pm \left( \begin{array}{cc} 1 & z \\ 0 & 1 \end{array}
\right) \; | \; z \in {\mathbb C}\} \subset {\cal T}_+ = \{\pm \left( \begin{array}{cc} z & w \\ 0 & z^{-1} \end{array}
\right) \; | \; z, w \in {\mathbb C}, z \ne 0\}$$ Under the natural action of $PSL_2(\mathbb C)$ on $\mathbb CP^1$, the fixed point sets of ${\cal T}_+$ and ${\cal P}_+$ coincide and consist of a single line $L_+$. That of ${\cal D}$ consists of two lines $\{L_+, L_-\}$.
The centraliser of a subset $E$ of $PSL_2(\mathbb C)$ wil be denoted by $Z_{PSL_2}(E)$ and the component of the identity of $Z_{PSL_2}(E)$ will be denoted $Z_{PSL_2}^0(E)$. For $\pm I \ne A \in PSL_2(\mathbb C)$ we have $$Z_{PSL_2}^0(A) \mbox{ is conjugate to }
\left\{ \begin{array}{ll}
{\cal D} & \mbox{if $A$ is diagonalisable} \\
{\cal P}_+ & \mbox{if $A $ is parabolic}
\end{array} \right.$$ Thus when $E \ne \{\pm I\}$, $Z_{PSL_2}^0(E)$ is abelian and reducible. For a group $\pi$ and representation $\rho \in R_{PSL_2}(\pi)$, we use $Z_{PSL_2}(\rho), Z_{PSL_2}^0(\rho)$ to denote, respectively, the centraliser and the component of the identity of the centraliser of $\rho(\pi)$.
$\Gamma = \Gamma_1 *_{\Gamma_0} \Gamma_2$ {#fpa}
-----------------------------------------
Fix $\rho: \Gamma \to PSL_2(\mathbb C)$ a homomorphism such that $\rho(\Gamma_0)$ is abelian but not isomorphic to $\mathbb Z/2 \oplus \mathbb Z/2$. Denote by $\rho_j$ the restriction of $\rho$ to $\Gamma_j $. For each $S \in Z_{SL_2}(\rho_0)$ define $\rho_S: \Gamma \to PSL_2(\mathbb C)$ to be the homomorphism determined by the push-out diagram
& & \_1 & &\
& & & \^[\_1]{} &\
\_0 & & & & PSL\_2(C)\
& & & \_[S \_2 S\^[-1]{}]{}\
& & \_2 &
We say that the character $\chi_{\rho_S}$ is obtained by [*bending*]{} $\chi_\rho$ by $S$. The [*bending function*]{} $$\beta_\rho: Z^0_{PSL_2}(\rho_0) \to X_{PSL_2}(\Gamma), \; S \mapsto \chi_{\rho_S}.$$ We say that $\rho$ can be [*bent non-trivially*]{} if $\beta_\rho$ is non-constant. Our next result determines necessary and sufficient conditions for this to occur.
\[constbendsep\] Suppose that $\rho \in R_{PSL_2}(\Gamma$ is such that $\rho(\Gamma_0)$ is abelian but not isomorphic to $\mathbb Z/2 \oplus \mathbb Z/2$. The bending function $\beta_\rho: Z^0_{PSL_2}(\rho_0) \to X_{PSL_2}(\Gamma)$ is constant if and only if one of the following two situations arises:\
$(a)$ $\rho_0(\Gamma_0) = \{\pm I\}$ and either $\rho_1(\Gamma_1) = \{\pm I\}$ or $\rho_2(\Gamma_2) = \{\pm I\}$.\
$(b)$ $\rho_0(\Gamma_0) \ne \{\pm I\}$ and either $\rho_1(\Gamma_1)$ is abelian and reducible, or $\rho_2(\Gamma_2)$ is abelian and\
reducible, or $\rho$ is reducible.
Suppose that the correspondence $S \mapsto \chi_{\rho_S}$ is constant. We leave the justification of the following claim to the reader.
\[constconj\] [Let $A, B, C \in SL_2(\mathbb C)$. Then $\mbox{trace}(ASBS^{-1}) =\mbox{trace}(AB)$ for all $S \in Z_{SL_2}(C)$ if and only if one of the following two situations arise:\
$(a)$ $C = \pm I$ and either $A = \pm I$ or $B = \pm I$.\
$(b)$ $C \ne \pm I$ and either $[A,C] = I$ or $[B,C] = I$ or $A,B$ and $C$ have a common\
eigenvector. ]{}
Set $G_j = \mbox{ image}(\rho_j)$ for $j = 0,1,2$. The claim shows that if $G_0 = \{\pm I\}$, then either $G_1 = \{\pm I\}$ or $G_2 = \{\pm I\}$, and we are done. Assume then that $G_0 \ne \{\pm I\}$ and that both $G_1$ and $G_2$ are either irreducible or non-abelian subgroups of $PSL_2(\mathbb C)$. It follows that neither $G_1 \subset Z^0_{PSL_2}(G_0)$ nor $G_2 \subset Z^0_{PSL_2}(G_0)$. We will show that $\rho$ is reducible.
Fix $C \in G_0 \setminus \{\pm I\}$ and observe that our hypotheses show $$Z^0_{PSL_2}(C) = Z^0_{PSL_2}(G_0) = \left\{\begin{array}{ll} {\cal D} & \mbox{if } G_0 \subset {\cal D} \\
{\cal P}_+ & \mbox{if } G_0 \subset {\cal P}_+ \end{array} \right.$$ If there is some $A_0 \in G_1 \setminus Z^0_{PSL_2}(G_0)$, then Claim \[constconj\] implies that for each $B \in G_2$, either $B \in Z^0_{PSL_2}(G_0) \subset {\cal T}_+$ or $A_0, B$ and $C$ have a common fixed point in $\mathbb CP^1$. It follows that each element of $G_2$ fixes at least one of $L_+$ and $L_-$ and it is simple to deduce from this that $G_2$ fixes one of these lines. A similar argument shows that $G_1$ fixes one of them as well. If $G_1$ and $G_2$ have a common fixed point, then $\rho$ is reducible, so suppose that they do not. One of them, say $G_1$, fixes $L_+$ and not $L_-$, while $G_2$ fixes $L_-$ and not $L_+$. Since $G_0 \subset G_1 \cap G_2$, it fixes both $L_+$ and $L_-$ and therefore we must have $G_0 \subset {\cal D}$. By choice, $A_0 \in G_1 \setminus {\cal D}$ and so its fixed point set is $L_+$. On the other hand we have assumed that there is some $B_0 \in G_2 \setminus Z^0_{PSL_2}(G_0) = G_2 \setminus {\cal D}$. Its fixed point set is $L_-$. But this is impossible as Claim \[constconj\] implies that $A_0, B_0$ and $C$ have a common fixed point in $\mathbb CP^1$. Thus $G_1$ and $G_2$ do have a common fixed point, and so $\rho$ is reducible.
Conversely if either $G_1 \subset Z^0_{PSL_2}(G_0)$, or $G_2 \subset Z^0_{PSL_2}(G_0)$, or $G_0 \ne \{\pm I\}$ and $\rho$ is reducible, then Claim \[constconj\] implies that the correspondence $S \mapsto \chi_{\rho_S}$, where $S \in Z^0_{PSL_2}(G_0)$, is constant. This completes the proof of the lemma.
$\Gamma = (\Gamma_1)_{\Gamma_0}$ {#hnn}
--------------------------------
In this case there is an injective homomorphism $\varphi: \Gamma_0 \to \Gamma$ such that $$\Gamma = \langle \Gamma_1, \mu : \mu \gamma \mu^{-1} = \varphi(\gamma), \gamma \in \Gamma_0 \rangle.$$ Set $\Gamma_0' = \varphi(\Gamma_0)$ and for $\rho \in R_{PSL_2}(\Gamma)$ we take $\rho_1, \rho_0, \rho_0'$ to be its restriction to $\Gamma_1, \Gamma_0, \Gamma_0'$ respectively. The correspondence $\rho \in R_{PSL_2}(\Gamma) \mapsto (\rho_1, \rho(\mu)) \in R_{PSL_2}(\Gamma_1) \times PSL_2(\mathbb C)$ determines an identification $$R_{PSL_2}(\Gamma) = \{(\rho_1, A) \in R_{PSL_2}(\Gamma_1) \times PSL_2(\mathbb C) : A \rho_0(\gamma) A^{-1} = \rho_0'(\varphi(\gamma)) \mbox{ for all } \gamma \in \Gamma_0 \}.$$ Note that $(\rho_1, A), (\rho_1, B) \in R_{PSL_2}(\Gamma)$ if and only if $B = AS$ for some $S \in Z_{PSL_2}(\rho(\Gamma_0))$. In particular, if $\rho(\Gamma_0)$ is abelian but not $\mathbb Z/2 \oplus \mathbb Z/2$, we have a bending function $$\beta_{(\rho_1, A)}: Z^0_{PSL_2}(\rho(\Gamma_0)) \to X_{PSL_2}(\Gamma_0), S \mapsto \chi_{(\rho, AS)}.$$
\[constbendnonsep\] Suppose that $(\rho_1, A) = \rho \in R_{PSL_2}(\Gamma)$ is a representation with $\rho(\Gamma_0)$ is abelian but not $\mathbb Z / 2 \oplus \mathbb Z /2$. The bending function $\beta_{(\rho_1, A)}$ is constant if and only if $\rho(\Gamma_0) \ne \{\pm I\}$ and, after a possible conjugation, one of the following two situations arises:\
$(a)$ $\rho(\Gamma_1) \subset {\cal D}$ and $A = \pm \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]$.\
$(b)$ $\rho(\Gamma_1) \subset {\cal T}_+$ and $A \in {\cal T}_+$.\
In particular, $\rho_1$ is reducible and $\rho$ is either reducible or conjugate into ${\cal N}$.
Let $G_1, G_0, G_0'$ denote the images of $\rho_1, \rho_0, \rho_0'$ respectively. After possibly replacing $\rho = (\rho_1, A)$ by a conjugate representation, we may suppose that either $G_0 = \{\pm I\}$, or $\{\pm I\} \ne G_0 \subset {\cal D}$, or $\{\pm I\} \ne G_0 \subset {\cal P}$. We consider these three cases separately.
[**Case 1**]{}. $G_0 = \{\pm I\}$.
Then $Z_{PSL_2}^0(\rho_0) = PSL_2(\mathbb C)$ and so in general, $\beta_{(\rho_1, A)}(\mu) = \pm \mbox{trace}(A) \ne \pm \mbox{trace}(AS)= \beta_{(\rho_1, AS)}(\mu)$ for $S \in Z_{PSL_2}^0(\Gamma_0)$, $\beta_{(\rho_1, A)}$ is not constant.
[**Case 2**]{}. $\{\pm I\} \ne G_0 \subset {\cal D}$.
Then $Z_{PSL_2}^0(\rho_0) = {\cal D}$. If $\beta_{(\rho_1, A)}$ is constant, then $\pm \mbox{trace}(\rho(\gamma)A) = \beta_{(\rho_1, A)}(\gamma\mu) = \beta_{(\rho, AS)}(\gamma\mu) = \pm \mbox{trace}(\rho(\gamma)AS)$ for each $\gamma \in \Gamma_1$ and $S \in {\cal D}$. It follows that $\rho(\gamma)A \in {\cal N} \setminus {\cal D}$ for each $\gamma \in \Gamma_1$. Hence $A \in {\cal N} \setminus {\cal D}$ and $\rho(\Gamma_1) \subset {\cal D}$. After a further conjugation we may suppose that $A = \pm \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]$.
Conversely suppose that $A = \pm \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]$ and $\rho(\Gamma_1) \subset {\cal D}$. Consider a word $w = \Pi_j \mu^{a_j} x_j$ where $a_j \in \mathbb Z$ and $x_j \in \Gamma_1$. Set $D_j = \rho(x_j) \in {\cal D}$. Then for any $S \in {\cal D}$ we have $$(\rho, AS): w \mapsto \Pi_{j} (AS)^{a_j} D_j = (AS)^{a_1 + a_2 + \ldots + a_n}
\Pi_j D_j^{(-1)^{a_{j+1} + a_{j+2} + \ldots + a_n}}.$$ The trace of the right-hand side of this identity is independent of $S$, so that $\beta_{(\rho_1, A)}$ is constant.
[**Case 3**]{}. $\{\pm I\} \ne G_0 \subset {\cal P}$.
Then $Z_{PSL_2}^0(\rho_0) = {\cal P}$. If $\beta_{(\rho_1, A)}$ is constant, then $\mbox{trace}(\rho(\gamma)A) = \beta_{(\rho_1, A)}(\gamma \mu) = \beta_{(\rho_1, AS)}(\gamma \mu) = \mbox{trace}(\rho(\gamma)AS)$ for each $\gamma \in \Gamma_1$ and $S \in {\cal P}$. It follows that $\rho(\gamma)A \in {\cal T}_+$ for each $\gamma \in \Gamma_1$. Hence $A \in {\cal T}_+$ and $\rho(\Gamma_1) \subset {\cal T}_+$.
Conversely suppose that $A \in {\cal T}_+$ and $\rho(\Gamma_1) \subset {\cal T}_+$. Consider a word $w = \Pi_{j} \mu^{a_j} x_j$ where $a_j \in \mathbb Z$ and $x_j \in \Gamma_1$. Set $U_j = \rho(x_j) \in {\cal T}_+$. Then for any $S \in {\cal P}$ we have $$(\rho, AS): w \mapsto \Pi_{j} (AS)^{a_j} U_j.$$ The trace of the right-hand side of this identity is independent of $S$, so that $\beta_{(\rho_1, A)}$ is constant.
This completes the proof of the lemma.
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Michel Boileau, Laboratoire Émile Picard, Université Paul Sabatier, Toulouse Cedex 4, France\
e-mail: [email protected]
Steven Boyer, Dépt. de math., UQAM, P. O. Box 8888, Centre-ville, Montréal, Qc, H3C 3P8, Canada e-mail: [email protected]
[^1]: [ Partially supported by the Laboratoire de Mathématiques Émile Picard, UMR CNRS 5580]{}
[^2]: [ Partially supported by NSERC and FQRNT]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a class of mean-field models for the isostatic transition of systems of soft spheres, in which the contact network is modeled as a random graph and each contact is associated to $d$ degrees of freedom. We study such models in the hypostatic, isostatic, and hyperstatic regimes. The density of states is evaluated by both the cavity method and exact diagonalization of the dynamical matrix. We show that the model correctly reproduces the main features of the density of states of real packings and, moreover, it predicts the presence of localized modes near the lower band edge. Finally, the behavior of the density of states $D(\omega)\sim\omega^\alpha$ for $\omega\to 0$ in the hyperstatic regime is studied. We find that the model predicts a nontrivial dependence of $\alpha$ on the details of the coordination distribution.'
author:
- 'Fernanda P.C. Benetti'
- Giorgio Parisi
- Francesca Pietracaprina
- Gabriele Sicuro
bibliography:
- 'biblio.bib'
title: 'Mean-field model for the density of states of jammed soft spheres'
---
Introduction
============
While the vibrational behavior of crystalline solids — the density of states and heat capacity, for example — is well known, the disorder present in the structure of non-crystalline systems such as glasses, granular materials, and foams leads to intriguing anomalies that are still not completely understood. Both in crystals and in disordered solids in $d$ dimensions the (vibrational) density of states (DOS) $D(\omega)$ in the low frequency regime — i.e., on large scales — is given by the Debye law, $D(\omega)\sim\omega^{d-1}$. However, disordered systems present a nontrivial deviation from Debye’s theory at higher frequencies. This motivated a large amount of literature on the general properties of $D(\omega)$ and of the structure factor in the disordered case, from both the numerical and experimental point of view [@Phillips1981; @*Malinovsky1986; @OHern2003].
For example, the so-called “Boson peak” — an excess of modes with respect to Debye’s prediction — is a common feature of the DOS of disordered solids. It has been interpreted as a precursor of instability in harmonic regular lattices with spatially fluctuating elasticity [@Schirmacher1998; @*Schirmacher2006; @*Marruzzo2013; @*Schirmacher2015] and it seems to be linked to the Ioffe-Regel crossover frequency [@Xu2009; @Beltukov2013]. It has also been suggested that the Boson peak is simply a smeared version of the van Hove singularity, a well known feature of crystals [@Chumakov2011; @*Chumakov2014]. A different point of view on this topic came from the study of the dynamic structure factor in supercooled liquids, which has been successfully tackled using Euclidean random matrix theory [@Mezard1999; @MartinMayor2001; @*Ciliberti2003; @MartinMayor2000; @*Grigera2002; @*Grigera2011; @Cavagna1999; @*Cavagna2000]. From this perspective, the Boson peak phenomenon can be interpreted as a phonon-saddle transition [@Parisi2002a; @*Grigera2003]. The relation between disorder and the Boson peak is, however, still a matter of debate, alongside other spectral properties of disordered solids.
![Pictorial representation of a small overjammed system ($N=256$) of soft spheres in $d=3$ dimensions with its corresponding contact network. The overjammed configuration has been obtained assuming periodic boundary conditions. To simplify the network figure the contacts across the boundary are not shown.\[fig:contatti\]](sfere256.png "fig:"){height="22.00000%"}![Pictorial representation of a small overjammed system ($N=256$) of soft spheres in $d=3$ dimensions with its corresponding contact network. The overjammed configuration has been obtained assuming periodic boundary conditions. To simplify the network figure the contacts across the boundary are not shown.\[fig:contatti\]](contatti256.png "fig:"){height="22.00000%"}
In the present paper, we want to study the properties of $D(\omega)$ in a mean-field model for soft spheres near the jamming point. The simplest model of a disordered system of soft spheres is an elastic network with some kind of randomness in it. Random elastic networks have a long tradition in the literature. For example, an effective-medium theory (EMT) has been developed for the study of a system of oscillators on a regular lattice of springs with random stiffness [@Feng1985; @*Garboczi1985; @*Garboczi1985]. In all these models it emerged quite clearly that one of the essential features that strongly affects the properties of the DOS is the average degree of a node in the network, as first observed by Maxwell in his study on the stability of solids [@Maxwell1864; @Alexander1998]. By means of a constraint counting, Maxwell showed that, given a system of particles in $d$ dimensions, global mechanical stability requires an average number of contacts per particle given at least by $\bar z=2d$, despite the fact that $z=d+1$ contacts on each particle are enough to pin it in a given position.
Applying Maxwell’s argument and using a variational approach, a general qualitative picture of $D(\omega)$ in a disordered elastic solid has been obtained in the last decade [@Wyart2005; @*Wyart2005c; @*Xu2007; @*Yan2016]. In particular, assuming $\delta z\coloneqq \bar z-2d>0$, it is expected that $D(\omega)$ has a plateau for $\omega\geq \omega_*\propto\delta z$, and that the plateau extends up to the origin for $\delta z\to 0^+$ [@OHern2003; @Silbert2005]. The frequency $\omega_*$ increases with compression [@Wyart2005b], due to the fact that $\bar z$ increases by consequence as well. The value $\omega_*$ is directly connected to the Boson peak and to an Ioffe-Regel crossover [@Xu2009]. Indeed, using EMT, @DeGiuli2014 found that the Boson peak frequency scales as $\omega_\text{bp}\sim\sqrt{\omega_e\omega_*}$, where $\omega_e$ is a frequency at which strongly-scattered modes appear and which depends on the compressive strain. A numerical study of the contact network of an overjammed system of soft spheres near the jamming point shows that there is a relation between the average number of contacts $\bar z$ and the packing fraction $\varphi$, i.e., $\delta z\propto(\varphi-\varphi_c)^{\sfrac{1}{2}}$ for $\varphi\geq\varphi_c$, $\varphi_c$ being the jamming transition packing fraction [@Durian1995; @OHern2003; @vanHecke2010]. The two variables $\bar z$ and $\varphi$ therefore play an equivalent role. On the other hand, if $\bar z<2d$, the stability condition is violated and the system is hypostatic: an extensive number of zero (floppy) modes appears and $D(\omega)$ has a gap for $0<\omega<\omega_0$ for a certain frequency $\omega_0$ [@During2013]. These results suggest quite clearly that, independently from the amount of disorder, stability is determined by two parameters: the average coordination $\bar z$ and the compressive strain applied to the system. Furthermore, on the transition between stability and instability, the frequency of the Boson peak vanishes and its amplitude diverges.
Despite the fact that the general features of the DOS in the three regimes are well established, the low-frequency properties of $D(\omega)$ for $\delta z>0$ are still a matter of investigation. In this regime, using an EMT approach, @DeGiuli2014 predicted $$\label{domega}
D(\omega)\sim\begin{cases}
\sfrac{\omega^{d-1}}{\omega_*^{\sfrac{d}{2}}},&\omega\ll \omega_e\\
\sfrac{\omega^2}{\omega_*^2},&\omega_e\ll \omega\ll\omega_*\\
\text{constant},&\omega\gg\omega_*.
\end{cases}$$ The same behavior has been obtained in the study of the soft perceptron, the simplest possible mean-field model for jamming [@Franz2015; @*Franz2016]. The lowest frequency behavior in Eq. corresponds to the phonon contribution, which is absent for $d\to+\infty$. What happens in finite dimension if the phonons are removed, however, is a nontrivial question. Indeed, both EMT and the perceptron model, which are mean-field theories, suggest that, for $\omega\to 0$, $D(\omega)\sim\omega^2$ once the Goldstone modes are neglected. On the other hand, @Gurarie2003 and @Gurevich2003 predicted a $D(\omega)\sim \omega^4$ scaling as a general behavior of the DOS in random media for $\omega\to 0$ in finite dimension. Numerical evidence is available in favor of both the mean-field [@Charbonneau2016] and the finite-dimensional [@BaityJesi2015; @Lerner2016; @Lupo2017; @Mizuno2017; @Angelani2018] predictions. Different authors dealt with the presence of phonons in finite dimension in different ways, e.g., by a random external field in spin glasses to break translational invariance [@BaityJesi2015; @Lupo2017], or, in structural glasses, carefully tuning the system size [@Lerner2016], isolating the localized low-frequency modes [@Mizuno2017], or performing a random pinning [@Angelani2018]. How to recover the finite-dimensional-scaling from the infinite-dimensional one is still an open problem. [Moreover, recent studies suggest that the protocol adopted for cooling the system might be relevant in the final low-frequency power-law behavior. In Ref. , for example, it has been shown that $D(\omega)\sim\omega^3$ in glasses obtained quenching from temperatures much higher than the glass transition temperature.]{}
The low-frequency regime for $\delta z>0$ is interesting also for its localization properties. The presence of (quasi) localized modes in the lower edge of the spectrum, alongside the presence of localized modes in the upper edge [@Silbert2009], has been observed in systems of soft spheres [@Laird1991; @*Schober1991] but also in the instantaneous normal modes spectrum of low-density liquids [@Cavagna1999; @*Cavagna2000; @Ciliberti2004]. In these cases the localized low-frequency modes tend to hybridize with extended Goldstone modes, becoming weakly localized.
The presence of localized low-frequency modes is common in many disordered models. For example, localized states appear on the edge of the spectrum in models with disorder on random graphs and Bethe lattices [@Biroli1999b; @*Biroli2010]. Localized eigenstates have been found also on the spectrum edges of Euclidean random matrix models on random graphs [@Ciliberti2005]. However, this property is out of the reach of mean-field models for jamming having infinite connectivity [@Franz2015; @*Franz2016]. The presence of localized low-frequency modes is relevant, because they are precursors of instabilities in the unjamming transition and of local rearrangements in sheared glasses [@WidmerCooper2008; @*WidmerCooper2009; @Manning2011]. Once again, the frequency $\omega_*$ plays the role of crossover frequency between the region of extended modes and the region of modes that are localized on few particles, which typically have low coordination [@Wyart2010; @Xu2010]. Moreover, the delocalization of low-frequency modes increases as $\varphi\to\varphi_c$ and $d$ increases [@Charbonneau2016].
The complexity of the scenario above motivated us to study a mean-field model in which Goldstone modes are absent by construction, that can be treated by the cavity method and that is still reminiscent of the finite dimensionality of real amorphous packings. We consider a tree-like random graph, which is our model for the (equilibrium) contact network in an amorphous packing. The lack of an underlying lattice regularity automatically forbids Goldstone modes. Each vertex in the graph corresponds to a sphere, and each edge is associated to a $d$-dimensional random unit vector joining the centers of two spheres in contact. The Hessian matrix ${{\boldsymbol{\mathsf M}}}$ is therefore constructed using this set of random vectors on the graph, and the DOS is computed from the spectrum of ${{\boldsymbol{\mathsf M}}}$ and averaging over all realizations.
This model has been investigated by @Parisi2014 on random regular graphs, and explicit expressions for the first moments of the corresponding DOS on Erdős–Rényi random graphs are available [@Cicuta2018].
The model discussed above has been inspired by the one introduced by @Manning2015, the so-called “diagonal-dominant (DD) random matrix model”. In the DD model the Hessian matrix ${{\boldsymbol{\mathsf M}}}$ is constructed using an Erdős–Rényi random graph with average coordination $\bar z$, in such a way that a random scalar quantity is associated to each edge. This model is therefore intrinsically “one-dimensional”. Isostaticity corresponds to $\bar z=2$ and thus there is no hypostatic regime.
A similar “one-dimensional” model has been very recently considered in Ref. , where the DOS of a ring of springs with random cross bonds has been studied in the presence of disorder in the elastic constants.
Starting from the results of Ref. , in this paper we consider a more general class of graphs that are also locally tree like, a fact that allows us to apply the cavity method to obtain information about both the DOS and the localization properties of the model. The predictions of the cavity method will be compared with the results obtained through an exact diagonalization procedure and the method of moments.
The paper is organized as follows. In Section \[sec:model\] we describe in detail the model under investigation and the methods that we have adopted to solve it. In Section \[sec:risultati\] we present our results for three possible cases (hypostatic, isostatic, and hyperstatic regimes). We compare the results obtained with fixed and with fluctuating coordination, stressing the main differences between the two cases. Finally, in Section \[sec:conclusioni\] we give our conclusions.
Model and methods {#sec:model}
=================
Let us consider a system of $N$ soft spheres in $d$ dimensions, whose centers are in positions $\{\mathbf r_i\}_{i=1,\dots,N}$, $\mathbf r_i=(r_i^\mu)_{\mu=1,\dots,d}$ being a $d$-dimensional vector in the Euclidean space. We assume that the spheres interact by a finite-range repulsive potential $U(x)$ depending on the modulus of their Euclidean distance only. We also assume that there are $N_c$ total “contacts” among the spheres, two spheres being in contact if there is a nonzero interaction between them. A given configuration of the spheres can therefore be naturally associated to a contact network, i.e., a graph $\mathcal G=(\mathcal V,\mathcal E)$ with vertex set $\mathcal V$ of cardinality $N$, and edge set $\mathcal E$ of cardinality $N_c$, in such a way that the $i$th sphere corresponds to the vertex $i\in\mathcal V$ and the edge $e=(i,j)$ is an element of $\mathcal E$ if, and only if, the $i$th sphere and the $j$th sphere are in contact (see Fig. \[fig:contatti\]). The average coordination number of the graph, i.e., the average number of contacts of each sphere, is given by $$\label{zmedio}
\bar z\coloneqq\frac{2N_c}{N}.$$
Denoting by $\mathbf x_{ij}\coloneqq \mathbf r_i-\mathbf r_j$ the distance between the $i$th sphere and the $j$th sphere, the Hamiltonian of the system depends on the set of distances $\{\mathbf x_{ij}\}_{(i,j)\in\mathcal E}$ only, and it can be written as $$\label{Hamhat}
\hat{\mathcal{H}}=\sum_{\mathclap{(i,j)\in\mathcal E}}U\left(\|\mathbf r_{i}-\mathbf r_j\|\right)\equiv\sum_{\mathclap{(i,j)\in\mathcal E}}U\left(\|\mathbf x_{ij}\|\right).$$ Given a set of *equilibrium positions* of the spheres, we can easily write down a quadratic Hamiltonian function that describes the fluctuations of the system around the given minimum (an inherent structure) by means of a harmonic approximation of the Hamiltonian in Eq. . Let ${{\boldsymbol\delta}}_i$ be the fluctuation of the $i$th sphere around its equilibrium position $\mathbf r_i$. We assume that, at equilibrium, $\|\mathbf r_{i}-\mathbf r_j\|=\|\mathbf x_{ij}\|=1$ for all $(i,j)\in\mathcal E$ and, moreover, we will neglect the so-called “initial stress” contribution [@Alexander1998], that indeed vanishes at jamming. Up to an additive constant and a global multiplicative factor, a quadratic approximation of Eq. gives us
\[Hamiltonian\] $$\label{Hamiltoniana}
\mathcal H[{{\boldsymbol\delta}}]\coloneqq
\sum_{ij}\sum_{\mu,\nu=1}^d \delta_i^\mu M_{ij}^{\mu\nu}\delta_j^{\nu}.$$ The element $\mathbf M_{ij}$ of the Hessian matrix ${{\boldsymbol{\mathsf M}}}=(\mathbf M_{ij})_{ij}$ is a $d\times d$ matrix given by $$\label{proprieta}
\mathbf M_{ij}=
\begin{cases}
-|\mathbf x_{ij}\rangle\langle\mathbf x_{ij}|&\text{if $(i,j)\in\mathcal E$},\\
\sum\limits_{k\in\partial i}|\mathbf x_{ik}\rangle\langle\mathbf x_{ik}|=-\sum\limits_{k\in\partial i}\mathbf M_{ik}&\text{if $i=j$},\\
\mathbf 0&\text{otherwise}.
\end{cases}$$ In the expression above, $\partial i$ is the set of neighbors of the vertex $i$ in the graph, i.e., the set of all the spheres in contact with the sphere $i$. For the sake of brevity, here and in the following we use a bra-ket notation, representing, for example, by $|\mathbf x\rangle$ the vector $\mathbf x\in\mathds R^d$ and by $|\mathbf x\rangle\langle\mathbf y|=(x^\mu y^\nu)_{\mu\nu}$ the outer product. Observe that the translational invariance constraint $$\label{proprieta2}
\sum_{k=1}^N M_{ik}^{\mu\nu}=0\qquad\forall i\in\mathcal V,\ \forall \mu,\nu=1,\dots,d$$
is satisfied. The DOS $D({{\boldsymbol{\mathsf M}}};\omega)$ of the system can be obtained directly from the spectral density $\varrho({{\boldsymbol{\mathsf M}}};\lambda)$ of the Hessian matrix in Eqs. , by means of the change of variable $D({{\boldsymbol{\mathsf M}}};\omega)=2\omega\varrho({{\boldsymbol{\mathsf M}}};\omega^2)$. In particular, the vibrational DOS $D({{\boldsymbol{\mathsf M}}};\omega)$ is a comb of $Nd$ Dirac deltas, $$D({{\boldsymbol{\mathsf M}}};\omega)=\frac{1}{Nd}\sum_{k=1}^{Nd}\delta(\omega-\omega_k)\equiv 2\omega\varrho({{\boldsymbol{\mathsf M}}};\omega^2),$$ where $\omega_k=\sqrt{\lambda_k}$, $\lambda_k$ being the $k$th eigenvalue of the dynamical matrix ${{\boldsymbol{\mathsf M}}}$. Observe that in the system described by the Hamiltonian in Eqs. , $d$ zero modes are always present, due to the fact that the translational invariance allows ${{\boldsymbol\delta}}_i\mapsto {{\boldsymbol\delta}}_i+\boldsymbol\lambda$ for any $\boldsymbol\lambda\in\mathds R^d$, and therefore there will always be a $\delta(\omega)/N$ contribution in the DOS.
To introduce and study the effects of randomness, we adopt a mean-field approximation [@Parisi2014]. We first suppose that the $N_c$ quantities $\mathbf x_{ij}$ appearing in Eqs. are independently generated random $d$-dimensional Gaussian unit vectors. Moreover, we suppose that the graph $\mathcal G$ is a random graph in which the coordination $z$ is distributed according to certain probability distribution $p_k$ such that $\Pr(z=k)=p_k$ for $k\in\mathds N$. For each value of $d$, we require that the coordination number $z_i$ of the $i$th vertex always satisfies the local stability condition $z_i\geq d+1$, and therefore $p_k=0$ for $k<d+1$. The translational invariance constraint in Eq. appears to be crucial in a random matrix model for the vibrational DOS of a disordered solid [@Manning2015] and it will be preserved. In this way randomness is introduced both in the edge weights and in the topology of the graph. We are interested in the properties of the DOS in the thermodynamical limit $N\to+\infty$ and keeping $\bar z$ constant.
In this paper, we will study two different random graph ensembles, always assuming $d=3$, if not otherwise specified.
We will first consider random regular graphs, i.e., graphs having $p_k=\delta_{k,\bar z}$. We will denote this model by ${{\mathcal G}}_{\bar z,0}$. Following Ref. , we have analyzed the three cases $\bar z=5$, $\bar z=6$, and $\bar z=7$, corresponding to a hypostatic, isostatic, and hyperstatic system, respectively.
We have then considered a second, more realistic class of graphs, in which fluctuations in the coordination are allowed. In an element of this second class of graphs, the coordination of each vertex $i$ is given by $z_i=z_0+\zeta_i$, where $z_0\geq d+1=4$ is a constant and $\zeta_i$ is a Poisson random variable having mean $\bar\zeta$. It follows that, in this case, $p_k=\bar\zeta^{k-z_0}\frac{\operatorname{e}^{-\bar\zeta}}{(k-z_0)!}$ for $k\geq z_0$, and zero otherwise. An element of this class can be thought of as an Erdős–Rényi random graph “superimposed” on a random regular graph. We will denote this model by ${{\mathcal G}}_{z_0,\bar{\zeta}}$. In our analysis, we have chosen $z_0$ and $\bar \zeta$ in such a way that either $\bar{z}<6$, or $\bar{z}=6$, or $\bar{z}>6$, corresponding to the hypostatic, isostatic, and hyperstatic case, respectively. This model reproduces in a reasonable way the real coordination distribution of sphere packings near jamming [@Charbonneau2015; @Jin2018] and allows us to consider the effects of fluctuating coordination[^1].
Both types of random graphs are locally tree-like and the models combine a mean-field approximation (the random graph topology) with the finite number of degrees of freedom of each contact, which is reminiscent of a finite dimensionality.
Density of states and the cavity method approach
------------------------------------------------
As usual in the study of disordered systems, we are interested in the properties of our model averaged over disorder, and in particular in the average DOS, namely, $$\label{gomega}
D(\omega)\coloneqq{\mathbb{E}\left[{D({{\boldsymbol{\mathsf M}}};\omega)}\right]}=2\omega {\mathbb{E}\left[{\varrho({{\boldsymbol{\mathsf M}}};\omega^2)}\right]}\eqqcolon 2\omega\varrho(\omega^2),$$ where the average ${\mathbb{E}\left[{\bullet}\right]}$ is performed over all instances of the problem. After some manipulations of the Dirac deltas, it can be shown [@MartinMayor2001; @*Ciliberti2003] that the DOS $D(\omega)$ can be written as $$D(\omega)= -\lim_{\varepsilon\to 0}\lim_{N\to \infty}\frac{2\omega}{Nd\pi}{\mathbb{E}\left[{\operatorname{Tr}\operatorname{Im}\boldsymbol{\mathsf R}(\omega^2+i\varepsilon) }\right]}$$ where we have introduced the resolvent $$\boldsymbol{\mathsf R}(\lambda)\coloneqq\frac{1}{\lambda \boldsymbol{\mathsf I}_{Nd}-{{\boldsymbol{\mathsf M}}}}.$$ Here and in the following $\boldsymbol{\mathsf I}_{k}$ is the $k\times k$ identity matrix. In this approach we make, as usual, the assumption that the quantity $D(\omega)$ is self-averaging. We denote by $\mathbf R_{ij}$ the $d\times d$ submatrix of $\boldsymbol{\mathsf R}$ corresponding to the couple of (not necessarily distinct) sites $(i,j)$. Assuming that no vertex plays a special role in the ensemble of realizations, the DOS can be expressed in terms of the averaged trace of the local resolvent $\mathbf R_{ii}$, i.e., $$D(\omega)
=-\lim_{\varepsilon\to 0}\lim_{N\to \infty}\frac{2\omega}{d\pi}{\mathbb{E}\left[{\operatorname{Tr}\operatorname{Im}\mathbf{R}_{ii}(\omega^2+i\varepsilon)}\right]}.$$
Before proceeding further, let us comment on some properties of the Hessian matrix under analysis. For each realization of our system, the matrix ${{\boldsymbol{\mathsf M}}}$ has dimension $dN\times dN$, but it has rank $N_c=\sum_{i\in\mathcal V} z_i/2=N\bar z/2$, where $z_i$ is the coordination number of the $i$th vertex. Therefore, if $\bar z<2d$, there are $N\left(d-\sfrac{\bar z}{2}\right)$ zero modes. In that case, a contribution $\left(1-\sfrac{\bar z}{2d}\right)\delta(\omega)$ to the DOS $D(\omega)$ appears, corresponding to a singularity in the trace of the local resolvent for $\lambda\to 0$ of the type $$\label{trhatR}
{\mathbb{E}\left[{\operatorname{Tr}\mathbf R(\lambda)}\right]}=\frac{2d-\bar z}{2\lambda}+o\left(\frac{1}{\lambda}\right).$$
By the same argument, moreover, no singularity is expected for $\lambda\to 0$ for $\bar z=2d$. This is nothing other than Maxwell’s criterion, which implies instability for $\bar z<2d$ due to the presence of an extensive number of zero modes.
A typical approach for the solution of Eq. in the thermodynamical limit is the cavity method [@mezard1987spin; @AbouChacra1973; @Cizeau1994], which is exact on a Bethe lattice and can be applied when the underlying topology is a tree like graph. Using this approach, it can be proved (see Appendix \[app:ricorsiva\]) that the local resolvent $\mathbf R$ satisfies *in probability* the equation
\[ricorsive\] $$\label{ricorsiva2}
\mathbf R(\lambda)\stackrel{\text{prob}}{=}\left[\lambda{{\boldsymbol{\mathsf I}}}_d+\sum_{k=1}^{z}\frac{|\mathbf x_k\rangle\langle\mathbf x_k|}{1+\langle\mathbf x_k|\mathbf G_k(\lambda)|\mathbf x_k\rangle}\right]^{-1},$$ where $z$ is distributed according to $p_k$, the degree distribution of the graph, and $\{\mathbf x_k\}_{k=1,\dots,z}$ are $z$ random Gaussian unit vectors in $d$ dimensions. The $\{\mathbf G_k\}_{k=1,\dots,z}$ are $z$ local *cavity fields* satisfying a similar equation $$\label{ricorsiva}
\mathbf G(\lambda)\stackrel{\text{prob}}{=}\left[\lambda{{\boldsymbol{\mathsf I}}}_d+\sum_{k=1}^{\eta-1}\frac{|\mathbf x_k\rangle\langle\mathbf x_k|}{1+\langle\mathbf x_k|\mathbf G_k(\lambda)|\mathbf x_k\rangle}\right]^{-1},$$ the main difference being the fact that the random variable $z$ is replaced by the random variable $\eta$, which is distributed with probability distribution [@mezard2009information] $$\hat p_\eta=\frac{\eta p_\eta}{\sum_k kp_k}.$$
Eqs. provide a recipe for the numerical evaluation of ${\mathbb{E}\left[{\operatorname{Tr}\operatorname{Im}\mathbf R}\right]}$ through a population dynamics algorithm [@Mezard2001]. We therefore tackled the problem of the DOS of our model both numerically solving Eqs. , through exact diagonalization (ED) via the implicitly restarted Lanczos method [@arpack] and through the method of moments [@Cyrot-Lackmann1967; @*Gaspard1973; @*Lambin1982; @*Jurczek1985; @*Benoit1992; @*Villani1995] (see Appendix \[app:mom\]). In particular, using ED we obtained the lowest part of the average spectrum, calculating the 50 lowest eigenmodes (or 100 for the smaller system sizes), whereas the rest of it has been obtained from the matrix ${{\boldsymbol{\mathsf M}}}$ using the method of moments.
![An instance of the ${{\mathcal G}}_{4,3}$ model for $N=200$ with a low-frequency eigenmode represented on it. The intensity of the color is proportional to the amplitude of the corresponding eigenmode on each site. It is evident that the eigenmode is localized on a site with $z=4$, the lowest possible coordination. \[fig:soffice\]](softmode.png){width="0.7\columnwidth"}
Eigenvectors localization
-------------------------
We also investigate the localization phenomenon near the band edges in the model proposed above. Let us denote by $|{\boldsymbol{\mathsf k}}\rangle$ the eigenmode of the matrix ${{\boldsymbol{\mathsf M}}}$ corresponding to the eigenvalue $\lambda_k$, ${{\boldsymbol{\mathsf M}}}|{\boldsymbol{\mathsf k}}\rangle=\lambda_k|{\boldsymbol{\mathsf k}}\rangle$, and by $|{\mathbf k_i}\rangle$ its projection on the site $i$. In this paper, we will always assume that the eigenvectors are labeled in such a way that $k<k'\Rightarrow \lambda_k\leq \lambda_{k'}$. We use as an indicator for the localization of the eigenvector $|{\boldsymbol{\mathsf k}}\rangle$ the inverse participation ratio (IPR) $$Y_k\coloneqq\frac{\sum_{i=1}^N\left|\langle \mathbf k_i|\mathbf k_i\rangle\right|^2}{\left(\sum_{i=1}^N\langle \mathbf k_i|\mathbf k_i\rangle\right)^2}.$$ The IPR scales as $O(1)$ if the eigenvector $|{\boldsymbol{\mathsf k}}\rangle$ is localized, or $O\left(N\right)$ if it is delocalized. The quantity above can be evaluated once the eigenvectors are known from an ED procedure on a given instance of the problem. To average over disorder we calculate the quantity $$\label{eq:avIPR}
Y({\mathbb{E}\left[{\omega_k}\right]})\coloneqq{\mathbb{E}\left[{Y_k}\right]},$$ i.e., the average of the participation ratio of the $k$th eigenmode as a function of the corresponding average frequency.
We can also study the localization properties of the eigenvectors with the cavity method, introducing, among the many possibilities [@Parisi2014], the quantity $$\label{iprcm}
\begin{split}
\hat Y(\omega)&\coloneqq \frac{{\mathbb{E}\left[{\sum \limits_{k\colon \lambda_k\sim\omega^2}\left|\langle \mathbf k|\mathbf k\rangle\right|^2}\right]}}{\left\{{\mathbb{E}\left[{\sum \limits_{k\colon \lambda_k\sim\omega^2}\langle \mathbf k|\mathbf k\rangle}\right]}\right\}^2}\\
&=\lim_{\varepsilon\to 0}\left.\frac{{\mathbb{E}\left[{\operatorname{Tr}\left(\mathbf R^\dag(z)\mathbf R(z)\right)^2}\right]}}{\left\{{\mathbb{E}\left[{\operatorname{Tr}\operatorname{Im}\mathbf R(z)}\right]}\right\}^2}\right|_{z=\omega^2+i\varepsilon}.
\end{split}$$ The last equality allows us to estimate $\hat Y(\omega)$ using the cavity method. It shares the same properties of $Y(\omega)$, i.e., diverges in the localized region and it is $O(1)$ in the delocalized region.
The localization and delocalization properties can also be detected using a different approach. The expected value of the square of $$\eta_\omega\coloneqq\left.\operatorname{Im}\operatorname{Tr}\mathbf R(\omega^2+i\varepsilon)\right|_{\varepsilon\to 0}$$ should diverge in the localized regime and therefore it can be seen as a localization indicator as well. The divergence of ${\mathbb{E}\left[{\eta_\omega^2}\right]}$ can be evaluated either directly or, as we will see below, studying the probability density of $\eta_\omega$ [@Parisi2014].
Results {#sec:risultati}
=======
In this Section we present our results for the DOS $D(\omega)$ and the IPR near the lower band edge. The tools that we use are the ones described in Section \[sec:model\]. We also consider the cumulative function $$\Phi(\omega)\coloneqq\int_0^\omega D(u)\operatorname{d}u.$$ We will distinguish between the hypostatic, isostatic and hyperstatic cases. As previously stated, in all cases under consideration, we have assumed $d=3$.
The hypostatic case
-------------------
#### Density of states.
Let us start from the ${{\mathcal G}}_{5,0}$ model, and therefore in the hypostatic regime. A hypostatic network is a good model for the so-called “floppy materials”, such as dense suspensions, gels, and glasses of low valence elements, which show an abundance of zero modes. In Fig. \[fig:DOSfixed5all\] we compare the results of the cavity method and ED on the full spectrum for small sizes of the system for $\bar z=5$, finding an excellent agreement. Both the ED and the cavity results suggest that a gap is present for $\omega<\omega_0\approx 10^{-1}$, as expected in floppy materials [@During2013]. The detail of the small frequency regime is shown in Fig. \[fig:DOS5head\]. Note that, for $\omega<\omega_0$ and finite $\varepsilon$, a small, nonzero contribution is predicted by the cavity method (see the inset). This contribution is however related to the unavoidable finiteness of the value of $\varepsilon$ adopted in the numerical calculation to solve Eqs. . More specifically, the presence of zero modes implies that a Dirac delta appears in the origin in $D(\omega)$. The finiteness of $\varepsilon$ causes a smoothening of the Dirac function that, in absence of any other contribution — i.e., in the gap region for $\varepsilon\ll\lambda\ll\omega_0^2$ — gives a $\overline{\operatorname{Tr}\operatorname{Im}\mathbf R(\lambda)}\sim \varepsilon \lambda^{-2}$ scaling near the origin. The ED and cavity method results have been superimposed. As expected, a zero density is found for $\omega<\omega_0$, whereas the two methods are in agreement for $\omega>\omega_0$.
A qualitatively similar result has been obtained for the ${{\mathcal G}}_{4,1}$ model, where we find again a gap for $\omega<\omega_0\approx 10^{-1}$ (see Fig. \[fig:DOS5all\] and the detail in \[fig:DOS5head\]). The value of the frequency $\omega_0$ in the ${{\mathcal G}}_{4,1}$ model appears to be very close to the one found in the ${{\mathcal G}}_{5,0}$ model, showing a weak dependence on the details of the model other than the value $\bar z$. As anticipated, we expect that $\omega_0\to 0$ as $\bar z\to 2d=6$. Taking advantage of the fact that in the ${{\mathcal G}}_{4,\bar\zeta}$ model we can smoothly vary $\bar z$, we have computed by cavity method the DOS for $5<\bar z<6$ \[see Fig. \[fig:5to6\] (left)\] and indeed we have observed that $\omega_0$ decreases as $\bar z$ increases, and the gap closes for $\bar z\to 6$. Moreover, the scaling $\omega_0\propto 2d-\bar z$, predicted by @During2013, holds in our model \[see again Fig. \[fig:5to6\] (right)\].
![Detail of the DOS $D(\omega)$, the cumulative function $\Phi(\omega)$ and the participation ratio $Y(\omega)$ \[Eq. \] at low frequencies for the ${{\mathcal G}}_{5,0}$ model (left) and the ${{\mathcal G}}_{4,1}$ model (right) using the cavity method (black) and ED (color).\[fig:DOS5head\]](Dos_testa_5.pdf){width="\columnwidth"}
#### Localization properties.
Both the ${{\mathcal G}}_{5,0}$ model and the ${{\mathcal G}}_{4,1}$ model present the same localization features. With reference to Figs. \[fig:DOSfixed5all\] and \[fig:DOS5all\], the ED results suggest the presence of a localized region in the upper edge of the spectrum. The participation ratio $Y^{-1}(\omega)$ becomes indeed infinitesimal slightly before the DOS goes to zero (a fact that is more evident in the ${{\mathcal G}}_{4,1}$ model), and, moreover, it scales as $O(\sfrac{1}{N})$ for $\omega\gtrsim 2.4$. In the low-frequency regime, instead, $Y(\omega)$ remains $O(1)$ for all the considered sizes up to the lower band edge, suggesting that no mobility edge is present and all eigenstates in the lower part of the spectrum are delocalized.
![Cumulative function $\Phi$ obtained using the cavity method in the ${{\mathcal G}}_{4,\bar\zeta}$ model for different values of $\bar z=4+\bar \zeta<6$. We observe that the value of $\omega_0$ decreases and the gap closes as soon as $\bar z\to 6$. The smooth lines are represented as guides for the eye. On the right panel, the same data are plotted with the $x$ axis rescaled by $6-\bar z$ (the distance from the isostatic transition). \[fig:5to6\]](5to6.pdf){width="0.9\columnwidth"}
The isostatic case
------------------
#### Density of states.
Let us now consider our model on a random regular graph with $\bar z=6$. As expected from the constraint counting argument, in the ${{\mathcal G}}_{6,0}$ model there is no gap and $D(\omega)$ shows a plateau up to low values of $\omega$ (see Figs. \[fig:DOSfixed6all\] and \[fig:DOS6head\]). A constant $D(\omega)$ for small values of $\omega$ implies that $\varrho(\lambda)\sim \lambda^{-\sfrac{1}{2}}$ for $\lambda \to 0$ and that $\Phi(\omega)\propto\omega$ for $\omega\to 0$. These properties have been verified numerically, and the ED results are compatible with our cavity prediction, as shown in Fig. \[fig:DOS6head\]. Both in $D(\omega)$ and in $\hat Y^{-1}(\omega)$ there is, however, an anomalous behavior near $\omega=0$. Both the cavity and the ED results suggest the presence of an integrable singularity in the DOS that is compatible with a logarithmic divergence. Note that it can be proved that no singularity is present in the model for $d\to +\infty$ [@Parisi2014].
Similar results have been obtained in the ${{\mathcal G}}_{4,2}$ model, as we show in Fig. \[fig:DOS6all\] and in Fig. \[fig:DOS6head\]. This suggests that the isostaticity condition $\bar z=2d=6$ is enough to guarantee that there is no gap in the DOS, irrespective of the presence of local fluctuations in the value of $z$. As in the ${{\mathcal G}}_{6,0}$ model, for very small values of $\omega$ both methods indicate the presence of an integrable singularity in the origin in the DOS, which in this case appears to be of the type $D(\omega)\sim d_0+\sfrac{d_1}{\omega^\beta}$ for some constants $d_0$ and $d_1$ and with $\beta\approx 0.3$ (see Fig. \[fig:DOS6head\]).
A more precise analysis of this singularity is not possible with the quality of the data that we have in the $\omega>0.01$ range.
![Detail of the DOS $D(\omega)$, the cumulative function $\Phi(\omega)$ and the participation ratio at low frequency for the ${{\mathcal G}}_{6,0}$ (left) and the ${{\mathcal G}}_{4,2}$ (right) models. The results were obtained using ED (color) and the cavity method (black dots)\[fig:DOS6head\].](Dos_testa_6.pdf){width="\columnwidth"}
#### Localization properties.
As in the hypostatic case, the participation ratio $\sfrac{1}{Y}$ scales as $O(\sfrac{1}{N})$ for $\omega\gtrsim 2.5$ in the ${{\mathcal G}}_{6,0}$ and ${{\mathcal G}}_{4,2}$ models (see Figs. \[fig:DOSfixed6all\] and \[fig:DOS6all\]), suggesting that localized states are present above this threshold. Near the lower band edge we find a value of the IPR that is larger than that of the bulk, yet does not scale with the size of the system. In particular, in the ${{\mathcal G}}_{6,0}$ model the IPR increases by a factor $10$ for $\omega\to 0$ for all considered sizes of the system (see Fig. \[fig:DOS6head\]), whereas its growth is more evident in the ${{\mathcal G}}_{4,2}$ model, where it increases by three orders of magnitude (see Fig. \[fig:DOS6head\]) without, however, showing any scaling with $N$. The low-frequency eigenvalues are therefore still delocalized, but the larger IPR is a signal of an avoided localization transition at $\omega=0$.
![Distribution of the imaginary part of the resolvent $\theta_\omega\coloneqq\omega\eta_\omega$ in the isostatic case for both the ${{\mathcal G}}_{6,0}$ model (left) and the ${{\mathcal G}}_{4,2}$ model (right).\[fig:istogrammi6\]](Isto6.pdf){width="\columnwidth"}
The incipient localization at very low frequencies can be detected studying the distribution of the imaginary part of the local resolvent $\eta_\omega$ for different values of $\omega$, as described in Section \[sec:model\]. Due to the fact that in this case ${\mathbb{E}\left[{\eta_\omega}\right]}\sim\sfrac{1}{\omega}$ for $\omega\to 0$, in Fig. \[fig:istogrammi6\] we plot the distribution $p_\omega(\theta)$ for $\theta \coloneqq \omega\eta_\omega$ for different values of $\omega$. In the $\omega\to 0$ limit, a fat tail appears in the ${{\mathcal G}}_{6,0}$ model, and in particular we find $p_\omega(\theta)\sim\theta^{-3}$. Such a tail would imply a divergent ${\mathbb{E}\left[{\eta_\omega^2}\right]}$ and therefore localization. The exponent can be justified by means of a qualitative argument. In Eq. the imaginary part of the resolvent $\eta_\omega$ is related to the inverse of a Wishart matrix of the type $\boldsymbol{\mathsf W}=\sfrac{1}{d}\,\boldsymbol{\mathsf X}^T\boldsymbol{\mathsf X}$, where $\boldsymbol{\mathsf X}$ is a $z\times d$ matrix with random Gaussian entries [@Parisi2014; @Eynard2015; @Livan2018]. The probability density of the smaller eigenvalue $\lambda_0$ of $\boldsymbol{\mathsf W}$ scales as $\rho(\lambda)\sim\lambda^{\frac{z-d-1}{2}}$, i.e., in our case ($z=6$, $d=3$), as $\rho(\lambda)\sim\lambda$, that indeed corresponds to a $p_\omega(\theta)\sim\theta^{-3}$ scaling for $\theta\sim\sfrac{1}{\lambda}$.
A similar behavior is found in the ${{\mathcal G}}_{4,2}$ model, but with a different scaling, namely, $p_\omega(\theta)\sim\theta^{-2}$ for large values of $\theta$. This implies, again, a divergent $\mathbb E[\eta_\omega^2]$ for $\omega \to 0$. The different tail scaling in the ${{\mathcal G}}_{4,2}$ model can be explained analyzing the average coordination of the $k$th eigenvector $|\boldsymbol{\mathsf{k}}\rangle$, $$\label{zave}
\overline{\langle z_k\rangle} ={\mathbb{E}\left[{\frac{\sum\limits_{i=1}^N z_i\langle\mathbf k_i|\mathbf k_i\rangle}{\sum\limits_{i=1}^N \langle\mathbf k_i|\mathbf k_i\rangle}}\right]}$$ that in Fig. \[fig:istoz\] we plot as a function of the average frequency $\bar \omega_k\coloneqq{\mathbb{E}\left[{\omega_k}\right]}$ of the $k$th eigenvectors. The plot shows that low-frequency modes mostly occupy nodes with low coordination. Assuming that $\langle z\rangle\to 4$ as soon as $\omega \to 0$, the scaling argument proposed for the ${{\mathcal G}}_{6,0}$ model can be applied again, and it predicts $p_\omega(\theta)\sim\theta^{-2}$.
![Average coordination $\overline{\langle z\rangle}$ as a function of $\bar \omega$ in the low-frequency region for the ${{\mathcal G}}_{4,2}$ model (left) and the ${{\mathcal G}}_{4,3}$ model (right), obtained using ED.\[fig:istoz\]](Istoz6-7.pdf){width="\columnwidth"}
The considerations above suggest that in both the ${{\mathcal G}}_{6,0}$ and ${{\mathcal G}}_{4,2}$ models there is an (avoided) localization transition at $\omega=0$, and the low-frequency modes are extended states that, in the case of the ${{\mathcal G}}_{4,2}$ model, have low average coordination.
The hyperstatic case
--------------------
#### Density of states.
Finally, let us consider the hyperstatic case that, for $d=3$, corresponds to coordination values $\bar z>6$. In this case, a quasi gap opens, and $D(\omega)$ has a power-law behavior for $\omega\to 0$, i.e., $D(\omega)\propto\omega^\alpha$ for some value of $\alpha>0$. As anticipated in the Introduction, the properties that determine the value of $\alpha$ are still a matter of investigation and different results have been found in mean-field models and numerical simulations in finite dimension. Understanding how the finite dimensionality affects the mean-field behavior is of great interest.
The results for the DOS in the ${{\mathcal G}}_{7,0}$ model are shown in Fig. \[fig:DOSfixed7all\]. Once again, an excellent agreement between the theoretical prediction of the cavity method and the method of moments in the bulk of the spectrum is found. The low-frequency regime is numerically more difficult to evaluate: large system sizes are needed to approach zero frequency with ED. Furthermore, the cavity method itself intrinsically presents some limitations in resolution, due to the finite population in the population dynamics algorithm and the finite value of $\varepsilon$ in the numerical integration of Eqs. . Nevertheless, from the results in Fig. \[fig:DOS7head\] we can still find that, for $\omega<10^{-1}$, approximately $\Phi(\omega)\propto\omega^5$ and therefore $D(\omega)\propto\omega^4$, a result that is compatible with theoretical predictions and numerical evidences for finite-dimensional disordered systems [@Gurarie2003; @Gurevich2003; @BaityJesi2015; @Lerner2016] and spin glasses on sparse graphs [@Lupo2017]. Apart from fitting the low-frequency behavior of $\Phi(\omega)$, the exponent $\alpha$ can be also extracted from the scaling with $N$ of the lowest eigenvalue of the spectrum. Indeed, given a power-law behavior $D(\omega)\sim\omega^\alpha$ for the DOS near the origin, and denoting by $\bar\omega_1\coloneqq{\mathbb{E}\left[{\omega_1}\right]}$ the average value of the first mode frequency, we have that $$\label{eq:minScale}
\int_0^{\bar\omega_1}D(\omega)\operatorname{d}\omega\sim\frac{1}{Nd}\Rightarrow \bar\omega_1\sim \frac{1}{N^{\frac{1}{\alpha +1}}}.$$ This relation has been verified on our data, as shown in Fig. \[fig:omegascaling\], and we find $\alpha=4.0(2)$.
A power law behavior, in the same regime, is also found for the ${{\mathcal G}}_{4,3}$ model. However, the power law exponent, extracted with the same methods discussed above, is different and we find, in this case, $\Phi(\omega)\propto\omega^2$, i.e., $D(\omega)\propto\omega$. This is also confirmed by the scaling of the first eigenvalue with respect to $N$, as shown in Fig. \[fig:omegascaling\], which gives $\alpha=1.049(8) $. The differences in these results suggest that there is a strong dependence on the topological details of the model, and especially on the lowest admissible coordination, despite the fact that $\bar z$ is the same. Indeed, the lowest part of the spectrum is populated by eigenstates having low average coordination. In Fig. \[fig:istoz\] we show that for $\omega<0.1$ the average eigenvalue coordination $\overline{\langle z\rangle}$ — evaluated using the formula in Eq. — is below $5$, and asymptotically approaches $4$ as $\omega\to 0$ (see also Fig. \[fig:istocoordinazione\]).
![Fraction of eigenvalues with different $\overline{\langle z\rangle}$ as a function of $\omega$ in the ${{\mathcal G}}_{4,3}$ model. Here we consider the system size $N=5000$. \[fig:istocoordinazione\]](Istocoor.pdf){width="0.9\columnwidth"}
To stress the role of the lowest accessible coordination in the power-law exponent $\alpha$, we have also considered the ${{\mathcal G}}_{5,2}$ model and the ${{\mathcal G}}_{6,1}$ model, both having $\bar z=7$ but with different lowest possible coordination, i.e., $5$ and $6$ respectively. In these cases we observe an intermediate value of the exponent $\alpha$ (see Fig. \[fig:7vari\]). In Fig. \[fig:7vari\] we have also plotted the $\overline{\langle z\rangle}$ as a function of $\omega$, showing that low-frequency modes are characterized by a low average coordination $\overline{\langle z\rangle}$, close to the lowest coordination allowed by the topology of the graph.
To further exemplify this fact, let us consider a different value of $\bar z$ in the hyperstatic regime: the ${{\mathcal G}}_{4,2.1}$ and ${{\mathcal G}}_{6,0.1}$ models. Both have the same average coordination $\bar z=6.1$, but they are constructed on a different underlying random regular graph. In the ${{\mathcal G}}_{6,0.1}$ model the isostatic condition is realized for every node in the network. Repeating the usual analysis on both models, we obtain the results in Figs. \[fig:omegascaling\] and \[fig:DOS6.1\]. Similarly to what happens in the $\bar z=7$ case, the results of both the cavity method calculation and ED suggest a different value of $\alpha$ in the two cases: $\alpha=0.67(8)$ for the ${{\mathcal G}}_{4,2.1}$ model and $\alpha=4.3(8)$ for the ${{\mathcal G}}_{6,0.1}$ model. Note that in the ${{\mathcal G}}_{4,2.1}$ model the value of $\alpha$ is closer to the exponent value observed for the ${{\mathcal G}}_{4,3}$ model that indeed has the same lowest admissible coordination.
#### Localization properties.
We present our results on the localization properties of the eigenstates in the ${{\mathcal G}}_{7,0}$ model in Fig. \[fig:DOS7head\]. High frequency modes are localized and the IPR scales with the system size for $\omega\gtrsim 2.6$. We also find that there is a low-frequency mobility edge and that for $\omega\lesssim 10^{-1}$ the IPR $Y(\omega)$ scales with the system size. Similarly, localized states are found in the ${{\mathcal G}}_{4,3}$ model approximately below the same frequency (see Fig. \[fig:DOS7head\]). These results show that, in the hyperstatic regime, at low frequencies a localized region is present. Moreover, taking into account the behavior of $\overline{\langle z\rangle}$ discussed above, in all the analyzed models ${{\mathcal G}}_{z_0,\bar\zeta}$ having $\bar\zeta\neq 0$, soft modes appear to be localized on nodes which have very low coordination (see, e.g., Fig. \[fig:soffice\]), a fact that is compatible with results of soft sphere systems [@Charbonneau2015]. This fact clarifies why the low-frequency behavior of the DOS strongly depends on the lowest possible coordination allowed in the graph topology.
![Cumulative function $\Phi(\omega)$ for the ${{\mathcal G}}_{6,0.1}$ model and for the ${{\mathcal G}}_{4,2.1}$ model using the cavity method (black) and ED (color). The numerical integration of the cavity method equation has been performed using $\varepsilon=10^{-8}$ and a population of $10^6$ fields. The arrows indicate the value of the average of the first nonzero frequency for each system size.\[fig:DOS6.1\]](Dos_testa_rz61.pdf){width="\columnwidth"}
Higher dimensions
-----------------
![DOS for $d=4$ evaluated on a random regular graph topology for different values of coordination $\bar z$. In the inset, detail of the low-frequency regime. The data have been obtained using ED on the entire spectrum for small values of $N$. Smooth lines represent the cavity prediction in this case.\[fig:d4\]](DOS_d4.pdf){width="\columnwidth"}
The behavior of the vibrational DOS in higher dimensions can be studied by changing the dimension of the vector connecting two spheres in contact, $\mathbf{x}_{ij}$ in Eq. . Analyses for $d=4$ on a random regular graph topology show that the DOS follows the expected hypostatic, isostatic, and hyperstatic behavior (see Fig. \[fig:d4\]). Specifically, a gap is present in the hypostatic regime, $\bar z=7$, which disappears in the isostatic case $\bar z=2d=8$ and gives way to the expected plateau. In the hyperstatic regime, $\bar z=9$, there is a quasi-gap, and the density of states exhibits a power-law $D(\omega)\propto \omega^6$ (see inset of Fig. \[fig:d4\]).
Discussion and conclusions {#sec:conclusioni}
==========================
In the present work we have discussed a mean-field model for the isostatic transition of soft spheres. The model merges mean-field properties (a contact network defined on a random graph) with finite dimensionality (each contact is associated to a $d$-dimensional vector). We have correctly recovered the main features of the physical isostatic transition, namely, the fact that the average coordination of the graph $\bar z$ determines the general properties of the density of states of the system, $\bar z=2d$ being the isostatic point.
If $\bar z<2d$ we find a gap in the DOS, and we have verified the scaling of its width with the distance from the isostatic point. For $\bar z\to 2d$ the gap closes.
For $\bar z>2d$ a quasi gap opens. With respect to other mean-field models, such as the perceptron, the model introduced here is able to reproduce additional features that are deeply related to finite-dimensional effects. For example, a localized region is observed at low frequencies. Furthermore, the modes in this region have average coordination typically very close to the lowest possible coordination allowed in the graph, i.e., they are localized on weakly connected nodes.
The model has enabled us to study the power-law behavior of the DOS $D(\omega)\sim\omega^\alpha$ for $\omega\to 0$ in the hyperstatic regime, in the absence of Goldstone modes. Using both exact diagonalization techniques and the cavity method, we have observed that the exponent $\alpha$ strongly depends on the details of the coordination distribution of the underlying contact network. In particular, the power-law behavior is determined by the aforementioned localized modes and therefore by the lowest accessible coordination in the graph, and not by the average value $\bar z$. Indeed, different models with the same average coordination $\bar z$ but different minimum admissible coordination show different power-law behaviors near the origin. The effect of the finite dimensionality on $\alpha$, and therefore of the finite connectivity, is relevant. [It is, however, worth mentioning that in our model the initial stress contribution has been neglected. It has been very recently observed that this term might be crucial to obtain a $D(\omega)\sim\omega^4$ behavior in the overjammed phase [@Lerner2018]. In this sense, the fact that no universal exponent is found in our model might be related to the absence of this contribution.]{}
This model is an attempt to go beyond the infinite-dimensional models for sphere packings. In the spirit of previous contributions [@Parisi2002a; @*Grigera2003; @Franz2015; @*Franz2016; @Manning2015], it relates the spectral properties of disordered systems to a random matrix theory model, combining it with an underlying random graph topology. Moreover, it exemplifies the coordination effects in mean-field models with respect to the spectral properties of amorphous solids [@DeGiuli2014]. A large number of open problems remain, such as the precise relation between $\alpha$ and the coordination distribution in the contact network, and further investigations are needed to fill the gap between the finite-dimensional packing problem and the available mean-field models.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to F. Ricci-Tersenghi and F. Zamponi for useful discussions. The authors also thank M.L. Manning, P. Morse, and E. Stanifer for discussions and correspondence. F.B, G.P. and G.S. acknowledge the financial support of the Simons Foundation (Grant No. 454949, Giorgio Parisi). This work benefited from access to the University of Oregon high-performance computer, Talapas, as well as the Chimera Group cluster at Sapienza Università di Roma. The work presented in this paper was supported by the project “Meccanica statistica e complessità”, a research grant funded by PRIN 2015 (Agreement No. 2015K7KK8L). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694925) and benefited from the support of the project THERMOLOC ANR-16-CE30-0023-02 of the French National Research Agency (ANR).
Derivation of the cavity equations {#app:ricorsiva}
==================================
To derive Eq. on a sparse graph, let us follow the approach of Refs. . We consider a generic matrix ${{\boldsymbol{\mathsf M}}}$ of size $Nd\times Nd$, such that its element $\mathbf M_{ij}$ is a $d\times d$ submatrix. Pictorially, we can associate the matrix ${{\boldsymbol{\mathsf M}}}$ to a graph, in such a way that each Latin index corresponds to a node of the graph, and the submatrix $\mathbf M_{ij}$ is associated to the link $(i,j)$. We also assume that the coordination distribution of the graph is $p_k$. Assuming that ${{\boldsymbol{\mathsf M}}}$ is an element of a given ensemble, we are interested in the average DOS of ${{\boldsymbol{\mathsf M}}}$ with respect to this ensemble in the $N\to+\infty$ limit.
It is useful to consider a matrix obtained from ${{\boldsymbol{\mathsf M}}}$ creating a “cavity” in the graph, i.e., removing a node and/or a link. Let us start from the graph corresponding to ${{\boldsymbol{\mathsf M}}}$ and let us select, uniformly at random, one of its edges. We then select one of the endpoints of this edge, also at random. This is the node that will be removed. It is called the *cavity node*, and we label it by $0$. We say that the site $0$ is connected to the site $i$ if $\mathbf M_{0i}\neq\mathbf 0$ and/or $\mathbf M_{i0}\neq\mathbf 0$. It has coordination $\eta_0$, which is distributed as $$\label{disteta}
\hat p_\eta=\frac{\eta p_\eta}{\sum_{k=1}^\infty kp_k}.$$ Observe that if $p_k=\delta_{k,z}$, then $\hat p_\eta=p_\eta=\delta_{\eta,z}$. If instead the coordination follows a Poisson distribution with mean $\lambda$, $p_k=\frac{\lambda^k}{k!}\operatorname{e}^{-\lambda}$, then $\hat p_\eta=\frac{\lambda^{\eta-1}}{(\eta-1)!}\operatorname{e}^{-\lambda}$ with $\eta\geq 1$, i.e., $\sum_\eta \eta \hat p_{\eta}=\lambda+1$.
The cavity graph is simply the graph without the node $0$. Once the node is removed, its $\eta_0$ neighbors will have coordination $\eta_i-1$, $i=1,\dots, \eta_0$, where $\eta_i$ are random variables distributed again as in Eq. . This will be essential for writing down recursive equations. The matrix ${{\boldsymbol{\mathsf M}}}^c$ of the new graph has size $(N-1)d\times(N-1)d$. To proceed in full generality, we will also assume that the removal of the site affects the value of $\mathbf M_{ij}\to \mathbf M_{ij}^c$ for $i,j\neq 0$, due to some required properties of the global matrix that must be preserved, and so the new matrix is not simply a submatrix of the old one with $d$ rows and $d$ columns removed.
The cavity graph is useful due to the fact that we can find an equation for the elements $\mathbf G_{kk}$ with $k\in\partial 0$ of the cavity resolvent, $${{\boldsymbol{\mathsf G}}}(\lambda)\coloneqq \frac{1}{\lambda{{\boldsymbol{\mathsf I}}}_{(N-1)d}-{{\boldsymbol{\mathsf M}}}^c}$$ to be solved in probability.
Let us now assume that site $0$ is re-introduced but connected to only $\eta_0-1$ of its neighbors [^2]. Then, $d$ new rows and $d$ new columns are added to the matrix ${{\boldsymbol{\mathsf M}}}^c$, obtaining a new matrix ${{\boldsymbol{\mathsf M}}}^+$ that has the same dimension of the original matrix but still a “cavity”, i.e., a missing link. As before, the addition of a site affects in general the entire matrix. The coordination distribution of the site $0$ is now the same that its neighbors had before its insertion. The new resolvent can be calculated as
$$\begin{split}
\frac{1}{{\left[{{\boldsymbol{\mathsf G}}}^{+}\right]}^{\alpha\beta}_{00}(\lambda)}&=\frac{\left[\prod\limits_{k=0}^N\int\operatorname{d}^d\varphi_k\right] \exp\left(\!-\frac{1}{2}\sum\limits_{{k,l=0}}^N\sum\limits_{{\mu,\nu=1}}^d \varphi_k^\mu\left[\lambda{{\boldsymbol{\mathsf I}}}_{(N+1)d}-{{\boldsymbol{\mathsf M}}}^+\right]_{kl}^{\mu\nu}\varphi_l^\nu\right)}{\left[\prod\limits_{k=0}^N\int\operatorname{d}^d\varphi_k\right] \varphi_0^\alpha \varphi_0^\beta\exp\!\left(\!-\frac{1}{2}\sum\limits_{{k,l=0}}^N\sum\limits_{{\mu,\nu=1}}^d \varphi_k^\mu\left[\lambda{{\boldsymbol{\mathsf I}}}_{(N+1)d}-{{\boldsymbol{\mathsf M}}}^+\right]_{kl}^{\mu\nu}\varphi_l^\nu\right)}\\
&=\left[\lambda{{\boldsymbol{\mathsf I}}}_{d}-\mathbf M_{00}^+-\sum_{k,l\neq 0}\mathbf M^+_{0k}\cdot \frac{1}{\lambda\delta_{kl}\mathbf I_d-\mathbf M^+_{kl}}\cdot\mathbf M^+_{l0}\right]^{\alpha\beta}.
\end{split}$$
Remembering now that, for $k,l\neq 0$ $$\lambda{{\boldsymbol{\mathsf I}}}_d-\mathbf M^+_{kl}=\left[\frac{1}{{{\boldsymbol{\mathsf G}}}(\lambda)}\right]_{kl}-\left(\mathbf M^+_{kl}-\mathbf M_{kl}^c\right),$$ and denoting by $\boldsymbol\Delta_{kl}\coloneqq \mathbf M^+_{kl}-\mathbf M_{kl}^c$ we can write [@Ciliberti2003] $$\begin{gathered}
\frac{1}{{\left[{{\boldsymbol{\mathsf G}}}^{+}\right]}^{\alpha\beta}_{00}}=\\
=\left[\lambda{{\boldsymbol{\mathsf I}}}_{d}-\mathbf M^+_{00}-\sum_{k,l\neq 0}\mathbf M^+_{0k}\cdot \frac{1}{\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{kl}-\boldsymbol\Delta_{kl}}\cdot\mathbf M^+_{l0}\right]^{\alpha\beta}.\end{gathered}$$
Let us now specify the equations above to our problem. In the case of a symmetric dynamical matrix in the form in Eq. , due to the rule in Eq. , for $i,j\neq 0$, $\boldsymbol\Delta_{ij}=-\delta_{ij}\mathbf M^+_{i0}$. Using the fact that $\mathbf M_{00}^+=-\sum_{k\in\partial 0}\mathbf M_{k0}^+$, the recursive equation becomes $$\begin{gathered}
\frac{1}{{\left[{{\boldsymbol{\mathsf G}}}^{+}\right]}^{\alpha\beta}_{00}}=\\
=\left[\lambda{{\boldsymbol{\mathsf I}}}_{d}+\sum_{\mathclap{k\in\partial 0}}\mathbf M_{k0}^+-\sum_{\mathclap{k,l\in\partial 0}}\mathbf M^+_{0k}\!\cdot\!\frac{1}{\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{kl}\!\!+\delta _{kl}\mathbf M_{k0}^+}\!\cdot\!\mathbf M_{l0}^+\right]^{\alpha\beta}.\end{gathered}$$ The sums in the equation above run over the $\eta-1$ neighbors of the vertex $0$. In the case of a sparse random graph we have that any two neighbors of $0$, let us say $k$ and $l$, are almost surely *not* directly connected for $N\to+\infty$ and therefore $$\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{kl}=-\mathbf M^+_{kl}\equiv \boldsymbol 0.$$ Moreover, if we assume that the off diagonal submatrices $\mathbf G_{ij}$ are subleading for $i\neq j$, $$\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{kk}=\frac{1}{\mathbf G_{kk}-\sum_{l\in\partial k}\mathbf G_{kl}\cdot\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{ll}\cdot \mathbf G_{lk}}\approx\frac{1}{\mathbf G_{kk}}.$$ Using this observation, and the fact that $\mathbf M^+_{0k}$ is a projector, Eq. can be obtained from $$\begin{gathered}
\sum_{k\in\partial 0}\mathbf M_{k0}^+-\sum_{k,l\in\partial 0}\mathbf M^+_{0k}\cdot\frac{1}{\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{kl}+\delta _{kl}\mathbf M_{k0}^+}\cdot\mathbf M_{l0}^+\\
=\sum_{k\in\partial 0}\sum_{n=0}^{\infty}\left(-\mathbf M_{0k}^+\cdot \frac{1}{\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{kk}}\right)^n\cdot\mathbf M_{k0}^+\\
=\sum_{k\in\partial 0}\frac{\mathbf M^+_{0k}}{1+\operatorname{Tr}\left(\frac{1}{\left[\frac{1}{{{\boldsymbol{\mathsf G}}}}\right]_{kk}}\cdot \mathbf M_{0k}^+\right)}\\
\approx \sum_{k\in\partial 0}\frac{\mathbf M^+_{0k}}{1+\operatorname{Tr}\left(\mathbf G_{kk}\cdot \mathbf M_{0k}^+\right)}.\end{gathered}$$ Observe that the right-hand side of the previous equation depends only on the elements of ${{\boldsymbol{\mathsf G}}}$ corresponding to the neighbors of the cavity site $0$ before its insertion. Due to the randomness in the model, it is not true in general that a fixed point solution of Eq. exists. However, we expect that the equation is true in probability, and we can search for a fixed point in the space of probability distributions of $\mathbf G$, solving the equation by means of a population dynamics algorithm. The fixed-point population of $\mathbf G$ that is found corresponds to a resolvent evaluated on a node of the graph with $\eta-1$ neighbors. The “true” local resolvent $\mathbf R$ for a site with $z$ neighbors distributed with probability $p_z$ can be obtained performing one last step, given by Eq. , extracting the $z$ required elements $\mathbf G_k$ from the cavity field population.
The method of moments {#app:mom}
=====================
In this Appendix, we summarize the method of moments that we used to compute the DOS of the Hessian matrix in Eq. . We will give here the procedure only, without providing the necessary proofs that can be found in the literature[@Cyrot-Lackmann1967; @*Gaspard1973; @*Lambin1982; @*Jurczek1985; @*Benoit1992; @*Villani1995]. The method, as opposed to ED, does not determine the single eigenvalues if the number of moments used are less than the rank of the Hessian matrix. Instead, it gives the envelope of their density. This has the advantage of allowing access to the entire spectrum even when using a limited number of moments.
Let us start by assuming that an $N\times N$ matrix ${{\boldsymbol{\mathsf M}}}$ is given and that we want to evaluate a spectral density function of the form $$\phi_{{{\boldsymbol{\mathsf p}}}}(\lambda)\coloneqq\sum_{k=1}^N |\langle {{\boldsymbol{\mathsf p}}}|{{\boldsymbol{\mathsf k}}}\rangle|^2\delta(\lambda-\lambda_k).$$ In the equation above, $\lambda_k$ is the $k$th eigenvalue of the matrix ${{\boldsymbol{\mathsf M}}}$ with corresponding eigenvectors $|{{\boldsymbol{\mathsf k}}}\rangle$, ${{\boldsymbol{\mathsf M}}}|{{\boldsymbol{\mathsf k}}}\rangle=\lambda_k|{{\boldsymbol{\mathsf k}}}\rangle$, and $|{{\boldsymbol{\mathsf p}}}\rangle$ is a given vector. If we introduce the Stiltjes transform $$R(z)\coloneqq \int_{-\infty}^{\infty}\frac{\phi_{{\boldsymbol{\mathsf p}}}(\lambda)}{z-\lambda}\operatorname{d}\lambda,$$ then the following relation holds: $$\label{rphimom}
\phi_{{\boldsymbol{\mathsf p}}}(\lambda)=-\frac{1}{\pi}\lim_{\varepsilon\to 0}\operatorname{Im}R(\lambda+i\varepsilon).$$
The non-negative function $\phi_{{\boldsymbol{\mathsf p}}}(\lambda)$ can be used as a weight function to generate a sequence of orthogonal polynomials $p_{n}(z)$ by imposing $$\int \lambda^np_n(\lambda)\phi_{{\boldsymbol{\mathsf p}}}(\lambda)\operatorname{d}\lambda=0.$$ These polynomials satisfy the relation
$$\begin{aligned}
p_{-1}(\lambda)&=0,\\
p_{0}(\lambda)&=1,\\
p_n(\lambda)&=(\lambda-a_n)p_{n-1}(\lambda)-b_{n-1}p_{n-2}(\lambda),\ n=1,2,\dots\label{ricorsivamom}\end{aligned}$$
where $$\begin{aligned}
a_{n}\coloneqq\frac{\bar \nu_{n-1}}{\nu_{n-1}},\quad b_{n}\coloneqq\frac{\nu_{n}}{\nu_{n-1}},\end{aligned}$$ and $$\nu_n\coloneqq \int p_n^2(\lambda)\phi_{{\boldsymbol{\mathsf p}}}(\lambda)\operatorname{d}\lambda,\quad \bar\nu_n\coloneqq \int \lambda p_n^2(\lambda)\phi_{{\boldsymbol{\mathsf p}}}(\lambda)\operatorname{d}\lambda$$ are the generalized moments of $\phi_{{\boldsymbol{\mathsf p}}}(\lambda)$.
The method relies on the nontrivial fact that the coefficients $a_n$ and $b_n$ in the recurrence relation for the polynomials $p_n(\lambda)$ are the same as those in the representation of $R(z)$ as a continued Jacobi fraction, i.e., $$R(z) = \cfrac{1}{z-a_1-\cfrac{b_1}{z-a_2-\cfrac{b_2}{z-a_3+\dots}}}.$$ This implies that, truncating the continued fraction expansion for $R$ to some order $M$, we can estimate $\phi_{{\boldsymbol{\mathsf p}}}$ by means of a finite set of coefficients $\{a_n,b_n\}$, i.e., a finite set of generalized moments. Moreover, it turns out that the generalized moments can be evaluated very easily by a sequence of matrix multiplications. Starting from the normalized vector $$|{{\boldsymbol{\mathsf t}}}_0\rangle\coloneqq\frac{1}{\sqrt{\langle{{\boldsymbol{\mathsf p}}}|{{\boldsymbol{\mathsf p}}}\rangle}}|{{\boldsymbol{\mathsf p}}}\rangle,$$ we can apply to it the recursive relation $$|{{\boldsymbol{\mathsf t}}}_{n+1}\rangle=\left({{\boldsymbol{\mathsf M}}}-a_{n+1}{{\boldsymbol{\mathsf I}}}_N\right)|{{\boldsymbol{\mathsf t}}}_n\rangle-b_n|{{\boldsymbol{\mathsf t}}}_{n-1}\rangle,$$ and extract the coefficients using $$\nu_n=\langle{{\boldsymbol{\mathsf t}}}_n|{{\boldsymbol{\mathsf t}}}_n\rangle,\quad \bar\nu_n=\langle{{\boldsymbol{\mathsf t}}}_n|{{\boldsymbol{\mathsf M}}}|{{\boldsymbol{\mathsf t}}}_n\rangle.$$ By evaluating $\{a_n\}_{n=1,\dots M}$ and $\{b_n\}_{n=1,\dots M}$ up to a certain order $M$ we can finally reconstruct $R(z)$ and then $\phi_{{\boldsymbol{\mathsf p}}}(\lambda)$ by means of Eq. . The spectral density $$\rho(\lambda)=\frac{1}{N}\sum_{i=1}^N\delta(\lambda-\lambda_i)$$ can be obtained averaging $\phi_{{\boldsymbol{\mathsf p}}}(\lambda)$ over all possible vectors $|{{\boldsymbol{\mathsf p}}}\rangle$, being $\overline{|\langle{{\boldsymbol{\mathsf p}}}|{{\boldsymbol{\mathsf k}}}\rangle|^2}=\sfrac{1}{N}$.
When a high number of moments is used ($M\approx 100$) numerical stability is further improved by performing a Gram-Schmidt orthonormalization of the vectors $|{{\boldsymbol{\mathsf t}}}_n\rangle$ at every iteration step. Finally, a truncation term $T(z)$ can be added to take into account the neglected terms in the continued fraction, i.e., $$R(z) = \cfrac{1}{z-a_1-\cfrac{b_1}{z-a_2\dots-\cfrac{b_n}{z-a_n+T(z)}}}.$$ Assuming that $a_n\to a$ and $b_n\to b$ when $n\to\infty$, with at most small oscillations around these values, $T(z)$ can be estimated from $$T(z)=\frac{1}{z-a-bT(z)}.$$ For details on the stability and precision of the method, we refer to Refs. .
[^1]: [Observe here that, once the value $\bar z$ for the ensemble is fixed, for finite $N$, each graph has an average coordination $z_{\text{av}}=\sfrac{1}{N}\,\sum_{i=1}^N z_i$ that fluctuates around $\bar z$, due to finite-size effects. This is particularly important for $\bar z=6$, because in this case, if no constraint is imposed on the graph, at finite $N$ an instance will be in general either hypostatic or hyperstatic. To avoid this problem, we have accepted only graphs having $|z_\text{av}-\bar z|\leq \sfrac{1}{M}$ with $M\gg N$ for each analyzed $N$, in such a way that fluctuations were so small that the number of zero modes was exactly equal to the expected one.]{}
[^2]: Note that if the coordination follows a Poisson distribution with mean $\lambda$, the random variable $\eta_0-1$ follows exactly the same distribution.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We investigate multicolour imaging data of a complete sample of low redshift ($z<0.2$) QSO host galaxies. The sample was imaged in four optical (*BVRi*) and three near-infrared bands (*JHKs*), and in addition spectroscopic data is available for a majority of the objects.
We extract host luminosities for all bands by means of two-dimensional modeling of galaxy and nucleus. Optical and optical-to-NIR colours agree well with the average colours of inactive early type galaxies. The six independent colours are used to fit population synthesis models. We assess the presence of young populations in the hosts for which evidence shows to be very weak.
author:
- 'Knud Jahnke$^1$, Björn Kuhlbrodt$^2$, Eva Örndahl$^3$, Lutz Wisotzki$^4$'
---
Goals
=====
For an assessment of galaxy-formation timescales QSO hosts play a vital role, due to their obvious connection to black hole formation. Dating the nuclear activity and possibly connecting this to external events in the galaxy can help to decide on merger scenarios and the triggering mechanism for activity.
With this work we wanted to start an assessment of the stellar content of host galaxies. By decomposing the host into stellar components we will be able in the future to make detailed comparisons to inactive galaxies.
Why multicolour data?
=====================
In QSO host galaxy studies using single band or single optical–near infrared (NIR) colours is sufficient to characterise morphological properties like galaxy types, host and nuclear luminosities, apparent signs of interaction or to conduct environment studies (e.g. McLeod & Rieke 1995, Percival et al. 2001). Optical colours or spectra permit the characterisation of the dominant stellar population or allows assessment of black hole masses (McLure et al. 1999, Boisson et al. 2000). The NIR on the other hand yields, for low $z$, the best contrast of host against nucleus and allows to assess the mass-to-light ratio of a host.
For the separation of an SED into stellar populations of different ages, using only optical information becomes insufficient for a unique solution. In the NIR the emission of young populations rapidly decreases and old populations dominate. Thus for a study of the stellar components information about the entire SED from the optical to NIR wavelength range is needed.
For luminous AGN the spectral separation of nuclear and host components is very difficult and at the moment largely dependent on subjective or ad hoc criteria for the nuclear component. The S/N requirements limit studies to small redshifts and low nuclear luminosities, as the acquisition of spectra becomes very expensive, requiring 8m-class telescopes already at $z=0.2$. While the quality of the spectral separation methods might change in the future the now available two-dimensional modeling software for QSO hosts allows a detailed and solid assessment of host galaxy fluxes and thus colour information with a high degree of reliability.
Sample & observations
=====================
We have compiled a sample of 20 objects with $z<0.2$, drawn from the Hamburg/ESO survey (HES) for luminous QSOs (Wisotzki et al. 2000). The HES is a flux limited objective-prism survey, with a limiting nuclear magnitude *B*$_\mathrm{lim} \sim 17.5$ depending on the field, designed to detect QSOs solely on basis of their spectral properties. Thus unlike samples from many other QSO surveys, the sample is not biased against extended objects.
The sample used is a complete sample from a sky area of 611$\;$deg$^2$, a low-$z$ subsample of a sample defined by Köhler et al. (1997) to study the luminosity function of QSOs. Distribution in redshift and absolute magnitude are shown in Fig. \[z\_b\], the sample represents moderately luminous QSOs when compared to the total population at all redshifts. The radio properties of most of the objects in the sample are not yet known, but as a subset of the QSO population most will be radio-quiet.
For all 20 objects we have acquired *BVRiJHKs* broadband photometry to evenly sample the SED over the optical–NIR wavelength interval. With three NIR bands we get some redundancy in the NIR to stabilise the stellar population fits. In addition we can make comparisons of sample properties to samples at higher redshift without the need for K-corrections. The *B* band images were integrated 30$\,$s at the ESO 3.6m telescope (EFOSC2), *VRi* images 300–1200$\,$s at ESO Danish 1.5m (DFOSC), and *JHKs* 160–900$\,$s at ESO NTT (SOFI). In addition we have available optical spectra (3800–7500$\;$Å) for 14 of the objects, taken with the ESO 3.6m telescope.
Fitting stellar populations
===========================
The nuclear contribution of the total QSO light has to be separated from the stellar light. We have developed a package for simultaneous two-dimensional modelling of a parametrised host model and the nuclear contribution. Luminosities for the hosts are determined from radial flux growth curves of the images, after subtracting the nuclear model resulting from the best fit (Fig. \[sepa\]). More details about the modeling are given in the contribution by B. Kuhlbrodt et al.(these proceedings).
We could produce colours for 18 of the 20 objects. In the two remaining cases the separation was not yielding unique solutions due to highly disturbed morphological structure. We excluded these two from further analysis.
For the optical spectra we are currently developing a two-dimensional separation method similar to the imaging case. For some objects the current program already yields host spectra almost free of broad emission line components from the nucleus. Since this is not the case for all objects we use the optical spectra only for an independent cross-check of fit-results based on the broad-band colours derived from our imaging data.
To assess the primary stellar populations of the hosts, we fit stellar population synthesis model spectra to the multicolour data. For this we use single age, single metallicity population (SSP) spectra from the GISSEL96 library (Bruzual & Charlot 1996, Leitherer et al.1996). We chose models with a Salpeter initial mass function and solar metallicity, ages 0.01–14 Gyr.
These specra were converted to *BVRiJHKs* colours using ESO filter curves and fitted to the measured host galaxy colours via a least-$\chi^2$ fit in two steps: 1) fitting only one SSP, age as free parameter, 2) fitting two SSPs, ages and mass-ratio of the two components free.
Results & Discussion
====================
The general photometric properties of the sample comply well with values for inactive galaxies, but of course with a large object-to-object variation, *B–V*$=0.76$ (0.78 for an inactive Sab galaxy), *V–R*$=0.57$ (0.55 for intermediate type galaxies), and *V–K*$=3.2$ (3.2 for intermediate type galaxies), values taken from Fukugita et al. (1995) and Griersmith et al. (1982).
Of the 20 objects we could classify three as spheroidal, ten as disks from morphological analysis. Seven show signs of at least mild disturbance.
Fitting one or two SSPs is a strong simplification. At least for disks with a significant amount of onging continuous starformation this will surely oversimplify the picture.
Fitting one SSP in principle only compares general optical-to-NIR colours of host and SSP. Still the ages derived from the fits (Fig. \[1ssp\]) show a generally good agreement between with ages expected from the morphological classification. The three classified spheroidals correspond to old populations of 7–17 Gyr while the majority of the disks have a clear tendency towards younger SSPs.
When fitting two SSPs, we can distinguish contributions from old and young populations. For most objects though, contributions from a second population did not improve the fit by a great amount. If at all, the involved masses of a young population were small, only for two objects of the order of $\sim2\,\%$. The resulting spectra for one of them are shown in Fig. \[he1300\]. We see an excess of blue light in the data (points) compared to the dotted line (single SSP fit). Since both objects are morphologically classified as disks, models with continuous star formation will also be able to explain the blue component.
For the other objects no major second component was detected, and in fact all these objects are consistent with only one SSP and a uniform upper limit for the second, younger component of $\sim 0.5\,\%$ (by mass).
In total we find no signs for strong starburst activity, neither from the spectral fitting nor from general sample colours. The results are in favor of the idea that the parent population of QSO host galaxies is in fact the general field population of inactive galaxies.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The series of events, which occurred at high redshift and originated multiple stellar populations in Globular Clusters (GCs) are still poorly understood. Theoretical work suggests that the present-day dynamics of stars in nearby GCs, including the rotation and velocity dispersion, may retain important clues on the formation of multiple populations.
So far, the dynamics of multiple populations have been investigated either from radial velocities of a relatively-small sample of stars, or from relative proper motions of stars in the small field of view provided by the [*Hubble Space Telescope*]{}. In this context, Gaia provides the unique opportunity to investigate the dynamics of thousands GC stars over a wide field of view.
For the first time, we combine Gaia DR2 proper motions and multi-band photometry to study the internal motions of the two main stellar populations of 47Tucanae in a wide field of view. We confirm that this cluster exhibits high rotation on the plane of the sky and find that both stellar generations share similar rotation patters. Second-generation stars show stronger anisotropies and smaller tangential-velocity dispersion than the first generation, while there is no significant difference between their radial-velocity dispersion profiles. We discuss the impact of these results in the context of the formation scenarios for multiple stellar populations in GCs.
date: 'Accepted 2018 July 9. Received 2018 July 6; in original form 2018 May 16'
title: Gaia unveils the kinematics of multiple stellar populations in 47Tucanae
---
\[firstpage\]
Hertzsprung-Russell and colour-magnitude diagrams, stars: kinematics and dynamics, stars: Population II, globular clusters: individual 47 Tucanae (NGC104).
Introduction {#sec:intro}
============
[*N*]{}-body simulations have shown that the internal dynamics of the distinct stellar populations in Globular Clusters (GCs) provide strong constraints on the formation scenarios of their multiple populations. Specifically, the present-day rotation and the velocity-dispersion profile of first-generation (1G) and second-generation (2G) stars would be related to the formation of 2G stars and their initial configuration (Mastrobuono-Battisti & Perets 2013, 2016; Vesperini et al.2013; H[é]{}nault-Brunet et al.2015).
The rotation of multiple populations in GCs has been poorly investigated to date. Nearly all the previous studies were indeed based on radial velocities and were limited by small sample sizes. $\omega$Centauri is a remarkable exception. Indeed, from the analysis of [*Hubble Space Telescope*]{} ([*HST*]{}) proper motions, Bellini et al.(2018) show that 1G stars have excess systemic rotation in the plane of the sky with respect to 2G stars.
The largest stellar sample based on radial velocities was used by Pancino et al.(2007) who analyzed 650 stars in $\omega$ Centauri and studied the rotation along the line of sight of the three main sub-populations of metal-poor, metal-intermediate, and metal-rich stars. They concluded that the three populations are all compatible with having the same rotational pattern, in contrast with previous finding by Norris et al.(1997) who showed that, while the majority of stars in $\omega$ Centauri exhibits strong rotation, the most metal-rich stars do not show any sign of rotation.
Bellazzini et al.(2012) analyzed the radial velocities of 1,981 stars in 20 GCs and did not find any relation between the presence of multiple populations and the rotation within each cluster.
In contrast, evidence that stars with extreme abundance of light elements exhibit different rotational patters than the remaining cluster stars comes from the study of 113 red-giant branch (RGB) stars in M13 (Cordero et al.2017). These authors concluded that the 24 analyzed Na-enhanced and extremely O-depleted stars exhibit faster rotation than the other stars.
The velocity dispersion of multiple stellar populations was typically studied by using spectroscopy. Bellazzini et al.(2012), concluded that in 17 out of 20 analyzed clusters, the stars with different light-elements abundance have similar velocity-dispersion profiles. The massive GCs NGC6388, NGC6441, and NGC2808, where sodium-rich stars seem to have a slightly lower line-of-sight velocity dispersion, are possible exceptions to this rule. Similarly, Marino et al.(2014) show that the two main populations of stars with different slow neutron-capture element abundance of NGC1851 have similar radial-velocity dispersion.
In a few cases, high-precision proper motions from [*HST*]{} allowed to extend the investigation of the velocity dispersion to a large number of thousands stars. Four GCs have been analyzed to date with [*HST*]{}, namely NGC104 (47Tuc) NGC2808, and $\omega$Centauri (Richer et al.2013; Bellini et al.2015, 2018) and NGC362 (Libralato et al.2018). In all the cases the stars with extreme helium abundances have more radially-anisotropic velocity distribution in the plane of the sky. For NGC362 this result is significant at 2.2$\sigma$ level. Unfortunately, these studies are limited to the relatively-small field of view covered by the [*HST*]{} cameras.
In this context, 47Tuc is an interesting case, which reveals an high degree of dynamical complexity. Specifically, [*HST*]{} proper motions have revealed that this cluster exhibits high internal rotation (Anderson & King 2003; Bellini et al.2017; Gaia collaboration et al.2018a; Bianchini et al.2018) and significant radial anisotropy in the external region (Richer et al.2013; Bellini et al.2017).
Since the seventies, 47Tuc has been widely studied in the context of stellar populations both spectroscopically and photometrically (e.g.Norris & Freeman 1979, 1982; Anderson et al.2009; Cordero et al.2014; Marino et al.2016; Wang et al.2017). High-precision [*HST*]{} and ground-based photometry has revealed that its color-magnitude diagram (CMD) is formed of two main discrete sequences of stars that can be followed along the various evolutionary stages from the main sequence (MS) to the asymptotic giant branch (Milone et al.2012; Piotto et al.2015). These sequences correspond to a first stellar generation (1G), which is formed of stars with a chemical composition similar to that of halo field stars at similar metallicity, and to the a second generation (2G) of stars enhanced in helium, nitrogen and sodium and depleted in carbon and oxygen. Both groups of 1G and 2G stars host sub populations (e.g.Marino et al.2016; Milone et al.2017).
The two main populations of 47Tuc exhibit different radial distributions, with 2G stars being significantly more-centrally concentrated than 1G stars (Norris & Freeman 1979; Milone et al.2012; Cordero et al.2014). Clearly, the stellar populations of 47Tuc are not well mixed and could still retain information on their star-formation history.
The first investigation of the dynamics of stellar populations of 47Tuc was provided by Richer et al.(2013) who used [*HST*]{} photometry and proper motions. These authors have divided MS stars into four groups, which presumably correspond to stellar populations with different chemical abundances, and found that the anisotropy in the proper-motion distribution correlates with stellar colors. Specifically, the bluest stars exhibit the most-pronounced proper-motion anisotropy while red stars show isotropic proper motions. This finding corroborates similar conclusion by Ku[č]{}inskas et al.(2014), who analyzed the spectra of 101 stars of 47Tuc and detected a significant correlation between the velocity dispersion along the line of sight and the O and Na abundance.
In this work we combine wide-field ground-based photometry and stellar proper motions from Gaia data release 2 (DR2, Gaia collaboration et al.2018b) to further investigate the internal dynamics of multiple stellar populations in 47Tuc. For the first time, this analysis will be extended to a large field of view.
The paper is organized as follows. In Section \[sec:data\] we describe the data and present the photometric diagrams of 47Tuc. The dynamics of 1G and 2G stars are investigated in Section \[sec:dynamics\]. Finally, a summary of the results and a discussion is provided in Section \[sec:summary\].
Data {#sec:data}
====
We combined ground-based wide-field photometry and proper motions from Gaia DR2 to investigate the internal dynamics of stellar populations in 47Tuc. To identify multiple populations in the CMD we used $U$, $B$, $V$, $I$ photometry derived from 856 images collected with various facilities, including the Wide-Field Imager of the ESO/MPI telescope and the 1.5 m telescope at Cerro Tololo Inter-American Observatory (Stetson 2000). These images have been reduced by Peter Stetson by using the methods and the computer programs by Stetson (2005) and are calibrated on the photometric system by Landolt (2002). Details on this dataset and on the data reduction are provided by Bergbush & Stetson (2009).
This photometric catalog has been used in previous studies on multiple populations showing that two distinct groups of 1G and 2G stars are clearly visible along the RGB and the horizontal branch (HB) of 47Tuc (Milone et al.2012; Monelli et al.2013; Marino et al. 2016). The presence of two populations is evident in various diagrams involving the $U$, $B$, $I$ filters, like the $U$-$B$ vs.$B-I$ two-color diagram and the $B$ vs.($U-B+I$) or $B$ vs.$U-2 \cdot B+I$=$C_{\rm U,B,I}$ pseudo CMDs. Stetson’s catalog was matched with the Gaia DR2 one and only stars for which both $U$, $B$, $V$, $I$ photometry and Gaia proper motions are available are used in this paper.
Moreover, we excluded stars with poor Gaia astrometry. To do this, we first used the parameter released with Gaia DR2 ‘astrometric\_gof\_al’, which is indicative of the goodness of fit statistic of the astrometric solution for the source in the along-scan direction (see Gaia Collaboration et al.2018a for details). When we plot this parameter as a function of the g-band magnitude of Gaia, most of the stars exhibit a clear trend. To exclude the ouliers from the analysis we followed a procedure similar to the one described by Milone et al.(2009, see their Sect. 2.1). Briefly, we divided the magnitude interval covered by the RGB stars of 47Tuc into bins of 0.5 mag. For each bin, we calculated the median magnitude, the median value of the astrometric\_gof\_al parameter and the corresponding 68.27$^{o}$ percentile ($\sigma$). We excluded from the analysis all the stars that exceed the median values by $N=5$ times $\sigma$ and iterated this procedure until two subsequent values of the median and the $\sigma$ values differ by less than 0.01. In addition, we excluded stars with proper-motion uncertainties larger than 0.15 mas yr$^{-1}$. Indeed, we noticed that the bulk of our sample of relatively-bright RGB and HB stars have uncertainties below this value[^1]. The final sample comprises 3,276 cluster members between 0.8 and 18.0 arcmin from the cluster center, including 1,208 1G stars and 2,068 2G stars.
The left panels of Figure \[fig:cmd\] show the $V$ vs.$C_{\rm U,B,I}$ pseudo-CMD of 47Tuc zoomed around the RGB and the HB (bottom) and the vector-point diagram of proper motion (top). We used the black circle to separate bona-fide cluster members from field stars, which are represented with black dots and grey crosses, respectively. We verified that our criterion is consistent with the membership selection by Gaia Collaboration et al.(2018c) for the sample of stars analyzed in this paper. The two main RGBs and HBs of 47Tuc, which we have widely investigated in previous papers (e.g.Milone et al.2012; Monelli et al.2013; Marino et al.2016), are clearly visible in this diagram and we used aqua and magenta colors to mark 1G and 2G stars, respectively, in the right-panel diagrams.
In our analysis we exploit the position of the cluster center, the values of core and half-light radius and the cluster distance provided by the Harris (1996, updated as in 2010) catalog. Their values are listed in Table 1.
---------------------------- --------------------------------
R.A.(J2000) 00 24 05.67
DEC.(J2000) $-$72 04 52.6
Core radius 0.36 arcmin
Half-light radius 3.17 arcmin
Distance 4.5 kpc
$\mu_{\alpha} cos{\delta}$ 5.25$\pm$0.01 mas yr$^{-1}$
$\mu_{\delta}$ $-$2.49$\pm$0.01 mas yr$^{-1}$
---------------------------- --------------------------------
: Parameters of 47Tuc used in this paper. The average proper motions are derived by using Gaia DR2 data. The remaining parameters are taken from the catalog by Harris(1996, updates as in 2010).
\
\[tab:parametri\]
{width="11.0cm"}
Dynamics of multiple stellar populations {#sec:dynamics}
========================================
The high-precision proper motions from Gaia DR2 allow the investigation of stellar dynamics within 47Tuc. In this section, we use the proper motions of each group of 1G and 2G stars selected in Figure \[fig:cmd\] to analyze the rotation in the plane of the sky and the velocity dispersion of each sub-population.
Rotation on the plane of the sky
--------------------------------
To investigate the rotation of the two main populations of 47Tuc, we first show in the upper panels of Figure \[fig:rotazione\] the density of 1G and 2G stars in the $\mu_{\alpha} cos{\delta}$ vs.$\theta$ and $\mu_{\delta}$ vs.$\theta$ planes, where $\theta$ is the position angle. The sinusoidal patterns of 1G and 2G stars clearly indicate that both populations exhibit significant rotation on the plane of sky.
We defined a grid of 16 values of $\theta$, ranging from 0$^{\rm o}$ to 360$^{\rm o}$ in steps of 22.5$^{\rm o}$, and associated to each value of $\theta$ a circular sector with an arc length of 45$^{o}$. Then we calculated the median values of $\mu_{\alpha} cos{\delta}$ and $\mu_{\delta}$ of 1G and 2G stars in each circular sector. The median points are associated with the mean coordinates of the analyzed stars and are represented with aqua and magenta points for 1G and 2G stars, respectively, in the upper panels Figure \[fig:rotazione\].
In the panels c1 and c2 of Figure \[fig:rotazione\] we compare the median motions of 1G and 2G stars in the 16 circular sectors after subtracting the median motions of all the analyzed stars[^2]. Clearly the two populations exhibit similar rotation patters.
Panel d of Figure \[fig:rotazione\] shows the average position of the stars in the various circular sectors relative to the cluster center. As expected, 2G stars have smaller radial distances than 1G stars, as a consequence of the fact that the second generation is the most centrally concentrated (Norris & Freeman 1982; Milone et al.2012; Cordero et al.2014). The arrows, which correspond to the average two-dimensional velocity vector calculated in each sector, indicate the motion of 1G and 2G stars relative to the cluster center in 250,000 years. The fact that the vector directions are within a few degrees of the tangential directions confirms that 47Tuc exhibits a significant rotation on the plane of the sky (Anderson & King 2003; Bellini et al.2017). We note that the perspective effects induced by the large apparent size on the sky of the cluster and its spatial motion (e.g.van de Ven et al.2006, see their equation 6) would not affect the tangential component of the motion but will result in an apparent expansion of the cluster. However, this effect is small for 47Tuc, which has a slow motion along the line of sight. Moreover, we note that such phenomenon would affect in the same way the two populations and would not affect the conclusion of this work.
{height="7.5cm"} {height="7.5cm"}
To investigate the rotation of 1G and 2G stars as a function of the radial distance from the cluster center, we divided the field of view in various circular annulii. Each annulus is defined by using the method of the naive estimator (Silverman 1986) in the region with radial distance from the cluster center between 0.8 and 18.0 arcmin. Specifically, we defined a series of points separated by a distance of $d=$2.5 arcmin. The bins are defined over a grid of points, which are separated by steps of $d/2$ in distance. For each bin we calculated the tangential velocity of 1G and 2G stars and estimated the corresponding error as $\sigma_{\rm 1G (2G)}/\sqrt{N_{1G (2G)}-1}$, where $\sigma_{\rm 1G (2G)}$ is the tangential-velocity dispersion of 1G (2G) stars and $N_{\rm 1G (2G)}$ the number of 1G (2G) stars in each annulus.
Our results, illustrated in Figure \[fig:tan\], reveal that 1G and 2G stars exhibit similar rotation along the plane of the sky. 1G stars seem to exhibit slightly higher tangential velocity than 2G stars in the region between $\sim$7 and $\sim$11 arcmin, although the difference is significant at the $\sim$2-sigma level only.
![The tangential velocity, which is indicative of the cluster rotation, is plotted as a function of the radial distance from the cluster center. Aqua triangles and magenta dots refer to 1G and 2G stars, respectively. The dashed vertical lines mark the core and the half-mass radius. The scales on the left and right, indicate the tangential velocity in angular and linear units. The latter is calculated by assuming for 47Tuc a distance of 4.5 kpc (Harris 1996, updated as in 2010).[]{data-label="fig:tan"}](tan.ps){width="8.cm"}
Velocity dispersion of multiple populations
-------------------------------------------
We calculated the radial ($\sigma_{\rm RAD}$) and the tangential velocity dispersion ($\sigma_{\rm TAN}$) of 1G and 2G stars in each radial bin defined in the previous subsection and plot the corresponding velocity-dispersion profiles in Figure \[fig:disp\]. To derive the dispersions we adapted to 47Tuc, the procedure described in Mackey et al.(2013) and Marino et al.(2014) that accounts for the contribution of observational errors to the proper-motion dispersion. Briefly, we used a maximum-likelihood technique, assuming that the stellar proper motions are normally distributed around the average value according to their measurement uncertainties and the intrinsic dispersion. We estimated the intrinsic dispersion by maximizing the logarithm of the joint probability function for the observed proper motions. The uncertainties associated to each point are determined by bootstrapping with replacements performed 1,000 times. The error bars indicate one standard deviation (68.27$^{\rm th}$ percentile) of the bootstrapped measurements.
We find similar radial-velocity dispersion profiles for 1G and 2G stars and no significant difference between the radial-velocity dispersion of 1G and 2G stars in the analyzed radial interval. In contrast, 1G stars exhibit, on average, smaller tangential-velocity dispersion than the 2G ones. Such difference seems to increase when moving from the half-light radius towards the outermost cluster region and is maximum for radial distance of about 10 arcmin from the cluster center. The tangential-velocity dispersion of 1G stars is consistent with that of 2G stars at radii larger than $\sim$12 arcmin[^3].
![Tangential velocity dispersion (upper panel) and radial velocity dispersion (lower panel) as a function of the radial distance from the cluster center for 1G (aqua triangles) and 2G (magenta circles) stars.[]{data-label="fig:disp"}](disp.ps){width="8.0cm"}
To further investigate the motion of 1G and 2G stars, we calculated for each population the quantity $\sigma_{\rm TAN}/\sigma_{\rm RAD}-1$, which is indicative of deviation from the isotropy. The radial dependence of $\sigma_{\rm TAN}/\sigma_{\rm RAD}-1$ is illustrated in Figure \[fig:anisotropia\] where the horizontal dotted lines correspond to an isotropic stellar system. 2G stars significantly deviate from isotropy in the analyzed region with radial distance from the cluster center smaller than $\sim 12$ arcmin. The value of $\sigma_{\rm TAN}/\sigma_{\rm RAD}-1$ decreases from $\sim -0.1$ to less than $\sim -0.2$ when moving from the half-light radius to about 12 arcmin from the center. This quantity increases in the cluster outskirts, where it is consistent with zero.
The first generation exhibits a mild deviation from isotropy in the region with radial distance from the center between $\sim$5 and 8 arcmin, where the values of $\sigma_{\rm TAN}/\sigma_{\rm RAD}-1$ are smaller than zero and this difference is significant at $\sim$1.5- 2.0-$\sigma$ level. 1G stars are consistent with an isotropic system at radial distances larger than 7 arcmin. In the outermost bin both 1G and 2G stars have slightly positive $\sigma_{\rm TAN}/\sigma_{\rm RAD}-1$ but the difference from zero has low statistical significance.
![Tangential to radial isotropy for 1G stars (top) and 2G stars (bottom) against the radial distance from the cluster center.[]{data-label="fig:anisotropia"}](anisotropia.ps){width="8.0cm"}
Summary and Discussion {#sec:summary}
======================
In the last years, several scenarios of formation of multiple populations in GCs have been suggested (see Renzini et al.2015 and references therein for a critical review). Some of these scenarios suggest that GCs have experienced at least two bursts of star formation and that 2G stars formed out from material ejected from more-massive 1G stars. Asymptotic-giant branch stars (AGBs), fast-rotating massive stars (FRMSs) and supermassive stars have been proposed as possible polluters (e.g.Cottrell & Da Costa 1981; Ventura et al.2001; Decressin et al.2007; Denissenkov & Hartwick 2014). As an alternative, all the stars in GCs are coeval and multiple populations are the result of exotic phenomena that occurred in the unique environment of the proto GC (e.g.De Mink et al.2009; Bastian et al.2013; Gieles et al.2018).
The various 2G-formation scenarios predict different geometries for the gas from which the 2G form. The signatures left by the evolution of the two nested populations are an exquisite tool to constrain the origin of the younger population. In this context, 47Tuc is an ideal cluster. Indeed, the evidence that its populations are not fully mixed (Norris & Freeman 1979; Milone et al.2012; Cordero et al.2014) demonstrates that this cluster has not reached a complete relaxation. As a consequence, it is expected to have retained some of the initial differences between the populations (H[é]{}nault-Brunet et al.2015).
Unfortunately, the dynamical implications of the various scenarios have not yet been fully explored from the dynamical point of view except for few studies that assumed an initial spherical configuration for both populations (e.g.Decressin et al.2010).
So far, the AGB scenario (D’Antona et al.2016; D’Ercole et al.2008, 2010) is the most developed one in terms of dynamical predictions. In this scenario, the gas produced by slow stellar winds inflows to the centre of the cluster carrying a small amount of angular momentum inherited from the stars of origin, leading to the formation of a 2G gaseous disk (e.g.Bekki 2010, 2011). Once settled, the disk fragments and forms a 2G stellar disk which relaxes within the 1G component of the cluster. This process alters the initial structure of the system, leading to an elliptical, anisotropic and differentially-rotating GC (see Mastrobuono-Battisti & Perets 2013, 2016, H[é]{}nault-Brunet et al.2015 and references therein for details).
If the relaxation time of the cluster is long enough, the initial phase-space configuration of 2G stars would imprint long-lasting signatures on the structure of the cluster. As a consequence, the present-day internal dynamics of GCs are directly linked to their past dynamical history.
Recently, Mastrobuono-Battisti & Perets (2013, 2016) explored the long-term evolution of multiple populations in the context of the AGB scenario by using [*N*]{}-body simulations. In the following we qualitatively compare our observational findings with their predictions.
To constrain the internal dynamics of the subpopulations of 47Tuc, we first exploited the $V$ vs.$C_{\rm U,B,I}$ diagram of stars in this cluster to identify two groups of 1G and 2G stars along the RGB and the HB. Then we combined photometry and Gaia DR2 proper motions to analyze the rotation and the velocity dispersion along the plane of the sky in the selected 1G and 2G stars. This approach allowed us to study, for the first time, the internal dynamics of a large sample of more than 3,000 1G and 2G stars over a wide field of view. Specifically, the analyzed region ranges from 0.8 to 18 arcmin from the cluster center, i.e.between 0.3 and 5.7 half-light radii.
We discovered that both 1G and 2G stars exhibit a strong rotation on the plane of the sky and that there is no evidence for any difference in the rotation pattern of the two populations. This finding is apparently in contradiction with the predictions of Mastrobuono-Battisti & Perets (2016) who find that 2G stars exhibit stronger rotation than the first generation. However, it should be noted that dynamical processes such as relaxation and angular momentum diffusion due to two-body interactions, which act on a short time scale in flattened structures (Haas 2014), could have already acted in the cluster reducing the initial difference (which is of the order of 1-2km/s after one relaxation time), leaving with 1G and 2G stars that rotate at a similar speed. We also derived the velocity-dispersion profiles of 1G and 2G stars along the tangential and the radial directions. While there is no significant difference between the radial-velocity dispersion of the two populations, the second generation exhibits, on average, smaller tangential-velocity dispersion than 1G stars. Such difference is more-pronounced in the region with radial distance of about 8-12 arcmin from the cluster center (i.e.$\sim$2.5-3.8 times the half-light radius) and strongly decreases when moving towards the innermost or the external regions. Second-generation stars strongly deviate from isotropy in the analyzed cluster region with radial distance smaller than $\sim$12 arcmin, in contrast with the first generation, which shows a mild deviation from isotropy at radial distance between $\sim$5 and 7 arcmin.
Our results are consistent with the conclusion by Richer et al.(2013) based on [*HST*]{} proper motions of stars within a $\sim$3.4$\times$3.4-arcmin region with a distance of 1.9 half-light radii from the cluster center. These authors concluded that stars with bluest F606W$-$F814W colors, which likely belong to the 2G, exhibit the largest proper-motion anisotropy which is undetectable for the reddest stars. A similar behaviour is observed in NGC2808 (Bellini et al.2015), $\omega$Cen (Bellini et al.2017) and, likely, in NGC362 (Libralato et al.2018), where the result is significant at the $\sim 2 \sigma$-level. These facts would suggest that the high radial anisotropy is a common feature among GCs.
Noticeably, simulations by Mastrobuono-Battisti & Perets (2016) predict a large difference in the tangential component of the velocity dispersion of the two populations, which is qualitatively similar to what we observe in 47Tuc. On the other hand, difference radial-velocity dispersion between 1G and 2G stars is also expected, in contrast with what we observe. According to Mastrobuono-Battisti and Perets, the stronger radial anisotropy that characterizes the 2G stars, combined to the presence of rotation, is consistent with the spatial diffusion of a second generation that formed centrally concentrated in a disk configuration. As an alternative suggested by the referee, the 2G could born in a spherical-centrally-concentrated configuration in a cluster primordially rotating.
It is worth noting that, although our results partially match the predictions by Mastrobuono-Battisti & Perets (2016), 47Tuc is significantly different from the cluster modeled by these authors in terms of mass, relaxation time, and fraction of 2G stars. In conclusion, although a more detailed model, tailored to 47Tuc will be necessary to properly interpret our results, the internal dynamics of this cluster inferred from Gaia DR2 suggest that the second generation formed in a disk-like, centrally concentrated configuration inside the 1G component. Qualitatively, this work demonstrates how Gaia can contribute to constrain the formation scenarios of the still-eluding multiple populations in GCs.
acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous referee for several suggestions that improved the quality of the manuscript. We are grateful to Peter Stetson who provided the photometric catalog used in the paper. AFM acknowledges support by the Australian Research Council through Discovery Early Career Researcher Award DE160100851. This work has been supported by the European Research Council through the ERC-StG 2016 project 716082 ‘GALFOR’ (http://progetti.dfa.unipd.it/GALFOR) and by the MIUR through the FARE project R164RM93XW ‘SEMPLICE’. AMB acknowledges support by Sonderforschungsbereich (SFB) 881 ‘The Milky Way System’ of the German Research Foundation (DFG).
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[^1]: We carefully checked that our results are not affected by the adopted selection criteria. To do this, we verified that by using values of $2 \sigma$ and $10 \sigma$ and by excluding stars with with proper motion uncertainties larger than 0.08 mas yr$^{-1}$ and 0.30 mas yr$^{-1}$ the conclusions of the paper remain unchanged. Similarly,we repeated the analysis by excluding stars with correlation coefficients that differ by more than $\pm 1 \sigma$ from the median value. We find that all the conclusions remain unchanged, thus demonstrating that there is no evidence for any significant bias connected to the correlation coefficients between the proper motions along the right ascension and the declination.
[^2]: We find that the median proper motions of all the analyzed stars are $\mu_{\alpha} cos{\delta}$=5.25$\pm$0.01 mas yr$^{-1}$ and $\mu_{\delta}$=$-$2.49$\pm$0.01 mas yr$^{-1}$, and is consistent within 2-$\sigma$ with the determination by Gaia Collaboration et al.(2018c, see their Table C.1). There are no significant differences between the median motions of 1G and 2G stars.
[^3]: Bellini et al.(2017) used [*HST*]{} images to measure the rotation of 47Tuc in the plane of the sky and the velocity anisotropy profile from the cluster core out to about 13 arcmin. Although, our paper is focused on the relative motions of 1G and 2G stars and not on the overall cluster dynamics, we verified that the tangential-velocity profiles by Bellini and collaborators are consistent with those of our paper at $\sim$0.5-1.5 $\sigma$ level. The velocity dispersion average profile derived for 1G and 2G stars in this paper are in agreement with those provided by Bellini and collaborators within one $\sigma$. For radial distances larger than $\sim$10 arcmin from the cluster center both the tangential and the radial velocities dispersions derived by Bellini and collaborators are consistent with those derived in our paper at $\sim$1.5-$\sigma$ level with those of our paper, with Bellini et al. providing higher dispersion values. The investigation of such small discrepancy is beyond the purposes of our paper.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg-de Vries equation, discrete modified Korteweg-de Vries equation and the discrete Schwarzian Korteweg-de Vries equation. We will also describe a direct method for obtaining Lax representations for the reduced equations.'
author:
- 'Christopher M. Ormerod$^1$'
- 'Peter H. van der Kamp$^2$'
- Jarmo Hietarinta$^3$
- |
G. R. W. Quispel.$^2$\
[$^1$ Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA,\
$^2$ Department of Mathematics and Statistics, La Trobe University, 3086, VIC, Australia,\
$^3$ Department of Physics and Astronomy, University of Turku, 20014 Turku, Finland.]{}
title: |
Twisted reductions of integrable lattice equations,\
and their Lax representations
---
Introduction
============
A key property of integrable partial differential equations (PDEs) is the existence of multisoliton solutions describing the elastic scattering between the solitons. The single soliton can also be called a traveling wave solution, as its form is unchanged after some time, up to translation [@AblowitzClarkson:Solitons]. Such invariances are generalized and formalized in the symmetry approach [@BC74; @Olver:LieDiff], where one uses symmetries of the original equation to derive an additional equation, the similarity constraint, which is compatible with the original equation. One can then use this constraint equation to reduce the original integrable PDE to an integrable ordinary differential equation (ODE). For example, in the case of the Korteweg-de Vries equation $$\begin{aligned}
\label{kdv}
\partial_t u + \partial_x^3u+u\partial_x u=0,\end{aligned}$$ the constraint $v\partial_x u+\partial_t u=0$ leads to the traveling wave ansatz $u=f(x-vt)$ and elliptic equation for $f$, while the similarity constraint $2u+x\partial_xu+3t\partial_t
u=0$ leads to the similarity ansatz $u=t^{-2/3}\phi(z),\,
z=x/(t^{1/3})$ and then to an equation for $\phi$ that can be transformed, by letting $\phi = \partial_z y-y^2/6$, to $$\label{P2}
\partial_z^2 y = \dfrac{y^3}{18} + \dfrac{yz}{3} + \alpha,$$ (where $\alpha$ is an integration constant) which is the second Painlevé equation (see [@Olver:LieDiff] p. 195). Given a constraint, a method for obtaining a Lax pair for the reduced equation was given in [@FlaschkaNewell]. For further applications of symmetries of PDE’s in mathematical physics see [@PWZ:I; @PWZ:II] and references therein.
Integrable partial difference equations (P$\Delta$Es), or lattice equations, can be seen as discrete analogues of integrable PDEs, and they have been shown to possess many of the same characteristics as their continuous analogues, such as Lax representations [@QNCvdL:NSG; @Nijhoff:dSKdV; @Nijhoff:Linearisation], bilinear structures and N-soliton solutions [@Hirota:DKdV; @Hirota:DtToda; @Hirota:dSG; @Jimbo:discretesolitonI; @Jimbo:discretesolitonII; @Jimbo:discretesolitonIII]. Furthermore, several approaches have been developed to reduce P$\Delta$Es to ordinary difference equations (O$\Delta$Es) [@dKdVreds; @Quispel:Intreds; @TKQ09; @K09; @KRQ07; @KQ10; @SimilarityReds; @HKQT13]. Reductions of the kind of those presented here are obtained in [@LeviW; @QCS] using Lie group techniques in the case of differential-difference equations.
We consider equations defined on the Cartesian two-dimensional lattice. In this context a particularly interesting set of equations is given by the form $$\label{Q}
Q(w_{l,m},w_{l+1,m},w_{l,m+1},w_{l+1,m+1};\alpha_{l},\beta_{m}) =
0,\,\forall l,m,$$ where the subscripts, $l,m$, indicate a point in the Cartesian 2-dimensional lattice on which the dependent variable $w$ is defined, and $\alpha_l$ and $\beta_m$ are lattice parameters associated with the horizontal and vertical edges respectively. [^1] Such equations are often called quad equations, because the equation connects values of $w$ given at the corners of an elementary quadrilateral of the lattice, and if the parameters $\alpha_l$ and $\beta_m$ do not depend on the coordinates $l,m$, respectively, then the equation is said to be autonomous. We assume also that equation is multilinear so that we can solve for any particular corner value in terms of the other three.
For quadrilateral equations one definition of integrability is by “multidimensional consistency” [@Nijhoff:dSKdVP6; @NijWal]. This has turned out to be a very effective definition, and in its three-dimensional version (Consistency-Around-a-Cube, CAC) it has led (under some mild additional assumptions) to a classification of scalar integrable quadrilateral equations [@ABS:ListI; @ABS:ListII]. Our examples have been chosen from this class of equations. One very important consequence of the CAC property is that it immediately provides a Lax pair [@Nij02], which is a system of linear difference equations whose consistency is equivalent to the equation (\[Q\]).
One may consider the analogue of a traveling wave solution to be a solution on the lattice admitting the constraint[^2] $$\label{eq:cons1}
w_{l+s_1, m+s_2} =w_{l,m},$$ leading to what is known as an $(s_1,s_2)$-reduction [@K09]. In order to construct consistent evolution we have to consider initial values satisfying this constraint and make sure that the evolution does not break the constraint.
In a similar manner to the continuous case, where reductions of PDE’s lead to interesting ODE’s, many authors have identified reductions given by with interesting O$\Delta$E’s such as discrete analogues of elliptic functions, known as QRT maps [@SimilarityReds], discrete Painlevé equations [@Gramani:Reductions; @Hay; @OvdKQ:reductions; @Ormerod:qP6; @Gramani:Q4Ell] and many higher dimensional mappings [@dKdVreds; @Quispel:Intreds; @TKQ09; @KQ10; @HKQT13]. Of particular interest to this study are QRT maps and discrete Painlevé equations, which are both classes of integrable second order nonlinear difference equations. The QRT maps are autonomous mappings that preserve a biquadratic invariant [@QRT1; @QRT2] whereas discrete Painlevé equations are integrable non-autonomous difference equations admitting the classical Painlevé equations as continuum limits [@DPS] and also QRT maps as autonomous limits [@ramani2002autonomous]. For example, two discretizations of are $$\begin{aligned}
y_{n+1} + y_{n-1} = \dfrac{y_n(hn + a)+b}{1-y_n^2},\\
y_{n+1}y_ny_{n-1} = \dfrac{aq^ny_n(y_n - q^n)}{y_n-1},\end{aligned}$$ which are called multiplicative and additive difference equations in accordance with their dependence on $n$ [@DPS]. Their autonomous limits, when $h \to 0$ and $q\to 1$ respectively, are QRT maps [@QRT1; @QRT2].
Let us consider the simplest (nontrivial) case of a periodic reduction, determined by the constraint $w_{l+1,m-1}=w_{l,m}$. We can then give the initial values on the blue staircase given in Figure \[fig:intro\].
(-1.5,-.5) grid (4.5,3.5); (-1.5,3) – (-1,3) – (-1,2) – (0,2) – (0,1)– (1,1)– (1,0)–(2,0)–(2,-0.5); (-1,3.5) – (-1,3) – (0,3) – (0,2) – (1,2) – (1,1)– (2,1)– (2,0)–(3,0)–(3,-0.5);
at (-1.45,3.15) [$y$]{}; at (-1.45,2.15) [$x$]{}; at (-0.45,2.15) [$y$]{}; at (-0.45,1.15) [$x$]{}; at (0.55,1.15) [$y$]{}; at (0.55,0.15) [$x$]{}; at (1.55,0.15) [$y$]{};
at (-0.15,3.45) [$y'$]{}; at (-0.15,2.45) [$x'$]{}; at (0.85,2.45) [$y'$]{}; at (0.85,1.45) [$x'$]{}; at (1.85,1.45) [$y'$]{}; at (1.85,0.45) [$x'$]{}; at (2.85,0.45) [$y'$]{};
(0,3) circle (.05); (1,2) circle (.05); (2,1) circle (.05); (3,0) circle (.05);
In this case only two initial values are needed, $x$ and $y$. Solving for $w_{l+1,m+1}$ from we obtain $$w_{l+1,m+1}=f(w_{l,m},w_{l+1,m},w_{l,m+1};\alpha,\beta)$$ for some rational function $f$ (here we assume the parameters $\alpha,\beta$ are constants). From Figure \[fig:intro\] we then find that the initial values on the staircase evolve by the two dimensional map $$x'=y,\quad y'=f(x,y,y;\alpha,\beta),$$ and that in particular the periodicity is preserved. This result can also be written as a second order ordinary difference equation of the form $$x_{n+2}=f(x_n,x_{n+1},x_{n+1};\alpha,\beta).$$ What is important is that if the original P$\Delta$E is integrable and has a Lax pair then it is possible to construct a Lax pair for the resulting ordinary difference equation, which therefore is integrable as well.
Recently three of the authors presented a direct method for obtaining the Lax representations of equations arising as periodic reductions of non-autonomous lattice equations [@OvdKQ:reductions; @Ormerod:qP6], which can be consider the discretization of the method given in [@FlaschkaNewell].
In this paper we consider the generalization of in the form $$\begin{aligned}
\label{twist}
w_{l+s_1,m+s_2} = T(w_{l,m}),\end{aligned}$$ where the transformation $T$ (which we call the “twist”) is fractional linear, which is also known as a homographic transformation [@DuVal]. In the example discussed above we would impose $w_{l+1,m-1} = T(w_{l,m})$ and start with a sequence of initial values of the form $$\dots,T^{-2}(y), T^{-1}(x), T^{-1}(y), x,y,T(x),T(y),\dots$$ and after one step of evolution the new values should be similarly related, i.e., $$\dots,T^{-2}(y'), T^{-1}(x'), T^{-1}(y'), x',y',T(x'),T(y'),\dots$$ Thus on the $k$-th step of the staircase we would get the evolution $$T^{k}(x')=T^k(y),\quad T^k(y')=f(T^{k+1}(x),T^{k+1}(y),T^{k}(y)),$$ But since $y'=f(T(x),T(y),y)$ this makes sense only if $$T^k(f(T(x),T(y),y))=f(T^{k+1}(x),T^{k+1}(y),T^{k}(y)),$$ in other words, equation must be invariant under the transformation $T$, i.e. $$Q(\{T(w_{l,m})\};\alpha_{l},\beta_{m}) \propto Q(\{w_{l,m}\};\alpha_{l},\beta_{m}) .$$
The main result of the paper is a method for calculating Lax representations for these reductions, even in the non-autonomous case.
The paper is organized as follows: First in §\[consistency\] we review the reduction method for the $s_1=2,s_2=1$ reduction and then discuss the possible non-autonomous parameters of the equation. We distinguish the following cases, based on how the lattice parameters $\alpha_l$ and $\beta_m$ vary:
- the autonomous case, where the parameters are constant;
- the simply non-autonomous case, where the parameters depend only explicitly on the lattice position; and
- the fully non-autonomous case, where the parameters also depend on additional constants, which are not left invariant under a lattice shift.
Each of these three cases exists in a twisted and a non-twisted version. We will review these parameter choices in more depth in §\[consistency\].
In §\[sec:Twist\] we present the general method for constructing the Lax matrices. To illustrate our method we then perform $(2,1)$-reductions of three archetypical equations with distinct twists. The first equation of the form (\[Q\]), considered in §\[sec:dmKdV\], will be the discrete modified Korteweg-de Vries equation (dmKdV), also called $H3_{\delta=0}$, where $$\begin{aligned}
\label{dmKdV}
Q_{H3_{\delta=0}}=
\alpha_l(w_{l,m}&w_{l+1,m} - w_{l,m+1}w_{l+1,m+1})\\
&-\beta_m(w_{l,m}w_{l,m+1} - w_{l+1,m}w_{l+1,m+1}),\nonumber\end{aligned}$$ with twist $T_1:w\to w\lambda$. Here we will review the non-twisted autonomous case, the twisted autonomous case, the twisted simply non-autonomous case and the twisted fully non-autonomous case. We will also briefly study a second twist, $T_2:w\to \frac{\lambda}{w}$.
The second equation, considered in §\[sec:dpKdV\], will be the lattice potential Korteweg-de Vries equation, or $H1$, where $$\label{dpKdV}
Q_{H1}=(w_{l,m}-w_{l+1,m+1})(w_{l+1,m}-w_{l,m+1}) - \alpha_l+ \beta_m,$$ with twists $T_1: w \to w+ \lambda$ and $T_2 : w \to \lambda - w$. Here we will consider the twisted autonomous case, the twisted simply non-autonomous case and the twisted fully non-autonomous case. In §\[sec:dSKdV\], we will consider the lattice Schwarzian Korteweg-de Vries equation, or $Q1_{\delta=0}$, with $$\begin{aligned}
\label{dSKdV}
Q_{Q1_{\delta=0}}= \alpha_{l}[(w_{l,m} & - w_{l,m+1})(w_{l+1,m}-w_{l+1,m+1})]\\
&-\beta_{m}[(w_{l,m} - w_{l+1,m})(w_{l,m+1}-w_{l+1,m+1})],\nonumber\end{aligned}$$ where we will consider the twisted autonomous case and the twisted fully non-autonomous case. The twist will be an arbitrary Möbius transformation.
Finally in §\[22qpvi\] we will consider the (2,2)-reduction of , and obtain the full parameter $q\text{-P}_{VI}$. In §\[s1s2\] we treat the general ($s_1,s_2$)-reduction, and provide a list of twists for ABS-equations [@ABS:ListI; @ABS:ListII].
While this paper was being edited, the preprint [@HHS13] appeared on the arXiv, which presents a twisted version of the approach in [@Gramani:Q4Ell].
Symmetry invariance {#consistency}
===================
For pedagogical reasons we specialize our reduction, given by , to one of the simplest possible cases; where $s_1 =2$ and $s_2 = 1$. Contrary to the case $s_1=s_2=1$, in this case there is a difference between the simply non-autonomous case and the fully non-autonomous case. In this special case, our reduction may be specified by introducing two variables, $$\label{np}
n = 2m-l, \hspace{2cm} p = l-m.$$ We label the variables of the reduction in terms of $n$ and $p$ by specifying $$\label{labelling}
w_{l,m} \mapsto T^{l-m} u_{2m-l} = T^p u_n.$$ This extends the labeling of [@OvdKQ:reductions] to accommodate for the twist. With this constraint, it is sufficient to specify just three initial conditions. Their values, and the values obtained from the similarity constraint, , form a staircase which determines a solution on all of $\mathbb{Z}^2$. A small portion of the staircase in $\mathbb{Z}^2$ has been depicted in Figure \[fig:label\].
(-1.5,-.5) grid (4.5,3.5); (-1.5,0) – (0,0) – (0,1) – (2,1)– (2,2)– (4,2)– (4,3)–(4.5,3);
at (0,0) [$T^pu_n$]{}; at (1,1) [$T^pu_{n+1}$]{}; at (-.1,1) [$T^{p-1}u_{n+2}$]{}; at (2,1) [$T^{p+1} u_n$]{}; at (2,2) [$T^p u_{n+2}$]{}; at (3.1,2) [$T^{p+1} u_{n+1}$]{}; at (.8,2) [$T^{p-1}u_{n+3}$]{}; at (4.1,2) [$T^{p+2} u_{n}$]{}; at (3,3) [$T^pu_{n+3}$]{}; at (4.1,3) [$T^{p+1}u_{n+2}$]{};
(1,2) circle (.07); (3,3) circle (.07);
The shift $(l,m) \to (l+1,m+1)$ leaves $p$ invariant and induces, by , the shift $n \to n+1$, as one can see in Figure \[fig:label\]. On the top-right square in Figure \[fig:label\] we can solve the equation, $$Q(T^{p+1}u_{n+1}, T^{p+2}u_n, T^{p}u_{n+3}, T^{p+1}u_{n+2};\alpha_{l+3},\beta_{m+2}) = 0,$$ to find $u_{n+3}$, and hence the triple $(u_{n+1},u_{n+2},u_{n+3})$, from the triple $(u_{n},u_{n+1},u_{n+2})$ and the twist $T$. But this is not the only equation for $u_{n+3}$; considering the middle square in Figure \[fig:label\] we have $$\begin{aligned}
\label{reductioninQ}Q(T^{p}u_{n+1}, T^{p+1}u_n, T^{p-1}u_{n+3}, T^{p}u_{n+2};\alpha_{l+1},\beta_{m+1}) = 0.\end{aligned}$$ which may also be used to find $u_{n+3}$. In general, if $\alpha_{l+2}=\alpha_l$ and $\beta_{m+1}=\beta_m$, then the reduction is consistent if $T$ is chosen to be a symmetry of . In particular, if $\alpha_l = \alpha$ and $\beta_m = \beta$, are constants the resulting reductions are autonomous 3-dimensional mappings.
To pass to the non-autonomous case, we notice[^3] that equations and only depend on the ratio $\alpha_l/\beta_m$. For such [*multiplicative*]{} equations the reductions are consistent if $\alpha_{l+2}/\beta_{m+1}=\alpha_l/\beta_m$. Using separation of variables this yields $$\label{1storder:multfull}
\dfrac{\alpha_{l+2}}{\alpha_l} = \dfrac{\beta_{m+1}}{\beta_m} := q^2,$$ which is a second order equation in $\alpha_l$ and first order in $\beta_m$. The general fully non-autonomous solution to is $$\label{H3Q1fullvars}
\alpha_l = \left\{ \begin{array}{c p{2cm}}
a_0 q^l & if $l$ is even,\\
a_1 q^l & if $l$ is odd,\end{array}\right. \hspace{1cm} \beta_m = b_0q^{2m},$$ where we may absorb $b_0$ in $a_0,a_1$, or simply take $b_0=1$. The resulting reduction may be expressed in terms of $\beta_m/\alpha_l \propto q^n$.
Equation may be written explicitly as a function of $\alpha_l-\beta_m$. For such [*additive*]{} equations, separation of variables yields $$\label{1storder:addfull}
\alpha_{l+2} - \alpha_l = \beta_{m+1} - \beta_m := 2h.$$ The general fully non-autonomous solution to is $$\label{H1fullvars}
\alpha_l = \left\{ \begin{array}{c p{2cm}}
a_0 + lh & if $l$ is even,\\
a_1 + lh & if $l$ is odd,\end{array}\right. \hspace{1cm} \beta_m = b_0 + 2hm.$$ Here we may, without loss of generality, take $b_0=0$. In the additive case, the reduction will depend on $\alpha_l-\beta_m$ which depends linearly on the variable $n=2m-l$.
For both these additive and multiplicative equations, the special reductions where $a_i$ and $b_i$ do not depend on $i$ will be called simply non-autonomous. For the fully non-autonomous reductions the shift $n \to n+1$ has the effect of swapping the roles of $a_0$ and $a_1$. We have two options here: either to introduce a second root of unity, or to consider the second iterate of the map. We choose the second option in this paper.
Twist Matrices and Lax representations {#sec:Twist}
======================================
In this section we will provide a method to construct Lax representations for twisted reductions. Firstly, let us consider a Lax pair for a lattice equation given by a pair of linear difference equations
\[LaxQ\] $$\begin{aligned}
\label{LaxL}\Psi_{l+1,m}(\gamma) &= L_{l,m}(\gamma) \Psi_{l,m}(\gamma),\\
\label{LaxM}\Psi_{l,m+1}(\gamma) &= M_{l,m}(\gamma) \Psi_{l,m}(\gamma),\end{aligned}$$
where $\gamma$ is a spectral parameter. This is a Lax pair in the sense that the compatibility condition between and , which can be written as $$\label{CompLM}
L_{l,m+1}M_{l,m} - M_{l+1,m}L_{l,m} = 0,$$ is equivalent to imposing . For 3D-consistent equations of the form , cf. [@ABS:ListI; @ABS:ListII], the matrices $L_{l,m}$ and $M_{l,m}$ are explicitly given in terms of derivatives of the function $Q$ [@OvdKQ:reductions Equation 1.10]. Therefore, and the importance of this will be apparent later on, because and are functions of $\alpha_l/\beta_m$, the Lax matrices for and , $L_{l,m}$ and $M_{l,m}$, will be functions of $\alpha_l/\gamma$ and $\beta_m/\gamma$ respectively. Similarly the Lax matrices $L_{l,m}$ and $M_{l,m}$ for are functions of $\alpha_l - \gamma$ and $\beta_m - \gamma$ respectively.
To arrive at a particular form of the Lax pairs, we will sometimes perform a gauge transformation. For example, if the reduced equations can be dimensionally reduced by choosing special variables, one would like to also express the Lax pair in terms of these variables. Then one considers $$\label{gauge}
\Psi_{l,m}' = Z_{l,m} \Psi_{l,m}.$$ The linear system satisfied by $\Psi_{l,m}'$ is
\[guagetransformedLax\] $$\begin{aligned}
\Psi_{l+1,m}' = (Z_{l+1,m} L_{l,m} Z_{l,m}^{-1})\Psi_{l,m}' = L_{l,m}'\Psi_{l,m}',\\
\Psi_{l,m+1}' = (Z_{l,m+1} M_{l,m} Z_{l,m}^{-1}) \Psi_{l,m}'= M_{l,m}'\Psi_{l,m}'.\end{aligned}$$
In a slight abuse of notation, we will not distinguish between the pair $(L_{l,m},M_{l,m})$ and $(L_{l,m}',M_{l,m}')$.
Before we turn to the key ansatz we make in order to derive Lax pairs for the reduction, one must realise that if a solution to the lattice equation is known, $w_{l,m}$ for all $l,m \in \mathbb{Z}$, one can obtain a fundamental solution of the linear problem. Relating the behaviour of solutions of the nonlinear partial differential equation with its spectral problem plays a fundamental role in inverse scattering methods for partial differential equations [@AblowitzSegur]. The discrete analogue of this theory for systems of difference equations has also been studied [@DISPI; @DISPII] and applied to a system of the form by Butler et al. [@Butler:scatteringKdV]. Our key anstatz is based on a relation between the solutions of systems defined by and solutions of .
Let us start with the autonomous case. Given the fact that any solution, $w_{l,m}$, lifts to a solution of the linear problem, we may lift a solution satisfying to a system that is now dependent on the variables $u_n$. That is to say, we have a solution to some linear system $$\Psi_{l,m}(\gamma;\{w_{l,m}\}) \mapsto Y_n(\gamma;\{u_n\}).$$ We proceed as per usual, and construct operators, $A_n$ and $B_n$, which are equivalent to shifts in $l$ and $m$ given by $(l,m) \to (l+2,m+1)$, and $(l,m) \to (l+1,m+1)$, respectively. These are given by the products
$$\begin{aligned}
\label{prodA}A_n(\gamma) &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}L_{l+1,m+1}L_{l,m+1}M_{l,m},\\
\label{prodB}B_n(\gamma) &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}L_{l,m+1}M_{l,m}.\end{aligned}$$
The matrix $A_n$ is called the monodromy matrix. It corresponds to a path, in Figure \[fig:label\], from $u_n$ to $Tu_n$, going up one step and to the right two steps. We note that, in general, the matrix $B_n$ is a particular factor of $A_n$, namely the one that corresponds to the shift $n\mapsto n+1$. The function $Y_n(\gamma;\{u_n\})$ now satisfies the equation
$$\begin{aligned}
\label{defA}T Y_{n}(\gamma;\{u_n\}) &= A_n(\gamma) Y_n(\gamma;\{u_n\})\\
\label{defB}Y_{n+1}(\gamma;\{u_n\}) &= B_n(\gamma) Y_n(\gamma;\{u_n\}),\end{aligned}$$
where, from the above, we may lift our symmetry, $T$, to the level of the linear problem via application on the $w_{l,m}$ (or equivalently on $u_n$).
Our [*key ansatz*]{} is that there is the additional relation $$\label{twistmatrix}
Y_n(\gamma;\{T u_n\}) Y_n(\gamma;\{u_n\})^{-1} = S_n(\{u_n\}),$$ where $S_n$ does not depend on the spectral parameter. This rather innocuous looking relation implies that the singularities of $Y_n$, as a function of the spectral parameter $\gamma$, are independent of any particular solution of the lattice equation. That is, the singularities and poles of $Y_n(\gamma;\{T u_n\})$ are cancelled out by the poles and singularities of $Y_n(\gamma;\{u_n\})^{-1}$ to give a constant matrix, $S_n(\{u_n\})$, which we call the [*twist matrix*]{}. For all examples of twisted reductions provided, we have been able to obtain such twist matrices.
Now, combining the two equations and we obtain the first half of a standard Lax pair for an autonomous mapping $$\begin{aligned}
\label{DSA}Y_n(\gamma) = S_n^{-1} A_n(\gamma) Y_n(\gamma).\end{aligned}$$ where the other half of the Lax pair is . The compatibility between and , which is equivalent to the autonomous reduction, is given by $$\label{autcomp}
S_{n+1}^{-1} A_{n+1}(\gamma) B_n(\gamma) - B_n(\gamma) S_n^{-1} A_n(\gamma) = 0,$$ and integrals for this reduction can be obtained by taking the trace of the twisted monodromy matrix $S_n^{-1} A_n(\gamma)$.
While $A_n$ and $B_n$ are determined by and , the task of determining $S_n$ remains. As is typical in integrable systems, the linear system is overdetermined, which gives us a straightforward, albeit complicated, way of calculating $S_n$. The complication arises because one needs to simultaneously calculate the twist matrix and the evolution equation from the compatibility condition, thereby increasing the number of conditions that need to be satisfied without increasing the number of relations from the compatibility. However, there is a simpler way to calculate $S_n$; observe that when we use , and , we get $$\begin{aligned}
T Y_{n+1} = T(B_n)A_n Y_n = A_{n+1} B_n Y_n.\end{aligned}$$ Rewriting yields the relation $$A_{n+1} B_n = S_{n+1} B_n S_n^{-1} A_n.$$ By combining these equations, and by cancelling irrelevant factors, we obtain $$\label{calcS}
T(B_n)S_n = S_{n+1} B_n,$$ which gives us an elegant way of calculating the twist matrix $S_n$ and $S_{n+1}$, that does not rely explicitly on using the reduction. We will see that for the examples provided, the twist matrices are actually quite succinct. Furthermore, they tend to the identity matrix in the limit where the twists tends to the identity transformation.[^4]
The non-autonomous case is a simple generalisation of the above, since nothing we did relied upon any of the properties of $\alpha_l$ or $\beta_m$. We just need to specify a new spectral parameter for our reduced system. For the multiplicative equations, and , we know that the $L_{l,m}$ and $M_{l,m}$ matrices are functions of $\alpha_l/\gamma$ and $\beta_m/\gamma$ respectively, which for our choices of parameters , can both be written in terms of $q^l/\gamma$ and $q^n$ only. This motivates the choice $$\label{spectralchoice}
x = q^l/\gamma,$$ as our spectral parameter. This implies that the shifts $(l,m) \to (l+2,m+1)$ and $(l,m) \to (l+1,m+1)$ both have the effect of translating $x$. As in the autonomous case, we may write $A_n(x)$ and $B_n(x)$ as products of matrices $L_{l,m}$ and $M_{l,m}$:
\[nonautABprod\] $$\begin{aligned}
\label{prodnA}A_n(x) &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}L_{l+1,m+1}L_{l,m+1}M_{l,m},\\
\label{prodnB}B_n(x) &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}L_{l,m+1}M_{l,m}.\end{aligned}$$
where the linear problem, which is now in $x$, satisfies the equations
\[LaxYABm\] $$\begin{aligned}
\label{defnA}T Y_n(q^2x) = A_n(x) Y_n(x),\\
\label{defnB}Y_{n+1}(qx) = B_n(x) Y_n(x),\end{aligned}$$
where, for the same reasons as above, we have the additional relation $$\label{nonauttwistmatrix}
T Y_n(x) = S_n Y_n(x).$$ This means our compatibility may be written $$\label{compnonautmult}
S_{n+1}^{-1} A_{n+1}(qx) B_n(x) - B_n(q^2x) S_n^{-1} A_n(x) = 0,$$ where $S_n$ is actually the same twist matrix as in the autonomous case.
For the additive equation, , the Lax matrices, $L_{l,m}$ and $M_{l,m}$, are functions of $\alpha_l - \gamma$ and $\beta_m-\gamma$ respectively. For the non-autonomous parameter choice, , $L_{l,m}$ and $M_{l,m}$ are both functions of $hl-\gamma$ and $nh$. This motivates the definition $$\label{spectralchoiceadd}
x = hl - \gamma.$$ Using the same product formulas for $A_n(x)$ and $B_n(x)$, given by , the matrix $Y_n(x)$ satisfies the equations
$$\begin{aligned}
\label{defaA}T Y_n(x+2h) = A_n(x) Y_n(x),\\
\label{defaB}Y_{n+1}(x+h) = B_n(x) Y_n(x),\end{aligned}$$
and also equation . This means that the compatibility yields $$\label{compnonautadd}
S_{n+1}^{-1} A_{n+1}(x+h) B_n(x) - B_n(x+2h) S_n^{-1} A_n(x) = 0.$$ In the following three sections we will provide the details for the (2,1)-reductions of our three examples to demonstrate this theory. We postpone the theory for general ($s_1,s_2$)-reduction to §\[s1s2\].
Reductions of the lattice modified Korteweg-de Vries equation {#sec:dmKdV}
=============================================================
The discrete modified Korteweg-de Vries equation (aka $H3_{\delta =0}$) , given by , was one of the earliest known integrable lattice equations. It appeared as a discrete analogue of the sine-Gordon equation (equivalent under a transformation) in the work of Hirota [@Hirota:dSG] and its Lax pair was derived using direct linearisation [@QNCvdL:NSG]. Reductions of this equation have been considered by many authors [@SimilarityReds; @Quispel:Intreds; @Gramani:Reductions; @TKQ09; @KRQ07; @HKQT13; @Hay; @Ormerod:qP6]. The equation has a Lax representation given by where the Lax matrices are
\[LaxH3\] $$\begin{aligned}
L_{l,m} (\alpha_l/\gamma)=& \begin{pmatrix}
\dfrac{\gamma }{\alpha_l} & w_{l+1,m} \\[10pt]
\dfrac{1}{w_{l,m}} & \dfrac{\gamma w_{l+1,m}}{w_{l,m} \alpha_l}
\end{pmatrix} ,\\
M_{l,m} (\beta_m/\gamma)=& \begin{pmatrix}
\dfrac{\gamma }{\beta_m} & w_{l,m+1} \\[10pt]
\dfrac{1}{w_{l,m}} & \dfrac{\gamma w_{l,m+1}}{w_{l,m} \beta_m}
\end{pmatrix}.\end{aligned}$$
We will first recall how the autonomous periodic reductions are obtained, then proceed to generalise the reductions and their Lax representations to the twisted, and non-autonomous cases. In the periodic case, is still valid, as is all the theory contained in §\[consistency\] and §\[sec:Twist\], with the specialisation to $T(w_{l,m}) = w_{l,m}$. This means that the labelling, , is simply $$w_{l,m} \mapsto u_{2m-l} = u_n.$$ In this case, the equation governing the reduction, , is given by $$\label{H1autountwisted}
u_{n+3} = \dfrac{u_n \left(\alpha u_{n+1}+\beta u_{n+2}\right)}{\alpha u_{n+2}+\beta u_{n+1}}$$ Using and we obtain the two Lax matrices, $A_n$ and $B_n$, given by $$\begin{aligned}
A_n(\gamma) =& \begin{pmatrix} \dfrac{\gamma}{\alpha} & u_n \\[10pt]
\dfrac{1}{u_{n+1}} & \dfrac{\gamma u_n}{\alpha u_{n+1}}\end{pmatrix}
B_n(\gamma),\\
B_n(\gamma) =& \begin{pmatrix} \dfrac{\gamma}{\alpha} & u_{n+1} \\[10pt]
\dfrac{1}{u_{n+2}} & \dfrac{\gamma u_{n+1}}{\alpha u_{n+2}}\end{pmatrix}
\begin{pmatrix} \dfrac{\gamma}{\beta} & u_{n+2} \\[10pt]
\dfrac{1}{u_{n}} & \dfrac{\gamma u_{n+2}}{\beta u_{n}}\end{pmatrix}.\end{aligned}$$ We note that since $T(B_n) = B_n$, equation \[calcS\] becomes $B_nS_n=B_nS_{n+1}$, where $S_n$ is a priori unknown. We parameterize $S_n$ by letting $$\label{arbitrarySn}
S_n = \begin{pmatrix} s_{1,n} & s_{2,n} \\ s_{3,n} & s_{4,n} \end{pmatrix}.$$ At the coefficient of $\gamma^2$, we obtain $$\begin{aligned}
s_{1,n+1} = s_{1,n}, \hspace{.5cm} u_n s_{2,n} = u_{n+1}s_{2,n+1}, \hspace{.5cm} u_{n+1} s_{3,n} = u_n s_{3,n+1} , \hspace{.5cm} s_{4,n+1} = s_{4,n},\end{aligned}$$ and at the coefficient of $\gamma$, we then obtain $$s_{1,n} = s_{4,n}, \hspace{1cm} s_{2,n} = 0 , \hspace{1cm} s_{3,n} = 0.$$ This tells us that we may choose $S_n = I$. Thus the twisted monodromy matrix coincides with the standard monodromy matrix, which shouldn’t come as a surprise. Taking the trace of the monodromy matrix gives us $\alpha\beta\Tr(A_n)=\frac{2}{\alpha}\gamma^3+K_{\eqref{H1autountwisted}}\gamma$ where $$K_{\eqref{H1autountwisted}}=\alpha\left(\frac{u_n}{u_{n+2}}+\frac{u_{n+2}}{u_n}\right)
+\beta\left(\frac{u_n}{u_{n+1}}+\frac{u_{n+1}}{u_n}+\frac{u_{n+1}}{u_{n+2}}+\frac{u_{n+2}}{u_{n+1}}\right)$$ is an integral, or constant of motion, of equation . One can verify that is satisfied on solutions of . This equation, under the transformation $y_n = u_{n+1}/u_n$, takes the more familiar form of a second order difference equation $$\label{mfso}
y_{n+1} y_n y_{n-1} = \dfrac{\alpha + \beta y_n}{\beta + \alpha y_n},$$ which is more clearly a mapping of QRT type [@QRT1; @QRT2]. The integral $K_{\eqref{H1autountwisted}}$ is also invariant under scaling and hence can be also written in terms of the reduced variable $y_n$, $$K_{\eqref{mfso}}=\alpha\left(y_ny_{n+1}+\frac{1}{y_ny_{n+1}}\right) +\beta\left(y_n+\frac{1}{y_n}+y_{n+1}+\frac{1}{y_{n+1}}\right).$$ This reduction appeared in [@Quispel:Intreds]. We will now give a one-parameter integrable generalisation of this reduction by considering the twisted case.
The twist we apply is given by $T(w_{l,m}) = \lambda w_{l,m}$, which means that $$w_{l,m} \mapsto \lambda^{l-m} u_{2m-l} = \lambda^p u_n.$$ Under this identification, the reduction, , is given by $$\begin{aligned}
\label{QRT2}
u_{n+3} = \dfrac{\lambda ^2 u_n \left(\alpha u_{n+1}+\beta u_{n+2}\right)}{\alpha u_{n+2}+\beta u_{n+1}},\end{aligned}$$ To obtain a Lax pair, we construct the operators $A_n$ and $B_n$, using the product representation, and , to give $$\begin{aligned}
A_n(\gamma) =& \begin{pmatrix} \dfrac{\gamma}{\alpha} & \lambda u_n \\[10pt]
\dfrac{1}{u_{n+1}} & \dfrac{\gamma \lambda u_n}{\alpha u_{n+1}}\end{pmatrix}
B_n(\gamma),\\
B_n(\gamma) =& \begin{pmatrix} \dfrac{\gamma}{\alpha} & u_{n+1} \\[10pt]
\dfrac{\lambda}{u_{n+2}} & \dfrac{\gamma \lambda u_{n+1}}{\alpha u_{n+2}}\end{pmatrix}
\begin{pmatrix} \dfrac{\gamma}{\beta} & \dfrac{u_{n+2}}{\lambda} \\[10pt]
\dfrac{1}{u_{n}} & \dfrac{\gamma u_{n+2}}{\beta \lambda u_{n}}\end{pmatrix}.\end{aligned}$$ We play the same game, where $S_n$ is a priori unknown, hence, we let $S_n$ be given by . The coefficient of $\gamma^2$ in gives us the same conditions as in the periodic case, and at the coefficient of $\gamma$ we find $$s_{1,n} = \lambda s_{4,n}, \hspace{1cm} s_{2,n} = 0 , \hspace{1cm} s_{3,n} = 0.$$ This gives us our first non-trivial twist matrix, given by $$\label{H3Sn}
S_n = \begin{pmatrix} \lambda & 0 \\ 0 & 1 \end{pmatrix}.$$ Taking the trace of the twisted monodromy matrix gives us $$\alpha\beta\Tr(S_n^{-1}A_n)=\left(\frac{1}{\lambda\alpha}+\frac{\lambda}{\alpha}\right)\gamma^3+K_{\eqref{QRT2}}\gamma,$$ where $$K_{\eqref{QRT2}}=\alpha\left(\frac{\lambda u_n}{u_{n+2}}+\frac{u_{n+2}}{\lambda u_n}\right)
+\beta\left(\frac{\lambda u_n}{u_{n+1}}+\frac{u_{n+1}}{\lambda u_n}+\frac{\lambda u_{n+1}}{u_{n+2}}+\frac{u_{n+2}}{\lambda u_{n+1}}\right)$$ is an integral for . In the limit as $\lambda \to 1$, we retrieve the periodic case, making this a nice one-parameter family of reductions and their Lax pairs and integrals. This provides all the elements for to give . Once again, by identifying $y_n = u_{n+2}/u_{n+1}$, we have the classic QRT map $$\label{cqrt}
y_{n+1}y_ny_{n-1} = \dfrac{\lambda^2 (\alpha + \beta y_n)}{\beta + \alpha y_n},$$ with corresponding integral obtained from $K_{\eqref{QRT2}}$. Thus we have obtained a one parameter generalisation of the reduction, , found in [@Quispel:Intreds].
When we turn to the simply non-autonomous case, we obtain a version of $q$-$\mathrm{P}_{II}$. In taking $\alpha_l=aq^l$ and $\beta_m=q^{2m}$, we need to take into account the position of the square we use to evaluate the reduction. With respect to Figure \[labelling\], if the square whose lower left entry is $u_n$ denotes $(l,m)$, the relevant square used for is at $(l+1,m+1)$ Thus we obtain the reduction $$\label{qP2u}
u_{n+3}=\dfrac{\lambda ^2 u_n \left(a u_{n+1}+ q^{n+1} u_{n+2}\right)}{a u_{n+2}+ q^{n+1} u_{n+1}}.$$ We now use in our product representation for $A_n(x)$ and $B_n(x)$, to obtain $$\begin{aligned}
A_n(x) &= \begin{pmatrix} \dfrac{1}{qxa} & \lambda u_n \\[10pt]
\dfrac{1}{u_{n+1}} & \dfrac{\lambda u_n }{qxau_{n+1}} \end{pmatrix}
B_n(x) ,\\
B_n(x) &= \begin{pmatrix} \dfrac{1}{xa} & u_{n+1} \\[10pt]
\dfrac{\lambda}{u_{n+2}} & \dfrac{\lambda u_{n+1} }{xau_{n+2}} \end{pmatrix}
\begin{pmatrix} \dfrac{1}{xq^n} & \dfrac{u_{n+2}}{\lambda} \\[10pt]
\dfrac{1}{u_{n}} & \dfrac{u_{n+2} }{x\lambda q^n u_{n}} \end{pmatrix}.\end{aligned}$$ We use the form once more, and the calculations follow analogously to the previous case and give . With $A_n(x)$, $B_n(x)$ and $S_n$ defined, the compatibility, , gives . Furthermore, by letting $y_n = u_{n+2}/u_{n+1}$, we find a more direct correspondence with a $q$-analogue of the second Painlevé equation found in [@Gramani:coalescences], $$y_{n+1}y_ny_{n-1} = \dfrac{\lambda^2 (a + q^{n+1} y_n)}{q^{n+1} + a y_n},$$ which generalizes a reduction of Nijhoff and Papageorgiou [@SimilarityReds]. At this point, we note that we may use alternative Lax matrices to . By considering a transformation of the form , where $$Z_{l,m} = \begin{pmatrix} \dfrac{1}{w_{l,m}} & 0 \\ 0 & 1 \end{pmatrix}$$ we obtain a Lax pair given by $$\begin{aligned}
L_{l,m} =& \begin{pmatrix}
\dfrac{\gamma w_{l,m}}{\alpha_l w_{l+1,m}} & 1 \\[10pt]
1 & \dfrac{\gamma w_{l+1,m}}{\alpha_l w_{l,m}}
\end{pmatrix} ,\\
M_{l,m} =& \begin{pmatrix}
\dfrac{\gamma w_{l,m}}{\beta_m w_{l,m+1}} & 1 \\[10pt]
1 & \dfrac{\gamma w_{l,m+1}}{\beta_m w_{l,m}}
\end{pmatrix} .\end{aligned}$$ Notice that these matrices are actually invariant under the uniform application of the transformation $w_{l,m} \to T(w_{l,m})$. Since all the variables $w_{l,m}$ in $L_{l,m}$ and $M_{l,m}$ appear in ratios, the Lax pair may be expressed in the variables $y_n = u_{n+2}/u_{n+1}$. In this light, we write an alternative set of Lax matrices $$\begin{aligned}
A_n(x) =& \begin{pmatrix}
\dfrac{y_{n-1}}{qx\lambda a} & 1 \\[10pt] 1 & \dfrac{\lambda}{qxa y_{n-1}} \end{pmatrix}
B_{n}(x),\\
B_{n}(x) =& \begin{pmatrix}
\dfrac{y_{n}}{x\lambda a} & 1 \\[10pt] 1 & \dfrac{\lambda}{xa y_{n}} \end{pmatrix}
\begin{pmatrix}
\dfrac{\lambda}{q^nxy_n y_{n-1}} & 1 \\[10pt] 1 & \dfrac{y_n y_{n-1}}{q^nx\lambda} \end{pmatrix},\end{aligned}$$ and twist matrix $S_n = I$. This is an immediate consequence of the fact that $y_n$ is an invariant of $T$: $T(y_n) = T(u_{n+2})/T(u_{n+1}) = u_{n+2}/u_{n+1} = y_n$.
The last case to do is the fully non-autonomous generalisation, where $\alpha_l$ and $\beta_m$ are given by with $b_0=1$. It should be noted that the resulting equation governing $n \to n+1$ turns an even $l$ into an odd $l$, hence, the evolution equation incorporates a change in $a_0$ and $a_1$. With this in mind, the evolution equation is given by combined with a change in $a_0$ and $a_1$: in the case that $n$ (and hence, $l$) is even, $u_{n+3}$ is calculated from $$\begin{aligned}
\label{H3singleev}
u_{n+3} = \dfrac{\lambda ^2 u_n \left(a_1 u_{n+1}+q^{n+2} u_{n+2}\right)}{a_1 u_{n+2}+q^{n+2} u_{n+1}}, \hspace{1cm} a_0 \to \dfrac{a_1}{q}, \hspace{1cm} a_1 \to q a_0.\end{aligned}$$ This system possesses a Lax pair of the form , where the Lax matrices are given by products and . The shift $n \to n+2$ has an alternative deformation matrix, given by $B_n(x) {\mathrel{\reflectbox{\ensuremath{\mapsto}}}}M_{l,m}$, which simplifies the calculation. If we let $A_n(x)$ be given by the product , we obtain $$\begin{aligned}
A_n(x) &= \begin{pmatrix} \dfrac{1}{xa_1} & \lambda u_n \\[10pt] \dfrac{1}{u_{n+1}} & \dfrac{\lambda u_n }{xa_1u_{n+1}} \end{pmatrix}
\begin{pmatrix} \dfrac{1}{xa_0} & u_{n+1} \\[10pt] \dfrac{\lambda}{u_{n+2}} & \dfrac{\lambda u_{n+1} }{xa_0u_{n+2}} \end{pmatrix}
B_n(x) ,\\
B_n(x) &= \begin{pmatrix} \dfrac{1}{xq^n} & \dfrac{u_{n+2}}{\lambda} \\[10pt] \dfrac{1}{u_{n}} & \dfrac{u_{n+2} }{x\lambda q^n u_{n}} \end{pmatrix}.\end{aligned}$$ By using and we once again obtain . Here the compatibility condition is $$\label{compnonautmult2}
S_{n+1}^{-1} A_{n+1}(x) B_n(x) - B_n(q^2x) S_n^{-1} A_n(x) = 0,$$ which we use to obtain . However, this is not as obviously a two dimensional mapping. We employ a technique used in [@OvdKQ:reductions] to rewrite this equation. We take $A_n(x)$ and evaluate the root of the upper right entry (in $x^2$), denoting this $y$. The determinant of $A_n(\sqrt{y})$ factors nicely, and the factors of $\det A_n(\sqrt{y})$ appear in the diagonal entries, in addition to a simple multiplicative factor, which we denote $z_n$. Explicitly, modulo some scaling, these variables are
$$\begin{aligned}
y_n =& \dfrac{a_1u_{n+1}}{u_n} + \dfrac{a_0 u_{n+1}u_{n+2}+ q^n u_n u_{n+2} }{ \lambda^2 u_n^2} ,\\
z_n =&u_{n+1}\left(\dfrac{a_1\lambda u_{n+1}}{u_{n+2}}+\dfrac{a_0}{\lambda u_n}\right),\end{aligned}$$
which then satisfy the difference equations
\[H3nonautred\] $$\begin{aligned}
y_n y_{n+2}=&\left(\lambda q^{n+2}+z_n\right) \left(\dfrac{q^n}{\lambda}+
z_n\right)\\
z_n z_{n+2}=&\dfrac{\left(a_1 q^{n+2}+a_0 y_{n+2}\right) \left(a_0 q^{n+2}+a_1 y_{n+2}\right)}{\left(a_0 a_1+ q^{n+2} y_{n+2}\right)}.\end{aligned}$$
This equation first appeared in the work of Ramani et al. [@Quadratic] and is related, via a Miura transformation, to a version of $q$-$\mathrm{P}_{III}$ found in [@AsymmetricdPs]. This equation has a symmetry group which is of affine Weyl type $A_2^{(1)} + A_1^{(1)}$ [@Hunting; @Sakai:rational].
Another possible choice of twist is $T_2: w \to \lambda/w$, which is is not homotopic to the identity. The twist matrix associated with ($s_1,s_2)$-reductions of with fixed Lax representation, , is $$S_n = \begin{pmatrix} 0 & \lambda \\ 1 & 0 \end{pmatrix},$$ for a large class of $s_1$ and $s_2$. We remark that twist matrices are not gauge invariant.
Reductions of the lattice potential Korteweg-de Vries equation {#sec:dpKdV}
==============================================================
The lattice potential Korteweg-de Vries equation (aka $H1$), , was derived from the direct linearisation approach [@Nijhoff:dSKdV], and it yields the potential Korteweg-de Vries equation in a continuum limit. Periodic reductions of were considered by many authors [@dKdVreds; @SimilarityReds; @Quispel:Intreds; @Nijhoff:dSKdVP6; @KQ10; @HKQT13]. The $(2,1)$-periodic non-autonomous reduction and its Lax pair were recently given in [@OvdKQ:reductions].
The equation has a Lax representation given by where the Lax matrices are given by
\[LaxH1\] $$\begin{aligned}
L_{l,m}=& \begin{pmatrix} w_{l,m} & -\gamma + \alpha_l - w_{l,m} w_{l+1,m} \\
1 & - w_{l+1,m}
\end{pmatrix},\\
M_{l,m} =& \begin{pmatrix} w_{l,m} & -\gamma + \beta_m - w_{l,m} w_{l,m+1} \\
1 & - w_{l,m+1}
\end{pmatrix}.\end{aligned}$$
The twist that we seek to apply is the transformation $T(w_{l,m}) = w_{l,m} + \lambda$, which means our reduced variables are specified by $$w_{l,m} \mapsto u_{2m-l} + (l-m)\lambda = u_n + p\lambda.$$ For the twisted autonomous case, where $\alpha_l = \alpha$ and $\beta_m = \beta$ are constants, it is clear that we obtain the difference equation $$\label{QRT1}
(u_{n} - u_{n+3} + 2\lambda)(u_{n+1}-u_{n+2}) = \alpha-\beta.$$ The Lax pair for this autonomous equation may be specified by and , where the lattice variables take on their reduced values, giving $$\begin{aligned}
A_n(\gamma) =& \begin{pmatrix}
u_{n+1} & \alpha - \gamma - (\lambda+u_{n})u_{n+1}\\
1 & -(\lambda+u_n)
\end{pmatrix}B_n(\gamma),\\
B_n(\gamma) =& \begin{pmatrix}
-\lambda + u_{n+2} & \alpha - \gamma - (u_{n+2}-\lambda)u_{n+1}\\
1 & -u_{n+1}
\end{pmatrix}\begin{pmatrix}
u_n & \beta - \gamma - (u_{n+2}-\lambda)u_n\\
1 & \lambda - u_{n+2}
\end{pmatrix}.\end{aligned}$$ We now need to calculate $S_n$, which is once again, a priori, an unknown function of $n$, hence, we label the elements of $S_n$ by . By utilizing , at the level of the coefficient of $\gamma$, we find $S_{n+1} = S_n$. Solving for the constant coefficient of gives us that if $S_n$ is independent of $n$, then $s_{3,n} =0$ and $s_{2,n} = \lambda s_{1,n}$ and $s_{4,n} = s_{1,n}$, which we may simplify to give the second non-trivial twist matrix in this study, given by $$\label{H1Sn}
S_n = \begin{pmatrix}
1 & \lambda \\
0 & 1
\end{pmatrix} .$$ Knowing $A_n(\gamma)$, $B_n(\gamma)$ and $S_n$ gives us all the necessary ingredients for calculating the compatibility, , which gives us the required mapping, . Calculating the trace of the twisted monodromy matrix, $\Tr(S_n^{-1}A_n)=2\lambda\gamma+K_{\eqref{QRT1}}$, we obtain an integral, $$\begin{aligned}
K_{\eqref{QRT1}}=\alpha(u_n-u_{n+2})&+\beta(u_{n+2}-u_n-2\lambda)\label{K52}\\
&+(u_{n+1}-u_{n+2})(u_n-u_{n+1})(u_{n+2}-u_n-2\lambda).\notag\end{aligned}$$ Note that once again $S_n$ has the property that as $\lambda \to 0$, $S_n \to I$, giving the periodic case.
To simply de-autonomize the lattice equation and the Lax pair, we let $\alpha_l=a+lh$ and $\beta_m=2mh$, in which case the reduction, , becomes $$\label{dP1u}
u_{n+3} - u_n = \dfrac{a -hn -h}{u_{n+2}-u_{n+1}} + 2 \lambda,$$ which we may transform to be a function of $y_n = u_{n+2}-u_{n+1}$, giving $$\label{dP1}
y_{n+1} + y_n + y_{n-1} = \dfrac{a -hn -h}{y_n} + 2 \lambda.$$ This is a form of $\mathrm{d}$-$\mathrm{P}_{I}$ (see [@DPS]) and generalizes the reduction found in [@OvdKQ:reductions]. Furthermore, the method we present also gives us the Lax pair for this reduction. We specify our spectral parameter, given by , and construct $A_n(x)$ and $B_n(x)$ via their product representations, and , to give $$\begin{aligned}
A_n(x) =& \begin{pmatrix}
u_{n+1} & h+x+a - u_{n+1}(u_n+\lambda) \\
1 & -u_n-\lambda
\end{pmatrix} B_n(x),\\
B_n(x) =& \begin{pmatrix}
u_{n+2}-\lambda & x+a - u_{n+1}(u_{n+2}-\lambda) \\
1 & \lambda-u_{n+2}
\end{pmatrix}
\begin{pmatrix}
u_n & hn+x - u_{n}(u_{n+2}-\lambda) \\
1 & \lambda- u_{n+2}
\end{pmatrix}.\end{aligned}$$ Once again, we assume that $S_n$ is unknown, hence, we let $S_n$ be given by . Then, using , we find that $S_n$ is given by . This gives us all the required elements of , which in turn, gives us .
As in the modified Korteweg-de Vries reduction, it is possible to apply a transformation of the form of , where $$Z_{l,m} = \begin{pmatrix} 1 & w_{l,m} \\ 0 & 1 \end{pmatrix}$$ to give the alternative Lax matrices, $$\begin{aligned}
L_{l,m}&=
\begin{pmatrix}
w_{l,m}-w_{l+1,m} & (w_{l,m}-w_{l+1,m})^2+\alpha_l-\gamma \\
1 & w_{l,m}-w_{l+1,m}
\end{pmatrix},\\
M_{l,m} &=
\begin{pmatrix}
w_{l,m}-w_{l,m+1} & (w_{l,m}-w_{l,m+1})^2+\beta_m-\gamma \\
1 & w_{l,m}-w_{l,m+1}
\end{pmatrix},\end{aligned}$$ which have the desirable property of being expressed in terms of differences of the variables $w_{l,m}$. This means, these matrices admit a parameterisation in terms of Painlevé variables, $y_n =u_{n+2}-u_{n+1}$, $$\begin{aligned}
A_n(x) =& \begin{pmatrix} y_{n-1} - \lambda & a + x+ h+ (\lambda-y_{n-1})^2 \\
1 & y_{n-1} - \lambda \end{pmatrix}
B_n(x) ,\\
B_n(x) =&\begin{pmatrix} y_{n} - \lambda & a + x + (\lambda-y_{n})^2 \\
1 & y_{n} - \lambda \end{pmatrix}
\begin{pmatrix} \lambda - y_{n-1}-y_n & x + nh + (\lambda-y_{n-1}-y_n)^2 \\
1 & \lambda -y_n - y_{n-1}\end{pmatrix}.\end{aligned}$$ We note that the transformation, $T$, applied to $y_n$ is trivial, just as in the previous section. This gives us that $S_n = I$, and the compatibility gives us . This is not the first Lax pair known for equation , as a $3\times 3$ Lax pair was derived in the work of Papageorgiou et al. [@Gramani:Isomonodromic]. We do not know whether a $2\times 2$ Lax pair, such as the one presented, is known or not.
We now turn to the fully non-autonomous twisted periodic reduction, where the $\alpha_l$ and $\beta_m$ variables are given by , with $b_0=0$. It was recently noted that the fully non-autonomous periodic reduction may be identified as a special case of the discrete analogue of the fourth Painlevé equation [@OvdKQ:reductions]. We expect this to be the case again.
As before, the evolution equations must take into account the way in which the $n \to n+1$ shift changes $l$ from an even number to an odd number, because the roles of $a_0$ and $a_1$ change every single iteration. The evolution equation, , in this case is given by $$\label{eqPI}
u_{n+3}-u_n = \frac{- a_1+h n+ h}{u_{n+1}-u_{n+2}}+2 \lambda, \hspace{.7cm} a_0 \to a_1 - h, \hspace{.7cm} a_1 \to a_0 +h.$$ Once again, it is not obvious that the mapping associated with the shift $n \to n+2$ is a two dimensional mapping. But we can find reduced variables $y_n$ and $z_n$, by exploiting the Lax matrices for the equation, which are $$\begin{aligned}
A_n(x) =& \begin{pmatrix} u_{n+1} & x+a_1-u_{n+1}(u_n+\lambda) \\
1 & -u_n-\lambda \end{pmatrix}
B_n(x) ,\\
B_n(x) =&\begin{pmatrix} u_{n+2}-\lambda & x+a_0-u_{n+1}(u_{n+2}-\lambda) \\
1 & -u_{n+1} \end{pmatrix} \begin{pmatrix} u_n & x + nh - u_n(u_{n+2}-\lambda) \\
1 & \lambda-u_{n+2} \end{pmatrix}.\end{aligned}$$ The variables are explicitly given by $$\begin{aligned}
y_n =& -a_0+\left(u_n-u_{n+1}\right) \left(2 \lambda +u_n-u_{n+2}\right),\\
z_n =& \dfrac{a_0+y_n}{u_n-u_{n+1}}.\end{aligned}$$ These two functions of the lattice variables satisfy
$$\begin{aligned}
y_{n+2} + y_n=& z_n(z_n - 2 \lambda)-a_0-a_1,\\
z_{n+2}z_n =& -\dfrac{(y_{n+2}+a_0)(y_{n+2} + a_1)}{y_{n+2} + h(n+2)},\end{aligned}$$
which is a discrete version of the fourth Painlevé equation found in [@Gramani:coalescences; @Quadratic]. This is a one parameter family of reductions that generalizes the one presented in [@OvdKQ:reductions].
On the other hand, the equation (\[eqPI\]) is equivalent to asymmetric d-$\mathrm{P}_{\mathrm{I}}$, see [@Quadratic Equation 3.33], where a relation to $d$-$\mathrm{P}_{IV}$ was obtained through a quadratic transformation. In fact, taking $$\alpha_l=a_1+a_2(-1)^l+hl,$$ instead of (\[H1fullvars\]), the equation then becomes $$u_{n+3} - u_n = \frac{h(n+1)-a_1+a_2(-1)^n}{u_{n+1}-u_{n+2}} + 2\lambda,$$ or, if we let $y_n = u_{n+1} - u_n$, this becomes $$y_{n+1} + y_n + y_{n-1} = \dfrac{hn - a_1 -a_2(-1)^n}{y_n} + 2\lambda$$ which is the most general form of $d$-$\mathrm{P}_{I}$ [@DPS]. In the autonomous limit, taking $h=0$, which would correspond to the “fully autonomous case", the equation admits the following integral: $$K_{\eqref{QRT1}} + a_2(-1)^n(2u_{n+1}-u_n-u_{n+2}),$$ where $K_{\eqref{QRT1}}$ is given in (\[K52\]), taking $\alpha=a_1$, and $\beta=0$.
Just as we did for , we present a twist matrix for a twist that is not homotopic to the identity twist, namely the twist $T_2 : w \to \lambda - w$. The twist matrix associated with ($s_1,s_2)$-reductions of with a fixed Lax representation, , is $$S_n = \begin{pmatrix} -1 & -\lambda \\ 0 & 1 \end{pmatrix},$$ for a number of different choices of $s_1$ and $s_2$. This twist also yields a class of integrable mappings and their Lax representations.
Reductions of the lattice Schwarzian Korteweg-de Vries equation {#sec:dSKdV}
===============================================================
Periodic reductions of the lattice Schwarzian Korteweg-de Vries equation (aka $Q1_{\delta = 0}$), given by , have been the subject of a number of studies [@Hay:Q1; @Nijhoff:dSKdVP6; @TKQ09]. Most recently, three of the authors considered periodic reductions that gave rise to $q$-$\mathrm{P}_{VI}$ and $q$-$\mathrm{P}(A_2^{(1)})$ [@OvdKQ:reductions].
A Lax pair for equation is of the form where the Lax matrices are $$\begin{aligned}
L_{l,m} =& \begin{pmatrix}
1 & w_{l,m}-w_{l+1,m} \\[10pt]
\dfrac{\alpha }{\gamma (w_{l,m}- w_{l+1,m})} & 1
\end{pmatrix},\\
M_{l,m} =& \begin{pmatrix}
1 & w_{l,m}-w_{l,m+1} \\[10pt]
\dfrac{\beta }{\gamma(w_{l,m}- w_{l,m+1})} & 1
\end{pmatrix}.\end{aligned}$$ From our perspective, is of particular interest, as it is invariant under the full group of Möbius transformations, denoted $\mathrm{PGL}(2,\mathbb{C})$. We parameterise each Möbius transformation in terms of its fixed points, $\tau_1$ and $\tau_2$, and the eigenvalues, $\lambda_1$ and $\lambda_2$, of a corresponding matrix, as follows: $$T(w) = \dfrac{(\lambda_1\tau_1- \lambda_2\tau_2)w - (\lambda_1 -
\lambda_2)\tau_1\tau_2}{(\lambda_1-\lambda_2)w +
\lambda_2\tau_1-\lambda_1\tau_2}.$$ The reduced variables are given nicely in terms of $\tau_1$, $\tau_2$, $\lambda_1$ and $\lambda_2$ as $$w_{l,m} \mapsto T^{l-m}u_{2m-l} = T^p u_n = \dfrac{(\lambda_1^p\tau_1-
\lambda_2^p\tau_2)u_n - (\lambda_1^p -
\lambda_2^p)\tau_1\tau_2}{(\lambda_1^p-\lambda_2^p)u_n +
\lambda_2^p\tau_1-\lambda_1^p\tau_2}.$$ It will often be more notationally convenient to use the symbolic notation $T^p
u_n$ over the explicit expression for obvious reasons. In the autonomous case, where $\alpha_l = \alpha$ and $\beta_m = \beta$ are constants, the reduced equation may be expressed as $$\label{Q1Aut}
u_{n+3} = \dfrac{\alpha T^2u_n Tu_{n+1}-Tu_{n+2} ((\alpha
-\beta ) Tu_{n+1}+\beta T^2u_{n})}{(\alpha -\beta )
T^2u_{n}-\alpha
Tu_{n+2}+\beta Tu_{n+1}}.$$ We form the Lax pair in the usual manner, where and give us the following representations for $A_n(\gamma)$ and $B_n(\gamma)$: $$\begin{aligned}
A_n(\gamma) =& \begin{pmatrix} 1 & u_{n+1} - T u_n \\[10pt]
\dfrac{\alpha}{\gamma(u_{n+1} - T u_n)} & 1 \end{pmatrix} B_n,\\
B_n(\gamma) =& \begin{pmatrix} 1 & T^{-1}u_{n+2} - u_{n+1} \\[10pt]
\dfrac{\alpha}{\gamma(T^{-1}u_{n+2} - u_{n+1})} & 1 \end{pmatrix}
\begin{pmatrix} 1 & u_n - T^{-1}u_{n+2} \\[10pt] \dfrac{\beta}{\gamma(u_n
- T^{-1}u_{n+2})} & 1 \end{pmatrix}.\end{aligned}$$ The calculation of the twist matrix is algebraically more difficult than in the previous cases, but essentially follows the same logic. That is, we let $S_n$ be given by and use at the various coefficients. The calculations are much simpler if one assumes , but it is not necessary to do so. It is also useful to compare the iterates of the entries of $S_n$ with the calculated values for $S_{n+1}$. This gives us our third non-trivial twist matrix, associated with the Möbius transformation, given by $$\label{twistQ1}
S_n = \begin{pmatrix}
\dfrac{\lambda_1\lambda_2 (\tau_1-\tau_2)}{\lambda_1(u_n-\tau_2)
-\lambda_2(u_n-\tau_1)} & 0\\[10pt]
\dfrac{\lambda_1-\lambda_2}{\tau_1-\tau_2} &
\dfrac{\lambda_1(u_n-\tau_2) - \lambda_2(u_n-\tau_1)}{\tau_1 - \tau_2}
\end{pmatrix}.$$ The coefficient of $\gamma^{-1}$ in the trace of the twisted monodromy matrix provides the following integral for equation : $$\begin{aligned}
K_{\eqref{Q1Aut}}=&\frac{\alpha(Tu_{n+1}-T^2u_{n})(\lambda_{1}(\tau_{2}-u_{n+2})-\lambda_{2}(\tau_{1}-u_{n+2}))}{\lambda_{1}\lambda_{2}(\tau_{1}-\tau_{2})(Tu_{n+1}-u_{n+2})}\\
&+\frac{\beta(u_{n+2}-T^2u_{n})(\lambda_{1}(Tu_{n}-\tau_{2})-\lambda_{2}(Tu_{n}-\tau_{1}))}{\lambda_{1}\lambda_{2}(\tau_{1}-\tau_{2})(Tu_{n}-u_{n+2})}\\
&+\frac{\alpha(\tau_{1}-\tau_{2})(Tu_{n}T^2u_{n}-(Tu_{n+1})^2+u_{n+2}(2Tu_{n+1}-Tu_{n}-T^2u_{n}))}{(\lambda_{1}(Tu_{n}-\tau_{2})-\lambda_{2}(Tu_{n}-\tau_{1}))(Tu_{n+1}-T^2u_{n})(Tu_{n+1}-u_{n+2})}.\end{aligned}$$ We have determined the reduced variables to be $$\begin{aligned}
y_n &= \dfrac{\beta\left(Tu_n-u_n\right) \left(T^{-1}u_{n+2}-u_{n+1}\right)}{\alpha
\left(Tu_n-u_{n+1}\right) \left(T^{-1}u_{n+2}-u_n\right)},\\
z_n &= \dfrac{\lambda _1 \lambda _2 \left(\tau _1-\tau _2\right)
(\alpha y_n -1) (T^{-1}u_{n+2}-u_n)}{(Tu_n-T^{-1}u_{n+2})
\left(\lambda _2 \left(\tau _1-u_n\right)+\lambda _1 \left(u_n-\tau
_2\right)\right)},\end{aligned}$$ and hence we obtain the equation
\[Q1AutR\] $$\begin{aligned}
y_{n+1}y_n =& \dfrac{\beta\left(z_n-\lambda_1\right)\left( z_n
-\lambda_2\right)}{\alpha \lambda_1 \lambda_2},\\
z_{n+1}z_n =& (1-y_{n+1})\lambda_1\lambda_2,\end{aligned}$$
which is of QRT type and admits the integral $$K_{\ref{Q1AutR}}=\alpha \left( \frac{ y_n-1}{z_n}-\frac{z_n}{\lambda_1\lambda_2}\right)
+\beta\left(\frac{\left(z_n-\lambda _1\right) \left(z_n-\lambda _2\right)}{\lambda _1 \lambda _2 y_n
z_n}-\frac{1}{z_n}\right).$$
Let us jump right to the fully non-autonomous reduction, where the variables $\alpha_l$ and $\beta_m$ are given by , with $b_0=1$. If we assume $l$ (and hence $n$) is even, then the evolution equation is given by $$\begin{aligned}
&a_1 \to q a_0 ,\hspace{1cm} a_0 \to \dfrac{a_1}{q},\label{Q1nonautred}\\
&u_{n+3}=\frac{a_1 T u_{n+1} \left(T u_{n+2}-T^2 u_n\right)+ q^{n+2}
T u_{n+2} \left(T^2 u_n-T u_{n+1}\right)}{a_1
\left(T u_{n+2}-T^2u_n\right)+ q^{n+2}
\left(T^2u_n-Tu_{n+1}\right)}\nonumber.\end{aligned}$$ The Lax matrices are given by and , $$\begin{aligned}
A_n(x) =& \begin{pmatrix} 1 & u_{n+1} - T u_n \\[10pt] \dfrac{xa_1}{u_{n+1}
- T u_n} & 1 \end{pmatrix} B_n(x), \\
B_n(x) =& \begin{pmatrix} 1 & T^{-1}u_{n+2} - u_{n+1} \\[10pt] \dfrac{x
a_0}{T^{-1}u_{n+2} - u_{n+1}} & 1 \end{pmatrix} \begin{pmatrix}
1 & u_n - T^{-1}u_{n+2} \\[10pt] \dfrac{xq^n}{u_n - T^{-1}u_{n+2}} & 1
\end{pmatrix}.\end{aligned}$$ We use to deduce that $S_n$ is again given by . Using the compatibility, , we readily find . Once again, the task remains to find a second order system from this equation. We choose a similar combination of lattice variables as before, by letting $$\begin{aligned}
y_n &= \dfrac{(Tu_n-u_{n}) (T^{-1}u_{n+2}-u_{n+1})}{a_0
(Tu_{n}-u_{n+1}) (T^{-1}u_{n+2}-u_n)},\\
z_n &= \dfrac{(T^{-1}u_{n+2}-Tu_n) \left(\lambda _2 \left(\tau
_1-u_{n}\right)+\lambda _1 \left(u_{n}-\tau
_2\right)\right)}{\left(\tau
_1-\tau _2\right) (T^{-1}u_{n+2}-u_{n})}.\end{aligned}$$ Under this change of variables, we obtain another version of the system obtained in [@Quadratic], which generalizes ,
$$\begin{aligned}
y_{n+2} y_n =& \dfrac{(z_n-\lambda_1)(z_n-\lambda_2)}{\lambda_1\lambda_2 a_0 a_1},\\
z_{n+2} z_n =& \dfrac{\lambda_1\lambda_2\left(a_0 y_{n+2}-1\right)
\left(a_1 y_{n+2}-1\right)}{1- q^{n+2} y_{n+2}},\end{aligned}$$
modulo a certain scaling of variables. It is interesting to note that as a system admitting singularity confinement, the critical values of $z_n$ depend explicitly on the eigenvalues of the twist.
($2,2$)-reduction, and $q$-$\mathrm{P}_{VI}$ {#22qpvi}
============================================
Three of the authors have presented two versions of $q$-$\mathrm{P}_{VI}$, from in [@Ormerod:qP6] and from in [@OvdKQ:reductions]. Both of these reductions were subcases of the system described in the work of Jimbo and Sakai [@Sakai:qP6]; the version in [@OvdKQ:reductions] appeared with an interesting biquadratic constraint, which was similar to the work of Yamada [@Yamada:LaxqEs] but not present in [@Sakai:qP6], while the version in [@Ormerod:qP6] is a subcase of the version in [@OvdKQ:reductions]. Here we will present the fully non-autonomous $(2,2)$-reduction of , which we identify with the full parameter unconstrained version of the $q$-analogue of the sixth Painlevé equation as it appears in [@Sakai:qP6].
We start by specifying new $n$ and $p$ variables, which we assign to be $$n= m-l, \hspace{1cm} p = \left\lfloor \dfrac{l}{2} \right\rfloor,$$ where $\left\lfloor x \right\rfloor$ rounds $x$ down to the nearest integer. In this way, we label the variables $w_{l,m}$ so that $$\label{labelling22}
w_{l,m} \mapsto \left\{ \begin{array}{l p{4cm}} T^p u_n & if $l$ is
even,\\T^p v_n & if $l$ is odd. \end{array} \right.$$ This labeling is depicted in Figure \[fig:labelqP6\].
(-1.5,-.5) grid (3.5,3.5); (-1,-.5)–(-1,0) – (0,0) – (0,1) – (1,1)– (1,2)– (2,2)– (2,3)–(3,3)–(3,3.5);
at (-.1,0) [$T^pu_n$]{}; at (0,1) [$T^pu_{n+1}$]{}; at (0,2) [$T^pu_{n+2}$]{}; at (.9,1) [$T^pv_{n}$]{}; at (1,2) [$T^pv_{n+1}$]{}; at (1,3) [$T^pv_{n+2}$]{}; at (2,2) [$T^{p+1}u_{n}$]{}; at (2.1,3) [$T^{p+1} u_{n+1}$]{}; at (3,3) [$T^{p+1} v_{n}$]{};
(0,2) circle (.07); (1,3) circle (.07);
In order for this system to be consistent, we require $$\dfrac{\alpha_{l+2}}{\beta_{m+2}} = \dfrac{\alpha_{l}}{\beta_{m}},$$ which we solve by letting $$\alpha_l = \left\{ \begin{array}{c p{2cm}}
a_0 q^l & if $l$ is even,\\
a_1 q^l & if $l$ is odd,\end{array}\right. \hspace{1cm}
\beta_m = \left\{ \begin{array}{c p{2cm}}
b_0 q^m & if $m$ is even,\\
b_1 q^m & if $m$ is odd.\end{array}\right.$$ We now pick a spectral variable, $x= q^l/\gamma$, in which we have the system of linear equations $$\begin{aligned}
TY_{n}(q^2 x) &= A_n(x) Y_n(x),\\
Y_{n}(x) &= B_n(x) Y_n(x),\end{aligned}$$ where the spectral matrix, $A_n(x)$, governs an operation that is equivalent to the shift $(l,m) \to (l+2,m+2)$ and the deformation matrix, $B_n(x)$, governs an operation that is equivalent to the shift $(l,m) \to (l,m+1)$. This gives us a linear system with Lax matrices $$\begin{aligned}
A_n(x) &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}L_{l+1,m+2}M_{l+1,m+1}L_{l,m+1}M_{l,m},\\
B_n(x) &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}M_{l,m}.\end{aligned}$$ explicitly given by $$\begin{aligned}
A_n(x) = &\begin{pmatrix} 1 & v_{n+1}-T u_n \\ \frac{x a_1}{v_{n+1}-T
u_n} & 1 \end{pmatrix} \begin{pmatrix} 1 & v_n-v_{n+1} \\ \frac{q^n x
b_1}{v_n-v_{n+1}} & 1\end{pmatrix}\\
& \times \begin{pmatrix}1 & u_{n+1}-v_n \\ \frac{x a_0}{u_{n+1}-v_n} & 1
\end{pmatrix} B_n(x), \\
B_n(x) = & \begin{pmatrix}1 & u_n-u_{n+1} \\ \frac{q^n x
b_0}{u_n-u_{n+1}} &1\end{pmatrix}.\end{aligned}$$ The twist matrix $S_n$ is the same as in the (2,1)-reduction, given by , and the compatibility condition $$S_{n+1}^{-1} A_{n+1}(x)B_n(x) - B_n(q^2x) S_n^{-1} A_n(x) =0$$ gives the system that fixes the $a_0$ and $a_1$, and induces the transformation $$\begin{aligned}
&b_0 \to \dfrac{b_1}{q}, \hspace{1cm} b_1 \to q b_0,\\
u_{n+2} &= \dfrac{a_0 u_{n+1} \left(v_n-v_{n+1}\right)+b_1 q^n v_{n+1}
\left(u_{n+1}-v_n\right)}{a_0 \left(v_n-v_{n+1}\right)+b_1 q^n
\left(u_{n+1}-v_n\right)},\\
v_{n+2} &= \dfrac{a_1 v_{n+1} \left(T u_n-T u_{n+1}\right)+b_0 q^{n+2} Tu_{n+1}
\left(v_{n+1}-T u_n\right)}{a_1 \left(T u_n-T u_{n+1}\right)+b_0 q^{n+2}
\left(v_{n+1}-T u_n\right)}.\end{aligned}$$ This system possesses a $2$-integral[^5], which we label $\kappa$, given by $$\kappa = \dfrac{b_0 q^n \left(u_{n+1}-v_n\right) \left(\lambda _2
\left(\tau _1-u_n\right)+\lambda _1 \left(u_n-\tau _2\right)\right)
\left(Tu_n-v_{n+1}\right)}{a_1 \lambda _1 \left(\tau _2-\tau
_1\right) \left(u_n-u_{n+1}\right)
\left(v_n-v_{n+1}\right)}.$$ The $A_n(x)$ may be identified with the parameterisation of the spectral matrix of Jimbo and Sakai (see [@Sakai:qP6]) with variables $y_n$ and $z_n$ specified by $$\begin{aligned}
y_n&= \dfrac{b_0 q^n \left(v_n-u_{n+1}\right) \left(\lambda _2
\left(\tau _1-u_n\right)+\lambda _1 \left(u_n-\tau _2\right)\right)
\left(T u_n-v_{n+1}\right)}{a_1 \lambda _1 \left(\tau _1-\tau
_2\right) \left(u_n-u_{n+1}\right)
\left(v_n-v_{n+1}\right)},\\
z_n &=\frac{\left(v_{n+1}-v_{n}\right) \left(T u_n-u_n\right)
\left(u_{n+1}-v_n\right)}{a_0 \left(u_n-u_{n+1}\right)
\left(v_n-v_{n+1}\right) \left(Tu_n-v_n\right)+b_1 q^n
\left(u_n-v_n\right) \left(u_{n+1}-v_n\right)
\left(Tu_n-v_{n+1}\right)}.\end{aligned}$$ Under this change of variables, the system takes the form
$$\begin{aligned}
y_{n+2} y_n &= \dfrac{\kappa (z_n-
\lambda_1)(z_n-\lambda_2)}{(a_1\kappa
z_n-q^n\lambda_2b_0)(a_0z_n-q^{n+2}\kappa \lambda_1b_1)},\\
z_{n+2}z_n &=\frac{\lambda _1 \lambda _2 \left(b_0 q^{n+2}
y_{n+2}-1\right) \left(b_1 q^{n+2} y_{n+2}-1\right)}{\left(a_0
y_{n+2}-1\right) \left(a_1 y_{n+2}-1\right)}.\end{aligned}$$
This is the $q$-analogue of the sixth Painlevé equation [@Sakai:qP6].
General $(s_1,s_2)$-reduction {#s1s2}
=============================
We have specified several cases of $(2,1)$-reductions and a single case of a $(2,2)$-reduction, however, this theory generalizes to an arbitrary $(s_1,s_2)$-reduction. Following [@OvdKQ:reductions], we let $s_1
= ag$ and $s_2 = bg$, with gcd($a,b$)=1. Then we specify two integers, $c$ and $d$, by $$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} =1.$$ From this we define the variables $$\label{nk}
n =n(l,m)= \det\begin{pmatrix} a & b \\ l & m \end{pmatrix},
\hspace{1cm} k =k(l,m)= \det \begin{pmatrix} l & m \\ c & d
\end{pmatrix} \mod g,$$ and $$\label{p}
p = p(l,m)= \left\lfloor \dfrac{1}{g} \det \begin{pmatrix} l & m \\ c
& d \end{pmatrix} \right\rfloor.$$ Now we perform the reduction in accordance with the rule $$w_{l,m} \mapsto T^p u_n^k.$$ We note that the $p$ variable is the power of the transformation, $T$, whereas the $k$ is a superscript. The general ($s_1,s_2$)-reduction of is given by the system of $g$ equations: $$Q(T^{p} u_n^k, T^{\tilde{p}}u_{n-b}^{k+d}, T^{\hat{p}}u_{n+a}^{k-c},
T^{\hat{\tilde{p}}}u_{n+a-b}^{k-c+d};\alpha, \beta) = 0, \hspace{1cm}
k = 0,1, \ldots, g-1,$$ where the superscripts are interpreted modulo $g$ and $\tilde{p} =
p(l+1,m)$ and $\hat{p} = p(l,m+1)$ are just the expressions for the $p$’s shifted in the $l$ and $m$ directions respectively. This choice of labels and powers of $T$ ensures that any two ways of calculating an iterate, $u_n^k$, coincide due to the invariance of $Q$ under the action of the twist, $T$.
We construct operators that govern the shifts $(l,m) \to (l+s_1,m+s_2)$ and $(l,m) \to (l+c,m+d)$, which have the effect
\[Laxaut\] $$\begin{aligned}
\label{LaxLn} T\Psi_n &= A_n \Psi_n,\\
\label{LaxMn}\Psi_{n+1} &= B_n \Psi_n,\end{aligned}$$
in which the matrices, $A_n$ and $B_n$, can be specified by
$$\begin{aligned}
\label{prodAl}A_n &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}\prod_{j=0}^{s_2-1}
M_{l+s_1,m+j}\prod_{i=0}^{s_1-1} L_{l+i,m},\\
\label{prodBl}B_n &{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}\prod_{j=0}^{d-1}
M_{l+c,m+j}\prod_{i=0}^{c-1} L_{l+i,m},\end{aligned}$$
and $n$ is given by , see also [@OR1; @OR2], where Lax matrices $A_n$ and $B_n$ are given in terms of a product along so called standard staircases. The determining equation, , that defines the twist matrix is also a valid ansatz for the general $(s_1,s_2)$-reduction.
A list of possible (Möbius) twists for the equations in the ABS-list appears in Table \[tableTlist\].
----------------------------------------------------------------------------------------
ABS Point Symmetries
---------------------- -----------------------------------------------------------------
$H1$ $T_1:w \to w+\lambda$, $T_2 : w \to \mu-w$,
$H3_{\delta=0}$ $T_1:w \to \lambda w,\,\, T_2 : w \to \dfrac{\mu}{w}$,
$H3_{\delta\neq0}$ $T_1:w \to -w$,
$Q1_{\delta\neq0}$ $T_1:w \to w+\lambda$, $T_2 : w \to \mu-w$,
$Q1_{\delta= 0}$ $T_1:w \to \dfrac{(\lambda_1\tau_1- \lambda_2\tau_2)w
- (\lambda_1 - \lambda_2)\tau_1\tau_2}{(\lambda_1-\lambda_2)w +
\lambda_2\tau_1-\lambda_1\tau_2}$,
$Q3_{\delta=0}$ $T_1 : w \to \lambda w$, $T_2 : w \to \mu/w$,
$Q3_{\delta\neq 0}$ $T_1 : w \to -w$,
$Q4$ $T_1 : w \to -w$, $T_2 : w \to 1/w$,
$A1_{\delta=0}$ $T_1 : w \to \lambda w$, $T_2 : w \to \mu/w$,
$A1_{\delta \neq 0}$ $T_1 : w \to -w$,
----------------------------------------------------------------------------------------
: A list of the Möbius point symmetries of the lattice equations that appear in the ABS list [@ABS:ListI; @ABS:ListII]. For Q4 we used the version given in [@Hie05].[]{data-label="tableTlist"}
Let us conclude by mentioning twisted reductions for non-autonomous multiplicative equations, i.e., those for which $Q$ and the Lax-matrices depend on $\alpha/\beta$ only. Under this assumption[^6] the reduction is consistent, provided $
\alpha_{l+s_1}/\beta_{m+s_2}= \alpha_l/\beta_m.
$ By separation of variables this gives $$\dfrac{\alpha_{l+s_1}}{\alpha_{l}} = \dfrac{\beta_{m+s_2}}{\beta_m} := q^{abg},$$ which is solved by $$\alpha_l = a_{l\,\, \mathrm{mod}\, s_1} q^{bl}, \hspace{2cm} \beta_m
= b_{m\,\, \mathrm{mod}\, s_2} q^{am}.$$ A simple choice of spectral variable is $x = q^l$, in which the product representations of $A_n$ and $B_n$, given above by and , depend on $x$, giving $A_n(x)$ and $B_n(x)$. These matrices define a linear system
$$\begin{aligned}
T Y_n(q^{abg} x) = A_n(x) Y_n(x),\\
Y_{n+1}(q^{cb} x) = B_n(x) Y_n(x),\end{aligned}$$
along with the definition of the twist matrix, , gives the compatibility $$S_{n+1}^{-1} A_{n+1}(q^{cb}x)B_n(x) - B_{n}(q^{abg}x)S_n^{-1}A_n(x) = 0.$$ This compatibility is equivalent to the system of $g$ equations that define the non-autonomous reductions, $$Q(T^{p} u_n^k, T^{\tilde{p}}u_{n-b}^{k+d}, T^{\hat{p}}u_{n+a}^{k-c},
T^{\hat{\tilde{p}}}u_{n+a-b}^{k-c+d};\alpha_l/\beta_m) = 0,
\hspace{1cm} k = 0,1, \ldots, g-1,$$ where we should recall that $\alpha_l/\beta_m$, as a function of $n$, is $$\dfrac{\alpha_l}{\beta_m} = \dfrac{a_{l\, \mathrm{mod}\, s_1}}{b_{m\,
\mathrm{mod}\, s_2}} q^{-n}.$$ This provides a Lax representation for the twisted $(s_1,s_2)$-reduction with general $s_1$ and $s_2$.
Conclusions
===========
We have presented a generalisation of periodic reductions, that would appear to be new. Applying this to integrable equations, the resulting reductions possess Lax representations. This method can be used to obtain many additional integrable mappings. This can be done either by considering other reductions, or by starting from other integrable equations on quads (both of ABS type or non-ABS type), or from (multi-component) equations on other stencils. The method proposed in this paper seems analogous to Sklyanin’s method for generalising periodic boundary conditions for integrable quantum systems [@S88]. Finally we note that twisted reductions may also apply to non-integrable equations (although in that case there will be no Lax representations).
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Dr V. Mangazeev for a question posed at the ANZAMP Inaugural Meeting in Lorne, Dec 2012, that let to this paper being written. We would also like to thank Dr. B. Grammaticos for some references on discrete Painlevé equations. This research is supported by Australian Research Council Discovery Grants \#DP110100077 and \#DP140100383.
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[^1]: We note that reductions from P$\Delta$Es to O$\Delta$Es have also been derived for equations depending on more points of the lattice, as well as for systems of equations [@K09; @KQ10].
[^2]: Another type of reduction, via a nonlinear similarity constraint, was given in [@SimilarityReds].
[^3]: See Table 1 in [@OvdKQ:reductions] for other equations of the ABS-list admitting such a representation.
[^4]: Note that we also consider some examples of twists that are not homotopic to the identity transformation.
[^5]: A $2$-integral is invariant under the second iterate of the map [@HBQC].
[^6]: The case for additive type twisted reductions can be formulated analogously to what is presented here.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Image recognition is an important topic in computer vision and image processing, and has been mainly addressed by supervised deep learning methods, which need a large set of labeled images to achieve promising performance. However, in most cases, labeled data are expensive or even impossible to obtain, while unlabeled data are readily available from numerous free on-line resources and have been exploited to improve the performance of deep neural networks. To better exploit the power of unlabeled data for image recognition, in this paper, we propose a semi-supervised and generative approach, namely the semi-supervised self-growing generative adversarial network (SGGAN). Label inference is a key step for the success of semi-supervised learning approaches. There are two main problems in label inference: how to measure the confidence of the unlabeled data and how to generalize the classifier. We address these two problems via the generative framework and a novel convolution-block-transformation technique, respectively. To stabilize and speed up the training process of SGGAN, we employ the metric Maximum Mean Discrepancy as the feature matching objective function and achieve larger gain than the standard semi-supervised GANs (SSGANs), narrowing the gap to the supervised methods. Experiments on several benchmark datasets show the effectiveness of the proposed SGGAN on image recognition and facial attribute recognition tasks. By using the training data with only $4\%$ labeled facial attributes, the SGGAN approach can achieve comparable accuracy with leading supervised deep learning methods with all labeled facial attributes.'
author:
- |
Haoqian Wang, Member, IEEE, Zhiwei Xu, Jun Xu, Member, IEEE\
Wangpeng An, Lei Zhang, Fellow, IEEE， and Qionghai Dai, Senior Member, IEEE [^1] [^2] [^3] [^4] [^5]
bibliography:
- 'main.bib'
title: 'Semi-Supervised Self-Growing Generative Adversarial Networks for Image Recognition'
---
Semi-supervised learning, generative adversarial network, self-growing technique, image recognition, face attribute recognition
Introduction {#sec:1}
============
In the past decade, we have witnessed the increasing interests in the image recognition problem solved by the deep learning approaches [@krizhevsky2012imagenet; @Simonyan14c; @he2016deep]. This interest is expanding quickly to many different fields ever since the advent of deep convolution neural networks [@krizhevsky2012imagenet; @Simonyan14c; @goodfellow2014generative; @he2016deep], resulting in many effective approaches in many different computer vision fields [@tip_video; @tip_track; @tip_reid; @tip_fcn; @tip_saliency]. However, despite these exciting progresses, most existing approaches are supervised learning based and largely limited by resorting to huge amounts of data with labels. Labeling these data will incur expensive costs on human labor. To alleviate the dependence of supervised learning approaches on the labeled data, many semi-supervised learning approaches [@semi-rgbd; @tip_small_data; @tip_semi_face_recognition; @cherniavsky2010semi; @XU2019679] have been developed to exploit the power of the numerous unlabeled data available in free on-line resources for the image recognition problem. On the other hand, with the successes of Deep Convolutional Generative Adversarial Networks (DCGAN) [@DBLP:journals/corr/RadfordMC15] on general pattern recognition tasks, Generative Adversarial Networks (GANs) have been widely applied into unsupervised learning problems [@improvegan].
It is well known that the GANs can hardly be trained deeply enough when compared to the other concurrently networks such as ResNet [@he2016deep]. This is because that the generator of the GANs are usually very shallow and can often drift to “model collapse” (a parameter setting where it always emits the same point), restricting the GANs to grow up to achieve promising performance on large scale datasets such as ImageNet [@imagenet_cvpr09]. In this paper, we propose a novel a self-growing GAN (SGGAN) for large scale image recognition tasks. The proposed SGGAN is a united semi-supervised GAN containing three self-growing groups. Each group contains a generator and a discriminator, which are trained at the same time and compete against each other to reach the Nash equilibrium of the game theory through an adversarial objective [@goodfellow2014generative]. The generator is trained to defeat the discriminator by creating virtually realistic images (maximize the loss), and the discriminator is trained to distinguish the images generated by the generator (minimize the loss). Through this min-max game, the loss of generator will become increased while the loss of the discriminator will becomes decreased. Finally, the two losses will become closer to each other, and reach an equilibrium in the end.
In semi-supervised learning (SSL) framework, label inferring is a major challenging to its success. Given an amount of labeled data and a larger amount of unlabeled data, the SSL framework can infer the latent label information of the unlabeled data from the labeled data by considering the structures and distributions of all these data as a whole. In order to guarantee the success of the semi-supervised learning approach, label inference of the unlabeled data is the most significant problem to address. For the labels assigned to the unlabeled data, the false positive rate of the label inference process is more important than the true negative rate for the recognition performance, since false positive labels would add noise into the training data and thus make the training unstable. Therefore, the confidence of the latent labeled data (i.e., unlabeled data with latent labels) should be accurate enough. Moreover, the semi-supervised classifiers may not improve if they perform well on the same types of data. Therefore, the two main obstacles in label inference are: how to measure the confidence of the unlabeled data, and how to generalize the semi-supervised learning classifier. In this paper, we propose to address the first problem through threshold setting techniques [@Chapelle:2010:SL:1841234], in which only the unlabeled data with recognition probability larger than a pre-set threshold will be assigned with a latent label. We solve the second problem by proposing a novel technique named convolution-block-transformation (CBT) proposed by us. In our proposed network, the depth is designed to be deep in order to generalize the classifier since deeper model enables the network to learn more information from the unlabeled data than the shallower one. It is difficult to directly train a deep network in our case, so we propose a simple yet effective convolution block transformation (CBT) technique to transfer weights from a shallower network to a deep one by shortcut and an adaptive scaling layer following the shallower convolution block. We evaluate our method on CIFAR10, SVHN and face attribute recognition dataset, which is more challenging due to complex face variations.
In summary, the major contributions of this paper are summarized as follows:
- We propose an semi-supervised self-growing generative adversarial network (SGGAN) for image recognition problem. We handle the semi-supervised learning problem via label inference to improve the performance of the training network.
- We introduce the minimum mean discrepancy (MMD) as the objective of the feature matching stage to replace the traditional $\ell_1$ distance objective. The employed MMD can help to stabilize the training of the proposed SGGAN model, and thus avoid the model collapse pitfall of traditional GANs.
- We propose a novel convolution block transformation (CBT) technique to harmonize the self-growing process of the proposed SGGAN model to address the generalization of its classifier. We prove it is easier to train a model growing from a shallow network to a deep one, and thus achieving better performance.
- We conduct extensive experiments on image and face attribute recognition problems to systematically evaluate our proposed SGGAN model. We demonstrate that MMD and CBT can separately and simultaneously stabilize the training of the proposed SGGAN. When compared with supervised methods, SGGAN can achieves competitive or even higher accuracies on various benchmark datasets when compared with state-of-the-art GAN based approaches such as the Improved GAN [@improvegan] and supervised learning networks such as VGG-16 [@Simonyan14c] and ResNet-50 [@he2016deep].
The rest of this paper is organized as follows: In Section \[sec:related\_work\], we briefly reviews related work on semi-supervised learning, generative adversarial networks, the optimization of GAN and face attribute recognition. In Section \[sec:method\], we introduces the architecture of our proposed semi-supervised self-growing generative adversarial network and how to train our SGGAN. Experiments and detailed analysis are introduced in Section \[sec:results\]. Finally, we conclude this paper in Section \[sec:conclusion\].
Related Work {#sec:related_work}
============
Semi-Supervised Learning
------------------------
Semi-supervised algorithm [@Chapelle:2010:SL:1841234] falls between unsupervised learning (e.g., clustering) and supervised learning (e.g., classification or regression) on providing the data labels [@newDNAProt; @cvdp; @mcwnnm; @twsc; @pgpd; @gid2018; @xu2018thesis; @xu2018real; @xu2019nac; @xuaccv2016; @An_2018_CVPR; @Liang_2018_CVPR; @rssc; @nrc; @tstss; @xu2019star; @hou2019nlh; @RANet2019]. Semi-supervised learning [@zhu2009introduction] contains multiple types of training strategy, such as self-training [@rosenberg2005semi] and co-training [@zhou2005semi]. Recently, Zhuang et al. [@tip_semi] considered the label information in the graph learning stage. Specifically, they enforce to be zero the weight of edges between every two labeled samples from different classes. To make use of the unlabeled data, one simple and effective way is to predict the labels of the unlabeled data by employing the model trained on existing labeled data. Indeed, the premise behind semi-supervised learning is that the learned statistics in the labeled examples contain information which is useful to predict the unknown labels. Self-training [@rosenberg2005semi] is one of the earliest semi-supervised learning strategy using unlabeled data to improve the training of recognition systems. The high confidence that the model predicts against a sample indicates the high probability of correct prediction.
[ccc]{} **Baby D** & **Junior D** & **Senior D**\
Input (32$\times$32$\times$3) & Input (128$\times$128$\times$3) & Input (512$\times$512$\times$3)\
Conv3-64S1 & Conv3-64S1 & Conv3-64S1\
Conv3-64S1 & Conv3-64S1 & Conv3-64S1\
Conv3-64S2 & Conv3-64S2 & Conv3-64S2\
Conv3-128S2**$\times2$** & Conv3-128S2**$\times2$** & Conv3-128S2**$\times2$**\
Conv3-128S1**$\times2$** & Conv3-128S1**$\times2$** & Conv3-128S1**$\times2$**\
Conv3-128S1**$\times1$** & Conv3-128S1**$\times1$** & Conv3-128S1**$\times1$**\
$- $ & Conv3-192S1**$\times2$** & Conv3-192S1**$\times2$**\
$- $ & Conv3-192S1**$\times2$** & Conv3-192S1**$\times2$**\
$- $ & Conv3-192S2**$\times1$** & Conv3-192S2**$\times1$**\
$- $ & $- $ & Conv3-256S1**$\times2$**\
$- $ & $- $ & Conv3-256S1**$\times2$**\
$- $ & $- $ & Conv3-256S2**$\times2$**\
\
\
\
\
\[tab:discriminator\]
[ ccc ]{} **Baby G** & **Junior G** & **Senior G**\
\
\
\
Deconv5-256S2 & Deconv5-256S2 & Deconv5-256S2\
Deconv5-128S2 & Deconv5-128S2 & Deconv5-128S2\
Deconv5-128S2 & Deconv5-128S2 & Deconv5-128S2\
- & Deconv5-128S1 & Deconv5-128S1\
- & Deconv5-128S2 & Deconv5-128S2\
- & Deconv5-128S2 & Deconv5-128S2\
- & Deconv5-128S1 & Deconv5-128S1\
$-$ & Deconv5-64S2 & Deconv5-64S2\
$-$ & Deconv5-64S2 & Deconv5-64S2\
$-$ & Deconv5-32S1 & Deconv5-32S1\
$- $ & $-$ & Deconv5-32S2\
$- $ & $- $& Deconv5-32S2\
Output(32$\times$32$\times$3)& Output(128$\times$128$\times$3) & Output(512$\times$512$\times$3)\
\
\[tab:generator\]
Generative Adversarial Networks
-------------------------------
The training objective of GANs is to find a Nash equilibrium between the discriminative and generator networks by a min-max game. Denote by the generative network in GAN by $G$ and the discriminative network in GAN by $D$. The purpose of the $G$ network is to generate virtually realistic images and the purpose of the $D$ network is to distinguish between the virtually generated and realistic unlabeled images through the min-max optimization problem. As described in the original paper [@goodfellow2014generative], the purpose of the generative modeling is to find a probabilistic model $Q$ that matches the true data distribution $P$. The training of GAN can be interpreted as minimizing the Jensen-Shannon divergence under some ideal conditions. The Jensen-Shannon divergence is not measured by the K-L divergence between $P$ and $Q$, i.e., $KL[P\|Q]$ or $KL[Q\|P]$, but is between the two extreme cases $KL[P\|Q]$ and $KL[Q\|P]$. And this property of the Jensen-Shannon divergence can push the generator to generate better samples than other methods [@goodfellow2014generative]. Actually, Nash equilibrium is difficult to achieve and the assumptions behind GANs maybe too strong to perfectly match the cases in real-world applications. In the work of DCGAN [@DBLP:journals/corr/RadfordMC15], there are several techniques proposed to stabilize the training of the GANs, i.e., using leaky-ReLUs and batch normalization for the training of the discriminator network, and convolution with stride $2$ instead of max-pooling layers for the training of the generator network. These techniques work very well and have become a standard setup in recent GAN based approaches. Recently, Wasserstein distance [@arjovsky2017towards] is introduced with theoretically proved effectiveness as the objective of the generative model to stabilize the training process of GAN. The main advantage of Wasserstein distance based GAN frameworks is that this distance can guarantee great stability for training the generative model, which is not limited to the DCGAN approach.
Existing GAN based approaches can be categorized into two types from the perspective of their motivations. The first type is the divergence minimization based approaches [@DBLP:journals/corr/RadfordMC15], which mainly focus on designing an effective generator network to produce virtually realistic images, and treat the discriminator network as an auxiliary model. And the second type is the contrast function based approaches [@improvegan], which attempt to enhance the discriminating power of the discriminator by simulating a large amount of fake samples. Our work can be categorized into the second type of approaches.
GAN based semi-supervised learning
----------------------------------
Donahue et al. [@adversarial_inference] introduced an adversarial formulation with a third component, which they call the “encoder”. While the generator maps a simple latent distribution to data space, the encoder attempts to encode real data to some latent space. They show that this encoder is capable of learning to invert the generator, and can be used as a feature for a supervised training. On the autoregressive side, Dai et al. [@semi_sequence] explored the idea of first “pretraining” a sequence model to perform a task on unlabeled text data. These pretrained weights are then used to train supervised models for text classification. Their results show improved learning stability and model generalization. Radford et al. [@generate_reviews] trained an mLSTM RNN on Amazon reviews to learn a language model and then used its internal cell state from the last time step as features for the subsequent supervised task of sentiment analysis of Amazon reviews. This enabled the authors to match the state-of-the-art in their sentiment analysis dataset with significantly less labeled data and to surpass it with the fully-supervised learning. Recently, Salimans, et al. [@improvegan] proposed a way to utilize GANs for a classification task with $k$ classes. Specifically, they propose an extension to the vanilla GAN where the labeled dataset is augmented with samples from the generator. The discriminator is also modified to predict $(k + 1)$th classes: the original $k$ classes and a new class for fake (generated) data. In a sense this helps the discriminative model by augmenting a smaller labeled dataset with larger unlabeled set of real examples and generated samples.
Face Attribute Recognition
--------------------------
Face attribute recognition in the wild is a challenging problem due to complex face variations such as varying lightings, scales, and occlusions, etc. Traditionally, previous attribute recognition approaches [@berg2013poof; @bourdev2011describing; @kumar2011describable] focus on extracting effective hand-crafted low-level features, e.g., edges, HSV, and gradients, etc, from the detected faces. Then the extracted features are fed into a standard classifier, such as SVM [@svm] and random forest [@random_forest]. For instance, the authors of FaceTracer [@kumar2008facetracer] split the whole face region into multiple sub-regions, extracted multiple types of features for each region, and train a SVM classifier on the concatenated features. Recently, deep learning (especially CNN based) methods [@krizhevsky2012imagenet] have achieved great success in face attribute recognition due to their ability to learn discriminative features from huge amount of labeled data. The authors in [@celeba] applied two CNNs (ANet and LNet) to the face attribute recognition task, on which the LNet is trained to locate the entire face region and the ANet is trained to extract high-level face representation. Finally, the extracted features are fed into a SVM classifier to produce the final recognition results. In [@DBLP:journals/corr/RuddGB16], the authors proposed a mixed objective to optimize $40$ face attributes together in a single CNN with $138$ million network parameters. However, these supervised deep learning methods are limited by largely depending on huge amount of labeled training data, which is very costly in real-world applications. This motivates us to utilize the large amount of freely available unlabeled data for the face attribute recognition in a semi-supervised manner.
Semi-Supervised Self-Growing Generative Adversarial Network {#sec:method}
===========================================================
In this section, we first reveal the mechanism of the proposed semi-supervised self-growing generative adversarial network (SGGAN) by presenting its architecture in details. Then the convolution block transformation strategy for network self-growing is illustrated. Finally, we introduce the MMD as an effective metric to stabilize the training of our model.
Architecture of SGGAN
---------------------
The flowchart of the proposed SGGAN is illustrated in Figure \[fig:framework\]. Our SGGAN network includes a group of GANs, in which the junior generator or discriminator grows from corresponding baby counterpart, and the senior generator or discriminator grows from corresponding junior counterpart. The detailed description of the structures of three generators and three discriminators are listed in Table \[tab:discriminator\] and \[tab:generator\], respectively. The convolutional layer parameters are denoted as “(convolution type)(kernel size)-(number of channels)-S(stride)”. The activation functions we employed for the generator and discriminator are ReLU and Leaky-ReLU, respectively. Batch normalization is used after each convolution layer. The self growing process will be discussed in the next subsection. In the whole network, the fundamental component is named as the GAN cell, which is composed of a generator network and a discriminator network. In the GAN cell, the discriminator is deeper and sometimes has more filters per layer than the corresponding generator. The reason is that it is important for the discriminator to be able to correctly estimate the ratio between the true data density and generated data density, but it may also be an artifact of the “mode collapse” since the generator tends not to use its full capacity with current training methods [@gan_tutorial]. We introduce each component of the proposed SGGAN model as follows.
### Generator
![The detailed architecture of Baby Generator.[]{data-label="fig:generator"}](./figs/tip_generator.pdf){width="50.00000%"}
The generator takes as input a random vector $z$ (drawn from a Gaussian distribution). After reshaping $z$ into a $4$-dimensional shape, it is fed to the generator that starts with a series of upsampling layers. Each upsampling layer represents a transposed convolution operation with a stride of $2$. The transposed convolution work by swapping the forward and backward passes of a convolution. The upsampling layers go from deep and narrow layers to wider and shallower ones. The stride of a transposed convolution operation defines the upsampling factor of the output layer. With the stride of $2$, the size of output features will be twice that of the input layer. After each transposed convolution operation, the reshaped $z$ becomes wider and shallower. All transposed convolutions use a $5\times5$ kernel with depths reducing from $512$ to $3$, which indicating a RGB color image. The output of the final layer is a $H\times W\times3$ tensor, squashed between values of $-1$ and $1$ through the Hyperbolic Tangent ($tanh$) function. The shape of the final output is defined by the size of the training image. Specifically, if we train the generator on the SVHN dataset [@svhn], it will produce an image of size $32\times32\times3$.
### Discriminator
![The detailed architecture of Baby Discriminator.[]{data-label="fig:generator"}](./figs/tip_discriminator.pdf){width="50.00000%"}
The baby discriminator has $9$ CNN layers with Batch Normalization [@batch_norm], followed by Leaky-ReLU activation function. It is the same with the deep neural networks used for image recognition [@tip_class], object detection [@tip_detect], and image segmentation [@tip_segm], etc. The difference is that the Leaky-ReLU [@leaky_relu] is used in our discriminator instead of the regular ReLU [@relu]. The reason we employ Leaky-ReLU instead of the regular ReLU is that, the regular ReLU function will truncate the negative values to $0$, which will block the gradients to flow through the generative networks. Instead of forcing the negative part to be $0$, the Leaky-ReLU allows a small negative value to pass through the activation layer. Theoretically, Leaky-ReLU represents an attempt to solve the dying ReLU problem [@p_relu]. This situation occurs when the neurons do not move in a state in which ReLU units always output $0$s for all inputs. For these scenarios, the gradients do not flow back through the network. This problem is especially important for GAN since the only way the generator learns is by receiving the gradients from the discriminator. Our baby discriminator takes into a $32\times32\times3$ image tensor as input. Being opposite to the generator, the discriminator contains a series of convolutions with a stride of $2$. Each layer reduces the spatial dimensions of feature vector by reducing its size by half, along with doubling the number of learned filters. Given the training data from $k$ classes, the discriminator will output $(k+1)$ neurons to represent these $k$ classes, where the $(k+1)$th class demonstrates the generated images. We employ the softmax activation function as the output of the final layer to generate the confidence for samples from each class. When the discriminator captures the difference between the generated image and realistic image, it will send a signal to the generator counterpart. This signal is the gradient that flows backward from the discriminator to the generator. Once receiving this signal, the generator is able to adjust its parameters accordingly to generate latent data whose distribution is closer to the true data distribution than the previous generated ones. In the final stage, the generator will produce data as good as that the discriminator hardly distinguishes them apart.
### Label inference by discriminator
The latent labels of our unlabeled images are firstly created via the baby discriminator, and then updated by the junior discriminator. The self-training approaches usually needs a threshold value to infer the latent labels. The threshold value of the confidence is determined on the validation dataset of CelebA dataset [@celeba]; we set the threshold value as $0.98$ in our experiments. In this way, we can utilize more unlabeled images, and hence train deeper neural networks for better recognition performance.
Self-Growing Network
--------------------
In this section, we propose a convolution block transformation (CBT) technique to transform an existing network into a deeper one. Our idea is motivated by the Net2Net model [@DBLP:journals/corr/ChenGS15]. In Net2Net, Chen et al. proposed to initialize a bigger model using the weights of a smaller model. However, they only initialize the weights of one layer in each cycle, and this operation has difficulties with the batch normalization (BN) layer. This is because that the BN layer requires running forward inference on the training data to calculate the mean and variance of activation function, which are then used to set the output scale and bias of the BN layer to disentangle the normalization of the statistics of this layer.
In Figure \[fig:cbt\], we show the flowchart of the proposed CBT technique. With the help of CBT, to train a deeper model, we initialize a newly added convolution block (instead of a single layer) with Gaussian noise to break symmetry and add identity shortcut to preserve the potential ability of shallow model. As one can see in Figure \[fig:cbt\], the weights of the shallow network are transferred to a consistent block of the deeper network. Some new convolution layers are added to the top of the shallow network. The output values of the newly added convolution block are scaled by an adaptive scaling layer. The adaptive layer is defined by the function $w(t)=1-e^{-t}$, where $t$ denotes the number of total iterations in one epoch divided by current iteration number. The adaptive scaled output is added with that of the shallow layer. Finally, the added results are fed into a global average pooling (GAP) layer (for more details about GAP, please refer to the Section \[sec:training\]). Here, we call the up-described operator as the convolution block transformation (CBT). Along with the training, the function $w(t)$ for the adaptive layer will gradually approach to $1$ and the newly added convolution block will becomes a part of the original shallow net.
{width="90.00000%"}
Feature Matching
----------------
Generative Adversarial Networks (GANs) are difficult to train since the generator is easy to collapse [@gan_tutorial] (we call it the “model collapse” phenomenon). In [@improvegan], in order to avoid mode collapse, Salimans et al. proposed the feature matching technique to improve the stability in training the GANs by employing a new objective for the generator. Instead of maximizing the output of the discriminator as the regular GAN training does, the feature matching requires the generator to create latent data that matches the statistics of the realistic data in the feature level of the discriminator network. Consequently, the generator updates its parameters by matching the expectation of the features on the next of the final layer of the discriminator network, which is the output of Global Average Pooling (GAP) layer in our case. This is a natural choice of statistics for the generator to match. Let $f(x)$ denote activations after GAP layer of the discriminator, the feature matching objective for the generator is proposed by [@improvegan] and defined by an $\ell_1$ distance as $||E_{x\in p_{data}}f(x)-E_{z\in p_z(z)} f(G(z))||_{1}$. In practice, we found that the above mentioned $\ell_1$ distance produces similar results with $\ell_2$ distance. The authors in [@mmd_gan] proved that the maximum mean discrepancy (MMD) using Gaussian kernels could match all moment’s mean, including the $\ell_1$ and $\ell_2$ distances between the generated features and unlabeled images. Therefore, in this work we employ the MMD metric as the feature matching objective to measure the distance between the features of generated images and unlabeled images. Then, we require generator to match the all levels statistics features of realistic data: $$\begin{aligned}
\begin{split}
\textrm{MMD}(\mathcal{F},p_{data},p_z)&=\sup_{f \in \mathcal{F}}(\mathbb{E}_{x\sim p_{data}}[f(x)\\
&-\mathbb{E}_{z\sim p_z}[f(G(z))]),
\label{eq:MMD}
\end{split}\end{aligned}$$ where $\mathcal{F}$ is a set of functions. When $\mathcal{F}$ is in a reproducing kernel Hilbert space (RKHS), the function approaching the supremum can be derived analytically and is called the witness function $$\begin{aligned}
f(x) = \mathbb{E}_{x\sim p_{data}}[\mathcal{K}(x,G(z))]
- \mathbb{E}_{z\sim p_z}[\mathcal{K}(x,G(z))],
\label{eq:witness}\end{aligned}$$ where $\mathcal{K}$ is the kernel of the RKHS. Here, we assume $\mathcal{K}$ is measurable and bounded. Then we substitute into and yield: $$\begin{aligned}
\begin{split}
\textrm{MMD}^2(\mathcal{F},p_{data},p_z) &=
\mathbb{E}_{x,x'\sim p_{data}}[\mathcal{K}(x,x')] \\
&- 2\mathbb{E}_{x\sim p_{data},z\sim p_z}[\mathcal{K}(x,G(z))] \\
&+ \mathbb{E}_{z,z\sim p_z}[\mathcal{K}(G(z),G(z'))].
\end{split}\end{aligned}$$ This expression only involves expectations of the kernel $\mathcal{K}$, which can be approximated by:
$$\begin{aligned}
\begin{split}
\textrm{MMD}_{sample}^2(\mathcal{F},p_{data},p_z) &=
\frac{1}{m^2}\sum_{i,j=1}^{m}\mathcal{K}(x_i,x_j)\\
- \frac{2}{mn}\sum_{i,j=1}^{m,n}\mathcal{K}(x_i,G(z_j))
&+ \frac{1}{n^2}\sum_{i,j=1}^{n}\mathcal{K}(G(z_i),G(z_j))
\label{eq:MMD_b}
\end{split}\end{aligned}$$
The MMD metric also depends on the choice of the kernel. We choose the inner product kernel for simplicity.
Generate images by using generator $G$. Feed generated, unlabeled, and labeled images into discriminator $D$ to obtain $Loss_{d}$. Compute $\frac{\partial Loss_{d}}{\partial W_{d}}$ and update $W_{d}$ with $W_{g}$ fixed. Feed unlabeled and generated images into $D$ to compute $Loss_{g}$. Compute $\frac{\partial Loss_{g}}{\partial W_{G}}$ through $D$ and update $W_{g}$ with $W_{d}$ fixed.\
**end for** Inference unlabeled images and create latent-labeled dataset by using discriminator $D$,\
**end for** Initialize a deeper model by using CBT preservation technique.
Learning Objective
------------------
A key challenging in semi-supervised GANs is how to construct the loss function. For the losses, we find that the cross-entropy with Adam is a good choice for the optimizer. In [@improvegan], Salimans et al. introduce an effective strategy to construct the discriminator loss function $Loss_d$. They regard the labeled and unlabeled data as one of $k$ classes and then classify the latently generated data into the $(k+1)$-th class. In this way, $Loss_d$ can be defined as follows: $$\begin{aligned}
Loss_d=&-\log(\frac{1}{\sum_{i=1}^m e^{g_i}+1})-\sum_{i=1}^m label_i\times\log(x_i)\\
&-\log(\frac{\sum_{i=1}^m e^{u_i}}{\sum_{i=1}^m e^{u_i}+1})
\label{eq:loss_d}
\end{aligned}$$ where $-\sum_{i=1}^m label_i\times\log(x_i)$, $-\log(\frac{1}{\sum_{i=1}^m e^{g_i}+1})$, and $-\log(\frac{\sum_{i=1}^m e^{u_i}}{\sum_{i=1}^m e^{u_i}+1})$ are the losses related to generated, labeled, and unlabeled images, respectively. Here, $m$ is the batch size, and $x_i$, $g_i$, $u_i$ represent the output (before softmax activation) of the labeled, generated, and unlabeled images, respectively. During the training of the generator, a simple feature matching method is introduced to measure the dissimilarity between two distributions of realistic and latently generated data as described in [@improvegan]. Motivated by the effectiveness of the maximum mean discrepancy (MMD) [@mmd_origin; @mmd_gan], in the proposed SGGAN we utilize MMD metric instead of the $\ell_1$ to measure the dissimilarity between latently generated data and the realistic data.
Training {#sec:training}
--------
A complete cycle of training the proposed SGGAN contains three iterative steps: 1) train the generator $G$ and discriminator $D$ on the labeled and unlabeled pool. Here, we employ the MMD metric for the updating of the weights of generator $G$; 2) apply the discriminator $D$ to predict the unlabeled pool, and then assign the most confident samples of all the $k$ classes to the labeled pool; 3) self-grow the discriminator $D$ and generator $G$ to be deeper and more powerful. The overall procedures of training the proposed SGGAN is summarized in Algorithm \[alg:selfgrow\].
### Pre-Training
The purpose of pre-training is to train the initial baby GAN cell. Inspired by the feature matching techniques introduced in the Improved GAN [@improvegan], the process of pre-training could solve the problems in training the discriminator. After this stage, we have a baby discriminator which achieves an accuracy of over $80\%$ on the testing set of the CelebA dataset [@celeba]. To this end, we can make use of the trained baby discriminator to infer the latent labels from the unlabeled images. We use Adam with initializing learning rate of $0.01$ to train both the generator $G$ and the discriminator $D$. The weights of baby generator and baby discriminator are initialized by using Xavier’s method [@szegedy2015going]. In all experiments, the pixel values of the images are normalized to $[-1, 1]$.
### Label Inference
As we mentioned in Section \[sec:1\], inferring the latent labels of the unlabeled images is the most significant step in training semi-supervised learning models. One typical way to obtain the latent labels of unlabeled data is to hypothesize that the labels predicted by the initial classifier is credible. Under this circumstance, the label inference problem is tackled. However, there are two issues in this approach. The first one is that the initial classifier can be inaccurate towards unlabeled data, and leading wrong absorption of inaccurate data and thus assimilating noise into the training data. The second one is that as the initial classifier does well on the same class of data, adding this type of unlabeled data as latently labeled samples may make the classifier only memorize this specific type of data and cannot be generalized to other data types. How to solve these two issues is crucial to the success of a semi-supervised self-growing network.
For the first issue, we can largely weaken its impact by only selecting the images in the unlabeled pool which have larger recognition probability than a pre-set threshold value $\alpha$, which can be determined by performing recognition experiments on the validation set of benchmark datasets (please refer to the experimental section for more details) through grid search strategy. In this work, we set $\alpha=0.98$. For the second issue, we initialize the junior network from the trained baby counterpart by introducing the proposed CBT preservation technique, and generalize the representational power of the classifier, accordingly. Comparing to the than the shallower network, a deeper network can potentially learn additional useful information from the latent labeled data. The improving performance of the Alexnet [@krizhevsky2012imagenet] to the VGG [@Simonyan14c], and finally to the ResNet [@he2016deep], all demonstrates the great successes in the ILSVRC [@ILSVRC15] challenge on the Imagenet Dataset [@imagenet_cvpr09]. For example, VGG [@Simonyan14c] uses $3\times3$ convolution to achieve deeper architecture and ResNet [@he2016deep] treats convolution added with shortcut as a basic unit and repeats that unit until the depth limit of the network is reached. Going deeper can really improve the capacity of network considerably. As the model grows up stronger (deeper), the network can learn useful information not only on labeled images, but also on the latently labeled images.
Experiments {#sec:results}
===========
In this section, we first evaluate the proposed semi-supervised self-growing GAN (SGGAN) approach and justify the effectiveness of each component in the SGGAN approach. Then we compare SGGAN with other state-of-the-art semi-supervised GAN based approaches on image recognition problem on two widely employed datasets. To demonstrate the broad applicability of the proposed SGGAN approach, we also compare it with the leading supervised deep learning approaches on two commonly used datasets for face attribute recognition.
Dataset Description
-------------------
![Samples from the CIFAR-10 dataset [@cifar].[]{data-label="fig:cifar_data"}](./figs/cifar10_shrink.png){width="45.00000%"}
**Image Recognition Datasets**. In this section, we compare the proposed SGGAN approach with state-of-the-art semi-supervised GAN based methods by using the widely used CIFAR-10 dataset [@cifar] and the Street View House Numbers (SVHN) dataset [@svhn].
The CIFAR10 dataset [@cifar] is introduced by A. Krizhevsky and G. Hinton in 2009, and has been a benchmark dataset for image classification problem ever since. This dataset contains $60,000$ color images of size $32\times32$ in 10 classes (i.e., airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck). Some samples of this dataset are shown in Figure \[fig:cifar\_data\]. Each class includes $6,000$ images, of which $5,000$ images are used for training and $1,000$ images are used for testing. It is a widely used dataset for evaluating both supervised and semi-supervised learning methods on the image recognition problem. We follow the same experimental setting as the previous work such as [@improvegan], in which only $100$, $200$, $400$ and $800$ samples along with their labels for each class are randomly selected as the training data for semi-supervised learning.
The SVHN dataset [@svhn] is a real-world image dataset for digit recognition problem. It is similar in flavor to the MNIST dataset [@mnist], but serves with a harder and real-world problem in the wild. This dataset contains over $600,000$ color digit images coming from the house numbers in Google Street View images. Some samples are listed in Figure \[fig:svhn\_data\]. Among these images, there are $73,257$ images in the training set, $26,032$ images in the testing set. Following the experimental settings as described in [@improvegan], in which only $50$, $100$ and $200$ samples along with their labels for each class are selected as the training data for semi-supervised learning.
![Samples from the SVHN dataset [@svhn].[]{data-label="fig:svhn_data"}](./figs/svhn.png){width="45.00000%"}
**Facial Attribute Recognition Datasets**. We also compare the proposed SGGAN approach with the leading supervised deep learning methods on facial attribute recognition problem with the CelebFaces Attributes Dataset (CelebA) dataset [@celeba] and the Labeled Faces in the Wild-a (LFW-a) dataset [@LFWTech].
The CelebA dataset [@celeba] is a large-scale face attributes dataset, which contains $202,599$ face images of $10,177$ identities in the wild, each of which includes $5$ landmark locations and $40$ binary attributes annotations. Among the $202,599$ face images in total, $19,962$ images are used as the testing set and the others are used as the training and validation set, respectively. In this article, we randomly select a small set of images as the training set and the others as the testing set.
![Samples from CelebA dataset [@celeba].[]{data-label="fig:celeba_data"}](./figs/celeba.pdf){width="46.00000%" height="25.00000%"}
The LFWA dataset [@LFWTech] has $13,233$ images of $5,749$ identities. Following the experimental settings as the previous work [@celeba], we employ $6,263$ images of $2,749$ peoples as the training set and the other $6,880$ images of $3,000$ peoples as the testing set. When we train the SGGAN model, the labeled images are randomly selected from the training set, and the final results on testing error are averaged by $10$ independent runnings. For the CelebA dataset [@celeba], the prediction threshold $\alpha$ is choose on the validation set.
In all these datasets (except the LFWA dataset [@LFWTech]), we train the model on the training set and select the model trained with the lowest recognition error on the validation set, and report the testing error with the selected training model accordingly. For the LFWA dataset [@LFWTech], we follow the experimental settings as described in [@celeba].
![Samples from LFWA dataset [@LFWTech].[]{data-label="fig:lfwa"}](./figs/lfwa.png){width="40.00000%"}
Ablation Study
--------------
In this section, we justify the influence of different components in our proposed SGGAN approach on the performance of recognition errors. The aspects we investigate here include the network self-growing route, the objective function of the feature matching, the generated samples, and the comparison with transfer learning approaches, etc. All these study is evaluated on the training set of the CelebA dataset [@celeba].
**Network Self-Growing Route**. In our SGGAN model, the model is designed to “grow up” from a small one to a big one. However, how to decide the route for our SGGAN model to achieve better performance (i.e., lower recognition error) is still a big problem. The routes for the model to “grow up” can be very different. For example, the model can be directly grow from a baby model to a junior model, or from a baby model to a senior model, or from a baby model to a junior model and finally to a senior model, etc. To th is end, we design a series of experiment to validate the most suitable “grow up” route for the proposed SGGAN model.
We compare the proposed SGGAN model with different routes of “grow up” on the CelebA dataset [@celeba]. The comparison is performed by using the “gender” attribute of $800$ labeled images. The experimental routes are summarized in the first three rows of the Table \[tab:grow\], while symbol “$\surd$” indicates that the corresponding baby/junior/senior model is employed as a part of the whole training model and “$-$” indicates that the we skip the corresponding model. The order in models with three models is to train the whole model from baby one, junior one, to the senior one. From the last row of the Table \[tab:grow\], one can see that the SGGAN model with the route of “grow up” along all the three models can achieve higher accuracy than with the other routes. Besides, the SGGAN model “grow up” with two models can achieve better performance than the SGGAN model with only one baby/junior/senior model. Similar findings can be found in the experiments on other attributes of the CelebA dataset [@celeba] as well as on other datasets such as LFWA [@LFWTech]. These results demonstrate that the network self-growing strategy can effectively improve the image recognition accuracy over the one with fixed single model. Specifically, using all these three models can significantly improve the recognition accuracy of the SGGAN model with only single individual baby/junior/senior network.
---------- --------- --------- --------- --------- --------- --------- ----------
Baby $\surd$ $-$ $-$ $\surd$ $\surd$ $-$ $\surd$
Junior $-$ $\surd$ $-$ $\surd$ $-$ $\surd$ $\surd$
Senior $-$ $-$ $\surd$ $-$ $\surd$ $\surd$ $\surd$
Accuracy 80.2 85.1 84.6 86.7 88.5 89.1 **89.6**
---------- --------- --------- --------- --------- --------- --------- ----------
: The accuracy (%) of the proposed SGGAN network with different self-growing routes by using the “gender” attribute in the CelebA dataset [@celeba].[]{data-label="results_different_routes"}
\[tab:grow\]
**Objective of Feature Matching**. The work of Wasserstein GAN [@wgan] discusses different distances between distributions adopted by existing generative adversarial algorithms, and show many of them are discontinuous, such as Jensen-Shannon divergence [@goodfellow2014generative] and Total Variation [@total_variation], except for Wasserstein distance. The discontinuity makes the gradient descent infeasible for training. Consequently, [@nips_mmd_gan] show Wasserstein GAN [@wgan] is a special case of the MMD, and hence MMD also has the advantages of being continuous and differentiable. We adopt the powerful MMD metric to our work to stabilize the training of generator.
We compare the different objective of the feature matching step, i.e., the Maximum Mean Discrepancy (MMD) and the $\ell_{1}$ distance as we have mentioned in Section \[sec:related\_work\]. Since the model collapse is a fundamental problem in the training of GAN, we use MMD to stabilize the GAN. The results are shown in Figure \[fig:MMD\], from which one can see that the generator trained with the MMD objective can achieve lower training loss than that trained with $\ell_1$ distance after several epochs. This demonstrates that MMD is more suitable than the $\ell_1$ distance to be the loss objective function during the training of the generator in GAN.
![The loss function of the SGGAN model trained with MMD v.s. with $l_1$ distance as the objective of feature matching.[]{data-label="fig:MMD"}](./figs/MMD.pdf){width="40.00000%"}
![The loss function of the SGGAN model trained with the CBT v.s. without CBT during training.[]{data-label="fig:CBT"}](./figs/CBT-eps-converted-to.pdf){width="40.00000%"}
**Convolution Block Transformation (CBT)**. Figure \[fig:CBT\] shows that the recognition accuracy (%) of the SGGAN model trained with the CBT technique are consistently higher than the model trained without CBT in different epochs. This demonstrate that CBT is more effective than its counterpart that simply copies the weights in the shallow model and initializes the newly added convolution-block layers randomly. This is due to the reason that the simple “training without CBT” strategy will wreck the weights in the shallow layers of the network. And evidence our proposed CBT technique will make the transfer of weights smoothly and hence preserve the function of shallower model at the beginning of training.
**Comparisons with Fine-tuned VGG and ResNet Networks**. In order to show the advantages of our algorithm on label effectiveness, we compare our SGGAN model with the state-of-the-art networks such as the VGG-16 network [@Simonyan14c] and the ResNet network [@he2016deep] in the deep learning field. For the two networks, we load the model provided by corresponding authors pre-trained on the ImageNet dataset [@imagenet_cvpr09] (which contains 1000 classes with 1.2 million images), and then carefully fine-tune these networks on the training set of the CelebA dataset [@celeba]. The fine-tuning procedure can usually help the original networks yield better performance than training those networks on small dataset directly. The proposed SGGAN model, the pre-trained VGG-16 and Resnet-50 networks are all fine-tuned with different numbers (i.e., 800, 1600, 3200, 4800, 6400, 7200) of labeled images in the CelebA dataset [@celeba] with “gender” attribute in the comparison experiments. We fine-tune the VGG-16 and ResNet-50 networks in a standard manner as described in corresponding paper. When the training set is of small scale, it is hard to train a very deep network from scratch. And the most frequently employed technique in literature is to fine-tune the off-the-shell networks, such as the famous VGG network [@Simonyan14c]. We compare the proposed SGGAN approach with the fine-tuned VGG-16 and ResNet-50 networks with different numbers of labeled training images.
The results on accuracy (%) are listed in Table \[tab:vgg\], from which one can see that when the numbers of labeled training images are $800$, $1,600$, $3,200$, and $4,800$, the proposed SGGAN approach can achieve higher recognition accuracies than the fine-tuned VGG-16 and ResNet-50 networks on the CelebA dataset with the “gender” attribute. Similar results can be found when we perform experiments on the other attributes of the CelebA dataset or the other datasets. When the numbers of the training samples increase to $6,400$ and $7,200$, the proposed SGGAN approach obtains slightly inferior (but still comparable) performance to the VGG-16 and ResNet-50 networks. All these results demonstrate the competing ability of the proposed SGGAN approach as a whole system over the leading VGG and ResNet networks on image recognition tasks such as face attribute recognition.
\[tab:vgg\]
800 1600 3200 4800 6400 7200
------------------------ ---------- ---------- ---------- ---------- ---------- ---------- --
VGG16 [@Simonyan14c] 89.3 92.4 94.8 95.9 97.6 98.1
resnet50 [@he2016deep] 88.6 91.9 94.6 96.2 **97.8** **98.3**
SGGAN **89.6** **94.3** **95.5** **96.4** 96.8 97.1
: Comparison with the VGG-16 and ResNet-50 networks fine-tuned with different numbers of labeled images from the CelebA dataset [@celeba] with the “gender” attribute.
Comparison with state-of-the-art semi-supervised learning approaches on image recognition {#sec:cifar}
-----------------------------------------------------------------------------------------
### Problem Description
Image recognition problem is the task of assigning one label to an input image from a fixed set of categories. It is a fundamental problem in computer vision community. Image recognition has a large variety of practical applications, and is related to many other computer vision tasks such as object detection and segmentation.
### Comparison Methods
We compare the proposed SGGAN approach with other competing semi-supervised learning approaches such as the Ladder Network [@ladder], which proposed to train the ladder network simultaneously minimize the sum of supervised and unsupervised cost functions by back-propagation, avoiding the need for layer-wise pre-training. And some leading GANs based approaches such as CatGAN [@catgan], which is based on an objective function that trades-off mutual information between unlabeled examples and their predicted categorical class distribution, against robustness of the classifier to an adversarial generative model. And the Improved GAN [@improvegan], which propose a technique called feature matching to address the instability of GANs by specifying a new objective for the generator to prevents it from overtraining on the current discriminator. Instead of directly maximizing the output of the discriminator, the new objective $\ell_1$ requires the generator to generate data that matches the statistics of the real data. We compare these competing methods on the CIFAR10 dataset and the SVHN dataset [@svhn].
### Results and Discussions
The experimental results are shown in Table \[tab:cifar\] and Table \[tab:svhn\]. It can be observed from Table \[tab:cifar\] that we achieve competitive results with the state-of-the-art on the two datasets. As the CIFAR10 dataset [@cifar], the SVHN dataset [@svhn] is used for validating semi-supervised learning methods. Table \[tab:svhn\] shows the testing error for SVHN experiment. One can see that the more labeled samples we use, the better the recognition performance the proposed SGGAN model will be.
\[tab:cifar\]
1000 2000 4000 8000
----------------------------- ----------- ----------- ----------- ----------- -- -- --
Ladder network [@ladder] N/A N/A 20.4 N/A
CatGAN [@catgan] N/A N/A 19.58 N/A
Improved GANs [@improvegan] 21.83 19.61 18.63 17.72
SGGAN **20.04** **18.43** **15.65** **16.51**
: Comparison test error with other semi-supervised learning methods on CIFAR10 dataset. The results are averaged by 10 runs. N/A is not available, which is not report in their papers.
\[tab:svhn\]
500 1000 2000
---------------------------------------------- ------- ------ ------ -- -- -- --
DGN [@DGN] 36.02 N/A N/A
Virtual Adversarial [@distributional] 24.63 N/A N/A
Auxiliary Deep Generative Model [@auxiliary] 22.86 N/A N/A
Skip Deep Generative Model [@auxiliary] 16.61 N/A N/A
Improved GANs [@improvegan] 18.44 8.11 6.16
SGGAN 17.31 6.53 5.13
: Comparison test error with other semi-supervised learning methods on SVHN dataset. All experiments are averaged by $10$ runs.
Comparisons with supervised learning approaches on face attribute recognition
-----------------------------------------------------------------------------
-- ----------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- --------
Face Tracer 85 76 78 76 89 88 64 74 70 80 81 60 80 86 88 98 93 90 85 84 91
PANDA-w 82 73 77 71 92 89 61 70 74 81 77 69 76 82 85 94 86 88 84 80 93
LNet+ANet(w/o) 88 74 77 73 95 92 66 75 84 91 80 78 85 86 88 96 92 93 85 84 94
LNet+ANet **91** **79** **81** **79** 98 **95** **68** 78 **88** **95** **84** 80 **90** **91** 92 **99** **95** **97** **90** **87** **98**
Virtual GAN 84 73 75 71 92 90 62 74 80 90 77 76 82 85 89 92 88 91 85 80 91
Auxiliary GAN 85 73 75 74 93 91 63 75 83 91 80 77 83 84 90 93 91 90 86 83 92
Cat GAN 87 72 76 72 93 92 62 77 81 91 78 75 85 86 90 93 89 91 84 83 93
Skip GAN 88 75 77 75 96 92 64 78 84 92 81 78 86 87 91 96 92 93 87 84 95
Improved GAN 87 76 78 76 95 91 65 79 85 91 82 79 87 88 91 95 90 92 88 86 93
SGGAN 90 77 79 77 **98** 94 66 **80** 86 94 83 **80** 88 89 **93** 98 94 95 89 86 97
FaceTracer 70 67 71 65 77 72 68 73 76 88 73 62 67 67 70 90 69 78 88 77 84
PANDA-w 64 63 70 63 82 79 64 71 78 87 70 65 63 65 64 84 65 77 86 75 86
LNets+ANet(w/o) 81 78 80 79 83 84 72 76 86 94 70 73 79 70 74 92 75 81 91 83 91
LNets+ANet 84 82 **83** 83 **88** **88** 75 81 **90** **97** 74 77 82 73 78 **95** 78 84 **95** 88 **94**
Virtual GAN 80 79 77 81 82 81 71 77 84 91 73 74 78 70 74 89 76 80 89 84 88
Auxiliary GAN 81 80 78 82 83 82 72 78 85 92 74 75 79 71 75 90 77 81 90 85 89
Cat GAN 80 81 79 81 82 83 73 79 86 91 75 76 80 73 76 89 79 82 89 83 87
Skip GAN 82 83 81 83 86 83 73 81 88 93 75 78 82 72 78 91 90 82 93 80 90
Improved GAN 83 82 80 84 85 84 74 80 87 94 76 77 81 73 77 92 79 83 92 87 91
SGGAN **85** **84** 82 **86** 87 86 **76** **82** 89 96 **77** **79** **83** **75** **79** 94 **81** **85** 94 **89** 93
-- ----------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- --------
\[tab:fully\]
### Problem Description
Face attributes recognition is to get descriptive attributes on faces (gender, sex, the presence of sunglasses etc). Kumar et al. [@kumar2009attribute] first introduced them as mid-level features for face verification [@kumar2011describable] and since then have attracted much attention. The recognition of face attributes has an important role in computer vision applications due to their detailed description of human faces. The applications of it include suspect identification [@klare2014suspect], face verification [@kumar2009attribute] and face retrieval [@kumar2011describable]. Predicting face attributes in the wild in challenging due to complex face variations. In facial attribute recognition field, labeled data are either expensive or unavailable to obtain. Consequently, the large number of unlabeled face images available on the Internet have attracted increasing interests of researchers to tackle facial attribute recognition problem by semi-supervised learning (SSL) [@Chapelle:2010:SL:1841234] methods.
### Comparisons methods
The proposed method is compared with four competitive fully-supervised approaches including FaceTracer [@kumar2008facetracer], PANDA-w [@zhang2014panda], LNet+ANet(w/o) and LNet+ANet [@zhang2014panda] on the two datasets mentioned above. Compared with the fully-supervised learning methods, our self-growing approach only uses $7200$ labeled images. The LFWA dataset is a standard benchmark for face attribute classification. However, the number of training and validation data of LFWA data set is small, which made it not suitable to our algorithm. So we report two patterns of result on LFWA dataset. The first one uses all the training/validation data in the LFWA dataset and the other uses the data of CelebA as the unlabeled data pool. Our algorithm runs ten times, and we report the average result.
### Results and Discussions
The comparison results on CelebA and LFWA datasets are shown in Table \[tab:fully\], from which one can see that the proposed SGGAN approach achieve comparable performance on the recognition accuracy when compared with the state-of-the-art supervised learning based deep learning methods. For example, the proposed SGGAN trained with the MMD objective and the CBT technique (i.e., SGGAN-MMD-CBT) achieves an accuracy of $86.22\%$, which is only slightly inferior to the LNet+ANet methods, but still superior to all the other methods. Note that the proposed SGGAN-MMD-CBT approach achieves such promising performance with only $4\%$ labeled training images.
Results on the LFWA dataset using external unlabeled data
---------------------------------------------------------
The LFWA dataset is a standard benchmark for face attribute recognition. However, the number of training images in LFWA dataset is small ($6,263$ images), which made it not well suitable for our algorithm. In order to achieve semi-supervised learning on LFWA dataset. We use all training images in CelebA dataset as the unlabeled pool for our algorithm to train a SGGAN on LFWA dataset. During training, the labels of CelebA dataset was not used. Table \[tab:lfwa\] show the results, in the first row of the table, the “LFWA (Outer data)” shows the result of SGGAN using CelebA as the unlabeled pool and “LFWA” is the result of SGGAN only use the images in LFWA. From the table one can see a large number of unlabeled images will improve around $6\%$ points for LFWA dataset. Which also demonstrate the effectiveness of our algorithm for the semi-supervised image recognition tasks.
Methods LFWA (Outer data) LFWA
------------------------------------------ ------------------- ------- --
Baseline: Feature Matching [@improvegan] 81.29 78.56
SGGAN-MMD 83.41 78.53
SGGAN-CBT 84.27 78.57
SGGAN-MMD-CBT 85.32 78.81
: Recognition accuracy (%) of SGGAN with different component settings on the LFWA dataset [@LFWTech] by using unlabeled data.
\[tab:lfwa\]
Generated Samples
-----------------
Feature matching is proved to help the GANs work much better if the goal is to obtain a strong classifier using the approach to semi-supervised learning [@improvegan]. It works well for semi-supervised learning approaches. However, the samples generated by the generator during semi-supervised learning using feature matching do not look visually appealing. The reason appears to be that the human visual system is strongly attuned to image statistics that can help infer what class of object an image represents, while it is presumably less sensitive to local statistics that are less important for interpretation of the image. This is supported by the high correlation between the quality reported by human annotators and the Inception score developed in the work [@improvegan].
We show the generated samples in Figure \[fig:generated\], from which one can see that the junior generator can produce samples with better image quality than those generated by the baby one, and the senior generator can further enhance the performance of the image quality of the produced samples than those generated by the junior one. This demonstrate that the proposed SGGAN network with “grow up” strategy can indeed make better generation during the training process, and hence implicitly help improve the performance of the GAN model on the recognition tasks.
Conclusion {#sec:conclusion}
==========
In this paper, we propose a simple yet effective semi-supervised self-growing generative adversarial network (SGGAN) for image recognition. We propose a convolution-block-transformation (CBT) preservation technique to promote the network self-growing and obtain deeper network. Meanwhile, we leverage a maximum mean discrepancy (MMD) metric to stabilize and improve the training of SGGAN. The experiments on CIFAR10 and SVHN dataset demonstrate effectiveness our methods. Extensive experiments on the CelebA and LFWA demonstrate the generalization of our method. With only around $4\%$ labeled training data, our SGGAN can achieve comparable performance with the fully-supervised convolutional neural network.
[^1]: This project is partially supported by the National Natural Scientific Foundation of China (NSFC) under Grant No. 61571259 and 61531014, in part by the Shenzhen Science and Technology Project under Grant (GGFW2017040714161462, JCYJ20170307153051701)
[^2]: H. Q. Wang, Zhiwei Xu and W. P. An are with the Graduate School at Shenzhen, Tsinghua University, and slso with Shenzhen Institute of Future Media Technology, Shenzhen 518055, China (e-mail: [email protected], [email protected]).
[^3]: J. Xu is with Media Computing Lab, College of Computer Science, Nankai University, Tianjin, China. (e-mail: [email protected]).
[^4]: Q. Dai is with TNLIST and Department of Automation, Tsinghua University, Beijing 100084, China (e-mail: [email protected]).
[^5]: L. Zhang is with the Department of Computing, The Hong Kong Polytechnic University, Hong Kong (e-mail: [email protected]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We introduce the community exploration problem that has many real-world applications such as online advertising. In the problem, an explorer allocates limited budget to explore communities so as to maximize the number of members he could meet. We provide a systematic study of the community exploration problem, from offline optimization to online learning. For the offline setting where the sizes of communities are known, we prove that the greedy methods for both of non-adaptive exploration and adaptive exploration are optimal. For the online setting where the sizes of communities are not known and need to be learned from the multi-round explorations, we propose an “upper confidence” like algorithm that achieves the logarithmic regret bounds. By combining the feedback from different rounds, we can achieve a constant regret bound.'
author:
- |
Xiaowei Chen$^1$, Weiran Huang$^2$, Wei Chen$^3$, John C.S. Lui$^1$\
$^1$The Chinese University of Hong Kong\
$^2$Huawei Noah’s Ark Lab, $^3$Microsoft Research\
`^1{xwchen, cslui}@cse.cuhk.edu.hk, ^[email protected]`\
`^[email protected]`
bibliography:
- 'ref.bib'
title: 'Community Exploration: From Offline Optimization to Online Learning'
---
Introduction
============
In this paper, we introduce the community exploration problem, which is abstracted from many real-world applications. Consider the following hypothetical scenario. Suppose that John just entered the university as a freshman. He wants to explore different student communities or study groups at the university to meet as many new friends as possible. But he only has a limited time to spend on exploring different communities, so his problem is how to allocate his time and energy to explore different student communities to maximize the number of people he would meet.
The above hypothetical community exploration scenario can also find similar counterparts in serious business and social applications. One example is online advertising. In this application, an advertiser wants to promote his products via placing advertisements on different online websites. The website would show the advertisements on webpages, and visitors to the websites may click on the advertisements when they view these webpages. The advertiser wants to reach as many unique customers as possible, but he only has a limited budget to spend. Moreover, website visitors come randomly, so it is not guaranteed that all visitors to the same website are unique customers. So the advertiser needs to decide how to spend the budget on each website to reach his customers. Of course, intuitively he should spend more budget on larger communities, but how much? And what if he does not know the user size of every website? In this case, each website is a community, consisting of all visitors to this website, and the problem can be modeled as a community exploration problem. Another example could be a social worker who wants to reach a large number of people from different communities to do social studies or improve the social welfare for a large population, while he also needs to face the budget constraint and uncertainty about the community.
In this paper, we abstract the common features of these applications and define the following community exploration problem that reflects the common core of the problem. We model the problem with $m$ disjoint communities $C_1,
\dots, C_m$ with $C=\cup_{i=1}^m C_i$, where each community $C_i$ has $d_i$ members. Each time when one explores (or visit) a community $C_i$, he would meet one member of $C_i$ uniformly at random.[^1] Given a budget $K$, the goal of community exploration is to determine the budget allocation $\bm k=(k_1, \dots, k_m)\in
{\mathbb{Z}}_+^m$ with $\sum_{i=1}^m k_i \le K$, such that the total number of distinct members met is maximized when each community $C_i$ is explored $k_i$ times.
We provide a systematic study of the above community exploration problem, from offline optimization to online learning. First, we consider the offline setting where the community sizes are known. In this setting, we further study two problem variants — the non-adaptive version and the adaptive version. The non-adaptive version requires that the complete budget allocation $\bm k$ is decided before the exploration is started, while the adaptive version allows the algorithm to use the feedback from the exploration results of the previous steps to determine the exploration target of the next step. In both cases, we prove that the greedy algorithm provides the optimal solution. While the proof for the non-adaptive case is simple, the proof that the adaptive greedy policy is optimal is much more involved and relies on a careful analysis of transitions between system statuses. The proof techniques may be applicable in the analysis of other related problems.
Second, we consider the online setting where the community sizes are unknown in advance, which models the uncertainty about the communities in real applications. We apply the multi-armed bandit (MAB) framework to this task, in which community explorations proceed in multiple rounds, and in each round we explore communities with a budget of $K$, use the feedback to learn about the community size, and adjust the exploration strategy in future rounds. The reward of a round is the the expected number of unique people met in the round. The goal is to maximize the cumulative reward from all rounds, or minimizing the regret, which is defined as the difference in cumulative reward between always using the optimal offline algorithm when knowing the community sizes and using the online learning algorithm. Similar to the offline case, we also consider the non-adaptive and adaptive version of exploration within each round. We provide theoretical regret bounds of $O(\log T)$ for both versions, where $T$ is the number of rounds, which is asymptotically tight. Our analysis uses the special feature of the community exploration problem, which leads to improved coefficients in the regret bounds compared with a simple application of some existing results on combinatorial MABs. Moreover, we also discuss the possibility of using the feedback in previous round to turn the problem into the full information feedback model, which allows us to provide constant regret in this case.
In summary, our contributions include: (a) proposing the study of the community exploration problem to reflect the core of a number of real-world applications; and (b) a systematic study of the problem with rigorous theoretical analysis that covers offline non-adaptive, offline adaptive, online non-adaptive and online adaptive cases, which model the real-world situations of adapting to feedback and handling uncertainty.
Problem Definition
==================
We model the problem with $m$ disjoint communities $C_1, \dots, C_m$ with $C=\cup_{i=1}^m C_i$, where each community $C_i$ has $d_i$ members. Each exploration (or visit) of one community $C_i$ returns a member of $C_i$ uniformly at random, and we have a total budget of $K$ for explorations. Since we can trivially explore each community once when $K \le m$, we assume that $K
> m$.
We consider both the offline setting where the sizes of the communities $d_1, \ldots, d_m$ are known, and the online setting where the sizes of the communities are unknown. For the offline setting, we further consider two different problems: (1) non-adaptive exploration and (2) adaptive exploration. For the non-adaptive exploration, the explorer needs to predetermine the budget allocation $\bm k$ before the exploration starts, while for the adaptive exploration, she can sequentially select the next community to explore based on previous observations (the members met in the previous community visits). Formally, we use pair $(i, \tau)$ to represent the $\tau$-th exploration of community $C_{i}$, called an [*item*]{}. Let ${\mathcal{E}}= [m]\times[K]$ be the set of all possible items. A [*realization*]{} is a function $\phi\colon {\mathcal{E}}\rightarrow C$ mapping every possible item $(i, \tau)$ to a member in the corresponding community $C_i$, and $\phi(i, \tau)$ represents the member met in the exploration $(i, \tau)$. We use $\Phi$ to denote a random realization, and the randomness comes from the exploration results. From the description above, $\Phi$ follows the distribution such that $\Phi(i,\tau) \in C_i$ is selected uniformly at random from $C_i$ and is independent of all other $\Phi(i',\tau')$’s.
For a budget allocation $\bm k=(k_1, \dots, k_m)$ and a realization $\phi$, we define the reward $R$ as the number of distinct members met, i.e., $R(\bm{k},
\phi) =\sum_{i=1}^{m}|\cup_{\tau=1}^{k_i}\{\phi(i,\tau)\}|$, where $|\cdot|$ is the cardinality of the set. The goal of the [*non-adaptive exploration*]{} is to find an optimal budget allocation $\bm{k}^{*} =(k^{*}_1,\ldots, k^{*}_m)$ with given budget $K$, which maximizes the expected reward taken over all possible realizations, i.e., $$\label{eq:offline_budget_allocation_problem}
\bm{k}^{*} \in \operatorname*{arg\,max}_{\bm{k}\colon {\| \bm k \|}_1 \leq K} \mathbb{E}_{\Phi}\left[ R(\bm{k}, \Phi) \right].$$
For the adaptive exploration, the explorer sequentially picks a community to explore, meets a random member of the chosen community, then picks the next community, meets another random member of that community, and so on, until the budget is used up. After each selection, the observations so far can be represented as a [*partial realization*]{} $\psi$, a function from the subset of ${\mathcal{E}}$ to $C = \cup_{i=1}^m C_i$. Suppose that each community $C_i$ has been explored $k_i$ times. Then the partial realization $\psi$ is a function mapping items in $\cup_{i=1}^m \{(i,1),\ldots, (i, k_i)\}$ (which is also called the domain of $\psi$, denoted as ${{\rm dom}}(\psi)$) to members in communities. The partial realization $\psi$ records the observation on the sequence of explored communities and the members met in this sequence. We say that a partial realization $\psi$ is consistent with realization $\phi$, denoted as $\phi \sim \psi$, if for all item $(i,\tau)$ in the domain of $\psi$, we have $\psi(i, \tau) = \phi(i,\tau)$. The strategy to explore the communities adaptively is encoded as a policy. The policy, denoted as $\pi$, is a function mapping $\psi$ to an item in ${\mathcal{E}}$, specifying which community to explore next under the partial realization. Define $\pi_K(\phi) =
(k_1,\ldots, k_m)$, where $k_i$ is the times the community $C_i$ is explored via policy $\pi$ under realization $\phi$ with budget $K$. More specifically, starting from the partial realization $\psi_0$ with empty domain, for every current partial realization $\psi_s$ at step $s$, policy $\pi$ determines the next community $\pi(\psi_s)$ to explore, meet the member $\phi(\pi(\psi_s))$, such that the new partial realization $\psi_{s+1}$ is adding the mapping from $\pi(\psi_s)$ to $\phi(\pi(\psi_s))$ on top of $\psi_s$. This iteration continues until the communities have been explored $K$ times, and $\pi_K(\phi) = (k_1,\ldots, k_m)$ denotes the resulting exploration vector. The goal of the adaptive exploration is to find an optimal policy $\pi^*$ to maximize the expected adaptive reward, i.e., $$\label{eq:adaptive_allocation_problem}
\pi^* \in \operatorname*{arg\,max}_{\pi} \mathbb{E}_{\Phi}\left[R(\pi_K(\Phi), \Phi) \right].$$ We next consider the online setting of community exploration. The learning process proceeds in discrete rounds. Initially, the size of communities $\bm{d}
= (d_1,\ldots, d_m)$ is unknown. In each round $t\ge 1$, the learner needs to determine an allocation or a policy (called an [*“action”*]{}) based on the previous-round observations to explore communities (non-adaptively or adaptively). When an action is played, the sets of encountered members for every community are observed as the [ *feedback*]{} to the player. A learning algorithm $A$ aims to cumulate as much reward (i.e., number of distinct members) as possible by selecting actions properly at each round. The performance of a learning algorithm is measured by the [*cumulative regret*]{}. Let $\Phi_t$ be the realization at round $t$. If we explore the communities with predetermined budget allocation in each round, the $T$-round (non-adaptive) regret of a learning algorithm $A$ is defined as $$\label{eq:non_adaptive_regret_definition}
\small
\text{Reg}^{A}_{\bm{\mu}}(T) = \mathbb{E}_{\Phi_1,\ldots, \Phi_T}\left[\sum_{t=1}^{T}R(\bm{k}^{*}, \Phi_t) - R(\bm{k}^{A}_t, \Phi_t) \right],$$ where the budget allocation $\bm{k}^{A}_t$ is selected by algorithm $A$ in round $t$. If we explore the communities adaptively in each round, then the $T$-round (adaptive) regret of a learning algorithm $A$ is defined as $$\label{eq:adaptive_regret_definition}
\small
\text{Reg}^{A}_{\bm{\mu}}(T) = \mathbb{E}_{\Phi_1,\ldots, \Phi_T}\left[\sum_{t=1}^{T}R(\pi^{*}_K(\Phi_t), \Phi_t) - R(\pi^{A,t}_K(\Phi_t), \Phi_t) \right],$$ where $\pi^{A,t}$ is a policy selected by algorithm $A$ in round $t$. The goal of the learning problem is to design a learning algorithm $A$ which minimizes the regret defined in and .
Offline Optimization for Community Exploration {#sec:offline}
==============================================
Non-adaptive Exploration Algorithms
-----------------------------------
If $C_i$ is explored $k_i$ times, each member in $C_i$ is encountered at least once with probability $1 - (1 - 1/d_i)^{k_i}$ . Thus we have $\mathbb{E}_{\Phi}[{\left| \{\Phi(i,1), \ldots, \Phi(i, {k_i})\} \right|}] = d_i (1 - (1 -
1/d_i)^{k_i})$. Hence $\mathbb{E}_{\Phi}\left[ R(\bm{k}, \Phi) \right]$ is a function of only the budget allocation $\bm{k}$ and the size $\bm{d} =
(d_1,\ldots, d_m)$ of all communities. Let $\mu_i = 1 / d_i$, and vector $\bm{\mu} = (1/d_1, \ldots, 1/d_m)$. Henceforth, we treat $\bm{\mu}$ as the parameter of the problem instance, since it is bounded with $\bm{\mu}\in
[0,1]^m$. Let $r_{\bm{k}}(\bm{\mu}) = \mathbb{E}_{\Phi}[R(\bm{k}, \Phi)]$ be the expected reward for the budget allocation $\bm{k}$. Based on the above discussion, we have $$\label{eq:expectedreward}
r_{\bm{k}}(\bm{\mu}) = \sum_{i=1}^m d_i (1 - (1 - 1/d_i)^{k_i}) = \sum_{i=1}^m (1 - (1 - \mu_i)^{k_i}) / \mu_i.$$ Since $k_i$ must be integers, a traditional method like [*Lagrange Multipliers*]{} cannot be applied to solve the optimization problem defined in Eq. . We propose a [ *greedy method*]{} consisting of $K$ steps to compute the feasible $\bm{k}^*$. The greedy method is described in Line \[line:budget\_allocation\_start\]-\[line:budget\_allocation\_end\] of Algo. \[algo:non\_adaptive\_exploration\].
\[line:budget\_allocation\_start\] \[line:budget\_allocation\_end\]
[theorem]{}[optimalitya]{}\[thm:nonadaptive\_greedy\_is\_optimal\] The greedy method obtains an optimal budget allocation.
The time complexity of the greedy method is $O(K\log m)$, which is not efficient for large $K$. We find that starting from the initial allocation $ k_i =
\ceil*{\frac{(K - m) / \ln (1 - \mu_i)}{\sum_{j=1}^m 1 / \ln (1 - \mu_j)}}$, the greedy method can find the optimal budget allocation in $O(m\log
m)$[^2].
Adaptive Exploration Algorithms
-------------------------------
With a slight abuse of notations, we also define $r_{\pi}(\bm{\mu}) =
\mathbb{E}_{\Phi}\left[ R(\pi_K(\Phi), \Phi) \right]$, since the expected reward is the function of the policy $\pi$ and the vector $\bm{\mu}$. Define $c_i(\psi)$ as the number of distinct members we met in community $C_i$ under partial realization $\psi$. Then $1 - c_i(\psi)/d_i$ is the probability that we can meet a new member in the community $C_i$ if we explore community $C_i$ one more time. A natural approach is to explore community $C_{i^*}$ such that $i^*\in \operatorname*{arg\,max}_{i\in[m]} 1 - c_i(\psi)/d_i$ when we have partial realization $\psi$. We call such policy as the greedy policy $\pi^g$. The adaptive community exploration with greedy policy is described in Algo. \[algo:adaptive\_exploration\]. One could show that our reward function is actually an [*adaptive submodular*]{} function, for which the greedy policy is guaranteed to achieve at least $(1 \!- \!1/e)$ of the maximized expected reward [@golovin2011adaptive]. However, the following theorem shows that for our community exploration problem, our greedy policy is in fact [*optimal*]{}.
\[line:adaptive\_explore\_start\] \[line:adaptive\_explore\_end\]
[theorem]{}[optimalityb]{}\[thm:greedy\_policy\_is\_optimal\] Greedy policy is the optimal policy for our adaptive exploration problem.
**Proof sketch.** Note that the greedy policy chooses the next community only based on the fraction of unseen members. It does not care which members are already met. Thus, we define $s_i$ as the percentage of members we have not met in a community $C_i$. We introduce the concept of [*status*]{}, denoted as $\bm{s} =
\left(s_1,\dots, s_m\right)$. The greedy policy chooses next community based on the current status. In the proof, we further extend the definition of reward with a non-decreasing function $f$ as $ R(\bm{k}, \phi)
=f\left(\sum_{i=1}^{m}{\left| \bigcup_{\tau=1}^{k_i}\{\phi(i,\tau)\} \right|}\right)$. Note that the reward function corresponding to the original community exploration problem is simply the identity function $f(x)=x$. Let $F_{\pi}(\psi, t)$ denote the expected [*marginal gain*]{} when we further explore communities for $t$ steps with policy $\pi$ starting from a partial realization $\psi$. We want to prove that for all $\psi$, $t$ and $\pi$, $F_{\pi^g} (\psi,t)\ge F_\pi(\psi,t)$, where $\pi^g$ is the greedy policy and $\pi$ is an arbitrary policy. If so, we simply take $\psi = \emptyset$, and $F_{\pi^g} (\emptyset,t)\ge F_\pi(\emptyset,t)$ for every $\pi$ and $t$ exactly shows that $\pi^g$ is optimal. We prove the above result by an induction on $t$.
Let $C_i$ be the community chosen by $\pi$ based on the partial realization $\psi$. Define $c(\psi) = \sum_i c_i(\psi)$ and $\Delta_{\psi, f} = f(c(\psi) + 1) -
f(c(\psi))$. We first claim that $F_{\pi^g} (\psi,1)\ge F_\pi(\psi,1)$ holds for all $\psi$ and $\pi$ with the fact that $F_{\pi}(\psi, 1) = (1 -
\mu_ic_i(\psi))\Delta_{\psi, f}$. Note that the greedy policy $\pi^g$ chooses $C_{i^*}$ with $i^*\in \operatorname*{arg\,max}_{i} (1 - \mu_i c_i(\psi))$. Hence, $F_{\pi^g}
(\psi,1)\ge F_\pi(\psi,1)$.
Next we prove that $F_{\pi^g} (\psi,t+1)\ge F_\pi(\psi,t+1)$ based on the assumption that $F_{\pi^g} (\psi,t')\ge F_\pi(\psi,t')$ holds for all $\psi$, $\pi$, and $t'\le t$. An important observation is that $F_{\pi^g}(\psi, t)$ has equal value for any partial realization $\psi$ associated with the same status $\bm{s}$ since the status is enough for the greedy policy to determine the choice of next community. Formally, we define $F_{g}(\bm{s}, t) =
F_{\pi^g}(\psi, t)$ for any partial realization that satisfies $\bm{s} = (1 -
c_1(\psi)/d_1,\dots, 1 - c_m(\psi)/d_m)$. Let $C_{i^*}$ denote the community chosen by policy $\pi^g$ under realization $\psi$, i.e., $i^*\in \operatorname*{arg\,max}_{i\in
[m]} 1 - \mu_ic_i(\psi)$. Let ${\boldsymbol{I}}_i$ be the $m$-dimensional unit vector with one in the $i$-th entry and zeros in all other entries. We show that $$\begin{aligned}
F_{\pi}(\psi, t + 1) &\leq c_i(\psi) \cdot \mu_i F_{g}(\bm{s}, t) + (d_i - c_i(\psi)) \cdot \mu_i F_{g}(\bm{s} - \mu_i {\boldsymbol{I}}_i, t) + (1 - \mu_ic_i(\psi))\Delta_{\psi, f} \\
& \leq \mu_{i^*}c_{i^*}(\psi)F_{g}(\bm{s}, t) + (1 - \mu_{i^*}c_{i^*}(\psi))F_{g}(\bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, t) + (1 - \mu_{i^*}c_{i^*}(\psi))\Delta_{\psi, f} \\
& = F_{g}(\bm{s}, t + 1) = F_{\pi^g}(\psi, t + 1).
\end{aligned}$$ The first line is derived directly from the definition and the assumption. The key is to prove the correctness of Line 2 in above inequality. It indicates that if we choose a sub-optimal community at first, and then we switch back to the greedy policy, the expected reward would be smaller. The proof is nontrivial and relies on a careful analysis based on the stochastic transitions among status vectors. Note that the reward function $r_{\pi}(\bm{\mu})$ is not necessary adaptive submodular if we extend the reward with the non-decreasing function $f$. Hence, a $(1 - 1/e)$ guarantee for adaptive submodular function [@golovin2011adaptive] is not applicable in this scenario. Our analysis scheme can be applied to any adaptive problems with similar structures.
Online Learning for Community Exploration
=========================================
The key of the learning algorithm is to estimate the community sizes. The size estimation problem is defined as inferring unknown set size $d_i$ from random samples obtained via uniformly sampling [*with replacement*]{} from the set $C_i$. Various estimators have been proposed [@finkelstein1998423; @bressan2015simple; @christman1994sequential; @katzir2011estimating] for the estimation of $d_i$. The core idea of estimators in [@bressan2015simple; @katzir2011estimating] are based on “[*collision counting*]{}”. Let $(u, v)$ be an [*unordered pair*]{} of two random elements from $C_i$ and $Y_{u, v}$ be a [*pair collision*]{} random variable that takes value 1 if $u =
v$ (i.e., $(u,v)$ is [*a collision*]{}) and $0$ otherwise. It is easy to verify that $\mathbb{E}[Y_{u,v}] = 1/ d_i = \mu_i$. Suppose we [*independently*]{} take $T_i$ pairs of elements from $C_i$ and $X_i$ of them are collisions. Then $\mathbb{E}[X_i/T_i] = 1 / d_i = \mu_i$. The size $d_i$ can be estimated by $T_i
/ X_i$ (the estimator is valid when $X_i > 0$).
\[line:online\_learning\_start\] \[line:radius\] \[line:feedback\] \[line:update\_pairs\] \[line:count\_collision\] \[line:online\_learning\_end\]
We present our CLCB algorithm in Algorithm \[algo:CLCB\_algorithm\]. In the algorithm, we maintain an unbiased estimation of $\mu_i$ instead of $d_i$ for each community $C_i$ for the following reasons. Firstly, $T_i/X_i$ is not an unbiased estimator of $d_i$ since $\mathbb{E}[T_i/X_i] \geq d_i$ according to the Jensen’s inequality. Secondly, the upper confidence bound of $T_i/X_i$ depends on $d_i$, which is unknown in our online learning problem. Thirdly, we need at least $(1 + \sqrt{8d_i\ln1/\delta + 1})/2$ uniformly sampled elements in $C_i$ to make sure that $X_i > 0$ with probability at least $1 - \delta$. We feed the lower confidence bound ${\underaccent{\bar}{\mu}}_i$ to the exploration process since our reward function increases as $\mu_i$ decreases. The idea is similar to CUCB algorithm [@CWYW16]. The lower confidence bound is small if community $C_i$ is not explored often ($T_i$ is small). Small ${\underaccent{\bar}{\mu}}_i$ motivates us to explore $C_i$ more times. The [*feedbacks*]{} after the exploration process at each round are the sets of encountered members ${\mathcal{S}}_1,\dots, {\mathcal{S}}_m$ in communities $C_1,\dots, C_m$ respectively. Note that for each $i\in[m]$, all pairs of elements in ${\mathcal{S}}_i$, namely $\{(x,
y)\mid x \leq y, x\in {\mathcal{S}}_i, y\in {\mathcal{S}}_i\backslash\{x\}\}$ are not mutually independent. Thus, we only use $\floor{{\left| {\mathcal{S}}_i \right|}/2}$ independent pairs. Therefore, $T_i$ is updated as $T_i +
\floor{{\left| {\mathcal{S}}_i \right|}/2}$ at each round. In each round, the community exploration could either be non-adaptive or adaptive, and the following regret analysis separately discuss these two cases.
Regret Analysis for the Non-adaptive Version
--------------------------------------------
The non-adaptive bandit learning model fits into the general combinatorial multi-armed bandit (CMAB) framework of [@CWYW16; @wang2017improving] that deals with nonlinear reward functions. In particular, we can treat the pair collision variable in each community $C_i$ as a base arm, and our expected reward in Eq. is non-linear, and it satisfies the monotonicity and bounded smoothness properties (See Properties \[pro:monotone\] and \[pro:bounded\_smoothness\]). However, directly applying the regret result from [@CWYW16; @wang2017improving] will give us an inferior regret bound for two reasons. First, in our setting, in each round we could have multiple sample feedback for each community, meaning that each base arm could be observed multiple times, which is not directly covered by CMAB. Second, to use the regret result in [@CWYW16; @wang2017improving], the bounded smoothness property needs to have a bounded smoothness constant independent of the actions, but we can have a better result by using a tighter form of bounded smoothness with action-related coefficients. Therefore, in this section, we provide a better regret result by adapting the regret analysis in [@wang2017improving].
We define the gap $\Delta_{\bm{k}} = r_{\bm{k}^*}(\bm{\mu}) -
r_{\bm{k}}(\bm{\mu})$ for all action $\bm{k}$ satisfying $\sum_{i=1}^m k_i = K$. For each community $C_i$, we define $\Delta^{i}_{\min} = \min_{\Delta_{\bm{k}} >
0, k_i > 1} \Delta_{\bm{k}}$ and $\Delta^{i}_{\max} = \max_{\Delta_{\bm{k}} > 0,
k_i > 1} \Delta_{\bm{k}}$. As a convention, if there is no action $\bm{k}$ with $k_i > 1$ such that $\Delta_{\bm{k}} > 0$, we define $\Delta^{i}_{\min} =
\infty$ and $\Delta^{i}_{\max} = 0$. Furthermore, define $\Delta_{\min} =
\min_{i\in [m]} \Delta^{i}_{\min}$ and $\Delta_{\max} = \max_{i\in
[m]}\Delta^{i}_{\max}$. Let $K^{\prime} = K - m + 1$. We have the regret for Algo. \[algo:CLCB\_algorithm\] as follows.
[theorem]{}[regretbounda]{}\[thm:regret\_bound\_non\_adaptive\_exploration\_1\] Algo. \[algo:CLCB\_algorithm\] with non-adaptive exploration method has regret as follows. $$\begin{aligned}
\label{eq:non_adaptive_regret_bound_a}
\text{Reg}_{\bm{\mu}}(T) &\leq \sum_{i=1}^{m}\frac{48{K'\choose 2}K\ln T}{\Delta^{i}_{\min}} + 2{K^{\prime}\choose 2}m + \frac{\floor*{\frac{K^{\prime}}{2}}\pi^2}{3}m\Delta_{\max}
= O\left( \sum_{i=1}^{m} \frac{K'^3\log T}{\Delta^{i}_{\min}} \right).\end{aligned}$$
The proof of the above theorem is an adaption of the proof of Theorem 4 in [@wang2017improving], and the full proof details as well as the detailed comparison with the original CMAB framework result are included in the supplementary materials. We briefly explain our adaption that leads to the regret improvement. We rely on the following monotonicity and 1-norm bounded smoothness properties of our expected reward function $r_{\bm{k}}(\bm{\mu})$, similar to the ones in [@CWYW16; @wang2017improving].
\[pro:monotone\] The reward function $r_{\bm{k}}(\bm{\mu})$ is monotonically decreasing, i.e., for any two vectors $\bm{\mu} = (\mu_1,\dots, \mu_m)$ and $\bm{\mu}^{\prime} =
(\mu^{\prime}_1, \dots,\mu^{\prime}_m)$, we have $r_{\bm{k}}(\bm{\mu})\geq
r_{\bm{k}}(\bm{\mu}^{\prime})$ if $\mu_i\leq \mu^{\prime}_i \ \forall i\in [m]$.
\[pro:bounded\_smoothness\] The reward function $r_{\bm{k}}(\bm{\mu})$ satisfies the 1-norm bounded smoothness property, i.e., for any two vectors $\bm{\mu} = (\mu_1,\cdots,
\mu_{m})$ and $\bm{\mu}^{\prime} =(\mu^{\prime}_1, \cdots, \mu^{\prime}_{m})$, we have $|r_{\bm{k}}(\bm{\mu}) - r_{\bm{k}}(\bm{\mu}^{\prime})| \leq \sum_{i =
1}^{m}{k_i\choose 2}|\mu_i - \mu^{\prime}_i|\leq {K' \choose 2}\sum_{i =
1}^{m}|\mu_i - \mu^{\prime}_i|$.
We remark that if we directly apply the CMAB regret bound of Theorem 4 in [@wang2017improving], we need to revise the update procedure in Lines \[line:update\_pairs\]-\[line:online\_learning\_end\] of Algo. \[algo:CLCB\_algorithm\] so that each round we only update one observation for each community $C_i$ if $|{\mathcal{S}}_i| > 1$. Then we would obtain a regret bound $O\left( \sum_i \frac{K'^4 m\log T}{\Delta^{i}_{\min}} \right)$, which means that our regret bound in Eq. has an improvement of $O(K'm)$. This improvement is exactly due to the reason we give earlier, as we now explain with more details.
For all the random variables introduced in Algo. \[algo:CLCB\_algorithm\], we add the subscript $t$ to denote their value at the [*end*]{} of round $t$. For example, $T_{i,t}$ is the value of $T_i$ at the end of round $t$. First, the improvement of the factor $m$ comes from the use of a tighter bounded smoothness in Property \[pro:bounded\_smoothness\], namely, we use the bound $\sum_{i = 1}^{m}{k_i\choose 2}|\mu_i - \mu^{\prime}_i|$ instead of ${K'
\choose 2}\sum_{i = 1}^{m}|\mu_i - \mu^{\prime}_i|$. The CMAB framework in [@wang2017improving] requires the bounded smoothness constant to be independent of actions. So to apply Theorem 4 in [@wang2017improving], we have to use the bound ${K' \choose 2}\sum_{i =
1}^{m}|\mu_i - \mu^{\prime}_i|$. However, in our case, when using bound $\sum_{i
= 1}^{m}{k_i\choose 2}|\mu_i - \mu^{\prime}_i|$, we are able to utilize the fact $\sum_{i=1}^{m}{k_{i}\choose 2} \leq {K^{\prime}\choose 2}$ to improve the result by a factor of $m$. Second, the improvement of the $O(K')$ factor, more precisely a factor of $(K'-1)/2$, is achieved by utilizing multiple feedback in a single round and a more careful analysis of the regret utilizing the property of the right Riemann summation. Specifically, let $\Delta_{\bm{k}_t} = r_{\bm{k}^*}({\bm{\mu}}) -
r_{\bm{k}_t}({\bm{\mu}})$ be the reward gap. When the estimate is within the confidence radius, we have $\Delta_{\bm{k}_t} \leq \sum_{i=1}^{m}\frac{c(k_{i,t}
- 1)}{2}/\sqrt{T_{i,t-1}}\leq c\sum_{i=1}^{m}\floor{k_{i,t} /
2}/\sqrt{T_{i,t-1}}$, where $c$ is a constant. In Algo. \[algo:CLCB\_algorithm\], we have $T_{i,t} = T_{i,t-1} +
\floor{k_{i,t}/2}$ because we allow multiple feedback in a single round. Then $\sum_{t\geq 1, T_{i,t}\leq L_i(T)}\floor{k_{i,t} / 2}/\sqrt{T_{i,t-1}}$ is the form of a right Riemann summation, which achieves the maximum value when $k_{i,t} = K^{\prime}$. Here $L_i(T)$ is a $\ln T$ function with some constants related with community $C_i$. Hence the regret bound $\sum_{t=1}^{T}\Delta_{\bm{k}_t} \leq c\sum_{i=1}^{m}\sum_{t\geq 1, T_{i,t}\leq
L_i(T)} \floor{\frac{k_{i,t}}{2}} /\sqrt{T_{i,t-1}} \leq
2c\sum_{i=1}^{m}\sqrt{L_i(T) }$. However, if we use the original CMAB framework, we need to set $T_{i,t} = T_{i,t-1} + {\mathbbm{1}}\{k_{i,t} > 1\}$. In this case, we can only bound the regret as $\sum_{t=1}^{T}\Delta_{\bm{k}_t} =
c\sum_{i=1}^m\sum_{t\geq 1, T_{i,t}\leq L_i(T)}(k_{i,t} -1)/ 2\sqrt{T_{i,t-1}}
\leq 2c {\frac{K^{\prime} - 1}{2}}\sum_{i=1}^{m}\sqrt{L_i(T)}$, leading to an extra factor of ${(K^{\prime}-1)/2}$.
**Justification for Algo. \[algo:CLCB\_algorithm\].** In Algo. \[algo:CLCB\_algorithm\], we only use the members in current round to update the estimator. This is practical for the situation where the member identifiers are changing in different rounds for privacy protection. Privacy gains much attention these days. Consider the online advertising scenario we explain in the introduction. Whenever a user clicks an advertisement, the advertiser would store the user information (e.g. Facebook ID, IP address etc.) to identify the user and correlated with past visits of the user. If such user identifiers are fixed and do not change, the advertiser could easily track user behavior, which may result in privacy leak. A reasonable protection for users is to periodically change user IDs (e.g. Facebook can periodically change user hash IDs, or users adopt dynamic IP addresses, etc.), so that it is difficult for the advertiser to track the same user over a long period of time. Under such situation, it may be likely that our learning algorithm can still detect ID collisions within the short period of each learning round, but cross different rounds, collisions may not be detectable due to ID changes.
**Full information feedback.** Now we consider the scenario where the member identifiers are fixed over all rounds, and design an algorithm with a constant regret bound. Our idea is to ensure that we can observe at least one pair of members in every community $C_i$ in each round $t$. We call such guarantee as [*full information feedback*]{}. If we only use members revealed in current round, we cannot achieve this goal since we have no observation of new pairs for a community $C_i$ when $k_{i} = 1$. To achieve full information feedback, we use at least one sample from the previous round to form a pair with a sample in the current round to generate a valid pair collision observation. In particular, we revise the Line \[line:radius\], \[line:update\_pairs\], and \[line:count\_collision\] as follows. Here we use $u_{0}$ to represent the last member in ${\mathcal{S}}_i$ in the previous round (let $u_{0}=\text{null}$ when $t=1$) and $u_x (x > 0)$ to represent the $x$-th members in ${\mathcal{S}}_i$ in the current round. The revision of Line \[line:radius\] implies that we use the empirical mean $\hat{\mu}_i = X_i / T_i$ instead of the lower confidence bound in the function ``. $$\label{eq:revision_2}
\begin{split}
& \text{Line~\ref{line:radius}:~~ For } i\in [m], \rho_{i} = 0;~~~\text{Line~\ref{line:update_pairs}:~~ For } i\in [m], T_i\leftarrow T_i + {\left| {\mathcal{S}}_i \right|} - {\mathbbm{1}}\{t = 1\}, \\
& \text{Line~\ref{line:count_collision}:~~ For } i\in [m], X_{i} \leftarrow X_i+ \sum\nolimits_{x=0}^{{\left| {\mathcal{S}}_i \right|} - 1}{\mathbbm{1}}\{u_{x} = u_{x+1}\}.\\
\end{split}$$
\[thm:non\_adaptive\_full\_information\] With the full information feedback revision in Eq. , Algo. \[algo:CLCB\_algorithm\] with non-adaptive exploration method has a constant regret bound. Specifically, $$\text{Reg}_{\bm{\mu}}(T) \leq \left(2 + 2me^2K'^2(K'-1)^2 / \Delta^2_{\min}\right)\Delta_{\max}.$$
Note that we cannot apply the Hoeffding bound in [@hoeffding1963probability] directly since the random variables ${\mathbbm{1}}\{u_{x} = u_{x+1}\}$ we obtain during the online learning process are not mutually independent. Instead, we apply a concentration bound in [@Dubhashi2009CMA] that is applicable to variables that have local dependence relationship.
Regret Analysis for the Adaptive Version {#sec:online_adaptive}
----------------------------------------
For the adaptive version, we feed the lower confidence bound ${\underaccent{\bar}{\bm{\mu}}}_{t}$ into the adaptive community exploration procedure, namely $\textproc{\texttt{CommunityExplore}}(\{{\underaccent{\bar}{\mu}}_1, \dots,
{\underaccent{\bar}{\mu}}_m\}, K, \textrm{adaptive})$ in round $t$. We denote the policy implemented by this procedure as $\pi^t$. Note that both $\pi^g$ and $\pi^t$ are based on the greedy procedure $\textproc{\texttt{CommunityExplore}}(\cdot , K, \textrm{adaptive})$. The difference is that $\pi^g$ uses the true parameter $\bm{\mu}$ while $\pi^t$ uses the lower bound parameter ${\underaccent{\bar}{\bm{\mu}}}_t$. More specifically, given a partial realization $\psi$, the community chosen by $\pi^t$ is $C_{i^*}$ where $i^*\in \operatorname*{arg\,max}_{i\in [m]} 1 - c_i(\psi){\underaccent{\bar}{\mu}}_{i,t}$. Recall that $c_i(\psi)$ is the number of distinct encountered members in community $C_i$ under partial realization $\psi$.
We first properly define the metrics $\Delta^{i, k}_{\min}$ and $\Delta^{(k)}_{\max}$ used in the regret bound as follows. Consider a specific full realization $\phi$ where $\{\phi(i, 1), \dots, \phi(i, d_i)\}$ are $d_i$ distinct members in $C_i$ for $i\in [m]$. The realization $\phi$ indicates that we will obtain a new member in the first $d_i$ exploration of community $C_i$. Let $U_{i,k}$ denote the number of times community $C_i$ is selected by policy $\pi^g$ in the first $k-1 (k > m)$ steps under the special full realization $\phi$ we define previously. We define $\Delta^{i, k}_{\min} = (\mu_iU_{i,k} -
\min_{j\in [m]}\mu_jU_{j,k})/U_{i,k}$. Conceptually, the value $\mu_iU_{i,k} - \min_{j\in
[m]}\mu_jU_{j,k}$ is gap in the expected reward of the next step between selecting a community by $\pi^g$ (the optimal policy) and selecting community $C_i$, when we already meet $U_{j,k}$ distinct members in $C_j$ for $j\in [m]$. When $\mu_iU_{i,k} = \min_{j\in [m]}\mu_jU_{j,k}$, we define $\Delta^{i,k}_{\min} = \infty$. Let $\pi$ be another policy that chooses the same sequence of communities as $\pi^g$ when the number of met members in $C_i$ is no more than $U_{i,k}$ for all $i\in [m]$. Note that policy $\pi$ chooses the same communities as $\pi^g$ in the first $k-1$ steps under the special full realization $\phi$. Actually, the policy $\pi$ is the same as $\pi^g$ for at least $k-1$ steps. We use $\Pi_k$ to denote the set of all such policies. We define $\Delta^{(k)}_{\max}$ as the maximum reward gap between the policy $\pi\in
\Pi_k$ and the optimal policy $\pi^g$, i.e., $\Delta^{(k)}_{\max} = \max_{\pi\in
\Pi_k} r_{\pi^g}(\bm{\mu}) - r_{\pi}(\bm{\mu})$. Let $D = \sum_{i=1}^{m}d_i$.
[theorem]{}[regretboundc]{}\[thm:regret\_bound\_adaptive\_exploration\] Algo. \[algo:CLCB\_algorithm\] with adaptive exploration method has regret as follows. $$\begin{aligned}
\label{eq:adaptive_regret_bound_a}
&\text{Reg}_{\bm{\mu}}(T) \leq \left( \sum_{i=1}^{m}\sum_{k = m + 1}^{\min\{K, D\}} \frac{6\Delta^{(k)}_{\max}}{(\Delta^{i,k}_{\min})^2}\right)\ln T +
\frac{\floor{\frac{K^{\prime}}{2}}\pi^2}{3}\sum_{i=1}^{m}\sum_{k = m + 1}^{\min\{K, D\}}\Delta^{(k)}_{\max}.\end{aligned}$$
\[thm:adaptive\_full\_information\] With the full information feedback revision in Eq. , Algo. \[algo:CLCB\_algorithm\] with adaptive exploration method has a constant regret bound. Specifically, $$\text{Reg}_{\bm{\mu}}(T) \leq \sum\nolimits_{i=1}^{m}\sum\nolimits_{k = m + 1}^{\min\{K, D\}} \left(2/\varepsilon^4_{i, k} + 1\right)\Delta^{(k)}_{\max}.$$ where $\varepsilon_{i, k}$ is defined as (here $i^*_k\in \operatorname*{arg\,min}_{i\in [m]} \mu_iU_{i,k}$) $$\varepsilon_{i, k} \triangleq (\mu_iU_{i,k} - \mu_{i^*_k}U_{i^*_k, k})/(U_{i,k} + U_{i^*_k, k}) \text{ for } i\neq i^*_k \text{ and } \varepsilon_{i, k} = \infty \text{ for } i = i^*_k.$$
@gabillon2013adaptive analyzes a general adaptive submodular function maximization in bandit setting. We have a regret bound in similar form as if we directly apply Theorem 1 in [@gabillon2013adaptive]. However, their version of $\Delta^{(k)}_{\max}$ is an upper bound on the expected reward of policy $\pi^g$ from $k$ steps forward, which is larger than our $\Delta^{(k)}_{\max}$. Their version of $\Delta^{i,k}_{\min}$ is the minimum $(\mu_ic_i(\psi) - \min_{j\in
[m]}\mu_{j}c_{j}(\psi))/c_i(\psi)$ for all partial realization $\psi$ obtained after policy $\pi^g$ is executed for $k$ steps, which is smaller than our $\Delta^{i, k}_{\min}$. Our regret analysis is based on counting how many times $\pi^g$ and $\pi^t$ choose different communities under the special full realization $\phi$, while the analysis in [@gabillon2013adaptive] is based on counting how many times $\pi^g$ and $\pi^t$ choose different communities under all possible full realizations.
**Discussion.** In this paper, we consider the online learning problem that consists of $T$ rounds, and during each round, we explore the communities with a budget $K$. Our goal is to maximize the [*cumulative reward*]{} in $T$ rounds. Another important and natural setting is described as follows. We start to explore communities with unknown sizes, and update the parameters every time we explore the community for *one step* (or for a few steps). Different from the setting defined in this paper, here [*a member will not contribute to the reward if it has been met in previous rounds*]{}. To differentiate the two settings, let’s call the latter one the “[*interactive community exploration*]{}”, while the former one the “[*repeated community exploration*]{}”. Both the repeated community exploration defined in this paper and the interactive community exploration we will study as the future work have corresponding applications. The former is suitable for online advertising where in each round the advertiser promotes different products. Hence the rewards in different rounds are additive. The latter corresponds to the adaptive online advertising for the same product, and thus the rewards in different rounds are dependent.
Related Work
============
@golovin2011adaptive show that a greedy policy could achieve at least $(1 - 1/e)$ approximation for the adaptive submodular function. The result could be applied to our offline adaptive problem, but by an independent analysis we show the better result that the greedy policy is optimal. Multi-armed bandit (MAB) problem is initiated by Robbins [@robbins1985some] and extensively studied in [@berry1985bandit; @sutton1998reinforcement; @bubeck2012regret]. Our online learning algorithm is based on the extensively studied [*Upper Confidence Bound*]{} approach [@auer2002finite]. The non-adaptive community exploration problem in the online setting fits into the general combinatorial multi-armed bandit (CMAB) framework [@gai2012combinatorial; @kveton2015tight; @CWYW16; @chen2016combinatorial; @wang2017improving], where the reward is a set function of base arms. The CMAB problem is first studied in [@gai2012combinatorial], and its regret bound is improved by [@CWYW16; @kveton2015tight]. We leverage the analysis framework in [@CWYW16; @wang2017improving] and prove a tighter bound for our algorithm. @gabillon2013adaptive define an adaptive submodular maximization problem in bandit setting. Our online adaptive exploration problem is a instance of the problem defined in [@gabillon2013adaptive]. We prove a tighter bound than the one in [@gabillon2013adaptive] by using the properties of our problem.
Our model bears similarities to the optimal discovery problem proposed in [@bubeck2013optimal] such as we both have disjoint assumption, and both try to maximize the number of target elements. However, there are also some differences: (a) We use different estimators for our critical parameters, because our problem setting is different. (b) Their online model is closer to the interactive community exploration we explained in \[sec:online\_adaptive\] , while our online model is on repeated community exploration. As explained in \[sec:online\_adaptive\], the two online models serve different applications and have different algorithms and analyses. (c) We also have more comprehensive studies on the offline cases.
Future Work
===========
In this paper, we systematically study the community exploration problems. In the offline setting, we propose the greedy methods for both of non-adaptive and adaptive exploration problems. The optimality of the greedy methods are rigorously proved. We also analyze the online setting where the community sizes are unknown initially. We provide a CLCB algorithm for the online community exploration. The algorithm has $O(\log T)$ regret bound. If we further allow the full information feedback, the CLCB algorithm with some minor revisions has a constant regret.
Our study opens up a number of possible future directions. For example, we can consider various extensions to the problem model, such as more complicated distributions of member meeting probabilities, overlapping communities, or even graph structures between communities. We could also study the gap between non-adaptive and adaptive solutions.
### Acknowledgments {#acknowledgments .unnumbered}
We thank Jing Yu from School of Mathematical Sciences at Fudan University for her insightful discussion on the offline problems, especially, we thank Jing Yu for her method to find a good initial allocation, which leads to a faster greedy method. Wei Chen is partially supported by the National Natural Science Foundation of China (Grant No. 61433014). The work of John C.S. Lui is supported in part by the GRF Grant 14208816.
**Supplementary Materials**
Improved Budget Allocation Algorithm {#app:improved_budget_allocation}
====================================
Let $r_i(j) = \mathbb{E}_{\Phi}[{\left| \{\Phi(i,1), \ldots, \Phi(i, j)\} \right|}]
= d_i (1 - (1 - 1/d_i)^{j})$ denote the expected reward when the community $i$ is explored $j$ times. Then we have that the marginal gain $r_i(j + 1) - r_i(j) = (1 - \mu_i)^j$ . Define a matrix $\bm{X}\in \mathbb{R}^{m\times K}$, where the $(i,j)$-th entry $X_{i,j}$ is $(1-\mu_i)^{j-1}$. When the budget allocation is $\bm{k} =
(k_1, \dots, k_m)$, the expected reward $r_{\bm{k}}(\bm{\mu})$ can be written as the sum of elements in $\bm{X}$, i.e., $r_{\bm{k}}(\bm{\mu})
= \sum_{i=1}^{m}\sum_{j=1}^{k_i}X_{i,j}$. A key property of $\bm{X}$ is that the value in each row is decreasing with respect to the column index $j$. Hence, for every $s \ge 1$, the $s$-th step of the greedy method chooses the $s$-th largest value in $\bm{X}$. At step $s = K$, the greedy method finds the largest $K$ values in matrix $\bm{X}$. We can conclude that the greedy method obtains a budget allocation that maximizes the reward $r_{\bm{k}}(\bm{\mu})$.
We propose a budget allocation algorithm which has time complexity $O(m\log m)$ in Algo. \[algo:improved\_budget\_allocation\]. The basic idea is to find a good initial allocation that is not far from the optimal allocation. Then starting from the initial allocation, we run our original greedy method.
\[line:initial\_allocation\]
\[lemma:basic\_property\] Let $\bm{k}^*$ be the optimal budget allocation when the parameter of the community is $\bm{\mu}$. For $i, j\in [m]$ , we have $$(1 - \mu_i)^{(k^*_i - 1)} \geq (1 - \mu_j)^{k^*_j}.$$
We define budget allocation $\bm{k}^{\prime}$ which is the same as $\bm{k}^{*}$ except that $k^{\prime}_i = k^{*}_i - 1$ and $k^{\prime}_j = k^*_j + 1$. If $(1 - \mu_i)^{(k^*_i - 1)} < (1 - \mu_j)^{k^*_j}$ and $i \neq j$, then we have $$r_{\bm{k}^{\prime}}(\bm{\mu}) = r_{\bm{k}^*}(\bm{\mu}) - (1 - \mu_i)^{(k^*_i - 1)} + (1 - \mu_j)^{k^*_j} > r_{\bm{k}^*}(\bm{\mu}),$$ which is contradict with the fact that $\bm{k}^*$ is the optimal solution. This proves the lemma.
\[lemma:allocation\_lower\_bound\] Let $\bm{k}^*$ be the optimal budget allocation when the parameter of the communities is $\bm{\mu}$. Define $\bm{k}^{-} = (k^{-}_1, \dots, k^-_m)$ where $$k^-_i = \frac{(K - m) / \ln (1 - \mu_i)}{\sum_{j=1}^m 1 / \ln (1 - \mu_j)}.$$ We have $k_i^* \geq k^-_i$.
According to the definition of $\bm{k}^-$, we have $k^-_i\ln (1 - \mu_i) =
k^-_j\ln (1 - \mu_j)$ for $i, j \in [m]$. If we can find $i$ such that $k^-_i + 1 \leq k^*_i $, then $$(1-\mu_j)^{k^-_j} = (1-\mu_i)^{k^-_i} \geq (1-\mu_i)^{k^*_i - 1} \geq (1 - \mu_j)^{k^*_j}.$$ Hence $k^-_j \leq k^*_j$. On the other hand, we can always find $k^-_i + 1 \leq k^*_i$ since $\sum_{i=1}^{m} (k^-_i + 1) = K$.
In Algo. \[algo:improved\_budget\_allocation\], we start with the lower bound $\bm{k}^{-}$ of the optimal allocation. Since $\sum_{i=1}^{m}k^{-}_i = K - m$, we have $\sum_{i=1}^{m}{\left| \ceil{k^{-}_i} - k^*_i \right|}
\leq\sum_{i=1}^{m}{\left| k^{-}_i - k^*_i \right|} = m$, which indicates Algo. \[algo:improved\_budget\_allocation\] obtains the optimal budget allocation within $m$ steps. We also provide an upper bound $\bm{k}^+$ in the following. The upper bound is also close to the optimal budget since $\sum_{i=1}^m {\left| \floor{k^+_i} - k^*_i \right|} \leq \sum_{i=1}^m {\left| k^+_i - k^*_i \right|}
= m$.
\[lemma:allocation\_upper\_bound\] Let $\bm{k}^*$ be the optimal budget allocation when the parameter of the communities is $\bm{\mu}$. Define $\bm{k}^{+} = (k^{+}_1, \dots, k^+_m)$ where $$k^+_i = \frac{K / \ln (1 - \mu_i)}{\sum_{j=1}^m 1 / \ln (1 - \mu_j)} + 1.$$ We have $k_i^* \leq k^+_i$.
According to the definition of $\bm{k}^+$, we have $(k^+_i - 1)\ln (1 - \mu_i) =
(k^+_j -1)\ln (1 - \mu_j)$ for $i, j \in [m]$. If we can find $i$ such that $k^+_i - 1 \geq k^*_i $, then $$(1-\mu_j)^{k^+_j - 1} = (1-\mu_i)^{k^+_i - 1} \leq (1-\mu_i)^{k^*_i} \leq (1 - \mu_j)^{k^*_j-1}.$$ Hence $k^+_j \geq k^*_j$. On the other hand, we can always find $k^+_i - 1 \geq k^*_i$ since $\sum_{i=1}^{m} (k^+_i - 1) = K$.
Properties of Greedy Policy
===========================
In the following, we show some important properties of the greedy policy. We further extend the definition of reward with a non-decreasing function $f$ as $
R(\bm{k}, \phi)
=f\left(\sum_{i=1}^{m}{\left| \bigcup_{\tau=1}^{k_i}\{\phi(i,\tau)\} \right|}\right)$.
Optimality of greedy policy {#app:optimality_greedy_policy}
---------------------------
In this part, we prove that the greedy policy is the optimal policy for our adaptive community exploration problem. To prove the optimality, we first rewrite the proof sketch of Theorem \[thm:greedy\_policy\_is\_optimal\], and then provide the supporting Lemma \[lemma:property\_greedy\_policy\]&\[lemma:concantenation\_policy\].
Let $F_{\pi}(\psi, t)$ denote the expected [*marginal gain*]{} when we further explore communities for $t$ steps with policy $\pi$ starting from a partial realization $\psi$. We want to prove that for all $\psi$, $t$ and $\pi$, $F_{\pi^g} (\psi,t)\ge F_\pi(\psi,t)$, where $\pi^g$ is the greedy policy and $\pi$ is an arbitrary policy. If so, we simply take $\psi = \emptyset$, and $F_{\pi^g} (\emptyset,t)\ge F_\pi(\emptyset,t)$ for every $\pi$ and $t$ exactly shows that $\pi^g$ is optimal. We prove the above result by an induction on $t$. Recall that $c_i(\psi)$ is the number of distinct members met in community $C_i$ under the partial realization $\psi$. Define $c(\psi) = \sum_i c_i(\psi)$ and $\Delta_{\psi, f} = f(c(\psi) + 1) - f(c(\psi))$.
For all $\psi$ and $\pi$, we first claim that $F_{\pi^g} (\psi,1)\ge
F_\pi(\psi,1)$ holds. Suppose that policy $\pi$ chooses community $C_i$ to explore based on the partial realization $\psi$. Since the exploration will return a new member with probability $1 - \mu_ic_i(\psi)$, the expected marginal gain $F_{\pi}(\psi, 1)$ is $(1 - \mu_ic_i(\psi))[f(c(\psi) + 1) - f(c(\psi))]$. Note that the greedy policy $\pi^g$ chooses community $C_{i^*}$ to explore with $i^*\in \operatorname*{arg\,max}_{j} (1 - \mu_j c_j(\psi))$, and $\Delta_{\psi, f}$ does not depend on the policy. Hence, $F_{\pi^g} (\psi,1)\ge F_\pi(\psi,1)$.
Assume $F_{\pi^g} (\psi,t')\ge F_\pi(\psi,t')$ holds for all $\psi$, $\pi$, and $t'\le t$. Our goal is to prove that $F_{\pi^g} (\psi,t+1)\ge F_\pi(\psi,t+1)$. Suppose that in the first step after $\psi$, policy $\pi$ chooses $C_i$ to explore based on partial realization $\psi$, and let $\pi(\psi)=(i,\tau)$. Define $E_{\psi}$ as the event that the member $\Phi(i,\tau)$ is not met in partial realization $\psi$, for $\Phi\sim \psi$. In the following, we represent partial realization $\psi$ equivalently as a relation $\{((i,\tau),
\psi(i,\tau)) \mid (i,\tau) \in {{\rm dom}}(\psi) \}$, so we could use $\psi\cup\{((i,\tau), \Phi(i,\tau))\}$ to represent the new partial realization extended from $\psi$ by one step with $(i,\tau)$ added to the domain and $\Phi(i,\tau)$ as the member met for this exploration of $C_i$. Then we have $$\begin{aligned}
&F_{\pi}(\psi, t + 1) = \sum\nolimits_{v\in C_i}\Pr\left( \Phi(i,\tau) = v\right)\mathbb{E}_{\Phi}[F_{\pi}(\psi, t + 1) \mid \Phi\sim \psi, \Phi(i,\tau) = v]\\
&= \sum_{v\in C_i}\mu_i\mathbb{E}_{\Phi}[F_{\pi}(\psi\cup\{((i,\tau), \Phi(i,\tau))\}, t) + f(c(\psi) + {\mathbbm{1}}\{E_{\psi}\}) - f(c(\psi)) \mid \Phi\sim \psi, \Phi(i,\tau) = v]\\
&\leq\sum\nolimits_{v\in C_i} \mu_i\mathbb{E}_{\Phi}[F_{\pi^g}(\psi\cup\{((i,\tau), \Phi(i,\tau))\}, t) \mid \Phi\sim \psi, \Phi(i,\tau) = v] + (1 - \mu_ic_i(\psi))\Delta_{\psi,f}.
\end{aligned}$$ The $2$nd line above is derived directly from the definition of $F_{\pi}(\psi,t)$. The $3$rd line is based on the induction hypothesis that $F_{\pi}(\psi', t) \leq F_{\pi^g}(\psi', t)$ holds for all $\psi'$. An important observation is that $F_{\pi^g}(\psi, t)$ has equal value for any partial realization $\psi$ associated with the same status $\bm{s}$ since the status is enough for the greedy policy to determine the choice of next community. Formally, we define $F_{g}(\bm{s}, t) =
F_{\pi^g}(\psi, t)$ for any partial realization that satisfies $\bm{s} = (1 -
c_1(\psi)/d_1,\dots, 1 - c_m(\psi)/d_m)$. Let $C_{i^*}$ denote the community chosen by policy $\pi^g$ under realization $\psi$, i.e., $i^*\in
\operatorname*{arg\,max}_{i\in [m]} 1 - c_i(\psi)\mu_i$. Let ${\boldsymbol{I}}_i$ be the $m$-dimensional unit vector with $1$ in the $i$-th entry and $0$ in all other entries. Therefore, $$\begin{aligned}
F_{\pi}(\psi, t + 1) &\leq c_i(\psi) \cdot \mu_i F_{g}(\bm{s}, t) + (d_i - c_i(\psi)) \cdot \mu_i F_{g}(\bm{s} - \mu_i {\boldsymbol{I}}_i, t) + (1 - \mu_ic_i(\psi))\Delta_{\psi, f} \\
& \leq \mu_{i^*}c_{i^*}(\psi)F_{g}(\bm{s}, t) + (1 - \mu_{i^*}c_{i^*}(\psi))F_{g}(\bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, t) + (1 - \mu_{i^*}c_{i^*}(\psi))\Delta_{\psi, f} \OnlyInFull{\tag{Lemma~\ref{lemma:concantenation_policy}}}\\
& = F_{g}(\bm{s}, t + 1) = F_{\pi^g}(\psi, t + 1). \OnlyInFull{\tag{Lemma~\ref{lemma:property_greedy_policy}}}
\end{aligned}$$ The key is to prove the correctness of Line 2 in above equation. It indicates that if we choose a sub-optimal community at first, and then we switch back to the greedy policy, the expected reward would be smaller. The proof is nontrivial and relies on a careful analysis based on the stochastic transitions among status vectors. The above result completes the induction step for $t+1$. Thus the theorem holds.
\[lemma:property\_greedy\_policy\] Let $\bm{s} = (s_1,\dots, s_m)$ be a status where each entry $s_i\in [0, 1]$. We have $$F_{g}(\bm{s}, t + 1) = (1 - s_{i^*})F_{g}(\bm{s}, t) + s_{i^*}F_{g}(\bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, t) + s_{i^*} (f(c(\psi) + 1) - f(c(\psi))),$$ where $i^* = \operatorname*{arg\,max}_{i\in [m]}s_i$. Here $\psi$ is any partial realization corresponding to status $\bm{s}$.
For any partial realization $\psi$ associated with status $\bm{s}$, $\pi^g$ would choose community $i^*$. With probability $\mu_{i^*}c_{i^*}(\psi) = 1 -
s_{i^*}$, we will obtain a member that is already met. If so, the communities stay at the same status. Hence, with probability $1 -s_{i^*}$, the expected extra reward is $F_g(\bm{s}, t)$ after the first step exploration. With probability $1 -\mu_{i^*}c_{i^*}(\psi) = s_{i^*}$, we will obtain an unseen member in $C_{i^*}$. The communities will transit to next status $\bm{s} -
\mu_{i^*}{\boldsymbol{I}}_{i^*}$. Therefore, with probability $s_{i^*}$, the expected extra reward is $F_g(\bm{s} - {\boldsymbol{I}}_{i^*}, t) + f(c(\psi) + 1) f(c(\psi))$ after the first step exploration.
\[lemma:concantenation\_policy\] Let $\bm{s} = (s_1,\dots, s_m)$ be a status where each entry $s_i\in [0, 1]$ and $\psi$ be any partial realization corresponding to $\bm{s}$. We have $$\label{eq:condition2}
\begin{split}
&(1 - s_i)F_{g}(\bm{s}, t) + s_i F_{g}(\bm{s} - \mu_i {\boldsymbol{I}}_i, t) + s_i\Delta_c \\
\leq\ \ &(1 - s_{i^*})F_{g}(\bm{s}, t) + s_{i^*}F_{g}(\bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, t) + s_{i^*}\Delta_c,
\end{split}$$ where $i^* \in \operatorname*{arg\,max}_{i\in [m]}s_i$, $s_i < s_{i^*}$ and $\Delta_c = f(c(\psi) +
1) - f(c(\psi))$.
Let $A(\bm{s}, i, t)$ denote the first line of Eq. , i.e., $$A(\bm{s}, i, t) = (1 - s_i)F_{g}(\bm{s}, t) + s_i F_{g}(\bm{s} - \mu_i {\boldsymbol{I}}_i, t) + s_i\Delta_c.$$ Note that $A(\bm{s}, i, t)$ is the expected reward of the following adaptive process.
1. At the first step, choose an arbitrary community $C_i$ (different from $C_{i^*}$) to explore.
2. From the second step to the $(t + 1)$-th step, explore communities with the greedy policy $\pi^g$.
Similarly, $A(\bm{s}, i^*, t)$ is the expected reward of the $t+1$ step community exploration via the greedy policy, i.e., $A(\bm{s}, i^*, t) = F_g(\bm{s}, t + 1)$. Eq. can be written as $A(\bm{s}, i, t) \leq F_g(\bm{s}, t + 1)$. We prove this inequality by induction. When $t = 0$, we have $A(\bm{s}, i, t) =
s_i\Delta_c$, and $A(\bm{s}, i^*, t) = s_{i^*}\Delta_c$. Hence, $A(\bm{s}, i,
t)\leq A(\bm{s}, i^*, t) = F_g(\bm{s}, t + 1)$ when $t = 0$. Assume $A(\bm{s}, i, t^{\prime}) \leq F_g(\bm{s}, t^{\prime} + 1)$ holds for any $0\leq t^{\prime} \leq t$, and any status $\bm{s}$. Our goal is to prove that $A(\bm{s}, i, t + 1) \leq
A(\bm{s}, i^*, t+1) = F_g(\bm{s}, t + 2)$. We expand $A(\bm{s}, i, t + 1)$ as follows. $$\begin{aligned}
A(\bm{s}, i, t + 1) &= (1 - s_i)F_{g}(\bm{s}, t + 1) + s_i F_{g}(\bm{s} - \mu_i {\boldsymbol{I}}_i, t + 1) + s_i\Delta_c\\
&= (1 - s_i)\left( (1 - s_{i^*})F_{g}(\bm{s}, t) + s_{i^*}F_{g}(\bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, t) + s_{i^*}\Delta_{c} \right)\\
& + s_i ((1 - s_{i^*})F_g(\bm{s} -\mu_i\bm{I}_{i}, t) + s_{i^*}F_g(\bm{s} - \mu_i\bm{I}_i - \mu_{i^*}\bm{I}_{i^*}, t) + s_{i^*}\Delta_{c + 1}) \\
& + s_i\Delta_c.
\end{aligned}$$ Here $\Delta_{c+1} = f(c(\psi) + 2) - f(c(\psi) + 1)$. Above expansion of $A(i,
t+1)$ is based on Lemma \[lemma:property\_greedy\_policy\]. We expand $A(\bm{s},
i^*, t+1)$ as follows. $$\begin{aligned}
A(\bm{s}, i^*, t + 1) &= (1 - s_{i^*})F_{g}(\bm{s}, t + 1) + s_{i^*}F_{g}(\bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, t + 1) + s_{i^*}\Delta_c \\
& \geq (1 - s_{i^*})\left( (1 - s_{i})F_{g}(\bm{s}, t) + s_{i}F_{g}(\bm{s} - \mu_{i} {\boldsymbol{I}}_{i}, t) + s_{i}\Delta_{c} \right)\tag{assumption $A(\bm{s}, i, t)\leq F_g(\bm{s}, t + 1)$}\\
& + s_{i^*} ((1 - s_i)F_g(\bm{s} - \mu_{i^*}\bm{I}_{i^*}, t) + s_{i}F_g(\bm{s} -\mu_{i^*}\bm{I}_{i^*} - \mu_i\bm{I}_i, t) + s_{i}\Delta_{c+1}) \tag{assumption $A(\bm{s} - \mu_{i^*}\bm{I}_{i^*}, i, t) \leq F_g(\bm{s} - \mu_{i^*}\bm{I}_{i^*}, t + 1)$}\\
& + s_{i^*} \Delta_{c}\\
&= A(i, t + 1).\end{aligned}$$ This completes the proof.
**Remarks.** During the rebuttal of this paper, we realized that @bubeck2013optimal applied similar inductive reasoning techniques to prove the optimality of the greedy policy for their optimal discovery problem (Lemma 2 of [@bubeck2013optimal]). To quantitatively measure how good is the greedy policy, we also give a formula to show the exact difference between $A(\bm{s},
i, t)$ and $A(\bm{s}, i^*, t)$ in Sec. \[sec:exact\_reward\_gap\].
Computation of expected reward {#app:expected_reward_greedy_policy}
------------------------------
Lemma \[lemma:property\_greedy\_policy\] indicates $r_{\pi^g}(\bm{\mu})$ can be computed in a recursive way. However, the recursive method has time complexity $O(2^K)$. It is impractical when $K$ is large. In the following we show that the expected reward of policy $\pi^g$ can be computed in polynomial time.
### Transition probability list of greedy policy {#app:transition_probability_list_greedy_policy}
Assume we explore the communities via the greedy policy when the communities already have partial realization $\psi$. Define $s_{i, 0} = 1 - \mu_ic_i(\psi)$ and $\bm{s}_0 = (s_{1, 0}, \dots, s_{m, 0})$. The greedy policy will choose community $i^*_0$ to explore, where $i^*_0 \in \operatorname*{arg\,max}_{i}s_{i, 0}$. After one step exploration, the communities stay at the same status $\bm{s}_0$ with probability $q_0 {\vcentcolon=}1 - s_{i^*_0}$. The communities transit to next status $\bm{s}_1 {\vcentcolon=}\bm{s}_0 -\mu_{i^*_0}{\boldsymbol{I}}_{i^*_0}$ with probability $p_0 {\vcentcolon=}s_{i^*_0}$. We recursively define $\bm{s}_{t+1}$ as $\bm{s}_{t} -
\mu_{i^*_{t}}{\boldsymbol{I}}_{i^*_{t}}$, where $i^*_t \in \operatorname*{arg\,max}_{i}s_{i, t}$. We call $p_t
{\vcentcolon=}\max_i s_{i,t}$ the [*transition probability*]{} and $q_t {\vcentcolon=}1 -
p_t$ the [*loop probability*]{}. Each time the communities transit to next status, a new member will be met. During the exploration, the number of different statuses the communities can stay is at most $1 + \sum_id_i -
c_i(\psi)$ since there are $D {\vcentcolon=}\sum_id_i - c_i(\psi)$ unseen members in total. Based on above discussion, we define a [*transition probability list*]{} $\mathcal{P}(\pi^g, \psi) {\vcentcolon=}(p_0, \dots, p_D)$, where $p_D \equiv 0$. The list $\mathcal{P}(\pi^g, \psi)$ is unique for any initial partial realization $\psi$. Fig. \[fig:illustration\_statuses\] gives an example to demonstrate statuses and the list $\mathcal{P}(\pi^g, \psi)$.
\[corollary:observation\_on\_probability\_list\] Let $\psi$ be any partial realization corresponding to the status $\bm{s} =
(s_1,\dots, s_m)$. The number of unseen members $\sum_id_i - c_i(\psi)$ is denoted as $D$. The probability list $\mathcal{P}(\pi^g, \psi) = (p_0,\dots,
p_{D})$ can be obtained by sorting $\cup_{i=1}^m\{s_i, s_i - \mu_i, \dots,
\mu_i\}\cup \{0\}$ in descending order.
Corollary \[corollary:observation\_on\_probability\_list\] is an important observation based on the definition of transition probability list.
![Illustration with $\bm{d} = (3, 4)$ and empty partial realization. The initial status is $(1, 1)$. The list $\mathcal{P}(\pi^g, \emptyset) = (1, 1, 3/4,
2/3, 1/2, 1/3, 1/4, 0)$.[]{data-label="fig:illustration_statuses"}](greedy_state_enumeration.pdf){width="90.00000%"}
### Compute the expected reward efficiently
\[lemma:expected\_reward\_probability\_list\] Let $\psi$ be a partial realization and $\bm{s}_0$ be the corresponding status. The number of unseen members is denoted as $D = \sum_i d_i - c_i(\psi)$. The transition probability list is $\mathcal{P}(\pi^g, \psi) = (p_0, \dots, p_{D})$. Then $$F_{\pi^g}(\psi, t) = F_g(\bm{s}_0, t) = \sum_{j=0}^{\min\{t, D\}} (f(j + c(\psi)) - f(c(\psi)))\times\left(\Pi_{l=0}^{j-1}p_j \right) \times \left(\sum_{I\in \mathcal{I}(j, t - j)} \Pi_{l\in I} q_l\right),$$ where $q_l = 1 - p_l$ and $\mathcal{I}(j, t-j)$ consists of subsets of multi-set $\{0, \dots, j\}^{t-j}$ with fixed size $t-j$.
When the communities ends at status $\bm{s}_j$, we meet $j$ distinct members. Let $\Pr(\bm{s}_j\square)$ be the probability for this event. We can the [*transition step*]{} as the communities transit to a new status, and the [*loop step*]{} as the communities stay at the same status. When the communities ends at status $\bm{s}_j$, we have $j$ transition steps and $t - j$ loop steps. The communities takes loops at statuses $\{\bm{s}_0,
\dots, \bm{s}_j\}$. Hence, $$\Pr(\bm{s}_j\square) = \sum_{I\in \mathcal{I}(j, t-j)} \Pi_{l=0}^{j-1}p_j \cdot \Pi_{l\in I} (1 - p_l) = \Pi_{l=0}^{j-1}p_j \times \sum_{I\in \mathcal{I}(j, t-j)} \Pi_{l\in I} q_l.$$ The reward $F_{\pi^g}(\psi, t) = \sum_{j=1}^{\min\{t, D\}} (f(j + c(\psi)) - f(c(\psi)))\times
\Pr(\bm{s}_j\square)$.
For later analysis, we define the [*loop probability*]{} $$L(\{q_0, \dots, q_j\}, t) {\vcentcolon=}\sum_{I\in \mathcal{I}(j, t)} \Pi_{l\in I} q_l$$ since $\sum_{I\in \mathcal{I}(j, t)} \Pi_{l\in I} q_l$ is just a function of $\{q_0, \dots, q_j\}$ and $t$ ($t \geq 1$). Actually, $L(\{q_0, \dots, q_j\},
t)$ aggregates the product of all possible $t$ elements in $\{q_0,\ldots,
q_{j}\}$. Note that each element in $\{q_0,\ldots, q_j\}$ can be chosen multiple times. W.l.o.g, we define $L(\{q_0, \dots, q_j\}, t) = 1$ and $\Pi_{l=0}^{t-1} p_l=
1$ when $t = 0$. Based on the definition, we can write $L(\{q_0, \dots, q_j\}, t)$ in a recursive way as follows. $$\label{eq:recursive_loop_probability}
L(\{q_0, \dots, q_j\}, t) = \sum_{s=0}^{t} q^s_aL(\{q_0, \dots, q_j\}\backslash\{q_a\}, t - s).$$ Here $a\in \{0, \dots, j\}$. According to Eq. \[eq:recursive\_loop\_probability\], the probability $\sum_{I\in
\mathcal{I}(j, t - j)} \Pi_{l\in I} q_l$ can be computed in $O((t - j)j^2)$ via [*dynamic programming*]{}. Hence $r_{\pi^g}(\bm{\mu}) = F_g((1,\dots, 1), K)$ can be computed in $O(K\min\{K, D\}^2)$ according to Lemma \[lemma:expected\_reward\_probability\_list\].
Reward gap between optimal policy and sub-optimal policy {#sec:exact_reward_gap}
--------------------------------------------------------
Recall that $A(\bm{s}, i, t)$ is the expected reward of the following adaptive process.
1. At the first step, choose an arbitrary community $C_i$ (different from $C_{i^*}$) to explore.
2. From the second step to the $(t + 1)$-th step, explore communities with the greedy policy $\pi^g$.
Here $\bm{s}$ is the initial status of the communities. Lemma \[lemma:concantenation\_policy\] only proves that $A(\bm{s}, i, t) \leq
F_g(\bm{s}, t + 1)$. In the following, we aim to answer the following question:
- How much is $F_g(\bm{s, t + 1})$ larger than $A(\bm{s}, i, t)$?
### Analysis of loop probability {#app:loop_probability}
The following two corollaries show the basic properties of the [*loop probability*]{}.
\[corollary:sum\_of\_probability\] For a transition probability list $\mathcal{P}(\pi^g, \psi) = (p_0, \dots,
p_{D})$, we have $$\sum_{j=0}^Mp_0\times\dots\times p_{j-1}\times L(\{q_0,\dots,q_j\}, t - j) = 1,$$ where $q_j = 1 - p_j$ and $M = \min\{t, D\}$.
Corollary \[corollary:sum\_of\_probability\] says the probabilities that the communities ends at status $\{\bm{s}_0, \ldots, \bm{s}_D\}$ sums up to 1.
\[corollary:loop\_probabilities\_switch\] For a transition probability list $\mathcal{P}(\pi^g, \psi) = (p_0,\dots, p_D)$ and $a, b\in \{0, \dots, j\}$ ($j\leq D, t\geq 1$), we have $$\begin{aligned}
& L(\{q_0, \dots, q_{j}\}\backslash\{q_a\}, t) - L(\{q_0, \dots, q_{j}\}\backslash\{q_{b}\}, t) = (q_b - q_a)L(\{q_0, \dots, q_{j}\}, t-1),\end{aligned}$$ where $D = \sum_i d_i - c_i(\psi)$ and $q_j = 1 - p_j$.
We prove the corollary according to Eq. . $$\begin{aligned}
& L(\{q_0, \dots, q_{j}\}\backslash\{q_a\}, t) - L(\{q_0, \dots, q_{j}\}\backslash\{q_{b}\}, t)\\
=& \sum_{s=0}^{t}(q^s_b - q^s_a) L(\{q_0, \dots, q_{j}\}\backslash\{q_a, q_b\}, t - s)\tag{by Eq.~\eqref{eq:recursive_loop_probability}}\\
=& \sum_{s=0}^{t-1}(q^{s+1}_{b} - q^{s+1}_a) L(\{q_0, \dots, q_{j}\}\backslash\{q_a, q_b\}, t - s - 1)\tag{replace $s-1$ as $s^{\prime}$}\\
=& (q_{b} - q_a)\sum_{s=0}^{t-1}\sum_{m=0}^{s}q^{s - m}_{b}q^{m}_{a}L(\{q_0, \dots, q_{j}\}\backslash\{q_a, q_{b}\}, t - 1 - s)\tag{sum of geometric sequence}\\
= & (q_{b} - q_a)L(\{q_0, \dots, q_{j}\}, t -1).\tag{by definition or expanding Eq.~\eqref{eq:recursive_loop_probability}}\end{aligned}$$ This completes the proof.
### Pseudo reward
\[lemma:small\_gap\] For a transition probability list $\mathcal{P}(\pi^g, \psi) = (p_0,\dots,
p_D)$ and a non-decreasing function $f(x)$, a pseudo reward $R(k)$ is defined as $$\begin{aligned}
R(k)
& = q_k\sum_{j=0}^Mf(j)\times p_0\times\dots\times p_{j-1}\times L(\{q_0,\dots, q_j\}, t - j)\\
& + p_k \sum_{j=0}^{k-1}f(j + 1)\times p_0\times \dots\times p_{j-1} \times L(\{q_0, \dots, q_j\}, t - j) \\
& + \sum_{j=k}^{M^{\prime}}f(j + 1)\times p_0\times \dots\times p_j \times L(\{q_0, \dots, q_{j+1}\}\backslash\{q_k\}, t - j),
\end{aligned}$$ where $M = \min\{D, t\}$ and $M^{\prime} = \{D - 1, t\}$. We claim that for $0\leq k \leq M - 1$, $$\begin{aligned}
R(k) - R(k+1)\!=\!(p_k\!-\!p_{k+1}) \left(\sum_{j=0}^{k}(f(j + 1) - f(j))p_0\times \dots\times p_{j-1} \times L(\{q_0, \dots, q_j\}, t - j)\right).\end{aligned}$$
We expand $R(k) - R(k+1)$ as follows using the definition. $$\begin{aligned}
& R(k) - R(k + 1)\\
& = -(p_k - p_{k+1})\sum_{j=0}^Mf(j)\times p_0\times\dots\times p_{j-1}\times L(\{q_0,\dots, q_j\}, t - j) \\
& + (p_k - p_{k+1}) \sum_{j=0}^{k-1}f(j + 1)\times p_0\times \dots\times p_{j-1} \times L(\{q_0, \dots, q_j\}, t - j) \\
& + f(k + 1)\times p_0\times \dots\times p_{k-1} \times p_k \times L(\{q_0, \dots, q_{k+1}\}\backslash\{q_k\}, t - k) \tag{from $R(k)$}\\
& - f(k + 1)\times p_0\times \dots\times p_{k-1} \times p_{k+1}\times L(\{q_0, \dots, q_k\}, t - k) \tag{from $R(k+1)$}\\
& + \sum_{j=k+1}^{M^{\prime}}f(j + 1)\times p_0\times \dots\times p_j \times (L(\{q_0, \dots, q_{j+1}\}\backslash\{q_k\}, t - j) \\
& \quad\underbrace{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad- L(\{q_0, \dots, q_{j+1}\}\backslash\{q_{k+1}\}, t - j))}_{(p_{k} - p_{k+1})\sum_{j=k+1}^{M-1}f(j + 1)\times p_0\times \dots\times p_j\times L(\{q_0, \dots, q_{j+1}\}, t - j -1)}.
\end{aligned}$$ The last line of above equation can be rewritten with the Corollary \[corollary:loop\_probabilities\_switch\]. The summation from $j =
k+1$ to $j = M-1$ in the last line cancels out with the second line when $j = k
+ 2$ to $j = M$. The summation from $j = 0$ to $j = k$ in the second line can be combined with the third line. We continue the computation of $R(k) - R(k+1)$ by rearranging its expansion. $$\begin{aligned}
& \quad R(k) - R(k + 1)\\
&\begin{aligned}
= & -(p_k - p_{k+1})f(k+1)\times p_0\times\dots\times p_{k}\times L(\{q_0,\dots, q_{k+1}\}, t - k - 1)\\
& -(p_k - p_{k+1}) f(k + 1)\times p_0\times \dots\times p_{k-1} \times L(\{q_0, \dots, q_k\}, t - k) \\
& +(p_k - p_{k+1}) \sum\nolimits_{j=0}^{k}(f(j + 1) - f(j))\times p_0\times \dots\times p_{j-1} \times L(\{q_0, \dots, q_j\}, t - j) \\
& + f(k + 1)\times p_0\times \dots\times p_{k-1} \times p_k \times L(\{q_0, \dots, q_{k+1}\}\backslash\{q_k\}, t - k) \\
& -f(k + 1)\times p_0\times \dots\times p_{k-1} \times p_{k+1}\times L(\{q_0, \dots, q_k\}, t - k). \\
\end{aligned}\end{aligned}$$ Define $\Delta_k$ as the sum of the 2nd, 3rd, 5th, 6th line in above equation. We have $$\begin{aligned}
& R(k) - R(k + 1)\\
& \begin{rcases}
= &-(p_k - p_{k+1})\times f(k + 1)\times p_0\times\dots\times p_{k-1}\times L(\{q_0,\dots, q_k\}, t - k) \\
& -(p_k - p_{k+1})\times f(k + 1)\times p_0\times\dots\times p_{k}\times L(\{q_0,\dots, q_{k+1}\}, t - k - 1) \\
& + f(k + 1)\times p_0\times \dots\times p_{k-1} \times p_k \times L(\{q_0, \dots, q_{k+1}\}\backslash\{q_k\}, t - k) \\
& - f(k + 1)\times p_0\times \dots\times p_{k-1} \times p_{k+1}\times L(\{q_0, \dots, q_k\}, t - k)
\end{rcases}\triangleq \Delta_k\\
&\quad +(p_k - p_{k+1}) \sum_{j=0}^{k}(f(j + 1) - f(j))\times p_0\times \dots\times p_{j-1} \times L(\{q_0, \dots, q_j\}, t - j).\end{aligned}$$ We rewrite $\Delta_k$ as follows. $$\begin{aligned}
\Delta_k / f(k + 1)= &\ p_0\times \dots\times p_{k-1} \times p_k \times L(\{q_0, \dots, q_{k+1}\}\backslash\{q_k\}, t - k) \\
&\begin{rcases}
& - p_0\times \dots\times p_{k-1} \times p_k \times L(\{q_0, \dots, q_{k}\}, t - k) \\
& + p_0\times \dots\times p_{k-1} \times p_k \times L(\{q_0, \dots, q_{k}\}, t - k) \\
\end{rcases}\textit{cancel each other}\\
&-(p_k - p_{k+1})\times p_0\times\dots\times p_{k-1}\times L(\{q_0,\dots, q_k\}, t - k) \\
& -(p_k - p_{k+1})\times p_0\times\dots\times p_{k}\times L(\{q_0,\dots, q_{k+1}\}, t - k - 1) \\
& - p_0\times \dots\times p_{k-1} \times p_{k+1}\times L(\{q_0, \dots, q_k\}, t - k)).\end{aligned}$$ According to Corollary \[corollary:loop\_probabilities\_switch\], the first line and the second line of above equation equals to $(p_k - p_{k+1})\times
p_0\times\dots\times p_{k}\times L(\{q_0,\dots, q_{k+1}\}, t
- k - 1)$, which cancels out with the fifth line. Hence, we have $$\begin{aligned}
\Delta_k / f(k+1) &= p_0\times \dots\times p_{k-1} \times p_k \times L(\{q_0, \dots, q_{k}\}, t - k) \\
&-(p_k - p_{k+1})\times p_0\times\dots\times p_{k-1}\times L(\{q_0,\dots, q_k\}, t - k) \\
& - p_0\times \dots\times p_{k-1} \times p_{k+1}\times L(\{q_0, \dots, q_k\}, t - k))\\
& = 0.\end{aligned}$$ With above result of $\Delta_k = 0$, we prove that $$\begin{aligned}
& R(k) - R(k+1) \\
& = (p_k - p_{k+1}) \left(\sum_{j=0}^{k}(f(j + 1) - f(j))p_0\times \dots\times p_{j-1} \times L(\{q_0, \dots, q_j\}, t - j)\right) \geq 0. \end{aligned}$$ This completes the proof.
### Reward gap
Let $\psi, \psi^{\prime}, \psi^{\prime\prime}$ be any partial realization corresponding to the status $\bm{s}, \bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, \bm{s} -
\mu_i {\boldsymbol{I}}_i$ respectively. Define $\mathcal{P}(\pi^g, \psi) = (p_0,\ldots,
p_D)$, where $D = \sum_id_i - c_i(\psi)$. Recalling Corollary \[corollary:observation\_on\_probability\_list\], we know that $(p_0,\dots, p_{D-1})$ can be obtained by sorting $\cup_{i=1}^m\{s_i, s_i - \mu_i, \dots, \mu_i\}$. Assume the first time $s_i$ appear in $(p_0, \dots, p_D)$ is the $k$-th entry, i.e., $k = \min\{k^{\prime}:
0\leq k^{\prime} \leq D, p_{k^{\prime}} = s_i\}$. According to Corollary \[corollary:observation\_on\_probability\_list\], we have the following. $$\begin{aligned}
\mathcal{P}(\pi^g, \psi^{\prime}) &= (p_1,\dots, p_{D}),\\
\mathcal{P}(\pi^g, \psi^{\prime\prime}) & = (p_0,\dots, p_{k-1}, p_{k+1}, \dots, p_D).
\end{aligned}$$ Note that $p_0 = s_{i^*}$ and $p_k = s_{i}$. Let $M = \min\{D, t\}$, $M^{\prime} = \min\{D - 1, t\}$, and $f^{\prime}(j) =
f(j + c(\psi)) - f(c(\psi))$. The second line of Eq. is $$\begin{aligned}
&R_1 = q_0\sum_{j=0}^{M} f^{\prime}(j)\times p_0\times \dots \times p_{j-1}\times L(\{q_0,\dots, q_j\}, t-j) \tag{$(1 - s_{i^*})F_{g}(\bm{s}, t)$}\\
& +p_0 \sum_{j=0}^{M^{\prime}}f^{\prime}(j + 1)\times p_1\times\dots \times p_{j}\times L(\{q_1,\dots, q_{j+1}\}, t - j).\tag{$s_{i^*}F_{g}(\bm{s} - \mu_{i^*} {\boldsymbol{I}}_{i^*}, t) + s_{i^*}\Delta_c$}
\end{aligned}$$ In fact, $R_1 = F_g(\bm{s}, t + 1)$ based on Lemma \[lemma:property\_greedy\_policy\]. The first line of Eq. is $$\begin{aligned}
R_2 & = q_k\sum_{j=0}^Mf^{\prime}(j)\times p_0\times\dots\times p_{j-1}\times L(\{q_0,\dots, q_j\}, t - j) \\
& + p_k \sum_{j=0}^{k-1}f^{\prime}(j + 1)\times p_0\times \dots\times p_{j-1} \times L(\{q_0, \dots, q_j\}, t - j) \\
& + \sum_{j=k}^{M^{\prime}}f^{\prime}(j + 1)\times p_0\times \dots\times p_j \times L(\{q_0, \dots, q_{j+1}\}\backslash\{q_k\}, t - j).
\end{aligned}$$ Our goal is to measure the gap $R_1 - R_2$. Let $\textrm{Prob}_{\bm{s}, t}(i)$ be the probability we can meet $i$ distinct members if we explore communities (whose initial status is $\bm{s}$) with greedy policy for $t$ steps. According to Lemma \[lemma:small\_gap\], we have $$\begin{aligned}
F_g(\bm{s}, t + 1) - A(\bm{s}, i, t) &= \sum_{j=0}^{k-1} \left(R(j) - R(j + 1)\right)\\
& = \sum_{j=0}^{k-1} (p_j - p_{j+1}) \left(\sum_{o=0}^{j}(f^{\prime}(o + 1) - f^{\prime}(o))\textrm{Prob}_{\bm{s},t}(o)\right)\\
& = \sum_{o=0}^{k-1}(f^{\prime}(o + 1) - f^{\prime}(o))\textrm{Prob}_{\bm{s},t}(o)\left( \sum_{j=o}^{k-1} p_j - p_{j+1}\right)\\
& = \sum_{j=0}^{k-1}(f^{\prime}(j + 1) - f^{\prime}(j))(p_j - p_k)\textrm{Prob}_{\bm{s},t}(j)
\end{aligned}$$ When the reward equals to the number of distinct members, we have $$F_g(\bm{s}, t + 1) - A(\bm{s}, i, t) = \sum_{j=0}^{k-1}(p_j - p_k)\textrm{Prob}_{\bm{s},t}(j).$$ Besides, the gap $F_g(\bm{s}, t + 1) - A(\bm{s}, i, t)$ increases as $k$ increases, which means the worse choice we have at first, the larger reward gap we have at end.
Basics of online learning problems
==================================
Set size estimation by collision counting
-----------------------------------------
Suppose we have a set $C_i = \{u_1, \cdots, u_{d_i}\}$ whose population $d_i$ is unknown. Let $u, v$ be two elements selected with replacement from $C_i$, and $Y_{u,v}$ denote a random variable that takes value 1 if $u = v$ ([*a collision*]{}) and $0$ otherwise. The expectation of $Y_{u, v}$ equals to $\frac{1}{d_i}$, i.e., $\mathbb{E}[Y_{u,
v}] = \frac{1}{d_i}$. Assume we sample $k_i$ elements [*with replacement*]{} uniformly at random from set $C_i$. Let ${\mathcal{S}}_i$ be the set of samples. With the sample ${\mathcal{S}}_i$, we compute the estimator for $d_i$ as $$\hat{d_i} = \frac{k_i(k_i - 1)}{2X_i},$$ here $X_i = \sum_{u\in {\mathcal{S}}_{i}, v\in {\mathcal{S}}_i\backslash\{u\}} Y_{u, v}$ is the number of collisions in ${\mathcal{S}}_i$. According to the Jensen’s inequality[^3], we have $d_i\leq \mathbb{E}[\hat{d}_i]$, i.e., $\hat{d}_i$ is a biased estimator. The estimator is invalid when $X_i = 0$. Since the equality only occurs when $\text{Var}[X_i] = 0$, which is not the case Here. We have $d_i <
\mathbb{E}[\hat{d}_i]$.
**Independence.** Let ${\mathcal{S}}_i = \{v_1,\cdots, v_{k_i}\}$. For the two random variable $Y_{v_x, v_y}$ ($1\leq x < y\leq k_i$) and $Y_{v_{x^{\prime}},
v_{y^{\prime}}}$ ($1\leq x^{\prime} < y^{\prime}\leq k_i$), we consider three difference cases.
1. There are ${k_i \choose 2}$ occurrences when $x = x^{\prime}, y =
y^{\prime}$. Here $\mathbb{E}[Y_{v_x, v_y}Y_{v_{x^{\prime}}, v_{y^{\prime}}}] = 1/d_i$.
2. There are $6{k_i \choose 3}$ occurrences when $x = x^{\prime}, y \neq
y^{\prime}$ or $x\neq x^{\prime}, y = y^{\prime}$. $\mathbb{E}[Y_{v_x, v_y}Y_{v_{x^{\prime}}, v_{y^{\prime}}}] = 1/d^2_i$.
3. There are $6{k_i \choose 4}$ occurrences when $x \neq x^{\prime}, y \neq
y^{\prime}$. Here $\mathbb{E}[Y_{v_x, v_y}Y_{v_{x^{\prime}}, v_{y^{\prime}}}] = 1/d^2_i$.
We say that pairs $(v_x, v_y)$ and $(v_{x^{\prime}}, v_{y^{\prime}})$ are different if $x\neq x^{\prime}$ or $y \neq y^{\prime}$. When $(v_x, v_y)$ and $(v_{x^{\prime}}, v_{y^{\prime}})$ are different, we have $\mathbb{E}[Y_{v_x, v_y}Y_{v_{x^{\prime}}, v_{y^{\prime}}}] = \mathbb{E}[Y_{v_x,
v_y}]\mathbb{E}[Y_{v_{x^{\prime}}, v_{y^{\prime}}}] = 1/d^2_i$. Above discussion indicates that the ${k_i\choose 2}$ pairs of random variables obtained from ${\mathcal{S}}_i$ are [*$2$-wise independent*]{}.
**Variance.** We compute the variance $\text{Var}[X_i] = \mathbb{E}[X^2_i]
- \mathbb{E}^2[X_i]$ in the following. $$\begin{split}
\text{Var}[X_i] & = \frac{k_i(k_i - 1)}{2d_i} + \frac{k_i(k_i-1)(k_i-2)}{d_i^2} + \frac{k_i(k_i-1)(k_i-2)(k_i-3)}{4d_i^2} - \frac{k_i^2(k_i-1)^2}{4d_i^2}\\
& = {k_i\choose 2}\frac{1}{d_i}(1 - \frac{1}{d_i}) = {k_i \choose 2}\text{Var}[Y_{u, v}].
\end{split}$$
**Collision** Since the estimator is based on the collision counting, we need to ensure that $X_i > 0$ with high probability. Let $B_{k_i}$ denote the event that the $k_i$ samples $\{v_{1}, \dots, v_{k_i}\}$ are distinct. We have $$\begin{split}
\Pr\{B_k\} = 1\cdot (1 - \frac{1}{d_i})(1 - \frac{2}{d_i}) \cdots (1 - \frac{k_i - 1}{d_i})&\leq e^{-1/d_i}e^{-2/d_i}\cdots e^{-(k_i-1)d_i}\\
&= e^{-\sum_{j=1}^{k_i-1}j/d_i} = e^{-k_i(k_i-1)/2d_i}.
\end{split}$$ To ensure that $X_i > 0$ with probability no less than $1 - \delta$, we have $$k_i \geq \left(1 + \sqrt{8d_i\ln \frac{1}{\delta} + 1}\right) / 2.$$
Concentration bound for variables with local dependence
-------------------------------------------------------
Note that the pairs $Y_{u, v}$ and $Y_{u^{\prime}, v^{\prime}}$ are not mutually independent. Actually, their dependence can be described with a [*dependence graph*]{} [@janson2004large; @Dubhashi2009CMA]. The Chernoff-Hoeffding bound in [@hoeffding1963probability] can not be used directly for our estimator of $\mu_i$. In the following, we present a concentration bound that is applicable to our problem.
Let $\xi_1, \dots, \xi_n$ be independent random variables, and let $$X {\vcentcolon=}\sum_{1\leq i_1\leq \dots\leq i_d} f_{i_1,\dots, i_d} (\xi_{i_1}, \dots, \xi_{i_d}).$$
\[lemma:local\_dependence\] If $a\leq f_{i_1, \dots, i_d}(\xi_{i_1}, \dots, \xi_{i_d})\leq b$ for every $i_1, \dots, i_d$ for some reals $a\leq b$, we have $$\Pr\left\{ |X - \mathbb{E}[X]| \geq \epsilon{n\choose d}\right\} \leq 2\exp\left( \frac{-2\floor{n/d}\epsilon^2}{(b-a)^2} \right).$$
In our problem, if we get $k_i$ samples from set $C_i$, then the number of collisions satisfies $$\Pr\left\{{\left| X_i - \mathbb{E}[X_i] \right|} \geq \epsilon {k_i\choose 2}\right\} \leq 2\exp\left( -2\floor{k_i/2}\epsilon^2 \right).$$ Above inequality indicates that the actual number of independent pairs is $\floor{k_i/2}$ when using collisions in $k_i$ samples to estimate $\mu_i$.
Regret Analysis for Non-Adaptive Problem
========================================
Supporting Corollaries
----------------------
\[corollary:action\_summation\] For action $\bm{k}$ with $\sum_{i=1}^{m}k_i = K$ and $k_i \geq 1$, we have $\sum_{i=1}^m{k_i\choose 2} \leq {K - m + 1\choose 2}$.
We prove the corollary by simple calculation. $$\begin{aligned}
\sum_{i=1}^m{k_i\choose 2} - {K - m + 1\choose 2} &= \frac{1}{2}\left( \sum_{i=1}^mk_i(k_i - 1) - \left(1 + \sum_{i=1}^m(k_i - 1)\right)\left(\sum_{i=1}^m(k_i - 1)\right)\right) \\
&= \frac{1}{2}\left( \sum_{i=1}^m(k_i-1)^2 - \left(\sum_{i=1}^m(k_i - 1)\right)^2\right) \leq 0. \qedhere
\end{aligned}$$
Basics
------
To compare with the CUCB algorithm introduced in [@wang2017improving] for general CMAB problem, we propose an revised Algo. \[algo:CLCB\_algorithm\] that is consistent with the CUCB algorithm in [@wang2017improving]. We revise the Line \[line:update\_pairs\]-\[line:online\_learning\_end\] in Algo. \[algo:CLCB\_algorithm\] as follows. $$\label{eq:revision_1}
\begin{split}
& \text{Line~\ref{line:update_pairs}:~~ For } i\in [m], T_i\leftarrow T_i + {\mathbbm{1}}\{{\left| {\mathcal{S}}_i \right|} > 1\},\\
& \text{Line~\ref{line:count_collision}:~~ For } i\in [m] \text{ and } {{\left| {\mathcal{S}}_i \right|} > 1}, X_{i,t} \leftarrow \sum_{x=1}^{\floor{{\left| {\mathcal{S}}_i \right|}} / 2}{\mathbbm{1}}\{u_{2x - 1} = u_{2x}\} / \floor{{\left| {\mathcal{S}}_i/2 \right|}},\\
& \text{Line~\ref{line:online_learning_end}: For } i\in [m] \text{ and } {{\left| {\mathcal{S}}_i \right|} > 1}, \hat{\mu}_i\leftarrow \hat{\mu}_i + (X_{i,t} - \hat{\mu}_i) / T_i.
\end{split}$$ Note that $\hat{\mu}_i$ in Eq. is also an unbiased estimator of $\mu_i$. Then we can obtain the regret bound of the revised Algo. \[algo:CLCB\_algorithm\] by applying the Theorem 4 in the extended version of [@wang2017improving] directly. $$\text{Reg}_{\bm{\mu}}(T) \leq \sum_{i=1}^{m} \frac{48{K - m + 1 \choose 2}^2 m\ln T }{\Delta^{i}_{\min}} + 2{K - m + 1 \choose 2}m + \frac{\pi^2}{3}\cdot m \cdot \Delta_{\max}.$$
We add superscript $r$ to differentiate the corresponding random variables in the revised Algo. \[algo:CLCB\_algorithm\] from the original ones. E.g., $T^{r}_{i, t}$ is the value of $T^{r}_i$ in the revised Algo. \[algo:CLCB\_algorithm\] at the end of round $t$. Recall that $K^{\prime} =
K - m + 1$, which is the maximum exploration times for a community in each round.
Proof framework
---------------
We first introduce a definition which describes the event that $\hat{\mu}_{i,t-1}$ ($\hat{\mu}^{r}_{i, t-1}$) is accurate at the beginning of round $t$.
We say that the sampling is nice at the beginning of round $t$ if for every community $i\in [m]$, $|\hat{\mu}_{i,t-1} - \mu_i| \leq \rho_{i, t}$ (resp. $|\hat{\mu}^{r}_{i,t-1} - \mu_i| \leq \rho^{r}_{i, t}$), where $\rho_{i, t} =
2\sqrt{\frac{3\ln t}{2T_{i,t-1}}}$ (resp. $\rho^{r}_{i, t} = 2\sqrt{\frac{3\ln
t}{2T^{r}_{i,t-1}}}$) in round $t$. Let $\mathcal{N}_t$ (resp. $\mathcal{N}^{r}_t$) be such event.
\[lemma:small\_probability\_for\_event\] For each round $t \geq 1$, $\Pr\left\{ \neg \mathcal{N}_t\right\}\leq
2m\floor{K^{\prime}/2}t^{-2}$ (resp. $\Pr\left\{ \neg \mathcal{N}^{r}_t\right\}\leq 2mt^{-2}$).
For each round $t \geq 1$, we have $$\begin{aligned}
\Pr\left\{\neg \mathcal{N}_t \right\} &= \Pr\left\{\exists i\in [m], |\hat{\mu}_{i, t-1} - \mu_i| \geq \sqrt{\frac{3\ln t}{2T_{i, t-1}}}\right\}\\
& \leq \sum_{i\in [m]} \Pr \left\{|\hat{\mu}_{i, t-1} - \mu_i| \geq \sqrt{\frac{3\ln t}{2K_{i, t-1}}} \right\}\\
& = \sum_{i\in [m]}\sum_{k=1}^{(t-1)\floor{K^{\prime}/ 2}} \Pr\left\{T_{i, t-1} = k, |\hat{\mu}_{i, t-1} - \mu_i| \geq \sqrt{\frac{3\ln t}{2T_{i, t-1}}}\right\}\\
&\leq \sum_{i\in [m]} \sum_{k=1}^{(t-1)\floor{K^{\prime}/ 2}} \frac{2}{t^3} < 2{m\floor{K^{\prime} /2}}t^{-2}.\tag{Hoeffding's inequality~\cite{hoeffding1963probability}}
\end{aligned}$$ When $T_{i,t - 1} = k$, $\hat{\mu}_{i,t}$ is the average of $k$ i.i.d. random variables $Y^{[1]}_i, \cdots, Y^{[k]}_i$, where $Y^{[j]}_i$ is a random variable that indicates whether two members selected with replacement from $C_i$ are the same. Since each community is explored at most $K^{\prime}$ times in each round, $T_{i,t-1}\leq (t-1)\floor{K^{\prime}/2}$. The last line leverages the Hoeffding’s inequality [@hoeffding1963probability]. By replacing the summation range $k\in [1, (t-1)\floor{K^{\prime}/ 2}]$ with $k\in [1, (t-1)]$ in the 3rd line of above equation, we have $\Pr\left\{ \neg
\mathcal{N}^{r}_t\right\}\leq 2mt^{-2}$.
Secondly, we use the monotonicity and bounded smoothness properties to bound the reward gap $\Delta_{\bm{k}_t} = r_{\bm{k}^*}(\bm{\mu}) - r_{\bm{k}_t}(\bm{\mu})$ between our action $\bm{k}_t$ and the optimal action $\bm{k}^*$.
\[lemma:bound\_gap\] If the event $\mathcal{N}_t$ holds in round $t$, we have $$\Delta_{\bm{k}_t} \leq\sum_{i=1}^m \binom{k_{i,t}}{2}\kappa_{T}(\Delta^i_{\min}, T_{i,t-1}).$$ Here the function $\kappa_T(M, s)$ is defined as $$\kappa_T(M, s) =
\begin{cases}
2 & \text{ if } s = 0, \\
2\sqrt{\frac{6\ln t}{s}} & \text{ if } 1 \leq s \leq l_{T}(M),\\
0 & \text{ if } s \geq l_{T}(M) + 1,
\end{cases}$$ where $$l_{T}(M) = \frac{24{K^{\prime}\choose 2}^2\ln T}{M^2}.$$
By $\mathcal{N}_t$ (i.e., ${\underaccent{\bar}{\bm{\mu}}}_t \leq \bm{\mu}$) and the monotonicity of $r_{\bm{k}}(\bm{\mu})$, we have $$r_{\bm{k}_t}({\underaccent{\bar}{\bm{\mu}}}_t) \geq r_{\bm{k}^*}({\underaccent{\bar}{\bm{\mu}}}_t) \geq r_{\bm{k}^*}(\bm{\mu}) = r_{\bm{k}_t}(\bm{\mu}) + \Delta_{\bm{k}_t}.$$ Then by the *bounded smoothness* properties of reward function, we have $$\Delta_{\bm{k}_t} \leq r_{\bm{k}_t}({\underaccent{\bar}{\bm{\mu}}}_t) - r_{\bm{k}_t}(\bm{\mu}) \leq \sum_{i=1}^{m}{k_{i,t}\choose 2}(\mu_i - {\underaccent{\bar}{\mu}}_{i, t}).$$ We intend to bound $\Delta_{\bm{k}_t}$ by bounding ${\mu}_i - {\underaccent{\bar}{\mu}}_{i,
t}$. Before doing so, we perform a transformation. Let $M_{\bm{k}_t} =
\max_{i\in [m], k_{i,t} > 1}\Delta^{i}_{\min}$. Since the action $\bm{k}_t$ always satisfies $\Delta_{\bm{k}_t} \geq \max_{i\in [m], k_{i,t} >
1}\Delta^{i}_{\min}$, we have $\Delta_{\bm{k}_t} \geq M_{\bm{k}_t}$. So $\sum_{i} {k_{i,t}\choose 2}(\mu_i - {\underaccent{\bar}{\mu}}_{i, t})\geq \Delta_{\bm{k}_t}
\geq M_{\bm{k}_t}$. Therefore, $$\begin{aligned}
\Delta_{\bm{k}_t} &\leq \sum_{i=1}^{m} {k_{i,t}\choose 2}(\mu_i - {\underaccent{\bar}{\mu}}_{i,t}) \leq - M_{\bm{k}_t} + 2\sum_{i = 1}^{m} {k_{i,t}\choose 2}(\mu_{i} - {\underaccent{\bar}{\mu}}_{i, t})\\
& \leq - \frac{\sum_{i=1}^{m} {k_{i,t} \choose 2}}{{K^{\prime}\choose 2}}M_{\bm{k}_t} + 2\sum_{i = 1}^{m} {k_{i,t}\choose 2}(\mu_{i} - {\underaccent{\bar}{\mu}}_{i, t})\tag{Corollary~\ref{corollary:action_summation}: $\sum_{i=1}^m{k_{i,t}\choose 2}\leq {K^{\prime}\choose 2}$}\\
& = 2\sum_{i=1}^{m}{k_{i,t}\choose 2}\left[(\mu_i - {\underaccent{\bar}{\mu}}_{i, t}) - \frac{M_{\bm{k}_t}}{K^{\prime}(K^{\prime}-1)}\right]\\
& \leq 2\sum_{i}{k_{i,t}\choose 2}\left[(\mu_i - {\underaccent{\bar}{\mu}}_{i, t}) - \frac{\Delta^{i}_{\min}}{K^{\prime}(K^{\prime}-1)}\right].\tag{by definition of $M_{\bm{k}_t}$}\end{aligned}$$ By $\mathcal{N}_t$, we have $\mu_i - {\underaccent{\bar}{\mu}}_{i, t}\leq \min\{2\rho_{i,t}, 1\}$. So $$\begin{split}
\mu_i - {\underaccent{\bar}{\mu}}_{i, t} - \frac{\Delta^i_{\min}}{K^{\prime}(K^{\prime}-1)} &\leq \min\{2\rho_{i,t}, 1\} - \frac{\Delta^i_{\min}}{K^{\prime}(K^{\prime}-1)} \leq \min\left\{ \sqrt{\frac{6\ln t}{T_{i,t-1}}} , 1\right\} - \frac{\Delta^i_{\min}}{K^{\prime}(K^{\prime}-1)}.
\end{split}$$ If $T_{i, t-1}\leq l_T(\Delta^i_{\min})$, we have $\mu_i - {\underaccent{\bar}{\mu}}_{i, t} - \frac{\Delta^i_{\min}}{K^{\prime}(K^{\prime}-1)}
\leq \min\left\{\sqrt{\frac{6\ln t}{T_{i,t-1}}} , 1\right\}\leq \frac{1}{2}\kappa_{T}(\Delta^i_{\min}, T_{i, t-1})$. If $T_{i, t-1} > l_T(\Delta^i_{\min}) + 1$, then $ \sqrt{\frac{6\ln t}{T_{i,t-1}}} \leq \frac{\Delta^i_{\min}}{K^{\prime}(K^{\prime}-1)}$, so $(\mu_i - {\underaccent{\bar}{\mu}}_{i,t} ) - \frac{\Delta^i_{\min}}{K^{\prime}(K^{\prime}-1)} \leq 0 =
\kappa_T(\Delta^i_{\min}, T_{i, t-1})$. In conclusion, we have $$\Delta_{\bm{k}_t} \leq \sum_{i=1}^{m}{k_{i,t}\choose 2}\kappa_{T}(\Delta_{\min}^i, T_{i, t-1}). \qedhere$$
Above result is also valid for the revised Algo. \[algo:CLCB\_algorithm\], i.e., $\Delta^{r}_{\bm{k}_t} \leq \sum_{i=1}^{m}{k_{i,t}\choose
2}\kappa_{T}(\Delta_{\min}^i, T^{r}_{i, t-1})$. Our third step is to prove that when $\mathcal{N}_{t}$ (resp. $\mathcal{N}^{r}_t$) holds, the regret is bounded in $O(\ln T)$.
![Demonstration of the regret summation $\sum_{t=2}^T\floor{k_{i,t}/
2}\kappa_T(\Delta^{i}_{\min}, T_{i,t-1})$. It is obvious that when $k_{i,t} = K^{\prime}$, then the shaded area (colored with orange) covered by the rectangles is maximized.[]{data-label="fig:regret_riemann_sum"}](non_adaptive_regret_riemann_sum.pdf)
We first prove the regret when the event $\mathcal{N}_t$ holds. In each run, we have $$\begin{aligned}
\sum_{t=1}^{T} \bm{1}(\left\{\Delta_{\bm{k}_t} \wedge \mathcal{N}_t\right\})\cdot \Delta_{\bm{k}_t} &\leq \sum_{t=1}^T \sum_{i=1}^{m} {k_{i,t}\choose 2}\kappa_T(\Delta^i_{\min}, T_{i, t-1})\\
&= \sum_{i=1}^{m}\sum_{t^{\prime}\in \{t \mid 1\leq t\leq T, k_{i,t} > 1\}} {k_{i,t^{\prime}}\choose 2}\kappa_T(\Delta^i_{\min}, T_{i, t^{\prime}-1}).
\end{aligned}$$ Hence, we just assume $k_{i, t} > 1$ for $t > 0$. $$\begin{aligned}
\small
\sum_{t=1}^{T} \bm{1}(\left\{\Delta_{\bm{k}_t} \wedge \mathcal{N}_t\right\})\cdot \Delta_{\bm{k}_t} &\leq \sum_{i=1}^{m}\sum_{t=1}^{T} {k_{i,t}\choose 2}\kappa_T(\Delta^{i}_{\min}, T_{i, t-1})\\
&\leq \sum_{i=1}^{m}2{k_{i,1}\choose 2} + K^{\prime}\sum_{i=1}^{m}\sum_{t=2}^{T} \frac{(k_{i,t}-1)}{2}\kappa_T(\Delta^{i}_{\min}, T_{i, t-1})\\
&\leq 2m{K^{\prime}\choose 2} + K^{\prime}\sum_{i=1}^{m}\sum_{t=2}^{T} \floor*{\frac{k_{i,t}}{2}}\kappa_T(\Delta^{i}_{\min}, T_{i, t-1}).\tag{Fig.~\ref{fig:regret_riemann_sum}}
\end{aligned}$$ To maximize the summation $\sum_{t=2}^{T} \floor{\frac{k_{i,t}}{2}}\kappa_T(\Delta^{i}_{\min}, T_{i, t-1})$, we just need to let $k_{i,t} =
K^{\prime}$ when $t > 1$. $$\begin{aligned}
\small
\sum_{t=1}^{T} \bm{1}(\left\{\Delta_{\bm{k}_t} \wedge \mathcal{N}_t\right\})\cdot \Delta_{\bm{k}_t} &\leq 2m{K^{\prime}\choose 2} + K^{\prime}\sum_{d=0}^{l_{T}(\Delta^{i}_{\min})/\floor{K^{\prime}/ 2}}\floor*{\frac{K^{\prime}}{2}}\kappa_T\left(\Delta^{i}_{\min}, 1 + d\floor{K^{\prime}/2}\right) \\
& \leq 2m{K^{\prime}\choose 2} + K^{\prime}\sum_{i=1}^{m}\sum_{d=0}^{l_{T, K}} \frac{\sqrt{24\ln T} \floor{K^{\prime}/ 2}}{\sqrt{1 + d\floor{K^{\prime}/ 2}}} \tag{$l_{T, K}{\vcentcolon=}\frac{l_{T}(\Delta^{i}_{\min})}{\floor{K^{\prime} / 2}}$}\\
& \leq 2m{K^{\prime}\choose 2} + K^{\prime}\sum_{i=1}^{m} \int_{x=0}^{l_{T, K}} \frac{\sqrt{24\floor{K^{\prime}/ 2}\ln T}}{\sqrt{x}}dx \\
& = 2m{K^{\prime}\choose 2} + K^{\prime}\sum_{i=1}^{m}\sqrt{96l_T(\Delta^{i}_{\min}, T)\ln T} \\
& = 2m{K^{\prime}\choose 2} + \sum_{i=1}^{m}\frac{48{K^{\prime}\choose 2}K^{\prime}\ln T}{\Delta^{i}_{\min}}. \end{aligned}$$ On the other hand, when $\mathcal{N}_t$ does not hold, we can bound the regret as $\Delta_{\max}$. Hence, $$\begin{aligned}
\mathbb{E} \left[ \sum_{t=1}^{T} \bm{1}(\left\{\Delta_{\bm{k}_t} \wedge \neg\mathcal{N}_t\right\})\cdot \Delta_{\bm{k}_t} \right] \leq\Delta_{\max}\sum^{T}_{t=1}2{m\floor{K^{\prime} / 2}}t^{-2}\leq \frac{m\floor{K^{\prime}/ 2}\pi^2}{3}\Delta_{\max}.\end{aligned}$$
Based on above discussion, we have $$\begin{aligned}
\text{Reg}_{\bm{\mu}}(T) &\leq \frac{m\floor{K^{\prime} / 2}\pi^2}{3}\Delta_{\max} + 2m{K^{\prime}\choose 2} + \sum_{i=1}^{m}\frac{48{K^{\prime}\choose 2}K^{\prime}\ln T}{\Delta^{i}_{\min}}. \qedhere\end{aligned}$$
[theorem]{}[regretboundb]{}\[thm:regret\_bound\_non\_adaptive\_exploration\_2\] The revised Algo. \[algo:CLCB\_algorithm\] has regret as follows. $$\begin{aligned}
\label{eq:non_adaptive_regret_bound_b}
\text{Reg}^{r}_{\bm{\mu}}(T) &\leq \sum_{i=1}^{m}\frac{48{K^{\prime} \choose 2}^2\ln T}{\Delta^{i}_{\min}} + 2{K^{\prime}\choose 2}m + \frac{\pi^2}{3} \cdot m \cdot \Delta_{\max}.\end{aligned}$$
We prove the regret when the event $\mathcal{N}^r_t$ holds. In each run, we have $$\begin{aligned}
\sum_{t=1}^{T} \bm{1}(\left\{\Delta^r_{\bm{k}_t} \wedge \mathcal{N}^r_t\right\})\cdot \Delta^r_{\bm{k}_t} &\leq \sum_{t=1}^T \sum_{i=1}^{m} {k_{i,t}\choose 2}\kappa_T(\Delta^i_{\min}, T^r_{i, t-1})\\
&= \sum_{i=1}^{m}\sum_{s = 0}^{T^r_{i, T}} {k_{i,s}\choose 2}\kappa_T(\Delta^i_{\min}, s)\\
& \leq 2m{K^{\prime}\choose 2} + {K^{\prime}\choose 2}\sum_{i=1}^{m}\sum_{s = 1}^{l_{T}(\Delta^i_{\min})} \sqrt{\frac{24\ln T}{s}}\\
& \leq 2m{K^{\prime}\choose 2} + \sum_{i=1}^{m}\frac{48{K^{\prime}\choose 2}^2\ln T}{\Delta^{i}_{\min}}.
\end{aligned}$$ On the other hand, $\Pr\{\neg \mathcal{N}^r_t\} \leq 2mt^{-2}$. Hence we have $$\begin{aligned}
\text{Reg}^r_{\bm{\mu}}(T) &= \mathbb{E} \left[ \sum_{t=1}^{T} \bm{1}(\left\{\Delta_{\bm{k}_t} \wedge \neg\mathcal{N}_t\right\})\cdot \Delta_{\bm{k}_t} \right] + \mathbb{E} \left[ \sum_{t=1}^{T} \bm{1}(\left\{\Delta_{\bm{k}_t} \wedge \mathcal{N}_t\right\})\cdot \Delta_{\bm{k}_t} \right]\\
& \leq \frac{m\pi^2}{3}\Delta_{\max} + 2m{K^{\prime}\choose 2} + \sum_{i=1}^{m}\frac{48{K^{\prime}\choose 2}^2\ln T}{\Delta^{i}_{\min}}.\qedhere
\end{aligned}$$
The bound in Eq. is tighter than the one obtained by directly applying [@wang2017improving].
Comparison
----------
**Estimator.** Let $\hat{\mu}_{i,t}$ be the estimator computed in Algo. \[algo:CLCB\_algorithm\] by end of round $t$ and $\hat{\mu}^{r}_{i,t}$ be the estimator computed with revision in Eq. by end of round $t$. Both of $\hat{\mu}_{i,t}$ and $\hat{\mu}^{r}_{i,t}$ are unbiased estimator of $\mu_i$. However, $\hat{\mu}_{i,t}$ is a [*more efficient*]{} estimator than $\hat{\mu}^{r}_{i,t}$. More specifically, ${{\rm Var}}[\hat{\mu}_{i,t}]
= \mu_i(1 - \mu_i) / (\sum_{t^{\prime} = 1}^{t}\floor{k_{i,t^{\prime}}/ 2})$ and ${{\rm Var}}[\hat{\mu}^{r}_{i,t}] = \mu_i(1 - \mu_i)\cdot (\sum_{t^{\prime} = 1}^t1 /
\floor{k_{i,t}/ 2}) / (T^r_{i,t})^2$. Here $k_{i,t}$ is the size of ${\mathcal{S}}_i$ in round $t$, and $T^{r}_{i,t} = \sum_{t^{\prime}=1}^t{\mathbbm{1}}\{k_{i,t^{\prime}} >
1\}$. Since the harmonic mean is always not larger than arithmetic mean, i.e., $
T^{r}_{i,t} / (\sum_{t^{\prime} = 1}^t1 / \floor{k_{i,t^{\prime}}/ 2}) \leq
(\sum_{t^{\prime} = 1}^{t}\floor{k_{i,t^{\prime}}/2}) / T^{r}_{i,t}$, we conclude that ${{\rm Var}}[\hat{\mu}_{i,t}]\leq {{\rm Var}}[\hat{\mu}^{r}_{i,t}]$.
**Regret Bound.** The regret bound in Eq. is tighter than the one in Eq. up to ${(K^{\prime} - 1)/ 2}$ factor in the $O(\ln T)$ term. The bound in Eq. has a larger constant term. That’s because we use a smaller confidence radius, which leads to earlier exploitation of Algo. \[algo:CLCB\_algorithm\] than the revised one.
Full information feedback
-------------------------
In the following, we prove the constant regret bound of the Algo. \[algo:CLCB\_algorithm\] with feeding the empirical mean in `` and making revision defined in Eq. .
We first bound $\Delta_{\bm{k}_t}$ by $\sum_{i=1}^m {\left| \mu_{i,t} - \mu_i \right|}$. $$\begin{aligned}
\Delta_{\bm{k}_t} = r_{\bm{k}^{*}}(\bm{\mu}) - r_{\bm{k}_t}(\bm{\mu}) &= r_{\bm{k}^{*}}(\bm{\mu}) - r_{\bm{k}_t}(\hat{\bm{\mu}}) + r_{\bm{k}_t}(\hat{\bm{\mu}})- r_{\bm{k}_t}(\bm{\mu})\\
& \leq r_{\bm{k}^{*}}(\bm{\mu}) - r_{\bm{k}^{*}}(\hat{\bm{\mu}}) + r_{\bm{k}_t}(\hat{\bm{\mu}})- r_{\bm{k}_t}(\bm{\mu})\tag{$r_{\bm{k}^{*}}(\hat{\bm{\mu}}) \leq r_{\bm{k}_t}(\hat{\bm{\mu}})$}\\
& \leq |r_{\bm{k}^{*}}(\bm{\mu}) - r_{\bm{k}^{*}}(\hat{\bm{\mu}})| + |r_{\bm{k}_t}(\hat{\bm{\mu}})- r_{\bm{k}_t}(\bm{\mu})|\\
&\leq \sum_{i=1}^{m}\left( {k^{*}_i\choose 2} + {k_{i,t}\choose 2}\right){\left| \hat{\mu}_{i,t-1} - \mu_i \right|}.\end{aligned}$$ Leverage the fact that $\sum_{i=1}^m {k_{i, t}\choose 2} \leq
{K^{\prime}\choose 2}$. If $|\hat{\mu}_{i,t-1} - \mu_i| <
\frac{\Delta_{\min}}{K^{\prime}(K^{\prime} - 1)}$, then $$\begin{aligned}
\Delta_{\bm{k}_t} \leq \sum_{i=1}^{m}\left( {k^{*}_i\choose 2} + {k_{i,t}\choose 2}\right) \frac{\Delta_{\min}}{K^{\prime}(K^{\prime} - 1)} < \Delta_{\min},\end{aligned}$$ which means $\Delta_{\bm{k}_t} = 0$. Hence, $$\begin{aligned}
\Pr\left(\Delta_{\bm{k}_t} > 0 \right) &\leq \sum_{i=1}^{m}\Pr\left( |\hat{\mu}_{i,t-1} - \mu_i| \geq \frac{\Delta_{\min}}{K^{\prime}(K^{\prime} - 1)}\right)\\
& \leq \sum_{i=1}^{m}2e^{-2(T_{i,t-1} / 2)\Delta^2_{\min} / (K^{\prime}(K^{\prime}-1))^2} \tag{Theorem 3.2 in~\cite{Dubhashi2009CMA}}. \end{aligned}$$ The second line of above inequality using Theorem 3.2 in [@Dubhashi2009CMA]. Note that the $T_{i, t-1}$ member pairs using for collision counting are not independent with each other. We need to construct a [*dependence graph*]{} $G$ to model their dependence. The dependence graph here is just a line with $T_{i,
t-1}$ nodes. Since the fractional chromatic number of the dependence graph is $2$, we have a $1/2$ factor for $T_{i,t-1}$ in the exponential. The regret is bounded as $$\begin{aligned}
\text{Reg}_{\bm{\mu}}(T) &\leq \sum_{t=1}^{T}\sum_{i=1}^{m}\Delta_{\bm{k}_t} 2e^{-T_{i,t-1}\Delta^2_{\min} / (K^{\prime}(K^{\prime}-1))^2}\\
&\leq 2\Delta_{\max} + \sum_{i=1}^{m}\sum_{t=3}^{T}\Delta_{\bm{k}_t} 2e^{-(t-2)\Delta^2_{\min} / (K^{\prime}(K^{\prime}-1))^2}\tag{$T_{i,t-1} \geq t - 2$}\\
& \leq 2\Delta_{\max} + 2m\Delta_{\max} \int_{t=0}^{\infty}e^{-t\Delta^2_{\min} / (K^{\prime}(K^{\prime}-1))^2}{\rm d}t\\
& \leq \left(2 + 8me^2{K^{\prime}\choose 2}^2 / \Delta^2_{\min}\right)\Delta_{\max}.\qedhere\end{aligned}$$
Regret Analysis for Adaptive Problem
====================================
Transition probability list of policy $\pi^t$ {#app:transition_probability_list_pit}
---------------------------------------------
Similar to the discussion in Section \[app:transition\_probability\_list\_greedy\_policy\], we define a transition probability list $\mathcal{P}(\pi^t,\psi)$ for the policy $\pi^t$ and write the reward function $r_{\pi^t}(\bm{\mu})$ with $\mathcal{P}(\pi^t,\emptyset)$.
**Definition.** Assume the initial partial realization is $\psi$. Let $\bm{s}_0$ be the status corresponding to $\psi$. Recall that $\bm{s}_0 = (s_{1, 0}, \dots, s_{m, 0}) = (1 - \mu_1c_1(\psi), \dots, 1 -
\mu_mc_m(\psi))$. At the first step, policy $\pi^t$ chooses community $i^*_0 =
\operatorname*{arg\,max}_{i\in [m]} 1 - c_i(\psi){\underaccent{\bar}{\mu}}_{i,t}$. With probability $q^{\pi^t}_0
{\vcentcolon=}c_{i^*_0}(\psi)\mu_{i^*_0}$, the communities stay at the same status $\bm{s}_0$. With probability $p^{\pi^t}_0 {\vcentcolon=}1 -
c_{i^*_0}(\psi)\mu_{i^*_0}$, the communities transit to next status $\bm{s}_1{\vcentcolon=}\bm{s}_0 -\mu_{i^*_0}\bm{I}$. Note that $$1 - c_i(\psi){\underaccent{\bar}{\mu}}_{i,t} = \frac{\mu_i - (1 - s_{i,0}){\underaccent{\bar}{\mu}}_{i,t}}{\mu_i} = \frac{{\underaccent{\bar}{\mu}}_{i,t}}{\mu_i}s_{i,0} + \frac{\mu_i - {\underaccent{\bar}{\mu}}_{i,t}}{\mu_i}.$$ We recursively define $\bm{s}_{k+1}$ as $\bm{s}_{k} - \mu_{i^*_k}\bm{I}_{i^*_k}$ where $i^*_k \in \max_{i\in [m]} {({\underaccent{\bar}{\mu}}_{i,t}/\mu_i)s_{i,k}} + (\mu_i -
{\underaccent{\bar}{\mu}}_{i,t})/\mu_i$. The transition probability $p^{\pi^t}_{k} {\vcentcolon=}s_{i^*_k, k}$. We define the transition probability list $\mathcal{P}(\pi^t,
\psi) = (p^{\pi^t}_0, \dots, p^{\pi^t}_D)$ where $D = \sum_{i=1}^{m}(d_i -
c_i(\psi))$ is the number of distinct member we haven’t meet under the partial realization $\psi$. Note that it is possible that $p^{\pi^t}_{k}= 0$. In this case, there is already no unmet members in $i^*_k$. The communities will be stuck in status $\bm{s}_{k}$ since the policy $\pi^t$ always chooses community $i^*_k$ to explore after the communities reach status $\bm{s}_{k}$. Hence, if $k$ is the smallest index such that $p^{\pi^t}_{k} = 0$, we define $p^{\pi^t}_{k^{\prime}} = 0$ for all $k^{\prime} > k$.
**Compute $\mathcal{P}(\pi^t, \psi)$**. Define $\mathcal{B}_i(\psi) = \{1 - c_i(\psi)\mu_i, 1 - (1 + c_i(\psi))\mu_i,
\dots, \mu_i, 0\}$ for $i\in [m]$. Let $b_i\in \mathcal{B}_i(\psi), b_j\in
\mathcal{B}_j(\psi), i, j\in [m]$. We define a [*sorting comparator*]{} as follows. $${\rm less}(b_i, b_j) =
{\mathbbm{1}}\{({\underaccent{\bar}{\mu}}_{i,t}/\mu_i)\cdot b_i + (\mu_i -
{\underaccent{\bar}{\mu}}_{i,t})/\mu_i < ({\underaccent{\bar}{\mu}}_{j,t}/\mu_i)\cdot b_{j} + (\mu_j - {\underaccent{\bar}{\mu}}_{j,t})/\mu_j\}$$ If $b_i \geq b_j$ and ${\rm less}(b_i, b_j) = 1$, we can infer that ${\underaccent{\bar}{\mu}}_{i, t}/\mu_i \geq {\underaccent{\bar}{\mu}}_{j,t}/\mu_j$, which means the size of community $j$ is more overestimated than the size of community $i$. The overestimation leads to wrong order between $b_i$ and $b_j$ when using the comparator ${\rm less}$. The list $\mathcal{P}(\pi^t, \psi)$ can be computed as follows. Firstly, we sort elements in $\cup_{i\in [m]}\mathcal{B}_i$ with the comparator ${\rm less}$. Secondly, we truncate the sorted list at the first zero elements. Thirdly, we paddle zeros at the end of list until the length is $D +
1$. All the arguments in Section \[app:expected\_reward\_greedy\_policy\]-\[app:optimality\_greedy\_policy\] about $\mathcal{P}(\pi^g, \psi)$ can be easily extended to $\mathcal{P}(\pi^t,
\psi)$.
**Expected reward.** In the following, we still use the extended definition of reward $$R(\bm{k}, \phi) =f\left(\sum_{i=1}^{m}{\left| \bigcup_{\tau=1}^{k_i}\{\phi(i,\tau)\} \right|}\right),$$ where $f$ is a non-decreasing function. We can write the reward function $r_{\pi^t}(\bm{\mu})$ as $$r_{\pi^t}(\bm{\mu}) = \sum_{j=0}^{\min\{K, \sum_{i=1}^m d_i\}} f(j)\times p^{\pi^t}_0\times\dots\times p^{\pi^t}_{j-1}\times L(\{q^{\pi^t}_0, \dots, q^{\pi^t}_{j}\}, K - j).$$ Here $p^{\pi^t}_j$ is element in $\mathcal{P}(\pi^t, \emptyset)$, $q^{\pi^t}_j{\vcentcolon=}1 - p^{\pi^t}_j$, and $K$ is the budget.
Proof framework {#app:proof_framework_for_adaptive_exploration}
---------------
**Notations.** Let $D = \sum_{i=1}^{m}d_i$ in this part. Let $\mathcal{P}(\pi^g, \emptyset) =
(p^{\pi^g}_0, \dots, p^{\pi^g}_{D})$ and $\mathcal{P}(\pi^t, \emptyset) =
(p^{\pi^t}_0, \dots, p^{\pi^t}_{D})$. According to Corollary \[corollary:observation\_on\_probability\_list\], we know that $\mathcal{P}(\pi^g, \emptyset)$ can be obtained by sorting $\cup_{i\in [m]}\{1,
1 - \mu_i, 1 - 2\mu_i, \dots, \mu_i\}\cup\{0\}$. Here we define another list $\tilde{\mathcal{P}}(\pi^g)$ which is obtained by sorting $\cup_{i\in [m]}\{(i,
1), (i, 1 - \mu_i), \dots, (i, \mu_i)\}$ via comparing the second value in the pair. Let $U_{i,k}$ denote how many times pair $(i, \cdot)$ appears in the first $k$ positions in the list $\tilde{\mathcal{P}}(\pi^g)$. The value $U_{i,k}$ satisfies that $p^{\pi^g}_k = \max_{i=1}^{m}1 - U_{i,k}\mu_i$. Note that the definition of $U_{i,k}$ are equivalent to the one defined in the main text.
When ${\underaccent{\bar}{\bm{\mu}}}_t$ is close to $\bm{\mu}$, the list $\mathcal{P}(\pi^t, \emptyset)$ is similar to the list $\mathcal{P}(\pi^g, \emptyset)$, which indicates the reward gap $r_{\pi^g}(\bm{\mu}) - r_{\pi^t}(\bm{\mu})$ is small. Let ${\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t)$ be the indicator that takes value 1 when $\mathcal{P}(\pi^g, \emptyset)$ and $\mathcal{P}(\pi^t, \emptyset)$ are the same for the first $k$ elements, and different at the $(k + 1)$-th elements (i.e., $p^{\pi^g}_j = p^{\pi^t}_j$ for $0\leq j \leq k-1$ and $p^{\pi^g}_k\neq
p^{\pi^t}_k$) with condition $p^{\pi^t}_k = 1 - U_{i, k}\mu_i$. Note that the first $m$ elements in $\mathcal{P}(\pi^t, \emptyset)$ and $\mathcal{P}(\pi^g, \emptyset)$ equal to 1. Then the reward gap at round $t$ is $$\Delta_{\pi^t} = r_{\pi^g}(\bm{\mu}) - r_{\pi^t}(\bm{\mu}) = \sum_{i=1}^m\sum_{k=m+1}^{\min\{K, D\}} {\mathbbm{1}}_{i,k}({\underaccent{\bar}{\bm{\mu}}}_t)\cdot \Delta^{i, k}_{\max},$$ where $\Delta^{i, k}_{\max}$ is the maximum reward gap among all possible ${\underaccent{\bar}{\bm{\mu}}}_t$ such that ${\mathbbm{1}}_{i,k}({\underaccent{\bar}{\bm{\mu}}}_t) = 1$, i.e., $$\Delta^{i, k}_{\max} = \max_{\forall {\underaccent{\bar}{\bm{\mu}}}_t, {\mathbbm{1}}_{i,k}({\underaccent{\bar}{\bm{\mu}}}_t) = 1} r_{\pi^{g}}(\bm{\mu}) - r_{\pi^t}(\bm{\mu}).$$ Note that $$\Delta^{i,k}_{\max} \leq \sum_{j = k}^{\min\{K, D\}} f(j) \times p^{\pi^g}_0 \times \cdots\times p^{\pi^g}_{j-1} \times L(\{1 - p^{\pi^g}_0, \cdots, 1 - p^{\pi^g}_j\}, K - j).$$
The expected cumulative regret can be expanded as $$\begin{aligned}
\text{Reg}_{\bm{\mu}}(T) = \mathbb{E}_{\Phi_1,\cdots, \Phi_{T}}\left[\sum_{t=1}^{T}\Delta_{\pi^t}\right] &\leq \sum_{t=1}^T\mathbb{E}_{\Phi_1,\cdots, \Phi_{t-1}}\left[\sum_{k = {m + 1}}^{\min\{K, D\}} \sum_{i=1}^{m} {\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t) \times \Delta^{i,k}_{\max}\right]\\
& = \sum_{i=1}^m \sum_{k = {m + 1}}^M \Delta^{i,k}_{\min}\mathbb{E}_{\Phi_1,\cdots, \Phi_{t-1}}\left[ \sum_{t=1}^T \mathbbm{1}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t)\right].\end{aligned}$$ Our next step is bound $\mathbb{E}_{\Phi_1,\cdots, \Phi_{t-1}}\left[
\sum_{t=1}^T {\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t)\right]$. We rewrite the indicator ${\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t)$ as: $${\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t) = {\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t){\mathbbm{1}}\{T_{i, t-1}\leq l_{i, k}\} + \mathbbm{1}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t){\mathbbm{1}}\{T_{i, t-1}> l_{i, k}\},$$ where $l_{i, k}$ is a problem-specific constant. In Lemma \[lemma:enough\_probing\_bound\], we show that the probability we choose a wrong community when community $i$ is probed enough times (i.e., $T_{i, t-1} >
l_{i, k}$) is very small. Based on the lemma, the regret corresponding to the event ${\mathbbm{1}}\left\{T_{i, t-1} > l_{i, k}\right\}$ is bounded as follows. $$\begin{aligned}
& \sum_{i=1}^{m}\sum_{k=m + 1}^{\min\{K, D\}} \Delta^{i,k}_{\min} \mathbb{E}_{\Phi_1,\cdots, \Phi_T}\left[\sum_{t=1}^T{\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t) {\mathbbm{1}}\left\{ T_{i, t-1} > l_{i,k}\right\}\right] \leq \frac{\floor*{\frac{K^{\prime}}{2}}\pi^2}{3}\sum_{i=1}^{m}\sum_{k = m + 1}^{\min\{K, D\}}\Delta^{i, k}_{\max}.\end{aligned}$$ On the other hand, the regret associated with the event $\mathbbm{1}\{T_{i, t-1}
\leq l_{i, k}\}$ is trivially bounded by $\sum_{i=1}^{m}\sum_{k = m +
1}^K\Delta^{i, k}_{\max}l_{i, k}$. In conclusion, the expected cumulative regret is bound as $$\begin{split}
&\text{Reg}_{\bm{\mu}}(T) \leq \sum_{i=1}^{m} \sum_{k=m + 1}^K \Delta^{i,k}_{\max}\mathbb{E}_{\Phi_1,\cdots, \Phi_T}\left[\sum_{t=1}^T{\mathbbm{1}}_{k, t}({\underaccent{\bar}{\bm{\mu}}}_t) \right]\\
& \leq \sum_{i=1}^{m}\sum_{k = m + 1}^K\Delta^{i, k}_{\max}l_{i, k} +
\frac{\floor{\frac{K^{\prime}}{2}}\pi^2}{3}\sum_{i=1}^{m}\sum_{k = m + 1}^{\min\{K, D\}}\Delta^{i, k}_{\max}\\
& \leq \left( \sum_{i=1}^{m}\sum_{k = m + 1}^K \frac{6\Delta^{i, k}_{\max}}{(\Delta^{i,k}_{\min})^2}\right)\ln T +
\frac{\floor{\frac{K^{\prime}}{2}}\pi^2}{3}\sum_{i=1}^{m}\sum_{k = m + 1}^{\min\{K, D\}}\Delta^{i, k}_{\max}.
\end{split}$$ Note $\Delta^{(k)}_{\max} \geq \max_{i\in [m]} \Delta^{i,k}_{\max}$. This completes the proof.
\[lemma:enough\_probing\_bound\] For all $k\leq \{M, \sum_{i=1}^{m}d_i\}$, we have $$\label{eq:enough_probing_bound}
\mathbb{E}_{\Phi_1, \dots, \Phi_T}\left[ \sum_{t=1}^{T} {\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t){\mathbbm{1}}\{T_{i,t-1} > l_{i, k}\} \right] \leq \frac{\floor{\frac{K^{\prime}}{2}}\pi^2}{3},$$ where $l_{i, k}$ is defined as $l_{i, k} {\vcentcolon=}6\ln T/(\Delta^{i,k}_{\min})^2$.
The following proof is similar to the that for the traditional Upper Confidence Bound (UCB) algorithm [@auer2002finite]. In the following, we define $i^*_k = \max_{i\in [m]} 1 - U_{i, k}\mu_i$. $$\begin{aligned}
&\sum_{t=1}^{T} {\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t){\mathbbm{1}}\{T_{i,t-1} > l_{i, k}\} = \sum_{t=l_{i, k}+1}^{T} {\mathbbm{1}}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t)\mathbbm{1}\{T_{i,t-1} > l_{i, k}\} \\
\leq &\sum_{t = l_{i,k} + 1}^{T} \mathbbm{1}\{(\hat{\mu}_{i,t-1} - \rho_{i,t-1})U_{i,k}) < (\hat{\mu}_{i^*_k, t-1} - \rho_{i^*_k, t-1})U_{i^*_k, k}, T_{i,t-1} > l_{i, k}\}.\\\end{aligned}$$
When $T_{i_k, t-1} > l_{i, k} \triangleq \frac{6\ln T}{(\Delta^{i,k}_{\min})^2}$, we have $$\rho_{i, t-1} = \sqrt{\frac{3\ln t}{2T_{i, t-1}}} < \frac{\Delta^{i,k}_{\min}}{2}\Rightarrow \underbrace{\mu_{i^*_k}U_{i^*_k, k}< (\mu_{i} - 2\rho_{i, t-1})U_{i, k}}_{\text{$i$ and $i^*_k$ are distinguishable with high prob.}}.$$ If $i\neq i^*_k$ exists such that $$\begin{aligned}
\hat{\mu}_{i^*_k,t-1} - \rho_{i^*_k,t-1} < \mu_{i^*_k}, \text{ and }\hat{\mu}_{i,t-1} + \rho_{i,t-1} > \mu_{i},\end{aligned}$$ we have $$\begin{split}
(\hat{\mu}_{i^*_k,t-1} - \rho_{i^*_k,t-1})U_{i^*_k, k}< \mu_{i^*_k}U_{i^*_k, k} < (\mu_{i} - 2\rho_{i, t-1})U_{i, k} < (\hat{\mu}_{i,t-1} - \rho_{i,t-1})U_{i,k},\\
\end{split}$$ which contradicts with $(\hat{\mu}_{i,t-1} - \rho_{i,t-1})U_{i,k} <
(\hat{\mu}_{i^*_k, t-1} - \rho_{i^*_k, t-1})U_{i^*_k, k}$. Hence when $T_{i, t-1} > l_{i, k}$, we have $$\begin{split}
&\left\{ (\hat{\mu}_{i,t-1} - \rho_{i,t-1})U_{i,k} < (\hat{\mu}_{i^*_k, t-1} - \rho_{i^*_k, t-1})U_{i^*_k, k}\right\} \\
& \subseteq \left\{ \hat{\mu}_{i,t-1} + \rho_{i,t-1}
\leq\mu_i \text{ or } \hat{\mu}_{i^*_k,t-1} - \rho_{i^*_k,t-1} \geq \mu_{i^*_k}\right\}
\end{split}$$ Using the union bound, we have $$\begin{split}
&\Pr\left( (\hat{\mu}_{i,t-1} - \rho_{i,t-1})U_{i, k} < (\hat{\mu}_{i^*_k, t-1} - \rho_{i^*_k, t-1})U_{i^*_k, k} \right)\\
\leq&\Pr\left( \hat{\mu}_{i,t-1} + \rho_{i,t-1}
\leq\mu_i \text{ or } \hat{\mu}_{i^*_k,t-1} - \rho_{i^*_k,t-1} \geq \mu_{i^*_k} \right)\\
\leq &\Pr\left( \hat{\mu}_{i,t-1} + \rho_{i,t-1} \leq\mu_i \right) + \Pr\left( \hat{\mu}_{i^*_k,t-1} - \rho_{i^*_k,t-1} \geq \mu_{i^*_k} \right). \\
\end{split}$$ Therefore, we can conclude that $$\begin{aligned}
&\mathbb{E}_{\Phi_1, \dots, \Phi_T}\left[ \sum_{t=1}^{T} \mathbbm{1}_{i, k}({\underaccent{\bar}{\bm{\mu}}}_t)\mathbbm{1}\{T_{i,t-1} > l_{i, k}\}\right] \\
\leq &\sum\nolimits_{t = l_{i,k} + 1}^{T} {\mathbbm{1}}\{(\hat{\mu}_{i,t-1} - \rho_{i,t-1})U_{i, k} < (\hat{\mu}_{i^*_k, t-1} - \rho_{i^*_k, t-1})U_{i^*_k, k}, T_{i,t-1} > l_{i, k}\}.\\
\leq &\sum\nolimits_{t = l_{i,k} + 1}^{T} \Pr\left\{\hat{\mu}_{i,t-1} + \rho_{i,t-1} \leq\mu_i \right\} +
\Pr \left\{ \hat{\mu}_{i^*_k,t-1} - \rho_{i^*_k,t-1} \geq \mu_{i^*_k} \right\} \\
\leq&\sum_{t = l_{i,k} + 1}^{T}\left( \sum\nolimits_{T_{i, t-1} = l_{i, k} + 1} ^{t\floor*{\frac{K^{\prime}}{2}}}\Pr\left\{ \hat{\mu}_{i,t-1} + \rho_{i,t-1} \leq\mu_i | T_{i, t-1} \right\}\right.\\
& \left.+ \sum\nolimits_{T_{i^*_k, t-1} = 1} ^{t\floor*{\frac{K^{\prime}}{2}}}\Pr\left( \hat{\mu}_{i^*_k,t-1} - \rho_{i^*_k,t-1} \geq\mu_{i^*_k} | T_{i^*_k, t-1}\right)\right)\\
& \leq \sum_{t=1}^{\infty} 2t\floor*{\frac{K^{\prime}}{2}} \times t^{-3} = 2\floor*{\frac{K^{\prime}}{2}}\sum_{t=1}^{\infty}t^{-2} =\frac{\floor*{\frac{K^{\prime}}{2}}\pi^2}{3}.\qedhere\end{aligned}$$
Full information feedback
-------------------------
If we feed the empirical mean in the exploration oracle, then the policy $\pi^t$ is determined by $\hat{\bm{\mu}}_t$. Similarly, we can define the event ${\mathbbm{1}}_{i,k}(\hat{\bm{\mu}}_t)$ by replacing ${\underaccent{\bar}{\bm{\mu}}}_t$ with $\hat{\bm{\mu}}$ in Section \[app:transition\_probability\_list\_pit\]-\[app:proof\_framework\_for\_adaptive\_exploration\].
\[lemma:full\_information\_feedback\_empirical\_mean\] If we make revisions defined in Eq. to Algo. \[algo:CLCB\_algorithm\] and feed the empirical mean in `` to explore communities adaptively, then for all community $C_i$ and $k\leq \{K, \sum_{i=1}^m d_i\}$, we have $$\label{eq:full_information_feedback_empirical_mean}
\mathbb{E}_{\Phi_1, \dots, \Phi_T}\left[ \sum_{t=2}^T{\mathbbm{1}}_{i, k}(\hat{\bm{\mu}}_t) \right] \leq\frac{2}{\varepsilon^4_{i, k}},$$ where $\varepsilon_{i, k}$ is defined as (here $i^*_k\in \operatorname*{arg\,min}_{i\in [m]} \mu_iU_{i,k}$) $$\varepsilon_{i, k} \triangleq \frac{\mu_iU_{i,k} - \mu_{i^*_k}U_{i^*_k, k}}{U_{i,k} + U_{i^*_k, k}} \text{ for } i\neq i^*_k \text{ and } \varepsilon_{i, k} = \infty \text{ for } i = i^*_k.$$
$$$$
We first bound the probability of the following event by relating $\mathbbm{1}_{i, k}(\hat{\bm{\mu}}_t)$ with the event that both $\mu_{i, t-1}$ and $\mu_{i_k, t-1}$ in the confidence interval $\varepsilon_{i,k}$. $$\mathbbm{1}_{i,k}(\hat{\bm{\mu}}_t) \leq \mathbbm{1} \left\{\hat{\mu}_{i, t-1}U_{i,k} < \hat{\mu}_{i^*_k, t-1}U_{i^*_k,k} \right\}.$$ If $i\neq i^*_k$ such that $$\begin{aligned}
\hat{\mu}_{i, t-1} > \mu_i - \varepsilon_{i, k}, \text{ and }\hat{\mu}_{i^*_k, t-1} < \mu_{i^*_k} + \varepsilon_{i, k},\end{aligned}$$ then $$\begin{aligned}
\hat{\mu}_{i, t-1}U_{i,k} &> (\mu_i - \varepsilon_{i, k}) U_{i,k} = (\mu_{i^*_k} + \varepsilon_{i, k})U_{i^*_k, k} > \hat{\mu}_{i^*_k, t-1}U_{i^*_k, k},\end{aligned}$$ which contradicts with that $\hat{\mu}_{i, t-1}U_{i,k} <
\hat{\mu}_{i^*_k, t-1}U_{i^*_k}$. Here $(\mu_i - \varepsilon_{i, k}) U_{i,k} = (\mu_{i^*_k} + \varepsilon_{i^*,
k})U_{i^*_k, k}$ can be derived from the definition of $\varepsilon_{i,k}$. Therefore $$\begin{aligned}
&\mathbbm{1}\left\{\hat{\mu}_{i, t-1}U_{i,k} < \hat{\mu}_{i^*_k, t-1}U_{i^*_k, k}\right\}\\
\leq\ &\mathbbm{1}\left\{\hat{\mu}_{i, t-1} \leq \mu_i - \varepsilon_{i, k} \text{ or } \hat{\mu}_{i^*_k, t-1} \geq \mu_{i^*_k} + \varepsilon_{i, k}\right\}.\end{aligned}$$ With above equation and the concentration bound in [@Dubhashi2009CMA], the expectation $\mathbb{E}_{\Phi_1, \dots, \Phi_T}\left[ \sum_{t=2}^{T}{\mathbbm{1}}_{i,k}(\hat{\bm{\mu}}_t)\right]$ can be bounded as $$\begin{aligned}
&\mathbb{E}_{\Phi_1, \dots, \Phi_T}\left[ \sum_{t=2}^T{\mathbbm{1}}_{i, k}(\hat{\bm{\mu}}_t)\right]\\
\leq &\sum_{t=2}^T\Pr\left\{\hat{\mu}_{i, t-1}U_{i,k} < \hat{\mu}_{i^*_k, t-1}U_{i^*_k,k} \right\} \\
\leq &\sum_{t=2}^T\Pr\left\{\hat{\mu}_{i, t-1} \leq \mu_i - \varepsilon_{i, k}\right\} + \Pr\left\{\hat{\mu}_{i^*_k, t-1} \geq \mu_{i^*_k} + \varepsilon_{i, k}\right\}\\
\leq &\sum_{t=2}^T \left( \sum_{T_{i,t-1} = t-1}^{t\floor{K^{\prime}/2}} e^{-\varepsilon^2_{i, k}T_{i, t-1}} + \sum_{T_{i^*_k,t-1} = t-1}^{t\floor{K^{\prime} / 2}} e^{-\varepsilon^2_{i, k}T_{i^*_k, t-1}}\right)\\
\leq &\ 2\sum_{t=1}^T \sum_{s=t}^{\infty} e^{-s\varepsilon^2_{i, k}} \leq 2\sum_{t=1}^T \frac{e^{-t\varepsilon^2_{i, k}}}{\varepsilon^2_{i, k}} \leq \frac{2}{\varepsilon^4_{i, k}}.\qedhere\end{aligned}$$
Experimental Evaluation
=======================
In this section, we conduct simulations to validate the theoretical results claimed in the main text and provide some insight for future research.
Offline Problems
----------------
In this part, we show some simulation results for the offline problems.
**Performance of Algorithm \[algo:non\_adaptive\_exploration\]**. In Fig. \[fig:allocation\_lower\_upper\_bound\], we show that the allocation lower bound $\bm{k}^{-}$ and upper bound $\bm{k}^{+}$ are close to the optimal budget allocation. From Fig. \[fig:allocation\_lower\_upper\_bound\], we observe that the $L1$ distance between $\bm{k}^{*}$ and $\bm{k}^{-}$ (or $\bm{k}^{+}$) is around $m/2$, which means the average time complexity of Algorithm \[algo:non\_adaptive\_exploration\] is $\Theta((m\log m) / 2)$.
**Reward v.s. Budget**. We show the relationship between the reward (i.e., the number of distinct members) and the given budget in Fig. \[fig:offline\_reward\_budget\]. From Fig. \[fig:offline\_reward\_budget\], we can draw the following conclusions.
- The performance of the four methods are ranked as: ’[“Adaptive Opt.”]{}, *“Non-adaptive Opt.”*, *“Proportional to Size”*, *“Random Allocation”*. This validate our optimality results in Sec. \[sec:offline\].
- The method “*Proportional to Size*” and “*Non-adaptive Opt.*” have similar performance. It is an intuitive idea to allocate budgets proportional to the community sizes. The simulation results also demonstrate the efficiency of such budget allocation method. In the following, we analyze the reason theoretically. Recall the definition of $\bm{k}^{-}$ as follows. $$k^-_i = \frac{(K - m) / \ln (1 - \mu_i)}{\sum_{j=1}^m 1 / \ln (1 - \mu_j)}.$$ When $\mu_i\ll 1$, we have $\ln (1 - \mu_i)\approx -\mu_i$. Hence, $$k^-_i \approx \frac{(K - m) d_i}{\sum_{j=1}^m d_j}.$$ Besides, the L1 distance between $\bm{k}^*$ and $\bm{k}^{-}$ is smaller than $m$. We can conclude that the budget allocation proportional to size is close to the optimal budget allocation. Fig. \[fig:offline\_budget\_allocation\] also validates this conclusion.
- The reward gap between “*Non-adaptive Opt.*” and “*Adaptive Opt.*” increases first and then decreases, as shown in Fig. \[fig:used\_budget\_greedy\_policy\].
\
![Reward v.s. Budget. In the first row, we show three different size distributions of $m = 100$ communities. In the second row, we show the reward of four different budget allocation methods. Here *“Random Allocation”* represents random budget allocation (sum up to $K$). *“Proportional to Size”* method allocates budget proportional to the community sizes. *“Non-adaptive Opt.”* corresponds to the optimal budget allocation obtained by the greedy method. *“Adaptive Opt.”* means we explore the communities with greedy adaptive policy $\pi^g$. The simulations are run for 200 times for each data point on the budget-reward curve.[]{data-label="fig:offline_reward_budget"}](reward_legend.pdf "fig:")\
**Budget Allocation Comparison**. Fig. \[fig:offline\_budget\_allocation\] and Fig \[fig:used\_budget\_greedy\_policy\] show the budget allocation of non-adaptive optimal method and adaptive optimal method. Fig. \[fig:used\_budget\_greedy\_policy\] shows that the adaptive optimal method use the budget more efficiently.
![Actually used budget. we only show the results for the community size configuration generated by $\mathcal{G}(0.1)$, as shown in the first row of Fig. \[fig:offline\_reward\_budget\]. The legend labels have the same meaning as in Fig. \[fig:offline\_budget\_allocation\][]{data-label="fig:used_budget_greedy_policy"}](used_budget_Geometric.pdf "fig:") ![Actually used budget. we only show the results for the community size configuration generated by $\mathcal{G}(0.1)$, as shown in the first row of Fig. \[fig:offline\_reward\_budget\]. The legend labels have the same meaning as in Fig. \[fig:offline\_budget\_allocation\][]{data-label="fig:used_budget_greedy_policy"}](used_budget_reward_Geometric.pdf "fig:") ![Actually used budget. we only show the results for the community size configuration generated by $\mathcal{G}(0.1)$, as shown in the first row of Fig. \[fig:offline\_reward\_budget\]. The legend labels have the same meaning as in Fig. \[fig:offline\_budget\_allocation\][]{data-label="fig:used_budget_greedy_policy"}](used_budget_reward_diff_Geometric.pdf "fig:")
![Comparison of different budget allocation methods. The distribution of community sizes generated by the geometric distribution with success probability $0.1$, as shown in the first row of Fig. \[fig:offline\_reward\_budget\]. The legend label “$\bm{k}^*$” represents the optimal budget allocation. The “truncated $\pi^g$” means we stop the greedy adaptive process if all the members are found. The “truncated $\bm{k}^*$” means we stop the non-adaptive exploration of community $C_i$ if all the members of $C_i$ are found. Each data point is an average of 1000 simulations. []{data-label="fig:offline_budget_allocation"}](allocation_scatter_Geometric_K500.pdf "fig:") ![Comparison of different budget allocation methods. The distribution of community sizes generated by the geometric distribution with success probability $0.1$, as shown in the first row of Fig. \[fig:offline\_reward\_budget\]. The legend label “$\bm{k}^*$” represents the optimal budget allocation. The “truncated $\pi^g$” means we stop the greedy adaptive process if all the members are found. The “truncated $\bm{k}^*$” means we stop the non-adaptive exploration of community $C_i$ if all the members of $C_i$ are found. Each data point is an average of 1000 simulations. []{data-label="fig:offline_budget_allocation"}](allocation_scatter_Geometric_K2500.pdf "fig:") ![Comparison of different budget allocation methods. The distribution of community sizes generated by the geometric distribution with success probability $0.1$, as shown in the first row of Fig. \[fig:offline\_reward\_budget\]. The legend label “$\bm{k}^*$” represents the optimal budget allocation. The “truncated $\pi^g$” means we stop the greedy adaptive process if all the members are found. The “truncated $\bm{k}^*$” means we stop the non-adaptive exploration of community $C_i$ if all the members of $C_i$ are found. Each data point is an average of 1000 simulations. []{data-label="fig:offline_budget_allocation"}](allocation_scatter_Geometric_K6000.pdf "fig:")
Online Problems
---------------
In the following, we show the simulation results for the online, non-adaptive problem. The simulation results for online, adaptive are similar. Hence, we only present the results for online, non-adaptive problems. Fig. \[fig:non\_adaptive\_toy\_regret\] shows the regret of three different learning methods. For illustration purpose, we set the community sizes as $\\bm{d} = (2, 3, 5, 6, 8, 10)$. From Fig. \[fig:non\_adaptive\_toy\_regret\], we can draw the following conclusions.
- If we feed the empirical mean into the oracle, the regret grows linearly.
- The regret of CLCB algorithm is bounded logarithmically, as proved in Thm. \[thm:regret\_bound\_non\_adaptive\_exploration\_1\].
- The regret under full information feedback setting is bounded as a problem related constant, as proved in Thm. \[thm:non\_adaptive\_full\_information\].
![Comparison of different learning algorithms. The sizes of communities are $\bm{d} = (2, 3, 5, 6, 8, 10)$. Here the label *Empirical mean* represents feeding the empirical mean into the oracle directly. The regret/error line plots are average of 100 simulations.[]{data-label="fig:non_adaptive_toy_regret"}](regret_legend "fig:")\
\
[^1]: The model can be extended to meet multiple members per visit, but for simplicity, we consider meeting one member per visit in this paper.
[^2]: We thank Jing Yu from School of Mathematical Sciences at Fudan University for her method to find a good initial allocation, which leads to a faster greedy method.
[^3]: If $X$ is a random variable, and $\varphi$ is a convex function, then $\varphi(\mathbb{E}[X])\leq \mathbb{E}[\varphi(X)]$.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
[**Etera R Livine**]{}[^1], [**Robert Oeckl**]{}[^2]\
[Centre de Physique Théorique, Campus de Luminy, Case 907, ]{}\
[13288 Marseille cedex 9, France ]{}
title: |
Three-dimensional Quantum Supergravity\
and Supersymmetric Spin Foam Models
---
Acknowledgements {#acknowledgements .unnumbered}
================
E. L. would like to thank Yi Ling and Daniele Oriti, with whom he had started the project of studying three-dimensional supergravity, for many discussions on this subject. R. O. thanks John Barrett for discussions. He was supported by a Marie Curie Fellowship of the European Union.
[^1]: email: [email protected]
[^2]: email: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand $C^{1}$ continuity of the shape functions. Here, recent development in discontinuous Galerkin methods are explored and exploited for the solution of a prototype nonlinear strain gradient dependent continuum model. A formulation is developed that allows the rigorous solution of a strain gradient damage model using standard $C^{0}$ shape functions. The formulation is tested in one-dimension for the simplest possible finite element formulation: piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed formulation.
\
Discontinuous Galerkin methods, gradient-dependent continua, damage.
author:
- 'Garth N. Wells$^{1}$[^1] Krishna Garikipati$^{2}$ Luisa Molari$^{3}$'
bibliography:
- 'papers\_full.bib'
- 'papers.bib'
- 'papers\_gradient.bib'
- 'dg.bib'
date: |
$^{1}$Faculty of Civil Engineering and Geosciences, Delft University of Technology\
Stevinweg 1, 2628 CN Delft, The Netherlands\
$^{2}$Department of Mechanical Engineering, University of Michigan\
Ann Arbor, Michigan 48109, USA\
$^{3}$DISTART, Università di Bologna\
Viale Risorgimento 2, 40136 Bologna, Italy\
title: ' A discontinuous Galerkin formulation for a strain gradient-dependent damage model'
---
Introduction
============
Strain gradient dependent continuum models have been developed for a wide range of problems. Strain gradient effects are included in continuum models to reproduce experimentally observed phenomena which cannot be captured with classical models [@aifantis:1984; @larsy:1988; @muhlhaus:1991; @peerlings:1996; @fleck:1997; @gao:1999; @fleck:2001]. The range of application is broad, from large geological problems to polycrystals. Typical phenomena which can be captured with strain gradient models include strain localisation in the presence of softening and size effects.
The development of strain gradient models has been hindered by the lack of a suitable numerical framework for their robust solution on arbitrary domains. The introduction of strain gradients into continuum models poses significant challenges in solving the ensuing equations. The finite element method, the dominant numerical method in solid mechanics, is ideally suited to the solution of second-order partial differential equations, such as classical elasticity. The solution of gradient-dependent continuum problems usually demands at least $C^{1}$ continuous basis functions, which are difficult to construct in spatial dimensions higher than one. Previous attempts to solve such problems with $C^{1}$ shape functions or *ad-hoc* measures have proven difficult [@larsy:1988; @borst:1992; @borst:1996]. More seriously, in numerous publications, basic continuity requirements are completely ignored. To avoid these difficulties, @askes:2000b applied the element-free Galerkin method, which can provide a high degree of continuity, for the solution of strain gradient dependent damage models. However, the element-free Galerkin method entails other difficulties, lacks the penetration in the solid mechanics community of the finite element method, and is generally less efficient. As a result of these difficulties, strain gradient dependent models are not widely applied, and many formulations are largely untested. The difficulties presented by continuity requirements has even lead to reformulations of strain gradient models that are driven by algorithmic convenience [@peerlings:1996; @engelen:2003].
In this work, a fresh perspective is taken on the solution of strain gradient dependent continuum problems in light of recent developments in discontinuous and continuous/discontinuous Galerkin methods for elliptic problems [@oden:1998dg; @arnold:2002; @engel:2002]. A summary of recent developments can be found in @arnold:2002. In the derivation of the Galerkin problem, potential discontinuities in the basis functions across internal surfaces are taken into account, resulting in a generalisation of the conventional Galerkin method.
To begin, a strain gradient-dependent damage model, which is used as a prototype example, is introduced. It is cast as a continuous Galerkin problem in a finite element framework and the difficulties with the conventional finite element method are highlighted. The Galerkin problem is then generalised to allow for discontinuities in the appropriate fields. The formulation is tested for the simplest possible finite element in one dimension. A series of test cases are computed and the results are compared to a benchmark solution.
Gradient-enhanced damage model: Preliminaries
=============================================
Consider a body $\Omega$ in $\mathbb{R}^{n}$, with boundary $\Gamma = {\partial}\Omega$. The strong form of the equilibrium equation for the body $\Omega$, in the absence of body forces, and associated standard boundary conditions, is: $$\begin{aligned}
{{\nabla}}\cdot {{\bm{\sigma}}}&= {{\bm{}}}{0} &{\rm in} \ \Omega
\label{eqn:strong_equil} \\
{{\bm{\sigma}}}\cdot {{\bm{}}}{n} &= {{\bm{}}}{h} &{\rm on} \ \Gamma_{h} \\
{{\bm{}}}{u} &= {{\bm{}}}{g} &{\rm on} \ \Gamma_{g}
\end{aligned}$$ where ${{\nabla}}$ is the gradient operator, ${{\bm{\sigma}}}$ is the stress tensor, ${{\bm{}}}{h}$ is the prescribed traction on $\Gamma_{h}$ and ${{\bm{}}}{g}$ is the prescribed displacement on the boundary $\Gamma_{g}$ ($\Gamma_{g} \cup \Gamma_{h} = \Gamma$, $\Gamma_{g} \cap \Gamma_{h} = \emptyset$). The outward normal to $\Gamma$ is denoted ${{\bm{}}}{n}$.
For an isotropic elasticity-based damage model, the stress at a material point is given by: $${{\bm{\sigma}}}= {\left( {1-\omega} \right)} \mathcal{C} {\mathbf{:}}{{\nabla}^{\rm s}}{{\bm{}}}{u}
\label{eqn:constit}$$ where $\mathcal{C}$ is the usual linear-elastic constitutive tensor and the damage variable ($0 \le \omega \le 1$) is a function of a scalar history parameter $\kappa$, $$\omega = \omega{\left( {\kappa} \right)}.
\label{eqn:damage}$$ The history parameter $\kappa$ is related to a gradient-dependent ‘equivalent strain’, ${\overline}{{\epsilon}}$. A common choice for ${\overline}{{\epsilon}}$ is: $${\overline}{{\epsilon}} = {\epsilon}_{\rm eq} + c^{2} \Delta {\epsilon}_{\rm eq}
\label{eqn:grad_strain}$$ where ${\epsilon}_{\rm eq}$ is an invariant of the local strain tensor ${{\bm{{\epsilon}}}}={{\nabla}^{\rm s}}{{\bm{}}}{u}$, $c$ is a length scale which reflects the strength of strain gradient effects and $\Delta$ is the Laplacian operator. This formulation is often named ‘explicit gradient damage’ [@peerlings:1996]. The chosen invariant for the local equivalent strain reflects the processes that drive damage growth in a given material. In one dimension, the obvious choice is that the equivalent strain is equal to the strain.
The history parameter $\kappa$ is equal to the largest positive value of ${\overline}{{\epsilon}}$ reached at a material point. Defining a loading function $f$, $$f = {\overline}{{\epsilon}} - \kappa$$ the evolution of $\kappa$ obeys the Kuhn-Tucker conditions, $$\begin{aligned}
\Dot{\kappa} \ge 0, && f \le 0, && \Dot{\kappa}f = 0.
\end{aligned}$$ A commonly adopted dependency is: $$\omega =
\begin{cases}
0 & {\rm if} \ \ \kappa \le \kappa_{0} \\
1- \dfrac{\kappa_{0} {\left( {\kappa_{c} - \kappa} \right)}}{\kappa
{\left( {\kappa_{c}-\kappa_{0}} \right)}}
& {\rm if} \ \ \kappa_{0} < \kappa < \kappa_{c} \\
1 & {\rm if} \ \ \kappa \ge \kappa_{c}
\end{cases}
\label{eqn:damage_evolution}$$ where $\kappa_{0}$ is the value of the history parameter at which damage begins to develop and $\kappa_{c}$ is the value at which $\omega=1$. The evolution of $\omega$ in equation yields a linear softening response for a uniaxial test in the absence of strain gradient effects. To make the dependency of $\omega$ on ${\overline}{{\epsilon}}$ clear, the expressions $\omega{\left( {\kappa} \right)}$ and $\omega{\left( {{\overline}{{\epsilon}}} \right)}$ will be used interchangeably.
Insertion of the constitutive model (see equations , and ) into the equilibrium equation leads to a non-linear fourth-order partial differential equation. This requires the prescription of boundary conditions on gradients of the displacement field higher than one. The physical implications of these boundary conditions are unclear and are the subject of debate. At this stage, the boundary condition $$\begin{aligned}
c^{2} {{\nabla}}{\epsilon}_{\rm eq} \cdot {{\bm{}}}{n}
= {{\epsilon}_{\rm bc}}&& {\rm on} \ \Gamma
\label{eqn:eps_bs}
\end{aligned}$$ is considered. A common choice is ${{\epsilon}_{\rm bc}}=0$, which is adopted for all examples in Section \[sec:examples\].
This elasticity-based damage model is convenient for preliminary developments as ${\epsilon}_{\rm eq}$ is calculated explicitly from the gradient of the displacement field, which is in contrast to the equivalent plastic strain in an elastoplastic model. However, equation is identical in form to the equation for the gradient-dependent equivalent plastic strain that is adopted in many strain gradient dependent plasticity models [@aifantis:1984; @muhlhaus:1991; @fleck:2001]. This model therefore provides a canonical formulation which can be extended to a broader class of models.
Galerkin formulation
====================
In developing a weak formulation for eventual finite element solution, the equilibrium equation and the equation for ${\overline}{{\epsilon}}$ are considered separately. The non-linear fourth-order equation resulting from insertion of the constitutive equations into the equilibrium equation could potentially be cast in a weak from. The formulation would inevitably be specific to the chosen dependency of damage on ${\overline}{{\epsilon}}$, a dependency which is potentially highly complex. Hence, for simplicity and generality, it is convenient to treat the two equations separately.
The body $\Omega$ is partitioned into $n_{el}$ non-overlapping elements $\Omega_{e}$ such that $${\overline}{\Omega} = \bigcup_{e=1}^{n_{el}} {\overline}{\Omega}_{e}.$$ where ${\overline}{\Omega}_{e}$ is a closed set (i.e., it includes the boundary of the element). The elements $\Omega_{e}$ (which are open sets) satisfy the standard requirements for a finite element partition. A domain ${\widetilde}{\Omega}$ is also defined $${\widetilde}{\Omega} = \bigcup_{e=1}^{n_{el}} \Omega_{e}$$ where ${\widetilde}{\Omega}$ does not include element boundaries. It is also useful to define the ‘interior’ boundary ${\widetilde}{\Gamma}$, $${\widetilde}{\Gamma} = \bigcup_{i=1}^{n_{b}} \Gamma_{i}$$ where $\Gamma_{i}$ is the $i$th interior element boundary and $n_{b}$ is the number of internal inter-element boundaries.
Consider now the function spaces $\mathcal{S}^{h}$, $\mathcal{V}^{h}$ and $\mathcal{W}^{h}$, $$\begin{aligned}
\mathcal{S}^{h} &={\left\{ {u^{h}_{i} \in H_{0}^{1}{\left( {\Omega} \right)} \ \left| \
u_{i}^{h}|_{\Omega_{e}} \in P_{k_{1}}{\left( {\Omega_{e}} \right)} \forall e,
\ u_{i} = g_{i} \ {\rm on}
\ \Gamma_{g} \right. } \right\}} \label{eqn:disp_trial} \\
\mathcal{V}^{h} &={\left\{ {w^{h}_{i} \in H_{0}^{1}{\left( {\Omega} \right)} \ \left| \
w_{i}^{h}|_{\Omega_{e}} \in P_{k_{1}}{\left( {\Omega_{e}} \right)} \forall e,
\ w_{i} = 0 \ {\rm on}
\ \Gamma_{g} \right. } \right\}} \label{eqn:disp_test} \\
\mathcal{W}^{h} &={\left\{ {q^{h} \in L_{2}{\left( {\Omega} \right)} \ \left| \
q^{h}|_{\Omega_{e}} \in P_{k_{2}}{\left( {\Omega_{e}} \right)} \forall e
\right. } \right\}} \label{eqn:space_q}
\end{aligned}$$ where $P_{k}$ represents the space of polynomial finite element shape functions (of polynomial order $k$). The spaces $\mathcal{S}^{h}$ and $\mathcal{V}^{h}$ represent usual $C^{0}$ continuous finite element shape functions. The space $\mathcal{W}^{h}$ can contain discontinuous functions.
Standard Galerkin weak form
---------------------------
The standard, continuous Galerkin problem for the equilibrium equation is of the form: Find ${{\bm{}}}{u}^{h} \in \mathcal{S}^{h}$ such that $$\begin{aligned}
\int_{\Omega} {{\nabla}}{{\bm{}}}{w}^{h} {\mathbf{:}}{\left( {1 - \omega{\left( {{\overline}{{\epsilon}}^{h}} \right)}} \right)} \mathcal{C} {\mathbf{:}}{{\nabla}^{\rm s}}{{\bm{}}}{u}^{h} { \ d}\Omega
- \int_{\Gamma_{h}} {{\bm{}}}{w}^{h} \cdot {{\bm{}}}{h} { \ d}\Gamma = 0
&& \forall {{\bm{}}}{w}^{h} \in \mathcal{V}^{h} \label{eqn:equil_cont}
\end{aligned}$$ where it was already assumed that ${{\bm{}}}{u}^{h}$ is $C^{0}$ continuous (see equation ). Note that the damage is a function of ${\overline}{{\epsilon}}$, which is in turn a function of displacement gradients, making the equation non-linear. It is presumed at this point that ${\overline}{{\epsilon}}^{h}$ is square-integrable over $\Omega$ (${\overline}{{\epsilon}}^{h} \in L_{2}{\left( {\Omega} \right)}$).
A second Galerkin problem is constructed to solve for ${\overline}{{\epsilon}}$ (equation ). It consists of: Find ${\overline}{{\epsilon}} \in \mathcal{W}^{h}$ such that $$\int_{\Omega} q^{h} {\overline}{{\epsilon}}^{h} { \ d}\Omega
-\int_{\Omega} q^{h} {\epsilon}_{\rm eq}^{h} { \ d}\Omega
+\int_{\Omega} {{\nabla}}q^{h}\cdot c^{2} {{\nabla}}{\epsilon}_{\rm eq}^{h}
{ \ d}\Omega
-\int_{\Gamma} q^{h} {{\epsilon}_{\rm bc}}{ \ d}\Gamma
= 0 \ \ \ \ \forall q^{h} \in \mathcal{W}^{h}
\label{eqn:bar_eps_weak}$$ where it is assumed that ${\epsilon}_{\rm eq}^{h}$ is known. Recall that discontinuities in $q^{h}$ and ${\overline}{{\epsilon}}^{h}$ are permitted.
Two difficulties exist in the preceding Galerkin formulation. The first is that the weight function $q^{h}$ can be discontinuous (cf. equation ), meaning that ${{\nabla}}q^{h}$ is not necessarily square-integrable on $\Omega$. This problem can be circumvented easily by requiring $C^{0}$ continuity of the functions in $\mathcal{W}^{h}$. The second problem, which is less easily solved, is that ${\epsilon}_{\rm eq}^{h}$ is computed from ${{\nabla}^{\rm s}}{{\bm{}}}{u}^{h}$. Therefore, calculating ${{\nabla}}{\epsilon}_{\rm eq}^{h}$ everywhere in $\Omega$ requires that the displacement field ${{\bm{}}}{u}^{h}$ be $C^{1}$ continuous if singularities are to be avoided. However, since ${{\bm{}}}{u}^{h} \in
H^{1}_{0}{\left( {\Omega} \right)}$ (see equation ), it is not necessarily $C^{1}$ continuous. To proceed with this formulation in a conventional manner, two possibilities present themselves. The first is to solve equations and using $C^{0}$ finite element shape functions to interpolate ${\overline}{{\epsilon}}^{h}$ and $q^{h}$, which is straightforward, and using $C^{1}$ shape functions for ${{\bm{}}}{w}^{h}$ and ${{\bm{}}}{u}^{h}$. The second approach is to interpolate ${\epsilon}_{\rm eq}$ using $C^{1}$ shape functions, from which the term $\Delta {\epsilon}_{\rm eq}$ can be evaluated everywhere in $\Omega$. The second approach may appear more attractive than the first as it requires a $C^{1}$ interpolation of a scalar field rather than a vector field. Both approaches pose significant difficulties as $C^{1}$ shape functions are difficult to construct, lack generality and lead to extremely complex element formulations. $C^{1}$ functions are difficult to construct in two dimensions, and to the authors’ knowledge, untried in three dimensions.
Discontinuous Galerkin form
---------------------------
The approach advocated here avoids the need for $C^{1}$ continuity of the displacement field by imposing the required degree of continuity in a weak sense. Before proceeding with the formulation, it is necessary to define jump and an averaging operations. The jump in a field ${{\bm{}}}{a}$ across a surface (which is associated with a body) is given by [@arnold:2002]: $${\left\llbracket {{{\bm{}}}{a}} \right\rrbracket} = {{\bm{}}}{a}_{1} \cdot {{\bm{}}}{n}_{1} +
{{\bm{}}}{a}_{2} \cdot {{\bm{}}}{n}_{2}$$ where the subscripts denote the side of the surface and ${{\bm{}}}{n}$ is the outward unit normal vector. This definition is convenient as it avoids introducing ‘+’ and ‘-’ sides of a surface. This is particularly so for arbitrarily-oriented surfaces in two and three dimensions. The average of a field ${{\bm{}}}{a}$ across a surface is given by: $${\left< {{{\bm{}}}{a}} \right>} = \frac{{\left( {{{\bm{}}}{a}_{1} + {{\bm{}}}{a}_{2}} \right)}}{2}.$$
Consider now equation for ${\overline}{{\epsilon}}$, which can be cast in a weak form using integration by parts and the divergence theorem on the boundary $\Gamma$ and on inter-element boundaries, ${\widetilde}{\Gamma}$. This yields: $$\begin{gathered}
\label{eqn:dg_1}
\int_{\Omega} q^{h} {\overline}{{\epsilon}}^{h} { \ d}\Omega
-\int_{\Omega} q^{h} {\epsilon}_{\rm eq}^{h} { \ d}\Omega
+\int_{{\widetilde}{\Omega}} {{\nabla}}q^{h} \cdot c^{2}{{\nabla}}{\epsilon}_{\rm eq}^{h}
{ \ d}\Omega
-\int_{\Gamma} q^{h} {{\epsilon}_{\rm bc}}{ \ d}\Gamma \\
-\int_{{\widetilde}{\Gamma}} {\left< {q^{h}} \right>}
\cdot c^{2}{\left\llbracket { {{\nabla}}{\epsilon}_{\rm eq}^{h}} \right\rrbracket} { \ d}\Gamma
-\int_{{\widetilde}{\Gamma}} {\left\llbracket {q^{h}} \right\rrbracket} \cdot c^{2}{\left< {{{\nabla}}{\epsilon}_{\rm
eq}^{h}} \right>} { \ d}\Gamma
= 0.
\end{gathered}$$ Note the distinction between $\Omega$ and ${\widetilde}{\Omega}$ for the volume integrals. It is chosen that the following weak statements of continuity should hold: $$\begin{aligned}
\int_{{\widetilde}{\Gamma}} {\left< {q^{h}} \right>} c^{2} {\left\llbracket {{{\nabla}}{\epsilon}_{\rm eq}^{h}} \right\rrbracket}
{ \ d}\Gamma &= 0
&& \forall q^{h} \in \mathcal{W}^{h} \\
- \int_{{\widetilde}{\Gamma}} {\left< {{{\nabla}}q^{h}} \right>} \cdot c^{2} {\left\llbracket {{\epsilon}_{\rm eq}^{h}} \right\rrbracket}
{ \ d}\Gamma &= 0
&& \forall q^{h} \in \mathcal{W}^{h}. \label{eqn:dg_symm}
\end{aligned}$$ Also, a ‘penalty-like’ term is introduced: $$\int_{{\widetilde}{\Gamma}} \frac{c^{2}}{h_{e}}
{\left\llbracket {q^{h}} \right\rrbracket}\cdot{{\left\llbracket {{\epsilon}_{\rm eq}^{h}} \right\rrbracket}} { \ d}\Gamma =0$$ where $h_{e}$ is a length scale which is required for dimensional consistency. Adding the additional equations to equation leads to the following Galerkin problem: Find ${\overline}{{\epsilon}}^{h} \in \mathcal{W}^{h}$ such that $$\begin{gathered}
\int_{\Omega} q^{h} {\overline}{{\epsilon}}^{h} { \ d}\Omega
-\int_{\Omega} q^{h} {\epsilon}_{\rm eq}^{h} { \ d}\Omega
+\int_{{\widetilde}{\Omega}} {{\nabla}}q^{h} \cdot c^{2}{{\nabla}}{\epsilon}_{\rm eq}^{h} { \ d}\Omega
-\int_{\Gamma} q^{h} {{\epsilon}_{\rm bc}}{ \ d}\Gamma \\
-\int_{{\widetilde}{\Gamma}} {\left\llbracket {q^{h}} \right\rrbracket} \cdot
c^{2}{\left< {{{\nabla}}{\epsilon}_{\rm eq}^{h}} \right>} { \ d}\Gamma
-\int_{{\widetilde}{\Gamma}} {\left< {{{\nabla}}q^{h}} \right>} \cdot
c^{2}{\left\llbracket {{\epsilon}_{\rm eq}^{h}} \right\rrbracket} { \ d}\Gamma \\
+ \int_{{\widetilde}{\Gamma}} \frac{c^{2}}{h_{e}}
{\left\llbracket {q^{h}} \right\rrbracket}\cdot{{\left\llbracket {{\epsilon}_{\rm eq}^{h}} \right\rrbracket}} { \ d}\Gamma
= 0 \ \ \ \ \forall q^{h} \in \mathcal{W}^{h}.
\label{eqn:eps_weak}
\end{gathered}$$ Adding the term in equation to the problem provides a degree of ‘symmetry’ with the term $\int_{{\widetilde}{\Gamma}}
{\left\llbracket {q^{h}} \right\rrbracket}\cdot c^{2}{\left< {{{\nabla}}{\epsilon}_{\rm eq}^{h}} \right>} { \ d}\Gamma$. The choice of $c^{2}/h_{e}$ may seem somewhat arbitrary considering that it appears as a penalty-like parameter. This choice will be justified later through an analogy between the proposed method and a finite difference scheme. No gradients of ${\epsilon}_{\rm eq}^{h}$ or $q^{h}$ appear in terms integrated over $\Omega$ (which includes interior boundaries) in equation , hence the continuity requirements on the spaces $\mathcal{S}^{h}$ and $\mathcal{W}^{h}$ are sufficient.
Equation reassembles the ‘interior penalty’ method for classical elasticity, which belongs to the discontinuous Galerkin family of methods [@arnold:2002]. Terms have been added to the weak form that for a conventional elasticity problem would lead to a symmetric formulation. Symmetry is however not of relevance here as the functions $q^{h}$ and ${\epsilon}_{\rm eq}^{h}$ will generally come from different function spaces. This formulation is general for the case in which the space $\mathcal{W}^{h}$ contains discontinuous functions. However, note if all functions in the space $\mathcal{W}^{h}$ are $C^{0}$ continuous, the formulation is still valid, with terms relating to the jump in ${\epsilon}_{\rm}^{h}$ remaining. The formulation would then resemble a continuous/discontinuous Galerkin method [@engel:2002].
The solution of the gradient enhanced damage problem requires the simultaneous solution of equations and , which are coupled. In summary, the problem is: Find ${{\bm{}}}{u}^{h} \in
\mathcal{S}^{h}$ and ${\overline}{{\epsilon}}^{h} \in \mathcal{W}^{h}$ such that $$\begin{aligned}
\int_{\Omega} {{\nabla}}{{\bm{}}}{w}^{h} {\mathbf{:}}{\left( {1 - \omega{\left( {{\overline}{{\epsilon}}^{h}} \right)}} \right)} \mathcal{C} {\mathbf{:}}{{\nabla}^{\rm s}}{{\bm{}}}{u}^{h} { \ d}\Omega
- \int_{\Gamma_{h}} {{\bm{}}}{w}^{h} \cdot {{\bm{}}}{h} { \ d}\Gamma = 0
&& \forall {{\bm{}}}{w}^{h} \in \mathcal{V}^{h} \label{eqn:equil_cont_b}
\end{aligned}$$ $$\begin{gathered}
\int_{\Omega} q^{h} {\overline}{{\epsilon}}^{h} { \ d}\Omega
-\int_{\Omega} q^{h} {\epsilon}_{\rm eq}^{h} { \ d}\Omega
+\int_{{\widetilde}{\Omega}} {{\nabla}}q^{h} \cdot c^{2}{{\nabla}}{\epsilon}_{\rm eq}^{h}
{ \ d}\Omega
-\int_{\Gamma} q^{h} {{\epsilon}_{\rm bc}}{ \ d}\Gamma \\
-\int_{{\widetilde}{\Gamma}} {\left\llbracket {q^{h}} \right\rrbracket} \cdot
c^{2}{\left< {{{\nabla}}{\epsilon}_{\rm eq}^{h}} \right>} { \ d}\Gamma
-\int_{{\widetilde}{\Gamma}} {\left< {{{\nabla}}q^{h}} \right>} \cdot
c^{2}{\left\llbracket {{\epsilon}_{\rm eq}^{h}} \right\rrbracket} { \ d}\Gamma \\
+ \int_{{\widetilde}{\Gamma}} \frac{c^{2}}{h_{e}}
{\left\llbracket {q^{h}} \right\rrbracket}\cdot{{\left\llbracket {{\epsilon}_{\rm eq}^{h}} \right\rrbracket}} { \ d}\Gamma
= 0 \ \ \ \ \forall q^{h} \in \mathcal{W}^{h}
\label{eqn:eps_weak_b}
\end{gathered}$$ where the nonlinear equations are coupled through the dependency of $\omega$ on ${\overline}{{\epsilon}}^{h}$ and the dependency of ${\overline}{{\epsilon}}^{h}$ on ${{\bm{}}}{u}^{h}$. Linearisation of these equations is straightforward, and is included in Appendix \[append:linearisation\].
In this work, the simplest possible finite element formulation is considered. It is chosen to interpolate the displacement field with linear piecewise continuous ($C^{0}$) functions and to use constant functions on elements for ${\overline}{{\epsilon}}$ ($k_{2} = 0$ in equation ). Also, the boundary condition ${{\nabla}}{\epsilon}_{\rm eq} \cdot
{{\bm{}}}{n} =0$ on $\Gamma$ is applied. As a consequence, several terms disappear from equation , leading to the problem: Find ${\overline}{{\epsilon}}^{h}
\in \mathcal{W}^{h}$ such that $$\begin{aligned}
\int_{\Omega} q^{h} {\overline}{{\epsilon}}^{h} { \ d}\Omega
-\int_{\Omega} q^{h} {\epsilon}_{\rm eq}^{h} { \ d}\Omega
+ \int_{{\widetilde}{\Gamma}} \frac{c^{2}}{h_{e}}
{\left\llbracket {q^{h} } \right\rrbracket}\cdot{{\left\llbracket {{\epsilon}_{\rm eq}^{h}} \right\rrbracket}} { \ d}\Gamma
= 0 &&\forall q^{h} \in \mathcal{W}^{h}.
\label{eqn:eps_weak_fe}
\end{aligned}$$ If $c=0$, ${\overline}{{\epsilon}} = {\epsilon}_{\rm eq}$ at all points in $\Omega$, and the model reduces to a local damage formulation (no gradient effects). In one dimension, $h_{e}$ is taken as ${\left< {h_{e}} \right>}$. A higher-dimension generalisation would be the distance between the centroid of the neighbouring elements.
A physical interpretation of equation is simple. The stronger the spatial variation in the strain field , the larger the jumps in the strain across element boundaries. Equation sets ${\overline}{{\epsilon}}$ equal to the local equivalent strain, and subtracts a component which is proportional to the equivalent strain jump and the material parameter $c^{2}$, effectively decreasing ${\overline}{{\epsilon}}$ (relative the ${\epsilon}_{\rm eq}$) in the presence of rapid spatial variation in the strain field, which is manifest in the form of jumps in the strain across element boundaries.
In practice, this finite element formulation is very simple. An element has three nodes. Displacement degrees of freedom are located at the two end-nodes, and a degree of freedom for ${\overline}{{\epsilon}}$ is located at the centre node of each element. The standard loop over all elements in a mesh is performed, and in addition all interior interfaces are looped over. Despite the node corresponding to ${\overline}{{\epsilon}}^{h}$ being internal to an element, it cannot be eliminated at the element level. The element stiffness matrices for this formulation are elaborated in Appendix \[append:fe\].
Consistency of the discontinuous formulation
--------------------------------------------
Having added non-standard terms to the weak form, it is important to prove consistency of the method. Applying integration by parts to the integral over ${\widetilde}{\Omega}$ in equation , $$\begin{gathered}
\int_{{\widetilde}{\Omega}} {{\nabla}}q^{h} \cdot c^{2} {{\nabla}}{\epsilon}_{\rm ep}^{h} { \ d}\Omega
=
-\int_{{\widetilde}{\Omega}} q^{h} c^{2}\Delta {\epsilon}_{\rm eq}^{h} { \ d}\Omega
+\int_{\Gamma} q^{h} c^{2} {{\nabla}}{\epsilon}_{\rm eq} \cdot {{\bm{}}}{n} { \ d}\Gamma \\
+\int_{{\widetilde}{\Gamma}} {\left< {q^{h}} \right>}
c^{2} {\left\llbracket {{{\nabla}}{\epsilon}_{\rm eq}} \right\rrbracket} { \ d}\Gamma
+\int_{{\widetilde}{\Gamma}} {\left\llbracket {q^{h}} \right\rrbracket} \cdot
c^{2} {\left< {{{\nabla}}{\epsilon}_{\rm eq}} \right>} { \ d}\Gamma.
\end{gathered}$$ Inserting this expression into the infinite-dimensional version of equation , and employing standard variational arguments, the following Euler-Lagrange equations can be identified: $$\begin{aligned}
{\overline}{{\epsilon}} - {\epsilon}_{\rm eq} - c^{2} \Delta {\epsilon}_{\rm eq} &= 0
&& {\rm in} \ \ {\widetilde}{\Omega} \label{eqn:EL_a}\\
c^{2} {\left\llbracket {{\epsilon}_{\rm eq}} \right\rrbracket} &= 0
&& {\rm on} \ \ {\widetilde}{\Gamma}
\label{eqn:EL_b} \\
c^{2} {\left\llbracket {{{\nabla}}{\epsilon}_{\rm eq}} \right\rrbracket} &= 0
&& {\rm on} \ \ {\widetilde}{\Gamma} \label{eqn:EL_c} \\
c^{2} {{\nabla}}{\epsilon}_{\rm eq} \cdot {{\bm{}}}{n} &= {{\epsilon}_{\rm bc}}&& {\rm on} \ \ \Gamma \label{eqn:EL_d}
\end{aligned}$$ Equation is the original problem over element interiors (see equation ). Equations and impose continuity of the corresponding fields across element boundaries and equation imposes the natural boundary condition on ${{\nabla}}{\epsilon}_{\rm eq} \cdot {{\bm{}}}{n}$. The Galerkin form (equation ) can therefore be seen as the weak imposition of these Euler-Lagrange equations.
Finite difference analogy
-------------------------
In one-dimension for equally spaced nodal points, it can be shown that the proposed formulation ($C^{0}$ linear $u^{h}$ and piecewise constant ${\overline}{{\epsilon}}^{h}$) is equivalent to a finite difference scheme for calculating ${\epsilon}_{{\rm eq},xx}$ (which is equal to $u_{,xxx}$ for ${\epsilon}_{\rm eq} = u_{,x}$) at the centre of each element.
Consider the two element configuration in figure \[fig:two\_elements\].
 \[fig:two\_elements\]
The displacement degrees of freedom are stored at the circular nodes and are denoted $a_{j}$. From the form of the finite element shape functions, the jump in the equivalent strain at element boundary $j$ is given by: $$\left. \frac{1}{h}{\left\llbracket {{\epsilon}^{h}_{\rm eq}} \right\rrbracket} \right|_{j}
= - \frac{a_{j-1} - 2a_{j} + a_{j+1}}{h^{2}}
= - u^{\prime \prime}|_{j}$$ which is equivalent to the second-order finite difference expression for the second derivative of the displacement field $j$. From equation , if the displacement field is known, ${\overline}{{\epsilon}}^{h}$ for an element is equal to: $$\begin{split}
q^{h}{\overline}{{\epsilon}}^{h}
&= q^{h}{\epsilon}^{h}_{\rm eq}
- \left. \frac{c^{2}}{h^{2}} {\left\llbracket {q^{h}} \right\rrbracket}{\left\llbracket {{\epsilon}^{h}_{\rm eq}} \right\rrbracket}
\right|_{j-1}
- \left. \frac{c^{2}}{h^{2}} {\left\llbracket {q^{h}} \right\rrbracket}{\left\llbracket {{\epsilon}^{h}_{\rm eq}} \right\rrbracket}
\right|_{j} \\[1ex]
&= q^{h} {\left( {{\epsilon}^{h}_{\rm eq}
+ \left. \frac{c^{2}}{h^{2}} {\left\llbracket {{\epsilon}^{h}_{\rm eq}} \right\rrbracket} \right|_{j-1}
- \left. \frac{c^{2}}{h^{2}} {\left\llbracket {{\epsilon}^{h}_{\rm eq}} \right\rrbracket} \right|_{j}} \right)}
\end{split}$$ This is equivalent to: $${\overline}{{\epsilon}}^{h} = {\epsilon}^{h}_{\rm eq}
+ \frac{c^{2}}{h} {\left( {\left. u^{\prime \prime} \right|_{j}
- \left. u^{\prime \prime} \right|_{j-1}} \right)}$$ which is a finite difference approximation of equation , showing that the proposed variational formulation is identical to a finite-difference procedure in one-dimension for the case of equally spaced nodal points.
Numerical examples {#sec:examples}
==================
Numerical examples is this section are intended to demonstrate the objectivity of the formulation with respect to mesh refinement for strain softening problems, and to compare the computed results against a known benchmark. It is well-known that classical, rate-independent continuum models are ill-posed when strain softening is introduced, which becomes evident in a severe sensitivity of the computed result to the spatial discretisation. One motivation for strain gradient dependent model is to provide regularisation in the presence of strain softening in order to avoid pathological mesh dependency.
For all examples, the evolution of damage is given by equation . The materials parameters are taken as: Young’s modulus $E = 20 \times 10^{3}$ MPa, $\kappa_{0}=0.0001$, $\kappa_{c} = 0.0125$ and $c=1$ mm. A Newton-Raphson procedure under displacement control is used to solve the problem and the governing equations have been linearised consistently.
Objectivity with respect to spatial discretisation
---------------------------------------------------
The first test is for objectivity of the load–displacement response with respect to mesh refinement. A tapered bar (figure \[fig:tapered\_bar\]) is tested in tension. The bar has a cross-sectional area of one square unit at each end, and tapers linearly towards the centre where the area is 0.8 square units. A displacement is applied incrementally at the right-hand end.
![Linearly tapering bar.[]{data-label="fig:tapered_bar"}](bar_tapered.eps){width="80.00000%"}
The response is examined for meshes with 100, 200 and 400 elements. For each mesh, all elements are of equal size.
![Load-displacement response for the tapered bar.[]{data-label="fig:tapered_pd"}](tapered_pd.eps)
Responses for the three meshes are shown in figure \[fig:tapered\_pd\] for both $c=1$ and $c=0$. Clearly, the introduction of strain gradient effects has regularised the problem, with the response for the three cases with $c=1$ being near identical. The response is further examined by comparing the damage profiles along the bar for the three regularised cases. The damage profiles, shown in figure \[fig:tapered\_dam\], are indistinguishable for the three meshes.
![Damage profiles for the tapered bar.[]{data-label="fig:tapered_dam"}](tapered_dam.eps)
Comparison with a high-order of continuity numerical method
-----------------------------------------------------------
The second test involves a bar with a narrow section at the centre, as shown in figure \[fig:bar\_narrow\]. This problem was previously computed for the same strain gradient dependent damage model using an element-free Galerkin method, which provides a high degree of continuity [@askes:2000b].
![Bar with narrow central section.[]{data-label="fig:bar_narrow"}](bar_narrow.eps){width="80.00000%"}
This problem is computed using meshes with the same number of elements as the previous example. For comparison, the computed load-displacement response from an element-free Galerkin method is also included [@askes:2000b] for this problem. It is clear from figure \[fig:bar\_narrow\_pd\] that the three meshes yield near-identical results and match the element-free Galerkin solution well.
![Load-displacement response.[]{data-label="fig:bar_narrow_pd"}](narrow_pd.eps)
The damage profiles along the bar are shown in figure \[fig:bar\_narrow\_dam\]. The damage profile from @askes:2000b is included as a reference.
![Damage profiles along the bar.[]{data-label="fig:bar_narrow_dam"}](narrow_dam.eps)
The computed results for all meshes are in excellent agreement with the benchmark.
Conclusions
===========
A discontinuous Galerkin formulation has been developed for a strain gradient-dependent continuum model. The problem is split into two fields – the displacement and a deformation measure – for generality. The scalar field, which is a measure of the deformation, is dependent on gradients of the strain field. Conventionally, this would require a $C^{1}$ finite element interpolation of the displacement field. By including element interface terms in the Galerkin formulation, the need for high-order continuity is circumvented.
The proposed formulation was tested for the simplest element configuration in one dimension – piecewise continuous linear displacement and discontinuous piecewise constant for the extra scalar field. For simple tests, the regularising properties of the strain gradient dependent model were demonstrated and the results compared excellently with a benchmark result computed using a numerical method with a high degree of continuity. These preliminary results are promising and should be extended for higher-oder elements and to multiple spatial dimensions. For the simple formulation adopted here, several terms in the weak from could be discarded. The importance of these terms must be assessed for higher-order interpolations. This work provides a first step towards a simple and well-founded finite element framework for modern strain gradient continuum models.
Acknowledgements {#acknowledgements .unnumbered}
================
G.N. Wells was partially supported by the J. Tinsley Oden Faculty Research Program, Institute for Computational Engineering and Sciences, The University of Texas at Austin for this work. The work of K. Garikipati at University of Michigan was supported under NSF grant CMS\#0087019. L. Molari was supported under Progetto Marco Polo, Università di Bologna. The authors are grateful to R.L. Taylor (University of California, Berkeley) for providing his program FEAP with discontinuous Galerkin capabilities.
Linearisation {#append:linearisation}
=============
Effective solution of problems requires the consistent linearisation of the Galerkin problem. For the formulation, the fundamental unknowns are the displacement ${{\bm{}}}{u}^{h}$ and ${\overline}{{\epsilon}}^{h}$. Linearisation requires expressing the problem in terms of increments of the two unknowns. Taking the directional derivative of equation , $$\int_{\Omega} {{\nabla}}{{\bm{}}}{w}^{h} {\mathbf{:}}{\left( {1 - \omega} \right)}\mathcal{C}
{\mathbf{:}}\Delta {{\bm{{\epsilon}}}}{ \ d}\Omega
-\int_{\Omega} {{\nabla}}{{\bm{}}}{w}^{h} {\mathbf{:}}\frac{{\partial}\omega}{{\partial}{\overline}{{\epsilon}}}
\mathcal{C}
{\mathbf{:}}{{\bm{{\epsilon}}}}\Delta {\overline}{{\epsilon}}^{h} { \ d}\Omega
= \int_{\Gamma_{h}} {{\bm{}}}{w}^{h} \cdot \Delta {{\bm{}}}{h} { \ d}\Gamma$$ where $\Delta {\left( {\cdot} \right)}$ indicates a change in ${\left( {\cdot} \right)}$. For brevity $\Delta {\left( {{{\nabla}^{\rm s}}{{\bm{}}}{u}^{h}} \right)}$ is expressed as $\Delta {{\bm{{\epsilon}}}}^{h}$. Since the gradient is a linear operator, $\Delta
{\left( {{{\nabla}^{\rm s}}{{\bm{}}}{u}} \right)} = {{\nabla}^{\rm s}}{\left( {\Delta {{\bm{}}}{u}} \right)}$. Similarly, equation is linearised by taking the directional derivative, $$\begin{gathered}
\int_{\Omega} q^{h} \Delta {\overline}{{\epsilon}}^{h} { \ d}\Omega
-\int_{\Omega} q^{h} \frac{{\partial}{\epsilon}_{\rm eq}}{{\partial}{{\bm{{\epsilon}}}}}
{\mathbf{:}}\Delta {{\bm{{\epsilon}}}}^{h} { \ d}\Omega
+\int_{{\widetilde}{\Omega}} {{\nabla}}q^{h} \cdot c^{2}
{{\nabla}}{\left( {\frac{{\partial}{\epsilon}_{\rm eq}}{{\partial}{{\bm{{\epsilon}}}}}
{\mathbf{:}}\Delta {{\bm{{\epsilon}}}}^{h}} \right)} { \ d}\Omega \\
-\int_{{\widetilde}{\Gamma}} {\left\llbracket {q^{h}} \right\rrbracket} \cdot c^{2}
{\left< { {{\nabla}}{\left( {\frac{{\partial}{\epsilon}_{\rm eq}}{{\partial}{{\bm{{\epsilon}}}}} {\mathbf{:}}\Delta {{\bm{{\epsilon}}}}^{h}} \right)}} \right>} { \ d}\Gamma
-\int_{{\widetilde}{\Gamma}} {\left< {{{\nabla}}q^{h}} \right>} \cdot c^{2}
{\left\llbracket { \frac{{\partial}{\epsilon}_{\rm eq}}{{\partial}{{\bm{{\epsilon}}}}} {\mathbf{:}}\Delta
{{\bm{{\epsilon}}}}^{h}} \right\rrbracket} { \ d}\Gamma \\
+\int_{{\widetilde}{\Gamma}} \frac{c^{2}}{h_{e}} {\left\llbracket {q^{h}} \right\rrbracket} \cdot
{\left\llbracket { \frac{{\partial}{\epsilon}_{\rm eq}}{{\partial}{{\bm{{\epsilon}}}}} {\mathbf{:}}\Delta
{{\bm{{\epsilon}}}}^{h}} \right\rrbracket} { \ d}\Gamma
= \int_{\Gamma} q^{h} \Delta {{\epsilon}_{\rm bc}}{ \ d}\Gamma.
\end{gathered}$$
Finite element formulation {#append:fe}
==========================
The finite element formulation is elaborated here for the case of piecewise continuous linear ${{\bm{}}}{u}^{h}$ and piecewise constant ${\overline}{{\epsilon}}^{h}$. It can be extended to the more general case of arbitrary interpolation orders.
Formulation of the stiffness matrix consists of two keys steps. The first is the usual loop over all elements. This yields a stiffness matrix for each element ${{\bm{}}}{k}_{e}$ of the form $${{\bm{}}}{k}_{e} =
\begin{bmatrix}
{{\bm{}}}{k}_{uu} & {{\bm{}}}{k}_{u{\overline}{{\epsilon}}} \\
{{\bm{}}}{k}_{{\overline}{{\epsilon}}u} & {{\bm{}}}{k}_{{\overline}{{\epsilon}}{\overline}{{\epsilon}}}
\end{bmatrix}$$ where the components of the matrix ${{\bm{}}}{k}_{e}$ are $$\begin{aligned}
{{\bm{}}}{k}_{uu}
&= \int_{\Omega_{e}} {\left( {1-\omega} \right)}{ {{{\bm{}}}{B}}^{\text{T}} } {{\bm{}}}{D} {{\bm{}}}{B}
{ \ d}\Omega \\
{{\bm{}}}{k}_{u{\overline}{{\epsilon}}}
&= - \int_{\Omega_{e}} { {{{\bm{}}}{B}}^{\text{T}} } \frac{{\partial}\omega}{{\partial}{\overline}{{\epsilon}}} {{\bm{}}}{D} {{\bm{{\epsilon}}}}{{\bm{}}}{N}_{{\overline}{{\epsilon}}}
{ \ d}\Omega \\
{{\bm{}}}{k}_{{\overline}{{\epsilon}}u}
&= -\int_{\Omega_{e}} { {{{\bm{}}}{N}}^{\text{T}} }_{{\overline}{{\epsilon}}}
{ {{\left( {\frac{{\partial}{\epsilon}_{\rm eq}}{{\partial}{{\bm{{\epsilon}}}}}} \right)}}^{\text{T}} }
{{\bm{}}}{B} { \ d}\Omega \\
{{\bm{}}}{k}_{{\overline}{{\epsilon}}{\overline}{{\epsilon}}}
&= \int_{\Omega_{e}} { {{{\bm{}}}{N}}^{\text{T}} }_{{\overline}{{\epsilon}}}
{{\bm{}}}{N}_{{\overline}{{\epsilon}}} { \ d}\Omega
\end{aligned}$$ where ${{\bm{}}}{B}$ is the usual finite element matrix containing spatial derivatives of the shape functions related to the displacement field, ${{\bm{}}}{D}$ is the elastic constitutive tensor in matrix form and ${{\bm{}}}{N}_{{\overline}{{\epsilon}}}$ contains the shape functions relating the ${\overline}{{\epsilon}}$. The strain is expressed in engineering column vector format. Once formed, an element element stiffness matrix is assembled into the global system of equations as usual.
The next, non-standard, step is a loop over all element interfaces. For this, ‘information’ is required for both the elements that are connect to the interface. The stiffness matrix at the interface two equal-order elements is twice the size of the stiffness matrix of a single element. It can be expressed as: $${{\bm{}}}{k}_{i} =
\begin{bmatrix}
{{\bm{}}}{k}_{u_{1}u_{1}} & {{\bm{}}}{k}_{u_{1}{\overline}{{\epsilon}}_{1}}
& {{\bm{}}}{k}_{u_{1}u_{2}} & {{\bm{}}}{k}_{u_{1}{\overline}{{\epsilon}}_{2}} \\
{{\bm{}}}{k}_{{\overline}{{\epsilon}}_{1}u_{1}}
& {{\bm{}}}{k}_{{\overline}{{\epsilon}}_{1}{\overline}{{\epsilon}}_{1}} &
{{\bm{}}}{k}_{{\overline}{{\epsilon}}_{1}u_{2}}
& {{\bm{}}}{k}_{{\overline}{{\epsilon}}_{1}{\overline}{{\epsilon}}_{2}} \\
{{\bm{}}}{k}_{u_{2}u_{1}} & {{\bm{}}}{k}_{u_{2}{\overline}{{\epsilon}}_{1}}
& {{\bm{}}}{k}_{u_{2}u_{2}} & {{\bm{}}}{k}_{u_{2}{\overline}{{\epsilon}}_{2}} \\
{{\bm{}}}{k}_{{\overline}{{\epsilon}}_{2}u_{1}}
& {{\bm{}}}{k}_{{\overline}{{\epsilon}}_{2}{\overline}{{\epsilon}}_{1}} &
{{\bm{}}}{k}_{{\overline}{{\epsilon}}_{2}u_{2}}
& {{\bm{}}}{k}_{{\overline}{{\epsilon}}_{2}{\overline}{{\epsilon}}_{2}}
\end{bmatrix}$$ where the subscripts ‘1’ and ‘2’ denote the element on either side of the surface. For the case of linear ${{\bm{}}}{u}^{h}$ and constant ${\overline}{{\epsilon}}$, only the terms ${{\bm{}}}{k}_{{\overline}{{\epsilon}}_{j}u_{k}}$ are non-zero. It is equal to: $${{\bm{}}}{k}_{{\overline}{{\epsilon}}_{j}u_{k}}
= \int_{{\widetilde}{\Gamma}_{i}} \frac{c^{2}}{h_{e}}
{ {{{\bm{}}}{N}}^{\text{T}} }_{{\overline}{{\epsilon}}j} { {{{\bm{}}}{n}}^{\text{T}} }_{j}
{{\bm{}}}{n}_{k}
{ {{\left( {\frac{{\partial}{\epsilon}_{\rm eq}}{{\partial}{{\bm{{\epsilon}}}}}} \right)}}^{\text{T}} }
{{\bm{}}}{B}_{k}
{ \ d}\Gamma$$ where the indices $j$ and $k$ run from one to two, corresponding to sides of the interface. Note that in the usual case of ${{\bm{}}}{n}_{1} = - {{\bm{}}}{n}_{2}$, ${ {{{\bm{}}}{n}}^{\text{T}} }_{i} {{\bm{}}}{n}_{j} = 1$ if $i=j$, and ${ {{{\bm{}}}{n}}^{\text{T}} }_{i} {{\bm{}}}{n}_{j} = -1$ if $i\ne j$.
[^1]: Corresponding author, email: [email protected], fax: +31 15 278 6383.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This is the second paper in a series following [@Tian_Xu], on the construction of a mathematical theory of the gauged linear $\sigma$-model (GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified $r$-spin curve.'
address:
- |
Department of Mathematics\
Princeton University\
Fine Hall, Washington Road\
Princeton, NJ 08544 USA
- |
Department of Mathematics\
University of California, Irvine\
Irvine, CA 92697 USA
author:
- Gang Tian
- Guangbo Xu
bibliography:
- 'symplectic\_ref.bib'
title: 'Correlation functions of gauged linear $\sigma$-model'
---
\[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Conjecture]{}
\[thm\][Definition]{}
\[thm\][Remark]{} \[thm\][Hypothesis]{} \[thm\][Example]{}
Introduction
============
The gauged linear $\sigma$-model (GLSM) was introduced by Witten in [@Witten_LGCY] in physics background towards the understanding of the Landau-Ginzburg/Calabi-Yau correspondence. In the A-model, the close-string Calabi-Yau theory is understood as counting holomorphic curves (Gromov-Witten theory). Motivated from Gromov’s pioneering work [@Gromov_1985] and Witten’s interpretation [@Witten_sigma_model], the foundation of Gromov-Witten theory were built up by [@Ruan_96], [@Ruan_Tian], [@Li_Tian], [@Fukaya_Ono] in the setting of symplectic geometry. Numerous work has appeared and it has become a fundamental tool in symplectic geometry as well as algebraic geometry. On the other hand, the close-string Landau-Ginzburg theory has been constructed just recently, by Fan-Jarvis-Ruan ([@FJR1], [@FJR3], [@FJR2]) following Witten’s idea (see [@Witten_spin]).
The GLSM unifies the two theories under one framework. It has been very influential in physics but is yet to be constructed rigorously in mathematics. This paper is the second input in a series in which we are trying to build a mathematical theory of GLSM following Witten’s proposal in [@Witten_LGCY].
The core in our construction is the analysis of the moduli spaces of the classical equation of motion, which we called the gauged Witten equation. In our first paper [@Tian_Xu], we set the gauged Witten equation under an appropriate framework. Suppose $X_0$ is a noncompact Kähler manifold admitting a holomorphic ${{\mathbb}C}^*$-action, and $Q: X_0 \to {{\mathbb}C}$ is a nondegenerate homogeneous holomorphic function. A typical example of $Q$ is a nondegenerate quintic polynomial on ${{\mathbb}C}^5$. Then the superpotential of the GLSM is $W = pQ : X_0 \times {{\mathbb}C} \to {{\mathbb}C}$, which we call a superpotential of Lagrange multiplier type. Then $W$ is invariant under a ${{\mathbb}C}^*$-action on $X = X_0 \times {{\mathbb}C}$. The triple $(X, W, {{\mathbb}C}^*)$ is the “target space” of GLSM. On the other hand, the domain of the GLSM is a rigidified $r$-spin curve, denoted by $\vec{{\mathcal}C}$, which is a punctured Riemann surface with some additional structures. Then the gauged Witten equation is an elliptic system about a gauge field $A$ and a matter field $u$ over $\vec{{\mathcal}C}$. The details are recalled in Section \[section3\].
In [@Tian_Xu], we also studied several crucial analytical properties of gauged Witten equation and its moduli spaces. Among them, the most crucial one is the compactness of the moduli space of solutions to the perturbed gauged Witten equation over any fixed smooth rigidified $r$-spin curve. The next crucial ingredient is the transversality of the moduli space, which can be stated as
For any strongly regular perturbation $\vec{P}$, and any asymptotic data $\vec\upkappa$, for any homology type $B$ of solutions (see Section \[section3\] for precise meanings), the moduli space ${{\mathcal}M}\left( \vec{{{\mathcal}C}}; B, \vec\upkappa \right)$ of gauge equivalence classes of solutions to the $\vec{P}$-perturbed gauged Witten equation over $\vec{C}$, whose asymptotics are described by $\vec\upkappa$ and whose homology classes are prescribed by $B$, is compact and admits a virtual fundamental class $$\begin{aligned}
\left[ {{\mathcal}M}\left( \vec{{{\mathcal}C}}; B, \vec\upkappa \right) \right]^{vir} \in H_* \left( {{\mathcal}M}\left( \vec{{\mathcal}C}; B, \vec\upkappa \right) ; {{\mathbb}Q} \right).\end{aligned}$$
The virtual fundamental class can be constructed by known techniques, such as the techniques developed in [@Fukaya_Ono] and [@Li_Tian]. It will be done in the incoming paper [@Tian_Xu_3]. We also remark that it is possible to use concrete perturbations of the gauged Witten equation to achieve transversality.
In the scope of the current series, we only consider the moduli space for a fixed smooth $r$-spin curve with a rigidification. We have the associated virtual count $$\begin{aligned}
\label{equation11}
\# {{\mathcal}M}\left( \vec{{\mathcal}C}; B, \vec\upkappa \right) \in {{\mathbb}Q}\end{aligned}$$ which is defined to be zero if the degree of the virtual fundamental cycle is nonzero.
The correlator is defined as a family of multi-linear maps on certain state space ${{\mathscr}H}_Q$ (see Section \[section2\]). ${{\mathscr}H}_Q$ can be viewed as a generalization of both the state space in Landau-Ginzburg A-model and the state space in gauged Gromov-Witten theory. If $\vec{C}$ has $m$ marked points, then certain linear combinations of virtual counts (\[equation11\]) give the correlation function (see Section \[section3\]). $$\begin{aligned}
\left\langle \ \cdot, \cdots, \cdot \ \right\rangle_{\vec{C}}^B: {{\mathscr}H}^{\otimes m}_Q \to {{\mathbb}Q}.\end{aligned}$$ The coefficients of the linear combinations as well as the virtual counts (\[equation11\]) depend on the choice of a strongly regular perturbation. However, we have
The correlation function is independent of the choice of “strongly regular ” perturbations and various other choices.
The proof is basically a cobordism argument. In many similar situation, such as Donaldson theory and Gromov-Witten theory, one can actually prove a direct cobordism of moduli spaces obtained by choosing different auxiliary data, such as the metric or the almost complex structure (cf. [@Donaldson_Kronheimer], [@McDuff_Salamon_2004]). In cases where virtual techniques are used, one can prove a cobordism in the virtual sense (cf. [@Fukaya_Ono], [@Li_Tian], [@Mundet_Tian_Draft]). In the current situation, the moduli spaces for different strongly regular perturbations may not be cobordant directly, but bifurcations (wall-crossings) happen in the interior of the cobordism. Such bifurcation analysis were carried out in many cases, such as [@Floer_intersection] and (closest to our situation) [@FJR3]. Then the difference between the virtual counts (\[equation11\]) for two sets of perturbations is given by a wall-crossing formula (see Theorem \[thm45\]). The coefficients of the linear combinations in defining the correlatino functions also differ by opposite wall-crossing terms, which exactly make the correlation function invariant. The detailed proof of the wall-crossing formula is given in [@Tian_Xu_3].
### Organization of the paper {#organization-of-the-paper .unnumbered}
In Section \[section2\] we define the state spaces in our formulation of GLSM. In Section \[section3\] we define the correlation function, assuming the existence of the virtual cycles. In Section \[section4\] we list the properties of the virtual cycles which are necessary to derive the well-definedness of the correlation functions.
### Acknowledgements {#acknowledgements .unnumbered}
We would like to thank Simons Center for Geometry and Physics for hospitality during our visit in summer 2013. We would like to thank Kentaro Hori, David Morrison, Edward Witten for useful discussions on GLSM. The second author would like to thank Chris Woodward for helpful discussions.
The state space for GLSM {#section2}
========================
The inputs of the correlation function we are going to define are classes (states) in certain cohomology groups (the state space). We have analogues in previously studied theories. In Gromov-Witten theory, the state spaces are the ordinary cohomology groups of symplectic manifolds; in gauged Gromov-Witten theory, they are equivariant cohomologies; in Fan-Jarvis-Ruan’s Landau-Ginzburg A-model (see [@FJR2]), they are certain cohomology groups naturally associated with the singularity.
### Lagrange multipliers {#lagrange-multipliers .unnumbered}
Let $(X_0, \omega, J)$ be a noncompact Kähler manifold. Assume that there is a holomorphic ${{\mathbb}C}^*$-action which restricts to a Hamiltonian $S^1$-action on $X_0$. For every $\upgamma \in S^1$, let $X_{0, \upgamma} \subset X_0$ be the fixed point set of $\upgamma$ and let $N_{0, \upgamma} \to X_{0, \upgamma}$ be the normal bundle. Suppose $Q: X_0 \to {{\mathbb}C}$ is a holomorphic function. We assume that $Q$ is homogeneous of degree $r$. This means that $$\begin{aligned}
\forall \xi \in {{\mathbb}C}^*,\ x\in X_0,\ Q(\xi x) = \xi^r Q(x).\end{aligned}$$ For any $a \in {{\mathbb}C}$, let $Q^a:= Q^{-1}(a) \subset X_0$. Then ${{\mathbb}Z}_r$ acts on $Q^a$. We make the following assumptions on $Q$.
1. $Q$ has a unique critical point $\bigstar \in X_0$ (the critical value must be zero).
2. There exist a constant $c_Q>1$ and $G$-invariant compact subset $K_0 \subset X_0$ such that $$\begin{aligned}
x\notin K_0\Longrightarrow {1\over c_Q} \left| \nabla^3 Q \right| \leq \left| \nabla^2 Q \right| \leq c_Q \left| \nabla Q \right|.\end{aligned}$$ Moreover, for every $\delta>0$, there exists $c_Q(\delta)>0$ such that $$\begin{aligned}
d(x, Q^0)\geq \delta,\ x\notin K_0\Longrightarrow |\nabla Q(x)| \leq c_Q(\delta) |Q(x)|.\end{aligned}$$
3. For every $\upgamma \in {{\mathbb}Z}_r$, it is easy to see that $dQ$ vanishes along the normal bundle $X_{0, \upgamma}$. We assume that $\nabla^2 Q$ vanishes along $N_{0, \upgamma}$.
The above hypothesis was assumed in [@Tian_Xu]. ([**Q2**]{}) is necessary to guarantee the compactness of the moduli space. ([**Q3**]{}) is not essential and can be removed. These conditions are satisfied nondegenerate quasi-homogeneous polynomials on ${{\mathbb}C}^N$.
### GLSM State space {#glsm-state-space .unnumbered}
The state space consists of narrow sectors and broad sectors.
$\upgamma \in {{\mathbb}Z}_r$ is [**broad**]{} (resp. [**narrow**]{}) if the restriction $Q_\upgamma:= Q|_{X_{0, \upgamma}}$ doesn’t (resp. does) vanish identically.
It is easy to prove that $\bigstar\in X_{0, \upgamma}$ for each $\upgamma$ and for broad $\upgamma$, still the unique critical point of $Q_\upgamma$.
From now on we will introduce many homology and cohomology groups. Whenever the coefficient ring is omitted, we mean integral homology or cohomology.
If $\upgamma$ is narrow, then the (reduced) $\upgamma$-sector of the GLSM state space is a 1-dimensional ${{\mathbb}Q}$-vector space, generated by an element $e_\upgamma$. If $\upgamma$ is broad, then the ([*reduced*]{}) $\upgamma$-sector of the GLSM state space is $$\begin{aligned}
{{\mathscr}H}_\upgamma:= H^{n_\upgamma-1} \left( Q_\upgamma^a; {{\mathbb}Q} \right)^{{{\mathbb}Z}_r}.\end{aligned}$$ Here $n_\upgamma = {\rm dim}_{{{\mathbb}C}} X_{0, \upgamma}$ and $a\in {{\mathbb}C}^*$ is an arbitrary regular value of $Q_\upgamma$. The total state space is $$\begin{aligned}
{{\mathscr}H}_Q:= \bigoplus_{\upgamma \in {{\mathbb}Z}_r} {{\mathscr}H}_\upgamma.\end{aligned}$$
Here, to see that the broad sectors are independent of the choice of $a$, we consider the [*monodromy action*]{}, which is a linear isomorphism $$\begin{aligned}
{{\mathfrak}m}: H^* \left( Q_\upgamma^a \right) \to H^* \left( Q_\upgamma^a \right).\end{aligned}$$ This map is defined as a straightforward extension of the monodromy action for isolated singularities. Namely, let $C$ be a simple closed curve in ${{\mathbb}C}$ which passes through $a$ and avoids the origin (the singular value of $Q_\upgamma$). Then $Q_\upgamma^{-1}(C) \to C$ is a locally trivial fibration. This gives the monodromy action ${{\mathfrak}m}$ on the integral homology of $Q_\upgamma^a$, which is independent of the choice of such simple closed curves.
${{\mathfrak}m}$ is equal to the action by the generator of ${{\mathbb}Z}_r$ on $H^*\left( Q^a \right)$.
Consider the path $e^{{\bm i} \theta}$ for $\theta \in [0, {2\pi \over r}]$. Let $a(\theta) = e^{{\bm i} r \theta} a$. Then $e^{{\bm i} \theta}$ induces an isomorphism $$\begin{aligned}
H^*\left( Q^{a(\theta)}; {{\mathbb}Z} \right) \to H^* \left( Q^a; {{\mathbb}Z}\right)\end{aligned}$$ which is equal to the parallel transport along the arc between $a$ and $a(\theta)$. Since $t({2\pi \over r}) = t$, we see that the action by the generator of ${{\mathbb}Z}_r$ is equal to the parallel transport along a loop, which is exactly the monodromy action.
Therefore, we see that the ${{\mathbb}Z}_r$-invariant part of $H^* \left( Q_\upgamma^a \right)$ is independent of the choice of $a\in {{\mathbb}C}^*$. Therefore the state space is well-defined.
There is an enrichment of the state space by including more states for the broad sectors. This enrichment, formally, will be closer to the state space in [@FJR2] and the state space used in gauged Gromov-Witten theory. It will be mentioned at the end of this section.
### Vanishing cycles and Lefschetz thimbles {#vanishing-cycles-and-lefschetz-thimbles .unnumbered}
We need a geometric description of generators of the homology group dual to the state space, i.e., vanishing cycles and Lefschetz thimbles. For each $\upgamma$, we can identify a neighborhood of $\bigstar$ in $X_{0, \upgamma}$ as a neighborhood of $0$ in ${{\mathbb}C}^{n_\upgamma}$. Denote this neighborhood by $B_\epsilon^{n_\upgamma}$. Then choose $\delta<< \epsilon$ and denote $U_\delta \subset {{\mathbb}C}$ the $\delta$-neighborhood of the origin. Then for $t \in \partial U_\delta$, denote $V_t:= B_\epsilon^{n_\upgamma} \cap Q_\upgamma^t$ and denote $V_T:= Q_\upgamma^{-1}(U_\delta) \cap B_\epsilon^{n_\upgamma}$. Then the classical result of Brieskorn [@Brieskorn_1970] says that the relative homology $$\begin{aligned}
H_{n_\upgamma} \left( V_T, V_t\right)\end{aligned}$$ is generated by Lefschetz thimbles.
Moreover, we have
The inclusion $(V_T, V_t) \to \left(X_{0, \upgamma} , Q_\upgamma^t \right)$ induces an isomorphism $$\begin{aligned}
H_{n_\upgamma} \left( V_T, V_t \right) \simeq H_{n_\upgamma} \left( X_{0, \upgamma}, Q_\upgamma^t\right).\end{aligned}$$
Since $\bigstar$ is the only critical point, we see that $Q_\upgamma^{-1}(U_\delta)$ is a deformation retract of $X_{0, \upgamma}$. Therefore it suffices to prove the isomorphism $$\begin{aligned}
H_{n_\upgamma} \left( V_T, V_t \right) \simeq H_{n_\upgamma} \left( Q_\upgamma^{-1}(U_\delta), Q_\upgamma^t \right).\end{aligned}$$ Let $\check{V}_T: Q_\upgamma^{-1}(U_\delta) \setminus V_T$, $\check{V}_t:= Q_\upgamma^t \setminus V_t$. Then because the restriction $Q_\upgamma:\check{V}_T \to U_\delta$ has no critical point, it is a trivial fibration. Therefore for $t \in B_\delta$, we have a trivialization $$\begin{aligned}
\check{V}_T \simeq U_\delta \times \check{V}_t.\end{aligned}$$ Therefore, we have homotopy equivalence $$\begin{aligned}
\left( Q_\upgamma^{-1}(U_\delta), Q_\upgamma^t \right) \sim \left( V_T \cup \check{V}_t, Q^t_\upgamma \right).\end{aligned}$$ Then by excision, we have $$\begin{aligned}
H_{n_\upgamma} \left( Q_\upgamma^{-1}(U_\delta), Q_\upgamma^t \right) \simeq H_{n_\upgamma} \left( V_T \cup \check{V}_t; Q_\upgamma^t \right) \simeq H_{n_\upgamma} \left( V_T, V_t \right).\end{aligned}$$
So we say that the relative homology $H_{n_\upgamma}\left( X_{0, \upgamma}, Q_\upgamma^t \right)$ is generated by Lefschetz thimbles. Indeed by the local triviality of the fibration $X_{0, \upgamma}\setminus \{\bigstar\} \to {{\mathbb}C}^*$, we see that this is true not just for $t$ close to $0$, but all nonzero $t$.
Now for each $a \in {{\mathbb}C}^*$, we have the exact sequence $$\begin{aligned}
H_{n_\upgamma} (X_{0, \upgamma}) \to H_{n_\upgamma} \left(X_{0, \upgamma}, Q_\upgamma^a \right) \to H_{n_\upgamma-1} \left( Q_\upgamma^a \right) \to H_{n_\upgamma-1} (X_{0, \upgamma}). \end{aligned}$$ We make the following simplifying assumption
\[hyp27\] For any broad $\upgamma \in {{\mathbb}Z}_r$, the map $H_{n_\upgamma}\left( X_{0, \upgamma}, Q_\upgamma^a \right) \to H_{n_\upgamma-1} \left( Q_\upgamma^a \right)$ is an isomorphism. In other words, the middle dimensional homology of $Q_\upgamma^a$ is generated by vanishing cycles.
This hypothesis is clearly satisfied by quasihomogeneous polynomials on ${{\mathbb}C}^N$.
### Intersection pairing {#intersection-pairing .unnumbered}
We need a natural perfect pairing between broad states in order to define the correlation function. For $M>>0$ sufficient large, denote $$\begin{aligned}
Q_\upgamma^\infty:= \left( {\rm Re} Q_\upgamma \right)^{-1} \left( [M, +\infty ) \right),\ Q_\upgamma^{-\infty}:= \left( {\rm Re} Q_\upgamma \right)^{-1} \left( (-\infty, -M]\right).\end{aligned}$$ Then we have a perfect pairing $$\begin{aligned}
\label{equation21}
H_{n_\upgamma} \left( X_{0,\upgamma}, Q_\upgamma^{-\infty} \right) \otimes H_{n_\upgamma} \left( X_{0, \upgamma}, Q_\upgamma^{+\infty} \right) \to {{\mathbb}Z}.\end{aligned}$$ This pairing is described in [@FJR2 Page 36] in the case of quasihomogeneous polynomials on ${{\mathbb}C}^N$ but for exactly the same reason we have it for the more general case.
On the other hand, via the parallel transport, we have canonical isomorphisms $$\begin{aligned}
\label{equation22}
H_{n_\upgamma}\left( X_{0, \upgamma}, Q_\upgamma^a \right)^{{{\mathbb}Z}_r} \simeq H_{n_\upgamma} \left( X_{0, \upgamma}, Q_\upgamma^{\pm\infty} \right)^{{{\mathbb}Z}_r}.\end{aligned}$$ Moreover, choose $\xi \in S^1$ such that $\xi^r = -1$. $\xi$ implies an isomorphism $$\begin{aligned}
\label{equation23}
H_{n_\upgamma} \left( X_{0, \upgamma}, Q_\upgamma^\infty \right)^{{{\mathbb}Z}_r} \to H_{n_\upgamma} \left( X_{0, \upgamma}, Q_\upgamma^{-\infty} \right)^{{{\mathbb}Z}_r},\end{aligned}$$ which is independence of the choice of $\xi$ because we have restricted to the monodromy invariant part. Then by Hypothesis \[hyp27\] and (\[equation21\])–(\[equation23\]), we have a perfect pairing $$\begin{aligned}
H_{n_\upgamma-1} \left( Q_\upgamma^a \right)^{{{\mathbb}Z}_r} \otimes H_{n_\upgamma-1} \left( Q_\upgamma^a \right)^{{{\mathbb}Z}_r} \to {{\mathbb}Z}.\end{aligned}$$ Therefore, by the duality between homology and cohomology we have a canonical identification $$\begin{aligned}
\label{equation24}
{{\mathscr}H}_\upgamma \simeq H_{n_\upgamma-1} \left( Q_\upgamma^a; {{\mathbb}Q} \right)^{{{\mathbb}Z}_r}.\end{aligned}$$
### $\infty$-relative cycles in $Q_\upgamma^a$ {#infty-relative-cycles-in-q_upgammaa .unnumbered}
The use of Lagrange multiplier requires us to consider the complex Morse theory of the hypersurfaces $Q_\upgamma^a$. If we have a holomorphic Morse function $F$ defined on $Q_\upgamma^a$, then critical points of $F$ (together with rays emitting from it) represent certain $\infty$-relative cycles. We will use the intersection between compact cycles and $\infty$-relative cycles, which is described as follows. For any compact subset $K \subset X$, we can consider the relative homology $H_* \left( Q_\upgamma^a, Q_\upgamma^a \setminus K\right)$. The inverse limit with respect to the direct system of compact subsets under inclusion is denoted by $$\begin{aligned}
H_* \left( Q_\upgamma^a, \infty \right).\end{aligned}$$ This is the dual space of $H^*_c\left( Q_\upgamma^a \right)$. Then we have the intersection pairing $$\begin{aligned}
\label{equation25}
\cap: H_* \left( Q_\upgamma^a \right) \otimes H_* \left( Q_\upgamma^a, \infty \right) \to {{\mathbb}Z}.\end{aligned}$$
There is certain enrichment of the GLSM state space. Take $X = X_0 \times {{\mathbb}C}$ with an additional $K:= S^1$-action by $$\begin{aligned}
e^{{\bm i} \theta} (x, p) = ( e^{{\bm i} \theta}x, e^{- {\bm i} r \theta} p ).\end{aligned}$$ We take $W: X\to {{\mathbb}C}$ to be $W(x, p) = p Q(x)$, which is invariant under the $K$-action. Then for $\upgamma \in {{\mathbb}Z}_r$ and any $a \in {{\mathbb}C}^*$, denote $W_\upgamma^a = W_\upgamma^{-1}(a)$. The [*enriched*]{} $\upgamma$-sector of GLSM state space is defined by $$\begin{aligned}
{\widetilde}{{\mathscr}H}_\upgamma:= H^*_K \left( X_\upgamma, W_\upgamma^a; {{\mathbb}Q} \right)^{{{\mathbb}Z}_r}.\end{aligned}$$ Here $(X_\upgamma, W_\upgamma^a)$ has the $K$-action and ${{\mathbb}Z}_r$-action commuting with each other. The enriched GLSM state space is defined as $$\begin{aligned}
{\widetilde}{{\mathscr}H}_Q:= \bigoplus_{\upgamma \in {{\mathbb}Z}_r} {\widetilde}{{\mathscr}H}_\upgamma.\end{aligned}$$ We see that formally the enriched state space generalizes the state space used in [@FJR2] (when $K$ is trivial), and the equivariant cohomology used in gauged Gromov-Witten theory (when $W \equiv 0$ and the ${{\mathbb}Z}_r$-action can be ignored) (see [@Cieliebak_Gaio_Mundet_Salamon_2002]). A more comprehensive correlation function can be defined over ${\widetilde}{{\mathscr}H}_Q$.
Definition of the correlation function {#section3}
======================================
In this section we define the correlation function, as a collection of ${{\mathbb}Q}$-valued multi-linear function on ${{\mathscr}H}_Q$. The definition depends on the construction of the virtual fundamental class of the moduli space of solutions to the perturbed gauged Witten equation. The construction will be provided in a separate paper.
Perturbed gauged Witten equation
--------------------------------
We recall the set-up of perturbed Witten equation given in [@Tian_Xu].
Let $({{\mathcal}C}, {{\mathcal}L}, \upvarphi)$ be a smooth $r$-spin curve, with orbifold markings ${\bm z} = (z_1, \ldots, z_m)$. Here ${{\mathcal}C}$ is a smooth orbifold Riemann surface, with possible nontrivial orbifold structures only at $z_1, \ldots, z_m$; ${{\mathcal}L} \to {{\mathcal}C}$ is a holomorphic orbifold line bundle; $\upvarphi$ is an isomorphism of orbifold line bundles $$\begin{aligned}
\upvarphi: {{\mathcal}L}^{\otimes r} \to {{\mathcal}K}_{\log}:= {{\mathcal}K}_{{\mathcal}C} \otimes {{\mathcal}O}(z_1) \otimes \cdots \otimes {{\mathcal}O}(z_m).\end{aligned}$$ $r$-spin structures are labelled by $$\begin{aligned}
\vec{\upgamma}:= \left( \upgamma_1, \ldots, \upgamma_m \right) \in ({{\mathbb}Z}_r)^m.\end{aligned}$$ $\upgamma_i$ is called the monodromy of the $r$-spin structure at the marking $z_i$. The notion of $\upgamma$ being narrow or broad has been defined in Section \[section2\]. A marking $z_i$ is called broad or narrow if its monodromy $\upgamma_i$ is broad or narrow respectively. In the current situation, we only consider a fixed $r$-spin curve, and we assume that the first $b$ markings are broad and the last $n = m- b$ markings are narrow.
Let $\Sigma$ be the smooth Riemann surface underlying ${{\mathcal}C}$. ${\bm z}$ is regarded as punctures on $\Sigma$ and $\Sigma^*:= \Sigma \setminus {\bm z}$. For each marking $z_i$, we fix a local holomorphic coordinate $w$ on $\Sigma$ centered at $z_i$. We assume these coordinate patches are disjoint from each other. A rigidification of the $r$-spin structure at $z_i$ is a choice of an element $e_j \in {{\mathcal}L}|_{z_i}$ such that $$\begin{aligned}
\upvarphi( e_j^{\otimes r}) = {dw \over w} \in {{\mathcal}K}_{\log}|_{z_i}.\end{aligned}$$ We fix rigidifications $\vec{\upphi}= ( \upphi_1, \ldots, \upphi_m )$ at all punctures. Now we fix $$\begin{aligned}
\vec{{\mathcal}C}:= \left( {{\mathcal}C}, {{\mathcal}L}, \upvarphi; \vec{\upphi} \right)\end{aligned}$$ as a rigidified $r$-spin curve, which is the domain of the gauged Witten equation.
The line bundle ${{\mathcal}L}$ descends to an ordinary line bundle $L \to \Sigma^*$. In [@Tian_Xu] we define the notion of adapted Hermitian metrics, which is a class of $W_{loc}^{2, p}$-Hermitian metrics on $L$ compatible with the $r$-spin structure. We choose a smooth adapted metric $H_0$ on $L$, and denote by $Q_0\to \Sigma^*$ the unit circle bundle.
On the other hand, choose another $K= S^1$-principal bundle $Q_1 \to \Sigma$, whose restriction to $\Sigma^*$ is still denoted by $Q_1$. Then in [@Tian_Xu], we considered a space ${{\mathpzc}A}$ of $G= S^1 \times S^1$-connections on $Q:= Q_0 \times_{\Sigma^*} Q_1$. Moreover, ${{\mathpzc}G}$ is the space of gauge transformations $g: \Sigma^* \to G$ of class $W_{loc}^{2, p}$ such that $g$ is asymptotic to the identity in a $W_\delta^{2, p}$-manner for some $\delta>0$ ($\delta$ is allowed to change).
$G= S^1 \times K$ acts on $X$. Denote $Y:= Q \times_G X \to \Sigma^*$ the fibre bundle. The $r$-spin structure induces a family of lifting $$\begin{aligned}
{{\mathcal}W}_A \in \Gamma \left( Y, \pi^* K_{\log} \right),\ A\in {{\mathpzc}A}.\end{aligned}$$ The vertical tangent bundle $T^\bot Y \to Y$ admits a natural Hermitian metric. With respect to this metric, we have the vertical gradient $$\begin{aligned}
\nabla {{\mathcal}W}_A \in \Gamma \left( Y, \pi^* \Omega_\Sigma^{0,1} \otimes T^\bot Y \right).\end{aligned}$$ Choose a biinvariant metric on the Lie algebra ${{\mathfrak}g}$ and an area form $\Omega$ on $\Sigma$. The gauged Witten equation reads $$\begin{aligned}
\label{equation31}
\left\{ \begin{array}{ccc}
{\overline}\partial_A u + \nabla {{\mathcal}W}_A (u) & = & 0,\\
* F_A + \mu(u) & = & 0.
\end{array} \right.\end{aligned}$$
### Perturbations {#perturbations .unnumbered}
Whenever there is broad punctures, the linearization of (\[equation31\]) (modulo gauge transformation) is not a Fredholm operator in a natural way. We have to perturb the equation near broad punctures. A perturbation is described as $$\begin{aligned}
\vec{P} = \left( \vec{a}, \vec{F} \right) = \left( a_i, F_i \right)_{i=1}^b.\end{aligned}$$ Here for each $i$, $a_i \in {{\mathbb}C}^*$ and $F_i: X_0 \to {{\mathbb}C}$ is a holomorphic function. Denote $$\begin{aligned}
W_i:= W - a_i p + F_i.\end{aligned}$$
$F_i$ is [**$\upgamma_i$-admissible**]{} if the following are satisfied.
1. There exist $r_l \leq {1\over 2} r$ ($l = 2, \ldots, s$) such that $$\begin{aligned}
F_i = \sum_{l=2}^s F_i^{(l)}\end{aligned}$$ and $F_i^{(l)}: X_0 \to {{\mathbb}C}$ is a holomorphic function of degree $r_l$.
2. There exists $c_i >0$ such that for $l = 2, \ldots, s$ $$\begin{aligned}
\left| F_i^{(l)} (x) \right| \leq c_i \left( 1 + |\mu_+(x)| \right)^{1\over 2},\ \left| d F_i^{(l)} (x) \right| \leq c_i.\end{aligned}$$
$P_i:= (a_i, F_i)$ is called [**$\upgamma_i$-regular**]{} if $F_i$ is $\upgamma_i$-admissible and
1. for every $\epsilon \in (0, 1]$, the restriction of $W_{i, \epsilon} = pQ - \epsilon^r a_i p + \epsilon^r F_i^\epsilon$ to $X_{\upgamma_i}$ is a holomorphic Morse function. Here $F_i^\epsilon (x) = F_i(\epsilon^{-1} x)$.
2. The perturbed functions $W_{i, \epsilon}$ has no critical points at infinity in the following sense: for every $T>1$, there is a $G$-invariant compact subset $K_T\subset X$ and $\epsilon_T>0$ such that $$\begin{aligned}
\epsilon \in \left[ T^{-1}, 1\right],\ \left| \nabla W_{i, \epsilon} (x, p) \right| \leq \epsilon_T \Longrightarrow (x, p) \in K_T. \end{aligned}$$
$P_i= (a_i, F_i)$ is called [**$\upgamma_i$-strongly regular**]{} if it is $\upgamma_i$-regular and all critical values of the restriction of $W_i$ to $X_{\upgamma_i}$ have distinct imaginary parts.
$\vec{P}= \left( \vec{a}, \vec{F}\right)$ is called regular (resp. strongly regular ) if $(a_i, F_i)$ is $\upgamma_i$-regular (resp. $\upgamma_i$-strongly regular) for each broad puncture $z_i$ whose monodromy is $\upgamma_i$.
\[lemma32\] If $(p, x)$ is a critical point of $W_i|_{X_{\upgamma_i}}$, then for every $\epsilon>0$, $(p, \epsilon x)$ is a critical point of $W_{i, \epsilon}|_{X_{\upgamma_i}}$ and $$\begin{aligned}
W_{i, \epsilon}(p, \epsilon x) = \epsilon^r W_i(p, x).\end{aligned}$$
If $(p, x) \in {\rm Crit} W_i|_{X_{\upgamma_i}}$, then $$\begin{aligned}
Q(x) = a_i,\ p dQ(x) + dF_i (x) = 0.\end{aligned}$$ Then $Q(\epsilon x) = \epsilon^r a_i$. Regard $\epsilon$ as the diffeomorphism of $X_0$ by multiplying $\epsilon$. Then $$\begin{aligned}
\left( \epsilon^* \left( pdQ + \epsilon^r dF_i^\epsilon \right) \right)(x) = \epsilon^r \left( pdQ(x) + dF_i(x) \right) = 0.\end{aligned}$$ Therefore $(p, \epsilon x) \in W_{i, \epsilon} |_{X_{0, \upgamma_i}}$. Moreover, the critical value is $$\begin{aligned}
W_{i, \epsilon}(p, \epsilon x) = \epsilon^r F_i^\epsilon (\epsilon x) = \epsilon^r F_i(x) = \epsilon^r W_i(p, x).\end{aligned}$$
In applications, we may assume that certain class of $\upgamma_i$-admissible perturbations form a finite dimensional, nonzero complex vector space. This is the case when we consider the GLSM for a quasihomogeneous polynomial $Q: {{\mathbb}C}^N \to {{\mathbb}C}$, where the space of $F$’s is the space of all linear functions on ${{\mathbb}C}^N$. So we assume the following conditions.
\[hyp33\] The space of all holomorphic functions $F: X_0 \to {{\mathbb}C}$ satisfying ([**P1**]{}) and ([**P2**]{}) is a finite dimensional nonzero complex vectors pace $V_F$. For fixed $a_i\in {{\mathbb}C}^*$, there is an analytic subset $V_F^{sing}(a_i)\subset V_F$ such that for every $F_i \in V_F \setminus V_F^{sing}(a_i)$, $(a_i, F_i)$ is $\upgamma_i$-regular.
Suppose $P_i:= (a_i, F_i)$ is $\upgamma_i$-regular. Then there is a one-to-one correspondence between critical points of $F_i|_{Q_{\upgamma_i}^{a_i}}$ and critical points of $W_i|_{X_{\upgamma_i}}$. We use both of the two perspectives. A critical point is denoted by $\upkappa_i$. Moreover, critical points of $W_{i, \epsilon}|_{X_{\upgamma_i}}$ exist in smooth families parametrized by $\epsilon \in (0, 1]$. We also use $\upkappa_i$ to denote a family $\upkappa_i(\epsilon)$ of critical points of $W_{i, \epsilon}|_{X_{\upgamma_i}}$. We denote ${\rm Crit} W_{P_i}$ the set of all such families, which is a finite set. Denote $$\begin{aligned}
{\rm Crit} W_{\vec{P}} = \prod_{i=1}^b {\rm Crit} W_{P_i}.\end{aligned}$$ An element of it is denoted by $\vec{\upkappa}= (\upkappa_1, \ldots, \upkappa_b)$.
In this section, we fix a strongly regular perturbation $\vec{P} = (\vec{a}, \vec{F})$. As in [@Tian_Xu], by choosing a cut-off function $\beta: \Sigma^* \to [0,1]$ supported near all broad punctures, we can lift the perturbation to $Y$. The lifting depends on the connection $A$, as well as choices of frames at broad punctures of $Q_1 \to \Sigma$. Denote by $$\begin{aligned}
\vec\uppsi:= \left( \uppsi_1, \ldots, \uppsi_b\right): K^b \to Q_1|_{\{z_1, \ldots, z_b\}}\end{aligned}$$ a choice of frames. The perturbed family of superpotentials is denoted by $$\begin{aligned}
{\widetilde}{{\mathcal}W}_A^{\vec\uppsi}\in \Gamma \left( Y, \pi^* K_{\log} \right).\end{aligned}$$
The perturbed gauged Witten equation is the following one on triples $(A, u, \vec{\uppsi})$ $$\begin{aligned}
\label{equation32}
\left\{ \begin{array}{ccc}
{\overline}\partial_A u + \nabla {\widetilde}{{\mathcal}W}_A^{\vec{\uppsi}}(u) & = & 0;\\
* F_A + \mu (u) & = & 0.
\end{array} \right.\end{aligned}$$ This equation transform naturally under the action of the group of gauge transformations ${{\mathpzc}G}$.
The energy of pairs $(A,u)$ is defined as $$\begin{aligned}
E(A, u) = {1\over 2} \left( \left\| d_A u \right\|_{L^2}^2 + \left\| F_A \right\|_{L^2}^2 + \left\| \mu(u) \right\|_{L^2}^2 \right) + \left\| \nabla {\widetilde}{{\mathcal}W}_A^{\vec{\uppsi}}(u) \right\|_{L^2}^2\end{aligned}$$ where the norms are taken with respect to the metric on $\Sigma^*$ determined by $\Omega$ and the complex structure.
We summarize the main results of [@Tian_Xu] in the following theorem.
1. For any solution $(A, u)$ to (\[equation32\]) with finite energy and $\left| \mu(u) \right|$ bounded on $\Sigma^*$ (such solutions are called bounded solutions), there exists $\upkappa_i \in X_{\upgamma_i}$ such that with respect to certain trivialization of $Y$ near $z_i$, $$\begin{aligned}
\lim_{z \to z_i} u(z) = \upkappa_i.\end{aligned}$$ Moreover, if $\upgamma_i$ is broad, then $\upkappa_i\in {\rm Crit} \left( W_{i, \epsilon}|_{X_{\upgamma_i}} \right)$ for some $\epsilon \in (0, 1]$.
2. Any bounded solution defines a homology class $[A, u] \in \Gamma_X^G:= H_G^2 \left(X; {{\mathbb}Z}[r^{-1}] \right)$. There exists a function $E: \Gamma_X^G \to {{\mathbb}R}_+$ such that for every bounded solution $(A, u)$ to (\[equation32\]) with $[A, u]= B \in \Gamma_X^G$, we have $$\begin{aligned}
E(A, u) \leq E(B).\end{aligned}$$
3. For every $E$, the moduli space of gauge equivalence classes of bounded solutions $(A, u)$ to (\[equation32\]) satisfying $E(A, u) \leq E$ is compact up to degeneration of solitons at broad punctures. In particular, if the perturbation $\vec{P}$ is strongly regular, then the moduli space itself is compact.
Therefore, for any $B \in \Gamma_X^G$ and $\vec\upkappa = (\upkappa_1, \ldots, \upkappa_b) \in {\rm Crit} W_{\vec{P}}$, denote by $$\begin{aligned}
\label{equation33}
{{\mathcal}M}_{\vec{P}}\left( \vec{{\mathcal}C}, B, \vec\upkappa \right)\end{aligned}$$ the moduli space of gauge equivalence classes of solutions to (\[equation32\]) which represent the class $B \in \Gamma_X^G$ and such that for each broad puncture $z_i$, $i = 1, \ldots, b$, the limit of $u$ at $z_i$ belongs to $\upkappa_i$. We say that such solutions have asymptotics [*prescribed*]{} by $\vec\upkappa$. Then in [@Tian_Xu], we proved that ${{\mathcal}M}_{\vec{P}}\left( \vec{{\mathcal}C}, B, \vec\upkappa \right)$ is the zero locus of a Fredholm section of certain Banach space bundle ${{\mathcal}E}$ over some Banach manifold ${{\mathcal}B}$. Moreover, the index of the Fredholm section is given by $$\begin{aligned}
\label{equation34}
\chi\left( \vec{{\mathcal}C}, B \right) = (2-2g) {\rm dim}_{{\mathbb}C} X_0 + 2 \left( c_1^G (B) - \sum_{j=1}^m \Theta_j \right)- \sum_{j=1}^b {\rm dim}_{{\mathbb}C} X_{0, \upgamma_i}.\end{aligned}$$ Here $c_1^G$ is the $G= S^1 \times S^1$-equivariant first Chern class of $X$, $\Theta_j \in {{\mathbb}Q}$ corresponds to certain degree shifting in Chen-Ruan cohomology. We remark that in the case of quasihomogeneous polynomials on ${{\mathbb}C}^N$, the above index coincides with the Fredholm index of the Witten equation calculated in [@FJR3 Section 5].
The correlation function
------------------------
The correlation function we considered is a collection of multi-linear maps $$\begin{aligned}
\label{equation35}
\left\langle \ \cdot, \cdots, \cdot\ \right\rangle_{\vec{{\mathcal}C}}^B: \bigotimes_{i=1}^n {{\mathscr}H}_{\upgamma_i} \to {{\mathbb}Q},\ B \in \Gamma_X^G.\end{aligned}$$ We can extend it trivially to a multi-linear map $$\begin{aligned}
\left\langle \ \cdot, \cdots, \cdot\ \right\rangle_{\vec{{\mathcal}C}}^B: \bigotimes_{i=1}^n {{\mathscr}H}_Q \to {{\mathbb}Q}.\end{aligned}$$ To define (\[equation35\]), we take a strongly regular perturbation $\vec{P}= \left( \vec{a}, \vec{F} \right)$. Consider all possible combinations $\vec{\upkappa} = \left( \upkappa_1, \ldots, \upkappa_b \right) \in {\rm Crit} W_{\vec{P}}$ and the moduli space (\[equation33\]). We claim
[@Tian_Xu_3]\[thm35\] If $\vec{P}$ is strongly regular, then there exists a virtual fundamental class $$\begin{aligned}
\left[ {{\mathcal}M}_{\vec{P}} \left( \vec{{\mathcal}C}, B, \vec{\upkappa} \right) \right]^{vir} \in H_{\chi\left( \vec{{\mathcal}C}, B \right)} \left( {{\mathcal}M}_{\vec{P}} \left( \vec{{\mathcal}C}, B, \vec{\upkappa} \right); {{\mathbb}Q}\right)\end{aligned}$$
So we have the virtual counts $\# {{\mathcal}M}_{\vec{P}} \left( \vec{{\mathcal}C}, B, \vec\upkappa \right)\in {{\mathbb}Q}$, which is zero if $\chi(\vec{{\mathcal}C}, B) \neq 0$. Certain linear combination of the virtual numbers gives the correlation function. The coefficients of the linear combination are described as follows.
Consider the negative gradient flow of the real part of $F_i$ restricted to $Q_{\upgamma_i}^{a_i} \subset X_{0, \upgamma_i}$, whose equilibria are all the $\upkappa_i$’s. Abbreviate $n_i= n_{\upgamma_i}$. Denote by $$\begin{aligned}
\left[ \upkappa_i^- \right] \in H_{n_i -1} \left( Q_{\upgamma_i}^{a_i}, F_i^{-\infty} \right)\ \left({\rm resp.}\ \left[ \upkappa_i^+ \right] \in H_{n_i -1} \left( Q_{\upgamma_i}^{a_i}, F_i^{\infty}\right) \right)\end{aligned}$$ the class of the unstable (resp. stable) manifold of this flow. Here $$\begin{aligned}
F_i^{\infty}= Q_{\upgamma_i}^{a_i} \cap \left( {\rm Re} F_i \right)^{-1} \left( [M, +\infty) \right),\ F_i^{-\infty}= Q_{\upgamma_i}^{a_i} \cap \left( {\rm Re} F_i \right)^{-1} \left( (-\infty, -M] \right)\end{aligned}$$ for some $M>>0$. We still use $\left[ \upkappa_i^\pm \right]$ to denote their images under the map $$\begin{aligned}
H_{n_i - 1} \left( Q_{\upgamma_i}^{a_i}, F_i^{\pm\infty} \right) \to H_{n_i-1} \left( Q_{\upgamma_i}^{a_i}, \infty \right).\end{aligned}$$ To define (\[equation35\]), we choose the last $n$ inputs (narrow states) to be the generators of the corresponding sectors $\theta_i= e_{\upgamma_i} \in {{\mathscr}H}_{\upgamma_i}$, $i = b+1, \ldots, b+n$. Suppose the first $b$ inputs (the broad states) are $\theta_i\in {{\mathscr}H}_{\upgamma_i}$, $i = 1, \ldots, b$. Then define $$\begin{aligned}
\label{equation36}
\left\langle \theta_1, \ldots, \theta_b, \theta_{b+1}, \ldots, \theta_m \right\rangle_{\vec{{\mathcal}C}}^B: = \sum_{\vec{\upkappa}} \# {{\mathcal}M}\left( \vec{{\mathcal}C}, B, \vec\upkappa \right) \prod_{i=1}^b \theta_i^* \cap \left[ \upkappa_i^- \right].\end{aligned}$$ Here $\theta_i^*$ is the image of $\theta_i$ under (\[equation24\]) and the $\cap$ is the intersection mentioned in (\[equation25\]). In general (\[equation35\]) is defined by taking linear extension of the above values.
In the future we would like to define descendant version of the correlation function. For this purpose we have to consider the variation of complex structures of the domain $\vec{{\mathcal}C}$. The moduli space of genus $g$, $m$-marked stable rigidified $r$-spin curve is a branched cover $$\begin{aligned}
{\overline}{{\mathcal}M}_{g, m}^r \to {\overline}{{\mathcal}M}_{g, m}\end{aligned}$$ over the Deligne-Mumford space (see [@FJR2 Section 2]). We can consider the universal moduli space $$\begin{aligned}
{{\mathcal}M}_{g, m}\left( B, \vec{\upkappa} \right)\end{aligned}$$ consists of gauge equivalence classes of solutions to all smooth rigidified $r$-spin curve of genus $g$ and $m$-marked points. We have to prove an extension of the compactness theorem of [@Tian_Xu] in which one allows the complex structure of the domain to vary and degenerate. In particular, when the complex structure degenerates, near the forming node the area form used for the vortex equation is exponentially small (in cylindrical coordinates); then we will be in a situation similar to what is considered in [@Mundet_Tian_2009]. When a broad node is forming, we have to include a strongly regular perturbation nearby as did in [@FJR2]. Nevertheless, we assume the existence of a good compactification of ${{\mathcal}M}_{g, m} \left( B, \vec{\upkappa} \right)$, denoted by ${\overline}{{\mathcal}M}_{g, m} \left( B, \vec{\upkappa} \right)$. We assume that the compactification has a virtual fundamental class $$\begin{aligned}
\left[ {\overline}{{\mathcal}M}_{g, m}(B, \vec\upkappa) \right]^{vir}\end{aligned}$$ whose degree is $6g-6$ more than the index in (\[equation34\]). Then by pulling back cohomology classes of the Deligne-Mumford space via the forgetful map, we can evaluate them against the above virtual fundamental class. So the descendant invariants are defined.
Invariance of the correlation function {#section4}
======================================
In this section we list the properties of the fundamental virtual class given in Theorem \[thm35\] should have, which will imply that the correlation functions are independent of the strongly regular perturbation $\vec{P}$. In the scope of the current series we only have to consider zero or one dimensional moduli spaces, so the properties can be stated in terms of the virtual counts $\# {{\mathcal}M}_{\vec{P}} \left( \vec{{\mathcal}C}, B, \vec\upkappa \right)$.
We briefly describe our argument. Suppose we have two strongly regular perturbation $\vec{P}_1$ and $\vec{P}_2$. It suffices to consider the case that $\vec{P}_1$ and $\vec{P}_2$ only differ at one broad puncture. Therefore we omit the dependence on perturbations at other punctures and suppose at this puncture, the monodromy is $\upgamma \in {{\mathbb}Z}_r$ and the two $\upgamma$-strongly regular perturbations are $P_0 = (a_0, F_0)$ and $(a_1, F_1)$. Then in Subsection \[subsection41\], using a homotopy argument, we show that there is another $\upgamma$-strongly regular perturbation $(a_1, F_0')$ for which the correlation functions defined by $(a_0, F_0)$ and $(a_1, F_0')$ are equal (indeed the corresponding virtual counts are equal). Therefore it remains to consider the case that $P_0$ and $P_1$ only differ in $F$. Then in Subsection \[subsection42\] we show that the correlation functions defined for different $F$’s are equal. This is more complicated than the case considered in Subsection \[subsection41\] because certain wall-crossing may happen during a homotopy of the perturbations.
We remark that both parts of the argument rely on constructing Kuranishi structures (with boundaries) on certain 1-dimensional moduli spaces parametrized by homotopies of the perturbation terms. The details are given in [@Tian_Xu_3].
Independence of $a$ {#subsection41}
-------------------
### Independence of the axial part {#independence-of-the-axial-part .unnumbered}
We consider a $\upgamma$-strongly regular perturbation $P = (a, F)$. For any $\lambda >0$, $P_\lambda := \left( \lambda^r a, \lambda^r F_\lambda \right)$ is also strongly regular (cf. Lemma \[lemma32\]). Moreover, the variation of $\lambda$ gives a homotopy $(P_t)_{t\in [0,1]}$ between $P$ and $P_\lambda$ and each $P_t$ is a $\upgamma$-strongly regular perturbation. Let $\vec{P}_t$ be the path of strongly regular perturbations for which the perturbations at all other broad punctures are fixed. They for each $\vec\upkappa \in {\rm Crit} W_{\vec{P}}$, the homotopy produces a smooth family $\vec\upkappa_t\in {\rm Crit} W_{\vec{P}_t}$. We consider the universal moduli space parametrized by this homotopy, denoted by $$\begin{aligned}
\cup_{t \in [0,1]} {{\mathcal}M}_{\vec{P}_t} \left( \vec{C}, B, \vec\upkappa_t \right).\end{aligned}$$ We can construct a Kuranishi structure with boundary on the above moduli space, where the boundary contributes to the difference of the correlation functions. Since each $\vec{P}_t$ is strongly regular, the oriented boundary is $$\begin{aligned}
{{\mathcal}M}_{\vec{P}_1} \left( \vec{C}, B, \vec\upkappa_1 \right) - {{\mathcal}M}_{\vec{P}_0} \left( \vec{C}, B, \vec\upkappa_0 \right).\end{aligned}$$ So the correlation functions defined by $\vec{P}_0$ and $\vec{P}_1$ are equal.
### Independence of the angular part {#independence-of-the-angular-part .unnumbered}
Now suppose $P = (a, F)$ is $\upgamma$-strongly regular. Then $$\begin{aligned}
P' = \left( e^{{\bm i} \alpha} a, F \circ e^{-{{\bm i} \alpha\over r}} \right)\end{aligned}$$ is also $\upgamma$-strongly regular. Let $\vec{P}$ and $\vec{P}'$ be the two strongly regular perturbations we want to compare, which coincide for every other broad puncture except for $P$ and $P'$. Then for each $\vec\upkappa\in {\rm Crit} W_{\vec{P}}$, there is a corresponding $\vec\upkappa' \in {\rm Crit} W_{\vec{P}'}$.
Choose a smooth gauge transformation $g^\alpha: \Sigma^* \to S^1 \subset G$ which is equal to $e^{{\bm i}{\alpha \over r}}$ near $z_1$ and equal to the identity away from a neighborhood of $z_1$. It is easy to see
For each $B \in \Gamma_X^G$ and each $\vec\upkappa\in {\rm Crti} W_{\vec{P}}$, the map $(A, u, \vec\uppsi)\mapsto ( ( g^\alpha)^* A, (g^\alpha)^* u, \uppsi)$ induces an orientation-preserving homeomorphism $$\begin{aligned}
{{\mathcal}M}_{\vec{P}} \left( \vec{{\mathcal}C}, B, \vec\upkappa \right)\to {{\mathcal}M}_{\vec{P}'}\left( \vec{{\mathcal}C}, B, \vec\upkappa' \right).\end{aligned}$$
Therefore, the corresponding virtual counts are equal. Moreover, $e^{{{\bm i} \alpha \over r}}$ induces a biholomorphism $Q_{\upgamma}^a\simeq Q_\upgamma^{e^{{\bm i} \alpha}a}$. The induced isomorphism $$\begin{aligned}
{\widetilde}{H}_* \left( Q_\upgamma^a \right) \simeq {\widetilde}{H}_* \left( Q_\upgamma^{e^{{\bm i} \alpha} a} \right)\end{aligned}$$ is compatible with the isomorphisms (\[equation24\]) for $a$ and $e^{{\bm i} \alpha} a$. $e^{{\bm i}\alpha \over r}$ also induces a one-to-one correspondence between the $\infty$-relative cycles in $Q_\upgamma^a$ and $Q_\upgamma^{e^{{\bm i} \alpha} a}$. Therefore, the coefficients in the linear combinations defining the correlation function are invariant. Therefore, the correlation functions defined for $\vec{P}$ and $\vec{P}'$ are equal.
Independent of the choice of strongly regular $F$ {#subsection42}
-------------------------------------------------
Now we need to compare two strongly regular perturbations which only differ at one broad puncture as $(a, F^0)$ vs. $(a, F^1)$. A generic homotopy $F^t$ connecting $F^0$ and $F^1$ may not be always strongly regular. For certain values of $t$ where the strong regularity is lost, wall-crossing happens. We first discuss the wall-crossing phenomenon in a general case.
### BPS solitons and intersection of vanishing cycles {#bps-solitons-and-intersection-of-vanishing-cycles .unnumbered}
Let $M$ be a noncompact Kähler manifold of complex dimension $m$ and $F: M \to {{\mathbb}C}$ be a holomorphic Morse function, which has finitely many critical points, listed as $\upkappa_1, \ldots, \upkappa_s$. If ${\rm Im} F(\upkappa_i)$ are distinct, we say that $F$ is strongly regular. In this case the unstable manifold of $\upkappa_i$ under the negative gradient flow of ${\rm Re} F$ defines a relative cycle $$\begin{aligned}
\left[ \upkappa_i^- \right] \in H_m \left( M, F^{-\infty} \right).\end{aligned}$$ More generally, if we have a path $\gamma$ connecting $F(\upkappa_i)$ with a regular value $a$ of $F$ such that the path avoids singular values except $F(\upkappa_i)$, then there is a well-defined vanishing cycle $$\begin{aligned}
\partial \left[ \upkappa_i^\gamma \right] \in H_{m-1} \left( F^a \right),\end{aligned}$$ which only depends on the homotopy class of such paths.
Now suppose we have a homotopy $F^\upnu$, $\upnu\in [0,1]$ between two strongly regular holomorphic Morse functions $F^0, F^1$ such that $F^\upnu$ is a holomorphic Morse function for every $\upnu$. Then there are continuous curves $\upkappa_i^\upnu\in M$ such that $$\begin{aligned}
\left\{ \upkappa_i^\upnu\ |\ i = 1, \ldots, s \right\} = {\rm Crit} F.\end{aligned}$$ On the other hand, there are canonical identifications $$\begin{aligned}
H_* \left( M, \left( F^0 \right)^{-\infty} \right) \simeq H_* \left( M, \left( F^1 \right)^{-\infty} \right). $$ This is because the critical values of $F^\upnu$ are uniformly bounded. We denote the space in common as $H_* \left( M, F^{-\infty} \right)$. We would like to compare $\left[ \left( \upkappa_i^0 \right)^-\right]$ with $\left[ \left( \upkappa_i^1\right)^- \right]$ as elements of $H_* \left( M, F^{-\infty} \right)$.
Suppose $F^0$ and $F^1$ are strongly regular holomorphic Morse functions on $M$ which are in the same path-connected components of the space of holomorphic Morse functions. A homotopy $F^\upnu$ ($\upnu \in [0,1]$) in the space of holomorphic Morse functions is called [**strongly regular**]{} if there exists $\upnu_1, \ldots, \upnu_k \in (0,1)$ such that
1. $F^\upnu$ is strongly regular for $\upnu \in [0,1]\setminus \{\upnu_1, \ldots, \upnu_k \}$.
2. For each $j \in \{1, \ldots, k\}$, $$\begin{aligned}
\# \left\{ {\rm Im} F^{\upnu_j} ( \upkappa_i^{\upnu_j})\ |\ i =1, \ldots, s \right\} = s-1,\end{aligned}$$ and there exist $i_j^-, i_j^+\in \{1, \ldots, s\}$, $\delta>0$ such that $$\begin{aligned}
\forall \upnu \in ( \upnu_j- \delta, \upnu_j +\delta),\ {\rm Re} F^\upnu \left( \upkappa_{i_j^-}^\upnu \right) < {\rm Re} F^\upnu \left( \upkappa_{i_j^+}^\upnu \right), \end{aligned}$$ and for $\upnu \in (\upnu_j - \delta, \upnu_j)$ and $\upnu \in (\upnu_j, \upnu_j + \delta)$, ${\rm Im} F^\upnu \left( \upkappa_{i_j^-}^\upnu \right) - {\rm Im} F^\upnu \left( \upkappa_{i_j^+}^\upnu \right)$ are of different signs.
Each $\upnu_j$ is called a [**crossing**]{} in this homotopy and we say that this crossing [*happens*]{} between $i_j^-$ and $i_j^+$. We say the crossing is positive (resp. negative), denoted by ${\rm sign} \upnu_j = 1$ (resp. ${\rm sign} \upnu_j = -1$), if the argument of $F^\upnu \left( \upkappa_{i_j^+}^\upnu \right) - F^\upnu\left( \upkappa_{i_j^-}^\upnu \right)$ rotates in the counterclockwise (resp. clockwise) direction as $\upnu$ moves from $\upnu_j - \delta$ to $\upnu_j + \delta$.
It is easy to see that we can obtain a strongly regular homotopy by perturbation. Then to compare $\left[ \left( \upkappa_i^0 \right)^-\right]$ with $\left[ \left( \upkappa_i^1\right)^- \right]$, it suffices to consider the case that there is only one crossing at $\upnu = {1\over 2}$ in the homotopy (in the case of zero crossing, the two relative cycles are equal). In this case, we use ${\widetilde}{F}$ to denote the homotopy $\{F^\upnu\}$ and use $(-1)^{{\widetilde}{F}}$ to denote the sign of the only crossing. Suppose $i^-$ and $i^+$ are the two indices between which the crossing happens. Then we have the following Picard-Lefschetz formula (see [@Singularity_I Chapter 2]).
\[thm43\]\[Picard-Lefschetz\] For each $i \in \{1, \ldots, s\}$, we have $$\begin{aligned}
\label{equation41}
\left[ \left( \upkappa_i^1 \right)^- \right] - \left[ \left( \upkappa_i^0 \right)^- \right] = \left( -1 \right)^{{\widetilde}{F}} \delta_{i, i^+} \left\langle \partial \left[ \upkappa_{i^+}^{\gamma_-} \right], \partial \left[ \upkappa_{i^-}^{\gamma_+} \right] \right\rangle_a \left[ \left( \upkappa_{i^-}\right)^-\right]\end{aligned}$$ Here $a$ is the mid point of $F^{1\over 2} \left( \upkappa_{i^-} \right)$ and $F^{1\over 2} \left( \upkappa_{i^+} \right)$, $\gamma_+$ (resp. $\gamma_-$) is the straight path connecting $F^{1\over 2} \left( \upkappa_{i^-} \right)$ (resp. $F^{1\over 2} \left( \upkappa_{i^+} \right)$) to $a$, and $\langle \cdot, \cdot \rangle_a$ means the intersection pairing in $F^a$.
The intersection number appeared in the Picard-Lefschetz formula can be intepreted as the number of BPS solitons. A BPS soliton is a nonconstant, finite energy solution $x: {{\mathbb}R} \to M$ to the ODE $$\begin{aligned}
x'(t) + \nabla F(x(t)) = 0.\end{aligned}$$ Here $\nabla F$ is the gradient of the real part of $F$. Then $\upkappa_\pm:= \displaystyle \lim_{t \to \pm\infty} x(t)$ are necessarily critical points of $F$ and $$\begin{aligned}
\label{equation42}
{\rm Im} F(\upkappa_+) = {\rm Im} F(\upkappa_-),\ {\rm Re} F(\upkappa_+) < {\rm Re} F(\upkappa_-).\end{aligned}$$ We identify two BPS solitons if they differ by a time translation. Then if (\[equation42\]) is satisfied and other critical values of $F$ have different imaginary part, then the number of BPS solitons between $\upkappa_-$ and $\upkappa_+$ is finite and is equal to the intersection number appeared in (\[equation41\]).
### Wall-crossing formula for the virtual counts {#wall-crossing-formula-for-the-virtual-counts .unnumbered}
Now we consider two strongly regular perturbations $\vec{P}^\pm = \left( \vec{a}, \vec{F}^\pm \right)$ where $\vec{F}^\pm$ only differ at one broad puncture $z_1$ as $F^-_1$ and $F^+_1$, whose monodromy is denoted by $\upgamma_1$. We consider smooth homotopies which connect $F^-_1$ and $F^+_1$. By Hypothesis \[hyp33\], the space of $F$ for which $(a_1, F_1)$ is $\upgamma_1$-regular is path-connected. Therefore we can find a path ${\widetilde}{F}_1 = \left\{F_1^\upnu\right\}_{\upnu \in [-1, 1]}$ such that for each $\upnu\in [-1, 1]$, $(a_1, F_1^\upnu)$ is $\upgamma$-regular. Moreover, it suffices to consider the case that $\left( a_1, F_1^\upnu \right)$ is $\upgamma_1$-strongly regular for all $\upnu$ except for $\upnu= 0$. Such a homotopy induces a homotopy $\vec{P}^\upnu$, and families $\vec\upkappa^\upnu\in {\rm Crit} W_{\vec{P}^\upnu}$.
If $(a_1, F^0_1)$ is strongly regular (i.e., there is no crossing), then $$\begin{aligned}
\# {{\mathcal}M}_{\vec{P}^-} \left( \vec{C}, B, \vec\upkappa^{-1} \right) = \# {{\mathcal}M}_{\vec{P}^+} \left( \vec{C}, B, \vec\upkappa^{+1} \right).\end{aligned}$$
The proof is a similar homotopy argument as used in Subsection \[subsection41\]. Note that the $\infty$-relative cycles persist under the homotopy in this case, hence have same intersection numbers with any broad states. Therefore in the case of no crossing, the correlation functions defined on the two sides of the homotopy are equal.
Now we consider the case that a crossing happens at $\upnu = 0$, between $ \left( \upkappa_1^\upnu\right)_-$ and $\left( \upkappa_1^\upnu\right)_+$, with $$\begin{aligned}
{\rm Re} F^\upnu \left( \upkappa_-^\upnu \right) < {\rm Re} F^\upnu \left( \upkappa_+^\upnu \right),\ \forall \upnu \in [-1,1].\end{aligned}$$ Then for each $\vec\upkappa^\upnu \in {\rm Crit} W_{\vec{P}^\upnu}$, we denote by $\vec\upkappa_\pm^\upnu \in {\rm Crit} W_{\vec{P}^\upnu}$ the asymptotic data obtained by replacing $\upkappa_1^\upnu$ by $\left(\upkappa_1^\upnu \right)_\pm$. Then we have the following wall-crossing formula.
\[thm45\] For any family $\vec\upkappa^\upnu \in {\rm Crit} W_{\vec{P}^\upnu}$, we have $$\begin{gathered}
\# {{\mathcal}M}_{\vec{P}^+} \left( \vec{{\mathcal}C}, B, \vec\upkappa^{+1} \right) - \# {{\mathcal}M}_{\vec{P}^-} \left( \vec{{\mathcal}C}, B, \vec\upkappa^{-1} \right)\\
= - (-1)^{{\widetilde}{F}_1} \delta_{\vec\upkappa, \vec\upkappa_-} \cdot \#_{BPS}\left( \left( \upkappa_1^0 \right)_-, \left( \upkappa_1^0 \right)_+ \right)\cdot \# {{\mathcal}M}_{\vec{P}^-} \left( \vec{{\mathcal}C}, B, \vec\upkappa_+ \right).\end{gathered}$$ Here $\delta_{\vec\upkappa^\upnu, \vec\upkappa_-^\upnu}$ is the Kronecker delta, and $\#_{BPS}\left( \left( \upkappa_1^0 \right)_-, \left( \upkappa_1^0 \right)_+ \right)$ is the (algebraic) counts of the number of BPS solitons in $Q_{\upgamma_1}^{a_1}$ for the function $F_1^0$ between the two critical points.
The proof uses a cobordism argument and the details are given in [@Tian_Xu_3]. In LG A-model, a similar wall-crossing formula was proved in [@FJR3 Theorem 6.16]. We consider the universal moduli space $$\begin{aligned}
{{\mathcal}N}:= \bigcup_{\upnu \in [-1,1]} {{\mathcal}M}_{\vec{P}^\upnu} \left( \vec{{\mathcal}C}, B, \vec\upkappa^\upnu \right).\end{aligned}$$ This space is not compact due to degeneration of solitons at the slice of $\upnu = 0$. The soliton appeared are connecting $\left( \upkappa_1^0 \right)_-$ and $\left( \upkappa_1^0 \right)_+$ and stable solutions with BPS solitons exist in a codimension 1 subset and stable solutions with non-BPS solitons exist in a higher codimensional subset. Therefore, in the virtual sense, $$\begin{aligned}
\partial {{\mathcal}N} \simeq \left( \bigcup_{\upnu = -1,1}{{\mathcal}M}_{\vec{P}^\upnu} \left( \vec{{\mathcal}C}, B, \vec\upkappa^\upnu \right)\right) \cup \left( {{\mathcal}M}_{\vec{P}^0} \left( \vec{{\mathcal}C}, B, \left( \vec\upkappa^0 \right)_+ \right)\times {{\mathcal}M}_{BPS} \left( \left( \upkappa_1^0 \right)_-, \left( \upkappa_1^0 \right)_+ \right) \right).\end{aligned}$$ Here ${{\mathcal}M}_{BPS}$ is the moduli of BPS solitons. Taking care of the orientation of the boundary, Theorem \[thm45\] can be proved.
One difference between the proof of Theorem \[thm45\] is that the BPS soliton used to compactify ${{\mathcal}N}$ are solutions $x: {{\mathbb}R} \to X_{\upgamma_1}$ to the equation $$\begin{aligned}
\label{equation43}
x'(s) + \nabla W_1 = 0\end{aligned}$$ but not for maps into $Q^{a_1}_{\upgamma_1}$. However, if the function $F_1$ is small, then solutions to (\[equation43\]) are geometrically very close to solutions to $$\begin{aligned}
y'(s) + \nabla \left( F_1|_{Q_{\upgamma_1}^{a_1}} \right) = 0.\end{aligned}$$ (See [@Lagrange_multiplier] for detailed treatment about the adiabatic limit of gradient flows in real Morse theory). Therefore the algebraic counting of BPS solitons will be the same as the intersection number between cycles in $Q^{a_1}_{\upgamma_1}$.
Therefore, to compare the correlation functions defined for $\vec{P}^+$ and $\vec{P}^-$, we see that the wall-crossing term appeared in the change of the virtual counts given in Theorem \[thm45\] and the wall-crossing term appeared in the change of the $\infty$-relative cycles given in Theorem \[thm43\] cancel each other. Similar situation happens for the LG A-model correlation function (see [@FJR3]). Therefore the correlation functions on the two sides of the homotopy are equal.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Reading comprehension QA tasks have seen a recent surge in popularity, yet most works have focused on fact-finding extractive QA. We instead focus on a more challenging multi-hop generative task (NarrativeQA), which requires the model to reason, gather, and synthesize disjoint pieces of information within the context to generate an answer. This type of multi-step reasoning also often requires understanding implicit relations, which humans resolve via external, background commonsense knowledge. We first present a strong generative baseline that uses a multi-attention mechanism to perform multiple hops of reasoning and a pointer-generator decoder to synthesize the answer. This model performs substantially better than previous generative models, and is competitive with current state-of-the-art span prediction models. We next introduce a novel system for selecting grounded multi-hop relational commonsense information from ConceptNet via a pointwise mutual information and term-frequency based scoring function. Finally, we effectively use this extracted commonsense information to fill in gaps of reasoning between context hops, using a selectively-gated attention mechanism. This boosts the model’s performance significantly (also verified via human evaluation), establishing a new state-of-the-art for the task. We also show promising initial results of the generalizability of our background knowledge enhancements by demonstrating some improvement on QAngaroo-WikiHop, another multi-hop reasoning dataset.'
author:
- |
Lisa Bauer$^*$ Yicheng Wang$^*$ Mohit Bansal\
UNC Chapel Hill\
[{lbauer6, yicheng, mbansal}@cs.unc.edu]{}\
bibliography:
- 'emnlp.bib'
title: 'Commonsense for Generative Multi-Hop Question Answering Tasks'
---
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the reviewers for their helpful comments. This work was supported by DARPA (YFA17-D17AP00022), Google Faculty Research Award, Bloomberg Data Science Research Grant, and NVidia GPU awards. The views contained in this article are those of the authors and not of the funding agency.
Supplemental Material
=====================
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
P. Düben$^{a}$, D. Homeier$^{a}$, K. Jansen$^{b}$, D. Mesterhazy$^{c}$ and $^{\, a}$\
$^{a}$ Westfälische Wilhelms-Universität, Intitut für Theoretische Physik\
Wilhelm-Klemm-Str. 9, 48149 Münster, Germany\
$^{b}$ NIC, DESY Zeuthen\
Platanenallee 6, 15738 Zeuthen, Germany\
$^{c}$ Humboldt-Universität zu Berlin, Institut für Physik\
Newtonstrasse 15, 12489 Berlin, Germany\
E-mail:
title: 'Monte Carlo approach to turbulence[^1]'
---
Introduction
============
Besides tremendous research having been done since Kolmogorov’s famous publication in 1941 [@1], hydrodynamic turbulence essentially remains an unsolved problem of modern physics. This is especially remarkable as the fundamentals seem to be fairly easy – the Navier-Stokes equations for the velocity field $u_{\alpha}$ and pressure $p$ $$\partial_{t}u_{\alpha} + u_{\beta} \partial_{\beta} u_{\alpha}
- \nu \nabla^{2} u_{\alpha}
+ \frac{1}{\rho}\partial_{\alpha}p = 0$$ with the additional constraint $$\partial_{\alpha}u_{\alpha} = 0$$ simply express the conservation of momentum in a classical, incompressible fluid of viscosity $\nu$ and density $\rho$. For laminar flows it is well known that the Navier-Stokes equations reproduce realistic flows very accurately; in the turbulent regime, it is still an open question how the universal characteristics of turbulent flow, characterized by the scaling exponents $\xi_p$ of structure functions $S_p$ of order $p$, defined by $$S_p(x) := \overline{ | u(r+x) - u(r) |^{p} } \sim |x|^{\xi_p},
\label{dh:sp}$$ can be extracted from first principles. Here the bar corresponds to a spatial averaging.
Monte Carlo simulations in the path integral formulation enable us to gain direct insight into the formation of localized structures and their behavior, and to measure observables as, e.g. structure functions and their scaling exponents [@2; @3; @4].
Burgers’ Equation
=================
We decided to elaborate the methods using the stochastically forced Burgers equation [@5] in 1+1 dimensions $$\partial_{t}u + u \partial_xu - \nu \partial_x^{2} u = f\,,
\label{dh:eqburg}$$ which may be interpreted as the flow equation for a fully compressible fluid. The stochastic force is modeled to be Gaussian with correlation $$\label{dh:eqchi}
\chi(x,t; x',t') := \big\langle f(x, t) f(x', t') \big\rangle
= \epsilon \, \delta(t-t') \exp\left(-\frac{|x-x'|}{\Lambda}\right),$$ where $\Lambda$ defines the correlation length of the forcing and the $\langle\, \cdots \,\rangle$ denotes the ensemble average. A finite viscosity $\nu$ and energy dissipation $\epsilon$ provide a dissipation length scale $\lambda$ corresponding to the Kolmogorov-scale in Navier-Stokes turbulence: $$\lambda := \left( \frac{\nu^{3}}{\epsilon}\right)^{\frac{1}{4}}.$$ We can furthermore identify the Reynolds-number as $$\mathit{Re} := (\epsilon \Lambda^4/\nu^3)^{1/3}.$$
The fundamental solutions to the Burgers equation are well-known – in the limit of vanishing viscosity (Hopf-equation) these form singular shocks. A finite dissipation scale $\lambda \sim \nu / U$, where $U$ is the characteristic velocity, provides an UV-regularization of the shock structures: $$u = -U \tanh\frac{U}{2\nu}x\,.$$ Most interestingly, the exponents $\xi_p$ as defined in (\[dh:sp\]) are non-trivial for Burgers turbulence; for the forcing (\[dh:eqchi\]) and in the regime $x \sim \lambda$ we have the analytic result [@6] $$\xi_p = \text{min} (1, p).$$
Path Integral Formulation
=========================
Following the method of Martin, Siggia and Rose [@7], we established a path integral for Burgers’ equation $$Z \propto \int \mathcal{D}u \, \exp\bigg(-\frac{1}{2}\int dt dx \,
\big(\partial_{t} u+ u \partial_x u
- \nu \partial_x^{2} u \big) \chi^{-1} \ast
\big(\partial_{t} u+ u\partial_x u
- \nu \partial_x^{2} u\big) \bigg),$$ where $\ast$ denotes the convolution.
It has been shown by Falkovich et al. [@Falkovich:1995fa; @Balkovsky:1997zz] on the basis of an equivalent sum of states that the fundamental solutions of Burgulence can be understood as instantons.
Monte Carlo Simulations
=======================
For 1+1 dimensional Burgulence, a large number of stable simulations could be performed; we are working on the final analysis. Typical lattice sizes range from $(N_x=16) \times (N_t=16)$ up to $(N_x = 4096) \times (N_t =
128)$ lattice points.
{width=".7\textwidth"}
Boundary Conditions
-------------------
To be in general agreement with literature and analytic calculations, we started with lattices periodic both in time and space direction. In an attempt to reduce autocorrelation times, we dropped these boundary conditions. While autocorrelation times did not change much, simulating with free boundaries effectively doubles the spatial lattice extent and gives access to excitations of the Burgers vacuum state.
Lattice Discretization
----------------------
Once having discretized the path integral on a Euclidean lattice of spacings $\Delta x$ and $\Delta t$, we get for $\nu$: $$\nu = \alpha \frac{(\Delta x)^2}{\Delta t}.
\label{dh:eqnu}$$ The continuum limit of our lattice theory is reached by holding $\nu$ and $\mathit{Re}$ constant while increasing the number of lattice sites. $\alpha$ is an a priori arbitrary constant that can be interpreted as a measure for $\Delta t$ depending on $\Delta x$ and also has to be kept fixed while performing the continuum limit.
Algorithms
----------
We use a local heat bath algorithm with successive over-relaxation (SOR) for the Monte Carlo evaluation of the partition function [@Adler:1981sn]. The use of certain acceleration techniques with SOR, specifically Chebyshev acceleration [@Varga:1962], significantly accelerates the thermalization process.
Though suiting our purposes so far, it poses certain restrictions on parallelization. We therefore started employing a Hybrid Monte Carlo algorithm that we expect to scale better with the number of parallel processors.
Autocorrelation Times
---------------------
With $\chi$ being a nonlocal operator one would expect long autocorrelation times in the simulation of stochastically forced differential equations. However, with the over-relaxed heat bath algorithm and an appropriate definition of structure functions on the lattice (where the reference point for evaluation is chosen randomly for each configuration) the integrated autocorrelation time is reduced to $\tau \sim O(1)$.
Resources
---------
For testing purposes small lattices may easily be simulated on desktop PCs. However, high resolution simulations on large lattices require massively parallel architectures. We have run our simulations on the IBM p690 cluster JUMP at FZ Jülich and on the Linux cluster at Humboldt University Berlin with up to 256 processors in parallel. In July 2009 we continued our simulations on the new supercomputer JUROPA at FZ Jülich.
First Results
=============
First results include further constraints that have to be imposed in order to ensure stable numerics. Most constricting is the need to resolve the Kolmogorov-length scale $\lambda$ on the lattice. We can in this way show the effect of $\lambda$ as UV-regularization of the otherwise singular shocks. This translates into a relation for the Reynolds-number:
$$\mathit{Re} < \frac{\Lambda}{\Delta x} .$$
This will become crucial for Navier-Stokes turbulence enforcing us to simulate big lattices.
Structure Functions
-------------------
From analytic calculations [@6] we have $$S_p(x) \sim C_p |x|^{p} + C_{p}' |x|,$$ for small seperations in the inertial range.
Though our results are in general agreement with this, the extraction of scaling exponents is far from trivial and very sensitive to statistical errors.
{width=".7\textwidth"}
Extended Self-Similarity (ESS)
------------------------------
Rather than measuring the scaling exponents $\xi_p$ directly, there have been attempts to measure the scaling behavior of ratios of structure functions [@9]. It was shown that this greatly enhances the inertial range not only at high but also moderate Reynolds numbers. However, we must stress that up to now it is not clear if there are any systematic effects in the evaluation of the structure function exponents via ESS.
![$\log[S_3(x)]$ as function of $\log[S_1(x)]$ clearly showing the linear ESS-dependence.](ess3.eps){width=".7\textwidth"}
Outlook
-------
After completing the analysis of 1+1 dimensional Burgulence, we will proceed to 3+1 dimensions. The ultimate challenge will be the simulation and analysis of 2+1 and 3+1 dimensional Navier-Stokes turbulence.
Acknowledgements
----------------
We thank the John von Neumann-Institute for Computing (NIC), Jülich, for computing time and support. D. M. thanks the IRZ Physik at HU Berlin for computing time spent on the local Linux cluster and their staff for technical support.
[99]{}
A. N. Kolmogorov, [*The local structure of turbulence in incompressible viscous fluid for very large Reynolds number*]{}, .
P. Düben, D. Homeier, K. Jansen, D. Mesterhazy, G. Münster, C. Urbach, [*Monte Carlo simulations of the randomly forced Burgers equation*]{}, .
D. Mesterhazy, [*Investigations of Burgers Equation in the Path Integral Formalism*]{} (Bachelor thesis, 2008).
P. Düben, [*Numerische Anwendungen des Pfadintegralformalismus in hydrodynamischer Turbulenz*]{} [(Diploma thesis, 2009)](http://pauli.uni-muenster.de/tp/fileadmin/Arbeiten/dueben.pdf).
J. M. Burgers, [*The nonlinear Diffusion Equation*]{} (Reidel, Dordrecht, 1974).
J. Bec, K. Khanin, [*Burgers Turbulence*]{}, .
P. C. Martin, E. D. Siggia, H. A. Rose, [*Statistical Dynamics of Classical Systems*]{}, .
G. Falkovich, I. Kolokolov, V. Lebedev and A. A. Migdal, [*Instantons and intermittency*]{}, .
E. Balkovsky, G. Falkovich, I. Kolokolov and V. Lebedev, [*Intermittency of Burgers’ Turbulence*]{}, .
R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, S. Succi, [*Extended self-similarity in turbulent flows*]{}, .
S. L. Adler, [*Over-relaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions*]{}, .
R. S. Varga, [*Matrix Iterative Analysis*]{} (Prentice Hall, 1962).
[^1]: DESY 09-174, MS-TP-09-23
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Thibaud Louvet
- Pierre Delplace
- 'Andrei A. Fedorenko'
- David Carpentier
date: 'March 13, 2015'
title: 'Minimal conductivity, topological Berry winding and duality in three-band semimetals'
---
=1
[**The physics of massless relativistic quantum particles has recently arisen in the electronic properties of solids following the discovery of graphene. Around the accidental crossing of two energy bands, the electronic excitations are described by a Weyl equation initially derived for ultra-relativistic particles. Similar three and four band semimetals have recently been discovered in two and three dimensions. Among the remarkable features of graphene are the characterization of the band crossings by a topological Berry winding, leading to an anomalous quantum Hall effect, and a finite minimal conductivity at the band crossing while the electronic density vanishes. Here we show that these two properties are intimately related: this result paves the way to a direct measure of the topological nature of a semi-metal. By considering three band semimetals with a flat band in two dimensions, we find that only few of them support a topological Berry phase. The same semimetals are the only ones displaying a non vanishing minimal conductivity at the band crossing. The existence of both a minimal conductivity and a topological robustness originates from properties of the underlying lattice, which are encoded not by a symmetry of their Bloch Hamiltonian, but by a duality.** ]{}
Electronic excitations which satisfy ultra relativistic quantum physics emerge around the band crossing of semimetals. While graphene constitutes the canonical example of such a phase, the discovery of topological insulators opened the route to other realizations [@Murakami:2007]: such semimetals exist at the transitions between topological and trivial insulators. Theoretical proposals were initiated by the identification of crystalline symmetries stabilizing these critical phases both in two and three dimensions [@Young:2012]. This idea turned out to be particularly successful: recently, a Weyl semimetal corresponding to two band crossings in three spatial dimensions was discovered in TaAs [@Lv:2015; @Xu:2015]. The existence of a three band crossing semimetal was proposed in a two dimensional carbon allotrope SG-10b [@Wang:2013b] and square MoS$_2$ sheet [@Li:2014], and discovered in three dimensions in HgCdTe [@Orlita:2014]. Stable four band crossing semimetals in three dimensions, denoted Dirac phases, were proposed theoretically and experimentally identified in Na$_3$Bi [@Wang:2012; @Liu:2014; @Xu:2015] and in Cd$_3$As$_2$ [@Wang:2013; @Neupane:2014; @Borisenko:2014], and predicted in $\beta$-cristobalite BiO$_2$ [@Young:2012] and distorted spinels [@Steinberg:2014].
In the case of graphene, the relativistic nature of the electronic excitations translates into remarkable transport properties including an anomalous half-integer Hall effect with half integer plateaus and a non vanishing minimal conductivity exactly at the band crossing [@Novoselov:2005; @Zhang:2005; @Twordzylo:2006; @Katsnelson:2006b]. The anomalous half-integer Hall effect is related to the topological properties of the band crossing: when winding around the crossing point, an electron picks up a quantized so-called Berry phase. This Berry phase is at the origin of the half-integer nature of the quantum Hall effect in graphene [@Zhang:2005; @Novoselov:2006], but also characterizes the topological property of the semi-metallic phase encoding its robustness towards gap opening perturbations [@Murakami:2007]. On the other hand the minimal conductivity at the band crossing was associated to the Zitterbewegung of Dirac particles, an intrinsic agitation characteristic of ultra relativistic particles which leads to diffusive motion even in perfectly clean samples [@Twordzylo:2006; @Katsnelson:2006].
In this article we consider these characteristic signatures of semimetals beyond the two-band crossing situation of graphene. When a third locally flat band is present at the crossing, none of these properties is necessarily enforced. Indeed, we find that the exact same models possess both a non vanishing minimal conductivity and a topological Berry winding at the crossing. Hence these properties are not hallmarks of a relativistic energy spectrum: they encode phase properties of the electronic wave functions. In the case of graphene, this phase winding originates from the hexagonal lattice of carbon atoms. We will show that the lattice properties at the origin of these remarkable transport signatures are encoded not by a standard symmetry constraints of Bloch Hamiltonians, but by a duality transformation. We identify two duality classes, corresponding to the two signatures: the existence or not of both a non-vanishing minimal conductivity and a topological Berry winding.
.
We consider three-band semimetals in two dimensions characterized by two linearly crossing energy bands $n=\pm $ and a third locally flat band $n=0$, represented in Fig. \[fig:cone\]. This corresponds to the situation of HgCdTe in three dimensions [@Orlita:2014], or the carbon allotrope SG-10b [@Wang:2013b] and square MoS$_2$ sheet [@Li:2014] in two dimensions. The energy spectrum $E(\vec{k})$ as a function of the momentum $\vec{k}$ exhibits the symmetry $E(\vec{k})\rightarrow -E(\vec{k})$ at least locally around the crossing point. This spectrum symmetry naturally originates from a chiral symmetry of the corresponding (low energy) Hamiltonian. This symmetry is represented by a unitary operator $C$ that *anticommutes* with the Hamiltonian: $\mathcal{H}=-C\mathcal{H} C$. An explicit chiral operator can be defined when considering the pedagogical examples of tight-binding Hamiltonians defined on lattices. Similarly to graphene, only nearest neighbor couplings can be kept when focusing around the band crossing points. In this case chiral symmetry corresponds to a sub-lattice symmetry: couplings are only present between the two sub-lattices $A$ and $B$ of a bipartite lattice. This is the case of the nearest neighbor description of graphene on the honeycomb lattice. In the case we consider in this article, three bands crossing implies the existence of three orbitals distributed on three Bravais lattices $A_1, A_2$ and $B$ of same geometry, as shown on Fig. \[fig:lattices\]. Chiral symmetry is satisfied if the only couplings $t_1, t_2$ relevant at low energy are between orbitals on the $B$ and the $A_1, A_2$ lattices whereas $A_1$ and $A_2$ stay uncoupled. The corresponding Bloch Hamiltonian in the orbital basis $(A_1,A_2,B)$ is written $$H(t_1,t_2;\vec{k})=\left( \begin{array}{ccc}
0 & 0 & t_1f_1 (\vec{k}) \\
0 & 0 & t_2f_2 (\vec{k}) \\
t_1f_1^* (\vec{k}) & t_2f_2^* (\vec{k}) & 0
\end{array} \right)\ .
\label{eq:general hamiltonian}$$ Such a Hamiltonian anti-commutes with a chirality operator $C = \textrm{diag}(1,1,-1)$. The complex functions $f_j(\vec{k})=|f_j(\vec{k})|e^{i\phi_j(\vec{k})}$ encode the geometry of the lattice of couplings. Their amplitudes determine the spectrum of the semimetal: $E_0(\vec{k})=0, E_{\pm}(\vec{k})=\pm
(t_1^2|f_1(\vec{k})|^2+t_2^2|f_2(\vec{k})|^2 )^{\frac12}$. A three band crossing occurs when $f_1$ and $f_2$ vanish simultaneously at a point $\vec{K}$ in the Brillouin zone. In this article, we focus on properties which depend on the phases $\phi_j(\vec{k})$ and which are thus *independent* of the spectrum provided a band crossing occurs.
Quite generally, we will consider chiral symmetric Bloch Hamiltonians describing a three-band semimetal with band crossing at point $\vec{K}$, and whose linear expansion around the crossing takes the form $$\begin{gathered}
H(\vec{K}+{\vec q})= \\
\left( \begin{array}{ccc}
0 & 0 & \Lambda_{11}q_x + \Lambda_{12}q_y\\
0 & 0 & \Lambda_{21}q_x + \Lambda_{22}q_y \\
\Lambda_{11}^*q_x + \Lambda_{12}^*q_y & \Lambda_{21}^*q_x + \Lambda_{22}^*q_y & 0
\end{array} \right) \ .
\label{eq:general linear hamiltonian}\end{gathered}$$ Such a Hamiltonian is entirely parametrized by a matrix $\Lambda=\{ \Lambda_{ij}\}$ of complex coefficients. The phase of the coefficients $\Lambda_{ij}$ encodes the geometry of the underlying lattice. The corresponding geometrical constraint on these phases must be independent of the amplitude of couplings between the orbitals. Hence it cannot result in a symmetry of the Hamiltonian: we show that it corresponds to a duality in a manner analogous to the Kramers-Wannier duality of statistical mechanics models which relates models with different coupling amplitudes on different lattices [@Kramers:1941].
. The geometry of the lattice can be described by two vectors $\vec{e}_1$ and $\vec{e}_2$ relating a vector of the lattice $B$ to neighbor sites of the $A_1$ and $A_2$ lattices, as shown on figure \[fig:lattices\]. The duality transformation $\mathcal{D}$ exchanges the lattices $A_1$ and $A_2$, or equivalently the vectors $\vec{e}_1$ and $\vec{e}_2$, while simultaneously exchanging the couplings between $B$ and $A_1$ orbitals with couplings between $B$ and $A_2$ orbitals. Quite generally, this duality which is an involutive transformation [*i.e.*]{} ${\cal D}^2=\mathbf{I}$, relates a Hamiltonian $\mathcal{H}$ on a lattice $\mathcal{L}$ to a Hamiltonian $\tilde{\mathcal{H}}$ on a *different* lattice $\tilde{\mathcal{L}}$. However, on symmetric lattices where initial and dual lattices $\mathcal{L},\tilde{\mathcal{L}}$ are related by a geometrical transformation $\mathcal{R}$, this duality translates into constraints on Hamiltonians defined on the same lattice (or same Hilbert space). In this case, and focusing for simplicity on nearest neighbor Hamiltonians, the duality transformation can be recast into the form $$(DU) H (t_2,t_1; \mathcal{R}\vec{k}) (DU)^{-1} = H (t_1,t_2;\vec{k})
\label{eq:geometric duality}$$ where $U$ is a unitary operator, $\mathcal{R}$ is the symmetry relating initial and dual lattices and $D$ the operator swapping $A_1$ and $A_2$ orbitals: $$D=\left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array} \right) \ .
\label{eq:dualityop}$$ A very special case, which we call the duality class $\mathcal{D}_{\textrm{I}}$ corresponds to the situation where two orbitals lie on the same geometrical lattice, [*i.e.*]{} when $\vec{e}_1 =\vec{e}_2\neq \vec{0}$. In this class, the duality transformation simplifies and $U$ and $\mathcal{R}$ reduce to the identity. In this case, Bloch Hamiltonians encode the geometrical properties of a bipartite lattice, whereas in the other case, which we denote the duality class $\mathcal{D}_{\textrm{II}}$, the underlying lattices are either Bravais lattices or possess three distinct sublattices. This duality restricts the form of the chiral tight-binding Hamiltonian (\[eq:general hamiltonian\]): in the class $\mathcal{D}_{\textrm{I}}$ we have $f_1 (\vec{k}) = f_2 (\vec{k})$ while a much weaker constrain $f_1 (\vec{k}) = f_2 (\mathcal{R}\vec{k})$ holds in class $\mathcal{D}_{\textrm{II}}$ for symmetric lattices. Generalizing this constrain to a local Bloch Hamiltonian (\[eq:general linear hamiltonian\]) around a three band crossing, the duality in class $\mathcal{D}_{\textrm{I}}$ implies the condition $$\frac{\Lambda_{21}}{\Lambda_{11}} = \frac{\Lambda_{22}}{\Lambda_{12}} \equiv \lambda \ ,
\label{eq:lambda}$$ while generically it only relates Hamiltonian at different crossing points in class $\mathcal{D}_{\textrm{II}}$. In the following we show that Hamiltonians belonging to the duality class $\mathcal{D}_{\textrm{I}}$ describe the only three-band semimetals possessing quantized topological Berry windings which are also those whose conductivity does not vanish at the band crossing and display a pseudo-diffusive regime.
To illustrate this relation, let us discuss two nearest-neighbor lattice models of semi-metals belonging to both classes. A natural model in class $\mathcal{D}_{\textrm{I}}$, inspired by graphene, corresponds to a honeycomb lattice with two orbitals on one of the two sub-lattices, shown on Fig. \[fig:lattices\] (c). The three bands of this model, which we call H$_3$, cross at points $\vec{K}$ and $\vec{K}'$ of the Brillouin zone. Around those points, the Bloch Hamiltonian takes the form with a matrix of coefficients $$\Lambda_{\textrm{H}_3}= \frac{3a}{2} \left(\begin{array}{cc}
t_1 & -{\mathrm{i}}t_1 \\
t_2 & -{\mathrm{i}}t_2
\end{array} \right),
\label{eq:lambda-3/2}$$ with $a$ being the honeycomb lattice spacing, while the characteristic energy scale of nearest neighbor couplings is encoded into $t=\sqrt{t_1^2+t_2^2}$. This model satisfies the condition with $\lambda = -{\mathrm{i}}$ and belongs to the duality class $\mathcal{D}_{\textrm{I}}$. The [T$_{3}$ ]{}model [@Raoux:2014] is again defined on a honeycomb lattice but with additional orbitals $A_2$ at the center of each hexagon as shown on Fig. \[fig:lattices\] (b). These orbitals are coupled by an amplitude $t_2$ only to the $B$ sub-lattice of the honeycomb lattice, while $A_1$ and $B$ orbitals on the honeycomb lattices are coupled by $t_1$. This model belongs to the duality class $\mathcal{D}_{\textrm{II}}$, with the inversion $\mathcal{R} \vec k = - \vec k$ relating initial and dual lattices. Indeed, the Bloch Hamiltonian linearized around the band touching point $\vec K$ is written in the form with $$\Lambda_{\textrm{{T$_{3}$ }}}= \frac{3a}{2}
\left(\begin{array}{cc}
t_1 & -{\mathrm{i}}t_1 \\
t_2 & {\mathrm{i}}t_2 \\
\end{array} \right)~,
\label{eq:lambda-t3}$$ which does not fulfill the condition . Note that when $t_1=t_2$, this linearized Hamiltonian can be written in the form $H_{\bf K} ({\bf q}) = \hbar v_F \bf S \cdot q $, where $S_x$, $S_y$ and $S_z \equiv \text{diag} (1,-1,0)$ satisfy the spin-$1$ algebra $[S_i,S_j]= {\mathrm{i}}\epsilon_{ijk}S_k$. Hence the [T$_{3}$ ]{}model realizes a continuous deformation of spin-$1$ massless fermions, which all belong to the duality class $\mathcal{D}_{\textrm{II}}$.
We now characterize both the topological properties of the band crossing, as well as the electronic transport properties around the crossing, which turn out to be associated to the duality class of the semi-metal.
[**Topological property of a band crossing**]{}.
{width="90.00000%"}
The topological properties of band crossings in two dimensional crystals can be described by a topological invariant associated to each band around the crossing point. These invariants are the Berry phases $\gamma_n$ (see eq. of Methods) acquired by Bloch eigenstates $\Psi_n$ upon winding around the crossing point $\vec{K}$. By definition this Berry winding is independent of the path winding around the crossing point: it is an homotopic invariant of a given Hamiltonian. It describes a *topological* property of the semimetal when it is robust against perturbations of the Hamiltonian which do not lift the band crossing. Such a robustness occurs when this Berry winding is quantized. This is indeed the case in graphene where $\gamma_{\pm} = \pm 1$. For the three-bands semi-metals, these Berry windings can be readily obtained from the diagonalisation of the Hamiltonian . We find that these windings are topological only when the condition is fulfilled: the only semimetals characterized by a topological Berry winding are those of the duality class $\mathcal{D}_{\textrm{I}}$, with values $$\gamma_{+} =\gamma_- = \text{sgn}\, \text{Im}\, \lambda \quad , \quad \gamma_0 = -2\, \text{sgn}\, \text{Im}\, \lambda \ .$$ These quantized values of the Berry windings are stable with respect to any perturbation compatible with the duality constraint, [*i.e.*]{} which does not break the geometry of the underlying lattice. Eigenstates of the H$_3$ model are characterized by Berry phases $\gamma_{\pm}=-1$, $\gamma_0=2$ as shown in Fig. \[fig:winding\]. These windings are in particular robust to variations of hopping amplitudes $t_1,t_2$. For any chiral symmetric semimetal which does not belong to the $\mathcal{D}_{\textrm{I}}$ class, the Berry phase can take any real value and depends continuously on deformations of the Hamiltonian. This result is illustrated for the [T$_{3}$ ]{}model in Fig. \[fig:winding\]: the Berry phases $\gamma_n$ are shown to vary continuously upon variation of the ratio $t_2/t_1$ of nearest neighbor couplings.
![ The setup for two-terminal transport measurements: a 2D sample of size $W \times L$ with leads attached on opposite sides. \[fig:cond3-1\] ](Figures/minimal-setup){width="8cm"}
. Transport measurements constitute a powerful tool to probe the physical properties in the vicinity of the Fermi energy. We will show that close to the band crossing electronic transport is related to the phase content of the Hamiltonian (the phases $\phi_{i}(\vec{k}$) of the amplitudes in ) and not to the spectrum. Remarkably, in graphene, when the Fermi level coincides with the twofold band crossing point, the conductivity of a clean sample was predicted to remain finite despite a vanishing density of states. This result was first derived by considering the conductivity of a narrow strip of graphene between two contact electrodes as shown on Fig. \[fig:cond3-1\]. Let us consider an analogous setup for a three band chiral semimetal, [*i.e.*]{} a finite sample of length $L$ and width $W$. Confinement of the sample between the leads gives rise to zero-energy evanescent states. At the band crossing, the conductivity depends entirely on the nature of these evanescent states. Prior to an explicit calculation of the conductivity (see Methods), it is instructive to consider the current operator $j_x (\vec k) = \langle \partial_{k_x} H(\vec k) \rangle_\psi$ defined from the tight-binding Bloch Hamiltonian . Introducing the amplitudes $( \psi_{A_1}, \psi_{A_2}, \psi_B)$ of the electronic wavefunction in the three sub-lattices the longitudinal current can be expressed as $ j_x (\vec k) = 2~ \text{Re}\, [ \psi_B (\vec k) (\partial_{k_x} f_i (\vec k)
\psi_{A_i}^* (\vec k) )] $ and is found to be proportional to the amplitude $\psi_B$ on the $B$ sub-lattice [@Hausler:2015]. This hints that electronic transport at the band crossing will occur provided the zero-energy evanescent modes have a non-vanishing component on the $B$ sub-lattice.
As expected from the previous qualitative argument, the existence of a finite minimal conductivity at the threefold band crossing point is uniquely determined by the non vanishing weight of the wave function on the hub lattice as we have checked using a Landauer description of transport (see Methods). From , this component is found to satisfy $$\Lambda .\left( \begin{array}{c}
q_x \\
q_y
\end{array} \right) \psi_B = 0 \ .
\label{eq:psiB}$$ Hence the condition of existence of a non-vanishing minimal conductivity at the band crossing is simply $ \text{det}\,\Lambda = 0 $. This constraint exactly coincides with the duality constraint defining the class $\mathcal{D}_{\textrm{I}}$: the only three band semi-metals with a non-vanishing minimal conductivity are exactly those belonging to this duality class $\mathcal{D}_{\textrm{I}}$. Moreover for the H$_{3}$ lattice model, this minimal conductivity corresponds exactly to the value $ \sigma^{\textrm{(min)}} = e^2/(\pi h)$ predicted for graphene. This result remains valid for any model in the dual class $\mathcal{D}_{\textrm{I}}$ with an isotropic dispersion relation. For more general three band semi-metals with an anisotropic dispersion relation, the minimal conductivity depends on the angle between the leads and the principal axes of the dispersion relation. However, the determinant of the corresponding conductivity tensor remains constant given by its isotropic value $\det \overline{\sigma} = (e^2/(\pi h))^2$. In contrast, any three-band chiral symmetric semimetal that does not belong to the $\mathcal{D}_{\textrm{I}}$ class possesses a vanishing conductivity $\sigma^{\textrm{(min)}} = 0$ in every directions. Beyond the minimal conductivity, the fluctuations of this conductivity can also be considered : their amplitude is encoded in the ratio between the shot noise power and the averaged current, the so-called Fano factor. This factor $F$ takes a constant value $F=1/3$ within the duality class $\mathcal{D}_{\textrm{I}}$, a value already encountered in graphene [@Twordzylo:2006] and characteristic of diffusive metals [@Beenakker:1997]. Such a result demonstrates that for all semi-metals in this class the transport through narrow perfectly clean junctions displays the characteristic features of diffusive metals.
We have evaluated the conductivity of different lattice models in the geometry of Fig. \[fig:cond3-1\]. We compare the analytical results to a numerical Landauer approach (see Methods) to check for possible inter-crossing point effects, neglected in the analytical approach. A perfect agreement is found between both approaches. The results for the H$_3$ model are shown as a function of the Fermi-energy, or gate potential $V_g$ on Fig. \[fig:cond3-2\]: the conductivity exhibits a plateau around the band crossing point $V_g=0$, corresponding to $\sigma = e^2/ (\pi h)$. The results of a similar study for the [T$_{3}$ ]{}model are also shown on Fig. \[fig:cond3-2\] and display a collapse of the conductivity around the band crossing point $V_g=0$ for three different values of the couplings between orbitals (Inset). Fig. \[fig:cond3-3\] displays the analytical results for the dependance of the Fano factor on the gate potential $V_g$. We show that for the [T$_{3}$ ]{}model $F(V_g=0)=1$, whereas for the H$_3$ model the Fano factor reaches the value $F=\frac13$ at the band crossing, characteristic of a disordered metal as expected for class $\mathcal{D}_{\textrm{I}}$. Finally, let us mention that for graphene the result of the Landauer formula for a narrow junction can be recovered for a long junction by an approach based on the Kubo formula [@Ryu:2007]. While this equivalence remains valid for three band semimetals in the dual class $\mathcal{D}_{\textrm{I}}$, it does not hold beyond it: we found that the Kubo conductivity for the [T$_{3}$ ]{}model diverges at the band crossing, as opposed to the vanishing Landauer conductivity for a narrow junction, in agreement with a previous result in the disordered limit [@Vigh:2013].
As follows from our results, the occurrence of a transport regime with a non vanishing conductivity at the crossing is not a generic property of linear dispersion relations near this crossing, nor a hallmark of relativistic physics of the associated electronic excitations such as the Zitterbewegung. It is indeed intimately related to the existence of a topological robustness of this band crossing, originating from lattice-properties encoded into the class $\mathcal{D}_{\textrm{I}}$ condition. This result opens the route to a direct probe of topological properties of semimetals through transport measurements around the band crossing. In graphene, a quantitative measurement of the conductivity exactly at the band crossing is hampered by fluctuations of the chemical potentials induced by a back gate. However the recent discovery of three dimensional semimetals changes the perspective: most of these new semi-metallic materials are stoichiometric and the Fermi-energy is expected to reside at the band crossing. In that situation the transport will be entirely dominated by the physics at the band crossing and will directly probe the topological properties associated with the band crossing points.
Let us put in perspective the association between minimal conductivity and topological properties raised by our results on three band semimetals in two dimensions. In two dimensions, we can extend our analysis for a band crossing described by the simple Hamiltonian $H(\vec{q})=S_x q_x + S_y q_y$, with $S_x,S_y$ satisfying a spin-$S$ algebra. Again, we find that the Berry topological winding [@Dora:2011] and the minimal conductivity are correlated. In particular for an integer spin $S$, corresponding to crossing of an odd number of bands, we find that the conductivity vanishes at the crossing, as does the Berry phase [@Dora:2011], while both are finite for half integer spins. In three dimensions, a two band crossing is denoted a Weyl semimetal and is characterized by a topological Chern number instead of a Berry phase. In that case, the conductance at the nodal point remains finite and scales as $G_{\textrm{3d}} = \ln 2 \ e^2/(2\pi h) (W/L)^2$ instead of $G_{\textrm{2d}} = e^2/(\pi h) (W/L)$ in two dimensions [@Baireuther:2014]. Similarly the value of the Fano factor is slightly modified from $F_{\textrm{2d}} = 1/3$ to $F_{\textrm{3d}} \simeq 0.57$. The existence of a finite minimal conductance scaling as $ (W/L)^2$ appears to be a robust property associated with the topological two band crossing as its existence is neither modified by a weak disorder [@Sbierski:2014], nor by an anisotropic deformation of the cone or a tilt of this cone which breaks the local chiral symmetry [@Trescher:2015].
Methods {#methods .unnumbered}
=======
[**Topological characterization.**]{} The topological characterization of a band crossing point is done by calculating the Berry phase associated to each eigenvector $|\Psi_n \rangle$ upon winding anticlockwise around the crossing point: $$\gamma_n (\vec{K}) = \frac{-i}{\pi} \oint d\vec{q}. \langle \Psi_n | \vec{\nabla}_{\vec{q}} | \Psi_n \rangle \ .
\label{eq:gamman}$$ [**Conductance and Fano factor**]{}. The conductance of a narrow sample is conveniently calculated from the set of the transmission probabilities $T_n$ of the conduction channel labeled by $n$ through the Landauer formula $G =\frac{e^2}{h} \sum_n T_n \ .$ The longitudinal conductivity $\sigma$ is related to this conductance as $ \sigma = LW^{-1} G$. The Fano factor is related to these transmission coefficients as $F = \sum_{n} T_{n}(1-T_{n})/\sum_{n} T_{n}$. The explicit calculation of the transmission probabilities $T_n$ requires solving the Schrödinger equation piecewise and matching the solutions at the boundaries of the sample.
Numerical calculations of transport were performed using the Kwant code [@Groth:2014], based on a Green function recursive technique to evaluate the transmission amplitude across a sample. Typical samples of dimensions $L=100$, $W=300$ in lattice units were used, using a large potential in the electrodes $e V_\infty W/(v_F \hbar) = 45$, with Fermi velocity $v_F=3at/(2\hbar)$.
[**Acknowledgments**]{}. We thank F. Pi[é]{}chon for stimulating discussions about the T$_{3}$ model, as well as A. Akhmerov and X. Waintal for helpful advice with the Kwant package. This work was supported by the grants ANR Blanc-2010 IsoTop and ANR Blanc-2012 SemiTopo from the french Agence Nationale de la Recherche.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We report studies of $B^0 \to \pi^+ \pi^- \pi^0$, using $78\ {\rm fb}^{-1}$ of data collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric $e^+e^-$ storage ring. We measure the branching fraction for $B^0 \to \rho^\pm \pi^\mp$ to be ${\mathcal B}\left( B^0 \to \rho^\pm \pi^\mp \right) =
\left( 29.1 ^{+5.0}_{-4.9}({\rm stat}) \pm 4.0({\rm syst}) \right)
\times 10^{-6}$, and find an untagged charge asymmetry ${\mathcal A} =
-0.38^{+0.19}_{-0.21} ({\rm stat}) ^{+0.04}_{-0.05} ({\rm syst})$. We find the first evidence for $B^0 \to \rho^0 \pi^0$ with $3.1\sigma$ statistical significance and with a branching fraction of ${\mathcal B}\left( B^0 \to \rho^0\pi^0 \right) =
\left( 6.0 ^{+2.9}_{-2.3} ({\rm stat}) \pm 1.2 ({\rm syst}) \right)
\times 10^{-6}$.
title: |
\
Studies of $B^0 \to \rho^\pm \pi^\mp$ and Evidence of $B^0 \to \rho^0 \pi^0$
---
There is a large amount of interest in $B^0 \to \pi^+\pi^-\pi^0$ decays [@cc]. The three body final state is expected to be dominated by the quasi-two body decays $B^0 \to \rho^{\pm} \pi^{\mp}$, the branching fraction of which has been previously measured to be ${\mathcal B}\left( B^0 \to \rho^{\pm} \pi^{\mp} \right) =
\left( 20.8 ^{+6.0+2.8}_{-6.3-3.1} \right) \times 10^{-6}$ by Belle [@ascelin], from a data sample of $29.4~{\rm fb}^{-1}$, corresponding to $31.9 \times 10^6$ $B\bar{B}$ pairs. Recently BaBar have announced a preliminary measurement of ${\mathcal B}\left( B^0 \to \rho^{\pm} \pi^{\mp} \right) =
\left( 22.6 \pm 1.8 \pm 2.2 \right) \times 10^{-6}$ [@babar_rhopi], from a data sample corresponding to $89 \times 10^6$ $B\bar{B}$ pairs. Additionally, they have performed a time-dependent analysis on the quasi-two body signal candidates. Such an analysis can obtain information about the Unitary Triangle angle $\phi_2$ [@phi2], which may be useful to help interpret the results of time-dependent $B^0 \to \pi^+\pi^-$ analyses [@pipi].
In addition to $\rho^{\pm} \pi^{\mp}$, the $\pi^+\pi^-\pi^0$ final state can be accessed via $B^0 \to \rho^0 \pi^0$ decay. Analogous to $B^0 \to \pi^0\pi^0$ amongst the $\pi\pi$ final states, ${\mathcal B} \left( B^0 \to \rho^0 \pi^0 \right)$ can be used to limit the possible contribution from penguin diagrams. Note that the penguin pollution in $\rho^{\pm} \pi^{\mp}$ is expected to be smaller than that in $\pi^+\pi^-$; a hypothesis which is supported by comparing the ratio ${\mathcal B} \left( B^0 \to \rho^{\pm}K^{\mp} \right) /
{\mathcal B} \left( B^0 \to \rho^{\pm}\pi^{\mp} \right)$ to ${\mathcal B} \left( B^0 \to \pi^{\pm}K^{\mp} \right) /
{\mathcal B} \left( B^0 \to \pi^{\pm}\pi^{\mp} \right)$. Currently the best upper limit is ${\mathcal B}\left( B^0 \to \rho^0 \pi^0 \right) <
5.3 \times 10^{-6}$ [@ascelin], whilst most theoretical estimates of this branching fraction are ${\mathcal O}\left(10^{-6}\right)$ or lower.
An unambiguous measurement of $\phi_2$ can, in principle, be made from the Dalitz plot of $B^0 \to \pi^+\pi^-\pi^0$ [@snyder_quinn]. It is also possible to observe direct $CP$ violation from a population asymmetry in the untagged Dalitz plot distribution [@gardner]. Furthermore, other resonant contributions to the $\pi^+\pi^-\pi^0$ final state are possible; recently there has been particular theoretical interest in $B^0 \to \sigma \pi^0$ [@sigma].
With very large statistics, and sophisticated analysis techniques, it would be possible to address all these open questions regarding $B^0 \to \pi^+\pi^-\pi^0$, using a time-dependent Dalitz plot analysis. While we cannot as yet achieve this ambitious goal, the results presented here represent milestones towards this objective. We present an updated measurement of the $B^0 \to \rho^{\pm}\pi^{\mp}$ branching fraction, using event selection that can provide a sample for a quasi-two body time-dependent analysis. We also present the first evidence for $B^0 \to \rho^0 \pi^0$.
The analysis is based on a 78 $\rm fb^{-1}$ data sample containing $85 \times 10^6$ $B$ meson pairs collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider [@KEKB]. KEKB operates at the $\Upsilon(4S)$ resonance ($\sqrt{s}=10.58~{\rm GeV}$) with a peak luminosity that exceeds $1\times 10^{34}~{\rm cm}^{-2}{\rm s}^{-1}$.
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a three-layer silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Čerenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a super-conducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect $K_L$ mesons and to identify muons (KLM). The detector is described in detail elsewhere [@Belle].
Charged tracks are required to originate from the interaction point and have transverse momenta greater than $100~{\rm MeV}/c$. To identify tracks as charged pions, we combine $dE/dx$ information from the CDC, pulse height information from the ACC and timing information from the TOF into pion/kaon likelihood variables ${\mathcal L}(\pi/K)$. We then require ${\mathcal L}(\pi)/\left( {\mathcal L}(\pi) + {\mathcal L}(K)\right) > 0.6$. Additionally, tracks which are consistent with an electron hypothesis are rejected.
Neutral pion candidates are reconstructed from photon pairs. Photon candidates are selected with a mininum energy requirement of $50~{\rm MeV}$ in the barrel region of the ECL, and $100~{\rm MeV}$ in its endcap. The $\pi^0$ candidates are required to have momenta greater than $200~{\rm MeV}$ in the lab frame, and a $\gamma\gamma$ invariant mass in the range $0.118 < M_{\gamma\gamma}/{\rm GeV}/c^2 < 0.150$. Additionally, we require $\left| \cos(\theta^{\pi^0}_{\rm hel}) \right| < 0.95$, where $\theta^{\pi^0}_{\rm hel}$ is defined as the angle between one photon’s flight direction in the $\pi^0$ rest frame and the flight direction of $\pi^0$ with respect to the lab frame, and make a loose requirement on the $\chi^2$ of a $\pi^0$ mass-constrained fit of $\gamma\gamma$.
$B$ candidates are selected using two kinematic variables: the beam-constrained mass $M_{bc}\equiv \sqrt{E^2_{\rm beam}-P^2_B}$ and the energy difference $\Delta E \equiv E_B - E_{\rm beam}$. Here, $E_B$ and $P_B$ are the reconstructed energy and momentum of the $B$ candidate in the center of mass (CM) frame, and $E_{\rm beam}$ is the average beam energy in the CM frame. $B$ candidates with $M_{bc} > 5.2~{\rm GeV}/c^2$ and $-0.30 < \Delta E / {\rm GeV} < 0.20$ are selected. We further define the signal regions: $M_{bc} > 5.27~{\rm GeV}/c^2$ and $-0.10 < \Delta E / {\rm GeV} < 0.08$.
To select $\rho^\pm\pi^\mp$ from 3-body $\pi^+\pi^-\pi^0$, we select candidates with an invariant $\pi \pi^0$ mass in the range $\left| M_{\pi \pi^0} - M_{\rho} \right| < 0.20~{\rm GeV}/c^2$, and $\rho$ helicity $\theta^{\rho}_{\rm hel}$, defined as the angle between the charged pion direction in the $\rho$ rest frame and the $\rho$ direction in the $B$ rest frame [@helicity], in the range $\left| \cos \theta^{\rho}_{\rm hel} \right| > 0.5$.
The dominant background to $B^0 \to \pi^+\pi^-\pi^0$ comes from continuum events, $e^+e^-\to q\bar{q}$ ($q = u, d, s, c$). Since these tend to be jet-like, whilst $B\bar{B}$ events tend to be spherical, we use event shape variables to discriminate between the two. We combine five modified Fox-Wolfram moments [@fox-wolfram] into a Fisher discriminant; the coefficients are then tuned to maximize the separation between signal and continuum events. We further define $\theta_B$ as the angle of the reconstructed $B$ candidate with respect to the beam direction. Signal events have a distribution proportional to $\sin^2(\theta_B)$, whilst continuum events are flatly distributed in $\cos(\theta_B)$. We combine the output of the Fisher discriminant with $\cos(\theta_B)$ into signal/background likelihood variables, ${\mathcal L}_{s/b}$. We find the optimum selection requirement by maximizing $S/\sqrt{S+B}$, where $S$ and $B$ are respectively the expected numbers of signal and background events in the signal region. We use our measured branching fraction of $B^0 \to \rho^\pm\pi^\mp$ [@ascelin] as input, and find the optimum requirement is ${\mathcal L}_s/\left( {\mathcal L}_s + {\mathcal L}_b \right) > 0.8$.
If more than one candidate remains in any event, that with the smallest $\chi^2_{\rm vtx} + \chi^2_{\pi^0}$ is selected, where $\chi^2_{\rm vtx}$ is the $\chi^2$ of a vertex-constrained fit of $\pi^+\pi^-$, and $\chi^2_{\pi^0}$ is that from a $\pi^0$ mass-constrained fit of $\gamma\gamma$.
We obtain the signal yield using a binned fit to the $\Delta E$ distribution, and cross-check the result by fitting the $M_{\rm bc}$ distribution. When fitting one variable, candidates are required to be in the signal region of the other. The signal probability density functions (PDFs) are obtained from Monte Carlo (MC); a Crystal Ball [@crystalball] lineshape plus a Gaussian is used for $\Delta E$, whilst the $M_{\rm bc}$ distribution is described by a Gaussian. The Gaussian in the $\Delta E$ PDF accounts for poorly reconstructed low momentum neutral pions. The $\Delta E$ width is calibrated using an inclusive $D^*$ sample ($D^{*+} \to D^0 \pi^+$, $D^0 \to K^-\pi^+\pi^0$) whilst the $\Delta E$ and $M_{bc}$ peak positions are adjusted according to a data sample of $B^+\to D^0 \pi^+$ with $D^0\to K^-\pi^+\pi^0$.
The dominant background is from continuum events. The $\Delta E$ distribution for these events is described by a Chebyshev polynomial, whilst the $M_{\rm bc}$ shape is given by the ARGUS function [@argus]. The parameters of these functions are determined from fitting a large continuum MC sample.
Background is also possible from generic $b \to c$ transitions; in this case the shape is hard to describe by a functional form and so we use a smoothed histogram. The $\Delta E$ distribution of background from the charmless decay $B^+ \to \rho^+ \rho^0$ has a similar shape to the generic $b \to c$, so we combine these components, with the relative normalization fixed according to our recent measurement of ${\mathcal B}\left( B^+ \to \rho^+ \rho^0 \right)$ [@jingzhi], and allow the overall normalization to float in the $\Delta E$ fit. In the $M_{\rm bc}$ fit, these backgrounds cannot be distinguished from signal, and the normalization is fixed.
There are other possible backgrounds from charmless $B$ decays, with distinctive $\Delta E$ shapes. $B^0 \to \rho^{\pm}K^{\mp}$ has a similar shape to the signal, but a shifted peak due to the misidentification of the kaon as a pion. The normalization of this component is fixed according to our recent measurement [@paoti]. Contributions from $\pi^+ \rho^0$, $hh$ and $h\pi^0$ ($h = \pi^{\pm},K^{\pm}$) final states are scaled according to the most recent measurements of their branching fractions [@ascelin; @hh; @pdg], then combined into a smoothed histogram. The normalization of this component is then fixed in the fit.
to
The results of the fits to $\Delta E$ and $M_{\rm bc}$ are shown in Fig. \[fig:rhopmpimp\_fit\]. From the $\Delta E$ fit we find a yield of $257.9^{+44.0}_{-43.2}$ signal events, with a significance of $6.3\sigma$.
The significance is defined as $\sqrt{-2\ln({\mathcal L}_0/{\mathcal L}_{\rm max})}$, where ${\mathcal L}_{\rm max}$ (${\mathcal L}_0$) denotes the likelihood with the signal yield at its nominal value (fixed to zero).
From the $M_{\rm bc}$ fit we find a signal yield of $177.7^{+24.7}_{-24.0}$. Taking into account the different efficiencies of the $\Delta E$/$M_{\rm bc}$ signal region selections, this result is consistent with the $\Delta E$ yield.
To check the events in the signal peak are $B^0\to \rho^\pm\pi^\mp$ events, and not from some other contribution to the $\pi^+\pi^-\pi^0$ final state, we relax the $M_{\pi \pi^0}$ and $\cos \theta^{\rho}_{\rm hel}$ criteria in turn, and perform the $\Delta E$ fit in bins of $M_{\pi \pi^0}$, and in bins of $\cos \theta^{\rho}_{\rm hel}$. The resulting distributions are shown in Fig. \[fig:rhopmpimp\_check\]. Clearly, the signal is consistent with being entirely due to $B^0 \to \rho^{\pm}\pi^{\mp}$ decays.
to
To extract the branching fraction, we measure the reconstruction efficiency from MC and correct for a discrepancy between data and MC in the pion identification requirement. This correction is obtained from an inclusive $D^*$ control sample ($D^{*+} \to D^0 \pi^+$, $D^0 \to K^- \pi^+$), and is applied in bins of track momentum and polar angle. After this correction, the reconstruction efficiency is $10.4\%$.
We calculate systematic errors from the following sources: PDF shapes $^{+11.7}_{-12.0}\%$ (by varying parameters by $\pm 1 \sigma$); continuum rejection $\pm 6.5\%$ (by comparing the efficiency of the selection between data and MC for the $B^+ \to D^0\pi^+$ control sample); $\pi^0$ reconstruction efficiency $\pm 4.8\%$ (by comparing the yields of $\eta \to \pi^0\pi^0\pi^0$ and $\eta\to \gamma \gamma$ between data and MC); tracking finding efficiency $\pm 2.0\%$ (from a study of partially reconstructed $D^*$ decays); pion identification efficiency $\pm 0.8\%$ (using the method described above). The contributions are summed in quadrature to obtain a total systematic error of $^{+13.6}_{-13.8}\%$.
We measure the branching fraction for $B^0 \to \rho^\pm \pi^\mp$ to be $$\nonumber
{\mathcal B}\left( B^0 \to \rho^\pm \pi^\mp \right) =
\left( 29.1 ^{+5.0}_{-4.9}({\rm stat}) \pm 4.0({\rm syst}) \right)
\times 10^{-6}.$$
A difference in the untagged decay rates to $\rho^+\pi^-$ and $\rho^-\pi^+$ would indicate direct $CP$ violation [@gardner]. In order to measure this untagged asymmetry, we remove candidate events with ambiguous $\rho$ charge assignment, [*i.e.*]{} events in the regions of the Dalitz plot where more than one of the combinations $\pi^+ \pi^0$, $\pi^-\pi^0$, $\pi^+\pi^-$ are consistent with having originated from a $\rho$ resonance. These are the regions where interference effects are strongest [@snyder_quinn]. We fit the $\Delta E$ distributions for $B^0 \to \rho^+ \pi^- + \bar{B}^0 \to \rho^+ \pi^-$ (denoted as $B \to \rho^+ \pi^-$) and $B^0 \to \rho^- \pi^+ + \bar{B}^0 \to \rho^- \pi^+$ ($B \to \rho^- \pi^+$) candidates separately. The results are shown in Fig. \[fig:rhopmpimp\_asym\]. We find $36.7^{+15.3}_{-14.3}$ $B \to \rho^+ \pi^-$ events, and $81.5^{+16.8}_{-16.0}$ $B \to \rho^- \pi^+$ events. The charge asymmetry is calculated as $$\nonumber
{\mathcal A} =
\frac
{ N(B\to \rho^+\pi^-) - N(B\to \rho^-\pi^+) }
{ N(B\to \rho^+\pi^-) + N(B\to \rho^-\pi^+) }
= -0.38^{+0.19}_{-0.21} ({\rm stat}) ^{+0.04}_{-0.05} ({\rm syst}),$$ where the systematic error is estimated by varying the PDFs used in the fit, allowing for asymmetry in the shape and normalization of the continuum component, and allowing for charge dependence in the efficiency.
to
In order to obtain good sensitivity to the as yet unobserved decay $B^0 \to \rho^0 \pi^0$, additional discrimination against the continuum background is required. Therefore, a slightly different analysis procedure is followed. The majority of selection requirements are similar to those described above, however, neutral pion candidates are selected with a wider invariant mass window of $0.100 < M_{\gamma\gamma}/{\rm GeV}/c^2 < 0.165$ to allow for the resolution of high momentum $\pi^0$s. One additional requirement is that possible contributions to the $\pi^+\pi^-\pi^0$ final state from $b \to c$ decays are explicitly vetoed. Another is that in order to reduce the event multiplicity before the best candidate selection, a smaller window in $\left( \Delta E, M_{\rm bc} \right)$ is allowed. The selection region is $-0.2 < \Delta E/{\rm GeV} < 0.4$, $M_{\rm bc} > 5.23~{\rm GeV}/c^2$. Whilst these changes do not significantly affect the $\rho^0 \pi^0$ final state, they will be important in retaining a clean and unbiased $\pi^+\pi^-\pi^0$ Dalitz plot in the future.
In order to select $\rho^0\pi^0$ from the three-body $\pi^+\pi^-\pi^0$ candidates, we require $0.50 < M_{\pi^+\pi^-}/{\rm GeV}/c^2 < 1.10$ and $\left| \cos \theta^{\rho}_{\rm hel} \right| > 0.5$. Contributions from $B^0 \to \rho^{\pm}\pi^{\mp}$ are explicitly vetoed by rejecting candidates which fall into the invariant mass window of $0.50 < M_{\pi^{\pm}\pi^0}/{\rm GeV}/c^2 < 1.10$. This requirement also vetoes the region of the Dalitz plot where the interference between $\rho$ resonances is strongest.
The most notable difference from the selection requirements described above for the $\rho^{\pm}\pi^{\mp}$ final state is in the procedure to reject continuum background. Here, we make use of the additional discriminatory power provided by flavour tagging. In our published time-dependent analyses [@cpv], we define the variables $q$ and $r$ as being, respectively, the more likely flavour of the other $B$ in the event ($B^0\ (q = +1)$ or $\bar{B}^0\ (q = -1)$), and a measure of the confidence that the $q$ prediction is correct. As a corollary, events with a high value of $\left|qr\right|$ are well-tagged as either $B^0$ or $\bar{B}^0$, and hence are unlikely to originate from continuum processes. Moreover, since the flavour tagging algorithm relies on particle identification information, it is unlikely that there is any strong correlation with any of the topological variables used above to separate signal from continuum.
We use a large statistics sample of $\rho^0\pi^0$ MC and data from a continuum dominated sideband region ($5.23 < M_{\rm bc}/{\rm GeV}/c^2 < 5.26$ and $0.2 < \Delta E/{\rm GeV} < 0.4$), in order to simultaneously find the optimum selection requirements on $\left|qr\right|$ and ${\mathcal L}_s/\left( {\mathcal L}_s + {\mathcal L}_b \right)$, as defined above. We find the sensitivity to $\rho^0\pi^0$ is maximized by requiring $\left|qr\right| > 0.74$ and ${\mathcal L}_s/\left( {\mathcal L}_s + {\mathcal L}_b \right) > 0.9$, where the branching fraction of $B^0 \to \rho^0\pi^0$ is taken to be $1 \times 10^{-6}$ as input to the optimization procedure. The effect of the $\left|qr\right|$ requirement is shown in Fig. \[fig:qr\]. Some modest improvement is seen in $S/\sqrt{S+B}$, whilst $S/B$ increases dramatically.
to
As for $\rho^{\pm}\pi^{\mp}$, we obtain the signal yield by fitting the $\Delta E$ distribution after requiring events be in the $M_{\rm bc}$ signal region ($M_{\rm bc} > 5.269~{\rm GeV}/c^2$), and cross-check the result by fitting the $M_{\rm bc}$ distribution after requiring events be in the $\Delta E$ signal region ($-0.135 < \Delta E/{\rm GeV} < 0.080$). We model the signal using a Crystal Ball lineshape, with parameters determined from MC, and calibrated using control samples of $B^- \to D^0 \rho^-$, with $D^0 \to K^-\pi^+$, $\rho^- \to \pi^-\pi^0$ and $\bar{B}^0 \to D^{*+}\rho^-$, with $D^{*+} \to D^0\pi^+$ and the same decays of the $D^0$ and $\rho$. In both cases the $\pi^0$ is required to have momentum in the CM frame greater than $1.8~{\rm GeV}/c$, in order to have similar dynamics to the $\rho^0\pi^0$ final state. We include additional components in the fit to accomodate continuum background (modelled by a Chebyshev polynomial with parameters obtained from a fit to the sideband region), $\rho^+\rho^0$ (shape from smoothed histogram of MC events) and $\rho^+\pi^0$ (shape from smoothed histogram of MC events). Contributions from $b \to c$ transitions are negligible in the fitted range of $\Delta E$, whilst the tiny contributions possible from other rare $B$ decays are accounted for in the systematic error. The only free parameters in the fit are again the signal and continuum normalizations; the $\rho^+\rho^0$ yield and polarization are fixed according to our recent measurements [@jingzhi], while the $\rho^+\pi^0$ yield is fixed based on theoretical expectations.
The fit result is shown in Fig. \[fig:rho0pi0\_fit\]. The $\Delta E$ fit gives a signal yield of $6.6^{+3.2}_{-2.6}$ with a significance of $3.1 \sigma$. The $M_{\rm bc}$ fit, in which only components for signal and continuum are included, gives a yield of $6.4^{+3.1}_{-2.4}$ with $3.6 \sigma$ significance.
to
In order to check that the signal candidates originate from $B^0 \to \rho^0\pi^0$ decays, and not from either non-resonant $\pi^+\pi^-\pi^0$ or $\sigma \pi^0$, we relax the criteria on $M_{\pi^+\pi^-}$ and $\cos \theta^{\rho}_{\rm hel}$ in turn and look at the distribution of the candidate events in both $\Delta E$ and $M_{\rm bc}$ signal regions in those variables. These distributions are shown in Fig. \[fig:rho0pi0\_check\]. Whilst the statistics are too small to make quantitative statements, there is no evidence for any contribution other than $\rho^0\pi^0$. We also consider possible contamination from $\rho^{\pm}\pi^{\mp}$; from a large MC sample in which interference between resonances in not simulated, we expect $<0.1$ events to pass the selection requirements.
to
To obtain the branching fraction, we measure the efficiency using MC, and correct for the pion identification efficiency as above. The systematic error due to pion identification is $\pm 3\%$. We also correct for possible differences between data and MC due to the $\left|qr\right|$ and ${\mathcal L}_s/\left( {\mathcal L}_s + {\mathcal L}_b \right)$ selections; the statistical errors of the control samples ($\bar{B}^0 \to D^{*+}\rho^-$ with $D^{*+} \to D^0\pi^+$, $D^0 \to K^-\pi^+$, $\rho^- \to \pi^-\pi^0$ and $B^- \to D^0\rho^-$ with $D^0 \to K^- \pi^+$, $\rho^- \to \pi^-\pi^0$ respectively) account for the largest contribution to the systematic error ($\pm 18\%$). We also calculate systematic errors due to: PDF shapes, by varying parameters by $\pm 1\sigma$ ($\pm 3\%$); $\pi^0$ reconstruction efficiency, from the inclusive $\eta$ study ($\pm 4\%$); track finding efficiency, from the partially reconstructed $D^*$ study ($\pm 2\%$). We repeat the fit after changing the normalization of the other $B$ decay components according to the error in their branching fractions, and obtain systematic errors from the change in the result. For the unobserved mode $\rho^+\pi^0$ we vary the normalization by a factor of two. The total systematic eror due to rare $B$ decays is $\pm 5\%$, we also verify that in the case that all rare $B$ contributions are simulaneously increased to their maximum values, the statistical significance remains above $3\sigma$. The total systematic error is $\pm 20\%$, and we measure the branching fraction of $B^0 \to \rho^0\pi^0$ to be $$\nonumber
{\mathcal B}\left( B^0 \to \rho^0\pi^0 \right) =
\left( 6.0 ^{+2.9}_{-2.3} ({\rm stat}) \pm 1.2 ({\rm syst}) \right)
\times 10^{-6}.$$
In order to test the robustness of this result, a number of cross-checks are performed. We vary the selection requirements on $\left|qr\right|$ and ${\mathcal L}_s/\left( {\mathcal L}_s + {\mathcal L}_b \right)$. In all cases, consistent central values for the branching fraction are obtained. We also select $\rho^{\pm}\pi^{\mp}$ candidates after using this continuum rejection technique, and measure a branching fraction for $B^0 \to \rho^{\pm}\pi^{\mp}$ which is consistent with that reported above. Furthermore, adopting the continuum rejection technique of our $\rho^{\pm}\pi^{\mp}$ analysis and selecting $\rho^0\pi^0$ candidates also results in a consistent central value for the branching fraction of $B^0 \to \rho^0\pi^0$, although the signal is insignificant above the large continuum background.
In summary, we have measured the branching fraction $$\nonumber
{\mathcal B}\left( B^0 \to \rho^\pm \pi^\mp \right) =
\left( 29.1 ^{+5.0}_{-4.9}({\rm stat}) \pm 4.0({\rm syst}) \right)
\times 10^{-6},$$ in agreement with previous measurements. We also measure the untagged asymmetry to be $$\nonumber
{\mathcal A} =
-0.38^{+0.19}_{-0.21} ({\rm stat}) ^{+0.04}_{-0.05} ({\rm syst}),$$ consistent with zero with the current statistical precision. In addition, we observe evidence, with $3.1\sigma$ statistical significance, for $B^0 \to \rho^0\pi^0$ with a branching fraction of $$\nonumber
{\mathcal B}\left( B^0 \to \rho^0\pi^0 \right) =
\left( 6.0 ^{+2.9}_{-2.3} ({\rm stat}) \pm 1.2 ({\rm syst}) \right)
\times 10^{-6}.$$
This is the first evidence for $B^0 \to \rho^0\pi^0$, with a branching fraction higher than most predictions [@snyder_quinn]. This may indicate that some contribution to the amplitude is larger than expected, which may complicate the extraction of $\phi_2$ from time-dependent analysis of $\rho^{\pm}\pi^{\mp}$.
We wish to thank the KEKB accelerator group for the excellent operation of the KEKB accelerator. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and the Japan Society for the Promotion of Science; the Australian Research Council and the Australian Department of Industry, Science and Resources; the National Science Foundation of China under contract No. 10175071; the Department of Science and Technology of India; the BK21 program of the Ministry of Education of Korea and the CHEP SRC program of the Korea Science and Engineering Foundation; the Polish State Committee for Scientific Research under contract No. 2P03B 01324; the Ministry of Science and Technology of the Russian Federation; the Ministry of Education, Science and Sport of the Republic of Slovenia; the National Science Council and the Ministry of Education of Taiwan; and the U.S. Department of Energy.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In this paper, security of practically decoy state quantum key distribution under fake state attack is considered. If quantum key distribution is insecure under this type of attack, decoy sources can not also provide it with enough security. Strictly analysis shows that Eve should eavesdrop with the aid of photon-number-resolving instruments. In practical implementation of decoy state quantum key distribution where statistical fluctuation is considered, however, Eve can attack it successfully with threshold detectors.
Keywords: fake state attack, decoy state, quantum key distribution, security
author:
- 'Yong-gang Tan'
title: Fake state attack on practically decoy state quantum key distribution
---
Quantum key distribution (QKD) is an important application of quantum information, with which two distant parties (the information sender, Alice and the information receiver, Bob) can share a string of secure key with the presence of an eavesdropper, Eve [@Bennett84; @Ekert91; @Gisin02]. It has been proven that quantum principles can provide it with unconditional security when it is implemented with ideal devices [@Mayers01; @Lo99; @Shor00]. In practical implementation of QKD, however, real-life devices are taken used. They are imperfect and apt to some sophisticated eavesdropping [@Huttner95; @Brassard00; @Lutkenhaus02; @Qi07; @Fung07; @Zhao08; @Makarov05; @Lydersen10; @Yuan10; @Lydersen101; @Gerhardt11], part of which have been realized with lab settings. Furthermore, in QKD realization, Alice and Bob’s experimental conditions are assumed to be based on present technology but Eve’s ability is only limited by quantum principles. Then those eavesdropping schemes still lacking experimental demonstration should also be considered seriously. Recently, fake state attack is experimentally proven to be fatal to some commercial quantum key distribution systems, with which the latent eavesdropper can obtain full information shared between Alice and Bob without been detected [@Makarov05; @Lydersen10; @Yuan10; @Lydersen101; @Gerhardt11].
The fake state attack is a type of intercept-resent attack, where Eve blocks all Alice’s pulses and measures them on randomly chosen bases. Then she prepares her measurement results on fresh pulses and transfers them to Bob. At the same time, she controls Bob’s detectors to work in linear mode: if Bob has the same basis choices as Eve, Eve’s pulses will provide enough power above the threshold value to generate triggers and Bob gets Eve’s bit values; when their basis choices are different, however, Eve’s pulses are split below the threshold intensity and unable to introduce click on Bob’s detectors. It is apparent that Eve’s intervention introduces tolerably error rate and generates identical key string as the legal users’. Then Alice and Bob will acknowledge the validity of their key and ignore Eve’s presence. With a half probability, Bob and Eve will have the same choices on their measurement bases. Thus Eve may eavesdrop on their communication successfully if the combining efficiency of the quantum channel and the measurement devices is less than $\frac{1}{2}$.
Suppose Alice has a weaken coherent source whose photon number obeys Poisson distribution $$p_{n}(\mu)=\frac{\mu^{n}}{n!}e^{-\mu},$$ where $\mu$ is average intensity of the source and $n$ is photon number of the incoming pulses. She randomly chooses her bases and prepares her bit values on the pulses, then she transfers them to Bob. To avoid being caught on line, Eve must make Bob’s gains and error rates identical to that when there is no eavesdropper. That is, Eve must make her eavesdropping satisfy $$\begin{array}{lll}
\frac{1}{2}\eta_{f}\sum_{n=1}^{\infty}p_{n}(\mu)&=&\sum_{n=0}^{\infty}p_{n}(\mu)[1-(1-p_{d})^{2}(1-\eta)^{n}],\\
e_{f}\eta_{f}\sum_{n=1}^{\infty}p_{n}(\mu)&=&\sum_{n=0}^{\infty}p_{n}(\mu)\{1-(1-p_{d})^{2}(1-\eta)^{n}-(1-p_{d})\\
&\times&[(1-{\eta}e_{d})^{n}-(1-\eta+{\eta}e_{d})^{n}]\}.
\end{array}$$ Here $\eta_{f}$ is the probability Eve will prepare fresh pulses according to her measurement results, $e_{f}$ is the probability she prepares wrong bit value on the fresh pulses. And $e_{d}$ is the probability of misalignment between Alice and Bob. If Eve can keep alignment between her measurement bases and Alice’s preparing bases, and that between her preparing bases and Bob’s measurement bases at the same time, she can control the error rate on Bob’s results at her will. Furthermore, practical QKD system is very lossy [@Gobby04], the relationships in Eq. (2) should be simulated by Eve easily. Thus if Eve can blind Bob’s detectors, the QKD system will be totally insecure.
In practical implementation of QKD, decoy sources are usually added in [@Hwang03; @Lo05; @Wang05]. They have the same characters with that of signal source apart from their average intensities, that is $$p_{n}(\nu)=\frac{\nu^{n}}{n!}e^{-\nu}.$$ Alice randomly encodes her bit value on signal source or decoy sources. Eve can not tell the decoy sources from the signal source, then she must treat all sources in the same way. Furthermore, she should mock the loss and noise in the quantum channel. Or else, her intervention will inevitably introduce different disturbances on Bob’s results from different sources. It is easy to verify that the relationship in Eq. (2) can not be met for the signal source and the decoy sources at the same time. In order to eavesdrop on the decoy state QKD, Eve should have the ability of differentiating photon number, with which she can treat pulses with the same photons similarly.
In photon-number-splitting (PNS) attack, Eve is assumed to have ability of differentiating photon number in pulses without affect their polarizations [@Huttner95; @Brassard00; @Lutkenhaus02]. Then quantum nondemolition (QND) measurement on photon number is required, and this is still missed with present technology. In the fake state attack, however, Eve can measure polarizations on the pulses directly, which means she can get photon number and polarization of the pulses at the same time with photon-number-resolving detectors [@Rosenberg05; @Hadfield09]. If she has the ability of differentiate photon number, she can treat pulses with the same photons in a similar way. In order to simulate the lossy and noise for all sources, Eve’s eavesdropping should satisfy $$\begin{array}{l}
\frac{1}{2}\eta_{n}^{f}=1-(1-p_{d})^{2}(1-\eta)^{n},\\
e_{n}^{f}\eta_{n}^{f}=1-(1-p_{d})^{2}(1-\eta)^{n}-(1-p_{d})[(1-{\eta}e_{d})^{n}-(1-\eta+{\eta}e_{d})^{n}].
\end{array}$$ Here $\eta_{n}^{f}$ and $e_{n}^{f}$ have similar definition as that in Eq. (2), but they corresponds to pulses with definite photon number $n$. Eve detects nothing from Alice when $n=0$, however, she should prepare random-bit-value pulses with probability $4p_{d}-2p_{d}^{2}$ in order to simulate dark count rate on Bob’s detectors. Noticing that Eq. (4) has nothing to do with $p_{n}(\mu)$ and $p_{n}(\nu)$, thus Eve can eavesdrop on the decoy state QKD successfully if she has a set of photon-number-resolving detectors.
Now it is interesting whether Eve can eavesdrop on the decoy state QKD protocol without photon-number-resolving detectors. However, as mentioned before, eavesdropping on the decoy sources should have similar relationships as those in Eq. (2), that is, $$\begin{array}{lll}
\frac{1}{2}\eta_{f}\sum_{n=1}^{\infty}p_{n}(\nu)&=&\sum_{n=0}^{\infty}p_{n}(\nu)[1-(1-p_{d})^{2}(1-\eta)^{n}],\\
e_{f}\eta_{f}\sum_{n=1}^{\infty}p_{n}(\nu)&=&\sum_{n=0}^{\infty}p_{n}(\nu)\{1-(1-p_{d})^{2}(1-\eta)^{n}-(1-p_{d})\\
&\times&[(1-{\eta}e_{d})^{n}-(1-\eta+{\eta}e_{d})^{n}]\}.
\end{array}$$ And one can find that there are not such $\eta_{f}$ and $e_{f}$ to meat the relationships in Eq. (2) and those in Eq. (5) at the same time. However, considering the imperfection of practical implementation of decoy state QKD, the relationships in Eq. (2) and Eq. (5) may be loosen.
Experimentally, Alice and Bob’s key is distributed within a finite period of time, generally in several hours [@Ma05]. Then the pulses generated by Alice should also be finite. We assume the number of pulses emitted from the sources is $N=10^{10}$ in the following discussion. If Bob’s expectingly detections is $p_{det}$ under ideal circumstance, and his detecting events with finite resources is $p_{det}^{\prime}$, $p_{det}^{\prime}$ should deviate from $p_{det}$ with a small fluctuation $\delta_{p_{det}}$. The probability $P(|p_{det}-p_{det}^{\prime}|>\delta_{p_{det}})$ can be estimated to be less than $q=\exp(-\frac{N\delta_{p_{det}}^{2}}{4p_{det}})$. If we require $q=\exp(-25)$, $\frac{\delta_{p_{det}}^{2}}{p_{det}}$ can be estimated to be $10^{-8}$ approximately, and $\delta_{p_{det}}$ can be calculated as $10^{-4}p_{det}^{\frac{1}{2}}$ accordingly. Thus in practical implementation of fake state attack, Eve should ensure Bob’s detecting events satisfy $$p_{det}-10^{-4}p_{det}^{\frac{1}{2}}<p_{det}^{\prime}<p_{det}+10^{-4}p_{det}^{\frac{1}{2}}.$$
Strictly speaking, the statistical fluctuations on different sources are usually not the same as the number of pulses generated from different sources may be not assigned to be the same. At the same time, the probabilities of detecting events for different sources are not identical because of their disparate intensities. As the total number of the pulses is $N$, the practical number of pulses assigned for different sources should be less than this value, the practically tolerant fluctuation should be greater than that in Eq. (6). It means Alice and Bob should accept the validity of their results if the statistical fluctuations on the detecting events from every sources satisfy the relationship in Eq. (6). And similar relationship can be obtained for the gain of QBER on every source, that is $$p_{err}-10^{-4}p_{err}^{\frac{1}{2}}<p_{err}^{\prime}<p_{err}+10^{-4}p_{err}^{\frac{1}{2}}.$$
It is apparent that Bob’s expecting detecting results should be $p_{det}(\mu)=1-(1-p_{d})^{2}e^{-\eta\mu}$ for signal source, and $p_{det}(\nu)=1-(1-p_{d})^{2}e^{-\eta\nu}$ for decoy sources. And the actual detecting events for them should be $p_{det}^{\prime}(\mu)=\frac{1}{2}\eta_{f}(1-e^{-\mu})$ and $p_{det}^{\prime}(\nu)=\frac{1}{2}\eta_{f}(1-e^{-\nu})$ respectively. Similarly, the expecting error rate for signal source and decoy sources are $p_{err}(\mu)=\frac{1}{2}\sum_{n=0}^{\infty}p_{n}(\mu)\{1-(1-p_{d})^{2}(1-\eta)^{n}-(1-p_{d})[(1-{\eta}e_{d})^{n}-(1-\eta+{\eta}e_{d})^{n}]\}$ and $p_{err}(\nu)=\frac{1}{2}\sum_{n=0}^{\infty}p_{n}(\nu)\{1-(1-p_{d})^{2}(1-\eta)^{n}-(1-p_{d})[(1-{\eta}e_{d})^{n}-(1-\eta+{\eta}e_{d})^{n}]\}$. And the actual error rate for them can be calculated as $p^{\prime}_{err}(\mu)=\frac{1}{2}e_{f}\eta_{f}(1-e^{-\mu})=e_{f}p^{\prime}_{det}(\mu)$ and $p^{\prime}_{err}(\nu)=\frac{1}{2}e_{f}\eta_{f}(1-e^{-\nu})=e_{f}p^{\prime}_{det}(\nu)$.
We can estimate the feasibility of Eve’s attack with the experimental parameters in [@Gobby04], that is, $p_{d}=8.5\times10^{-7}$, $\eta_{B}=4.5\%$, $e_{d}=3.3\%$ and loss coefficient $\alpha$ in the quantum channel is $0.21$ dB/km. If the transmission distance between Alice and Bob is $120$km, one can obtain $\eta=1.359\times10^{-4}$. When there are only two sources, that is, a signal source with intensity $\mu=0.479$ and a weaker decoy state with intensity $\nu=0.127$ [@Ma05]. As the statistical fluctuations on the the results of signal source satisfy the relationships in Eq. (6) and Eq. (7), its $\eta_{f}$ should range from $3.467\times10^{-4}$ to $3.553\times10^{-4}$, and its $e_{f}$ can range from $4.178\times10^{-2}$ to $4.806\times10^{-2}$. Similarly, statistical fluctuation on the the results of weaker decoy source should satisfy the relationships in Eq. (6) and Eq. (7), its $\eta_{f}$ can be calculated to range from $3.106\times10^{-4}$ to $3.252\times10^{-4}$, and its $e_{f}$ ranges from $6.705\times10^{-2}$ to $8.307\times10^{-2}$. As there is no overlap on the parameters of both sources, it seems that Eve can not eavesdrop on the decoy state QKD protocol with threshold detectors.
Noticing that the dark count rate functions importantly in practical implementation of decoy state QKD protocol when the transmission distance is comparably long. Furthermore, though threshold detectors can not tell the photon number in the incoming pulses, they can differentiate vacuum pulses from non-vacuum pulses. Then it may help Eve with her eavesdropping if she treats the vacuum pulses and non-vacuum pulses in different ways. She prepares random-bit-value pulses with probability $4p_{d}-2p_{d}^{2}$ for Bob when she detecting nothing from Alice. It is easily verified that Eve’s eavesdropping results on the vacuum pulses coincide well with what Bob expecting for. When there is nonvacuum pulses, she makes fresh pulses according to her results with probability $\eta_{f}$ and introduces error on them with probability $e_{d}$. Here Eve makes error on the nonvacuum pulses with probability $e_{d}$ because errors introduced on nonvacuum pulses are mainly introduced by misalignment between Alice and Bob. Then for signal source, one can obtain $$\begin{array}{lll}
\frac{1}{2}\eta_{f}\sum_{n=1}^{\infty}p_{n}(\mu)&=&\sum_{n=1}^{\infty}p_{n}(\mu)[1-(1-p_{d})^{2}(1-\eta)^{n}],\\
\eta_{f}e_{d}\sum_{n=1}^{\infty}p_{n}(\mu)&=&\sum_{n=1}^{\infty}p_{n}(\mu)\{1-(1-p_{d})^{2}(1-\eta)^{n}-(1-p_{d})\\
&\times&[(1-{\eta}e_{d})^{n}-(1-\eta+{\eta}e_{d})^{n}]\}.
\end{array}$$ Similar relationship can also be obtained for decoy source.
The probability of expecting detections $p_{det}$ both for signal source and decoy source can still be calculated as that above. However, the actual detections $p^{\prime}_{det}$ for them are altered slightly. That is, $p^{\prime}_{det}(\mu)=e^{-\mu}[1-(1-p_{d})^{2}]+\frac{1}{2}\eta_{f}(1-e^{-\mu})$ and $p^{\prime}_{det}(\nu)=e^{-\nu}[1-(1-p_{d})^{2}]+\frac{1}{2}\eta_{f}(1-e^{-\nu})$. Similarly, the expressions for $p_{err}(\mu)$ and $p_{err}(\nu)$ are still the same. And $p^{\prime}_{err}(\mu)$ and $p^{\prime}_{err}(\nu)$ should be recalculated as $\frac{1}{2}e^{-\mu}[1-(1-p_{d})^{2}]+\frac{1}{2}e_{d}\eta_{f}(1-e^{-\mu})$ and $\frac{1}{2}e^{-\nu}[1-(1-p_{d})^{2}]+\frac{1}{2}e_{d}\eta_{f}(1-e^{-\nu})$ respectively. As their statistical fluctuations should still be bounded with the relations in Eq. (6) and Eq. (7). With simple calculation, we find there is no such $\eta_{f}$ for signal source, and $\eta_{f}$ for decoy source ranges from $2.855\times10^{-4}$ to $3.001\times10^{-4}$. That is, this scheme is still inefficient in helping Eve to eavesdrop on decoy state QKD protocol with threshold detectors.
Eve takes control the whole quantum channel, however, she may not set her eavesdrop point adjacent to Alice’s lab. Her intervention site may be anywhere between Alice’s and Bob’s labs. We will show that this change will help Eve to Eavesdrop on the decoy state QKD protocol successfully with threshold detectors. Let the distance between Alice’s lab and Bob’s eavesdropping site be $l$ km, it is apparent smaller $l$ requires ability to discriminate photon number in the pulses, and larger $l$ may lead to failure of her blinding attack. Then the optimal site should have largest $l$ where Eve can carry out her eavesdropping successfully. The transmission efficiency at this point can be calculated as $\eta_{l}=10^{-\frac{\alpha{l}}{10}}$. And the statistical distribution in the incoming pulses can be represented as $$\begin{array}{lll}
p_{n}^{l}(\mu)&=&\frac{(\eta_{l}\mu)^{n}}{n!}e^{-\eta_{l}\mu},\\
p_{n}^{l}(\nu)&=&\frac{(\eta_{l}\nu)^{n}}{n!}e^{-\eta_{l}\nu}.
\end{array}$$
If Eve takes her eavesdropping scheme as that in Eq. (8), one can obtain $$\begin{array}{lll}
\frac{1}{2}\eta_{f}\sum_{n=1}^{\infty}p_{n}^{l}(\mu)&=&\sum_{n=1}^{\infty}p_{n}^{l}(\mu)[1-(1-p_{d})^{2}(1-\eta^{\prime})^{n}],\\
\eta_{f}e_{d}\sum_{n=1}^{\infty}p_{n}^{l}(\mu)&=&\sum_{n=1}^{\infty}p_{n}^{l}(\mu)\{1-(1-p_{d})^{2}(1-\eta^{\prime})^{n}-(1-p_{d})\\
&\times&[(1-{\eta^{\prime}}e_{d})^{n}-(1-\eta^{\prime}+{\eta^{\prime}}e_{d})^{n}]\},
\end{array}$$ with $\eta^{\prime}=\eta_{b}10^{-\frac{120-l}{10}}$ for signal source. And similar relationships can be obtained for decoy source $$\begin{array}{lll}
\frac{1}{2}\eta_{f}\sum_{n=1}^{\infty}p_{n}^{l}(\nu)&=&\sum_{n=1}^{\infty}p_{n}^{l}(\mu)[1-(1-p_{d})^{2}(1-\eta^{\prime})^{n}],\\
\eta_{f}e_{d}\sum_{n=1}^{\infty}p_{n}^{l}(\nu)&=&\sum_{n=1}^{\infty}p_{n}^{l}(\mu)\{1-(1-p_{d})^{2}(1-\eta^{\prime})^{n}-(1-p_{d})\\
&\times&[(1-{\eta^{\prime}}e_{d})^{n}-(1-\eta^{\prime}+{\eta^{\prime}}e_{d})^{n}]\}.
\end{array}$$ As Eve’s detecting efficiency is very lower, it is easy to verify that Eve can set $l=120$ km. We can then obtain $\eta_{f}$ ranging from $8.893\times10^{-2}$ to $9.120\times10^{-2}$ for signal source, and it ranges from $8.775\times10^{-2}$ to $9.229\times10^{-2}$ for decoy source. Then Eve can launch fake state attack at $l=120$ km with threshold detectors just by preparing what she have measured with probability $\eta_{f}$ ranging from $8.893\times10^{-2}$ to $9.120\times10^{-2}$, and she introduces error on them with probability $e_{d}$.
Then when statistical fluctuation is considered, Eve can eavesdropping on decoy state QKD even with threshold detectors. She may fail to eavesdrop successfully when her intervention site is closer to Alice’s lab, and numerical simulation shows it may be easier for her to attack on this protocol when her intervention site is farer away from Alice’s lab. This is because the nearer to Bob’s lab, the greater probability of single-photon pulses for nonvacuum pulses can be obtained. Then Eve can omit the effect of multi-photon pulses treat all pulses as single photons. In practical decoy state QKD protocol, Alice may introduce vacuum decoy state to estimate the dark counts on Bob’s detectors. [@Ma05]. As Eve prepares random-bit-value pulses with probability $4p_{d}-2p_{d}^{2}$ when she detects nothing, however, the statistical fluctuations on the vacuum decoy state can be verified to be met automatically. In order to understand Eve’s eavesdropping better, we give a simulation numerically on the relation between Eve’s $\eta_{f}$ and her intervention site $l$, as is plotted as that in Fig. 1. It shows that there is not suitable $\eta_{f}$ and for signal source when $l\le10$ km. Furthermore, Eve can not launch her fake state attack on this protocol when her intervention site $l$ is less than $30$ km as there is no overlap on $\eta_{f}$ for both sources. When $l$ is greater than $45$ km, one can find the suitable $\eta_{f}$ for signal source also suits for decoy source.
![Relationship between Eve’s probability for her to prepare fresh pulses according to her results, $\eta_{f}$ and her intervention site, $l$. It suitable value should be chosen from the area greater than the lower bounds and less than the upper bounds of $\eta_{f}$ for both sources.[]{data-label="fig:etaf"}](etaf.eps){width="100.00000%"}
It has been proven that some commercial QKD systems may be totally insecure under fake state attack [@Lydersen10; @Yuan10; @Lydersen101; @Gerhardt11], thus we hope that decoy states is efficiency in combatting against this brutal attack. As we have shown above, however, decoy states can not also provide them with enough security. We have shown that Eve can launch her fake state attack successfully. Especially, we have proven that she can eavesdrop on the decoy state QKD without any photon-number-resolving instrument when statistical fluctuation is considered on Bob’s results. With the presence of new technology, especially with improvement on Bob detecting efficiency, Alice and Bob may overcome this loophole. However, other loopholes still unknown to people may also threaten the security of practical QKD. And it has been claimed QKD is superior to classical cryptography as quantum principle provide it with physically secure. In order to avoid the mouse and cat game between legal users and eavesdropper in QKD, new protocols should be presented to combat all these loopholes in principle [@Mayers98; @Acin07; @Pironio09; @Lo11]. This work is sponsored by the National Natural Science Foundation of China (Grant No 10905028) and HASTIT.
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---
abstract: 'A permanent challenge in physics and other disciplines is to solve partial differential equations, thereby a beneficial investigation is to continue searching for new procedures to do it. In this Letter, a novel Monte-Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the paths space with a probability functional obtained by using the maximum caliber principle. The methodology was applied to the free particle and the harmonic oscillator problems, where the numerically-averaged path obtained from the Monte-Carlo simulation converges to the analytical solution from classical mechanics, in an analogous way with a canonical system where energy is minimized by sampling the state space and computing the average state for each system. Thus, we expect that this procedure can be general enough to solve other differential equations in physics and to be a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems.'
author:
- 'Diego González$^{1,2}$, Sergio Davis$^{3}$, Sergio Curilef$^{1}$.'
bibliography:
- 'maxcal.bib'
title: 'Solving equations of motion by using Monte Carlo Metropolis: Novel method via Random Paths and Maximum Caliber Principle'
---
Introduction
============
The main objective of this work is to show a new framework for the study of dynamical systems which are described by a Lagrangian, being a first approach for the understanding of non-equilibrium statistical mechanics (NESM) by using constraints, as performed in statistical mechanics.
Here we propose a technique capable of simulating deterministic, dynamical systems through a stochastic formulation. This technique is based on sampling a statistical ensemble of paths defined by having the maximum path entropy (also known as the *caliber*) available under imposed time-dependent constraints. This approach is known as the Maximum Caliber principle [@Jaynes1980], a generalization of Jaynes’ principle of maximum entropy [@Jaynes1957; @Cafaro2016; @General2018].
The principle of Maximum Entropy (MaxEnt) is a systematic method for constructing the simplest, most unbiased probability distribution function under given constraints, a conceptual generalization of Gibbs’ method of ensembles in Statistical Mechanics. The complete generality of the principle makes it widely used in several areas of Science, such as astronomy, ecology, biology, quantitative finance, image processing, electronics and physics among others. According to Jaynes, choosing a candidate probability distribution by maximizing its entropy is a rule of inferential reasoning far beyond its original application in Physics, which makes this rule a powerful tool for creating models in any context.
MonteCarlo Metropolis (MCM) is an computational algorithm for obtaining random samples drawn from a probability distribution. This probability distribution is usually constructed by using MaxEnt. MCM is used for the understanding of different systems in Physics [@Binder86; @Graham2013] such as Spin Models, Material Simulations, Termodynamics, among others. An analogous methodology is presented, by assigning probabilities to paths instead of probabilities of states, allowing the minimization of functional quantities such as the classical action instead of the energy.
It is possible to directly use MaxEnt instead of MaxCal as an inference methodology for dynamical systems, by constraining time-dependent functions, and this has been used to understand Newtonian dynamics [@Caticha2007] and the Schrödinger equation [@Caticha2011] in terms of traditional MaxEnt. A new approach has been explored recently for recovering frameworks for dynamical systems by using MaxCal and the paths space [@Gonzalez2014; @Davis2015; @Gonzalez2016; @Gonzalez2016b], and this work shows a numerical implementation of this new approach, exposing the capability for solving complex problems in NESM. In summary, this constitutes a novel approach to the study of dynamical systems by using Maximum Caliber for Monte Carlo simulation.
Creating a paths ensemble: The Maximum Caliber Principle {#sec_maxcal}
========================================================
The Maximum Caliber principle allows the definition of a unique paths ensemble given prior information and a number of dynamical constraints[@Jaynes1980; @Grandy2008; @Gonzalez2016].
MaxCal is similar to the Maximum Entropy Principle (MaxEnt)[@Jaynes1982; @Stock2008; @Presse2013; @Davis2015; @Gonzalez2016b] but defined over the paths space, allowing to define a probability functional $P[x]$ as follows. Consider a path $x \in \mathbb{X}$. In order to construct a probability functional for each path $P[x]$, given an initial probability (prior) $P_0[x]$ and an arbitrary constraint $$\Big<A[x]\Big> = a,
\label{action}$$ the Caliber (or path entropy) $$S[P_0 \rightarrow P] = -\int_{\mathbb{X}} D_x P[x] \ln \frac{P[x]}{P_0[x]},$$ must be maximized under the constraint in Eq. \[action\] and the requirement that probability is normalized, $$\int_{\mathbb{X}} D_x P[x] = 1.$$ Then, the probability functional obtained is, $$P[x|\beta]= \frac{1}{Z(\beta)}P_0[x]\exp(-\beta A[x]),
\label{probability}$$ where $Z(\beta)$ is the partition function and $\beta$ is the Lagrange multiplier, given by the constraint equation $$-\frac{\partial}{\partial \beta}\ln Z(\beta) = a. % Cambié f por a$$
In this formalism, $\beta$ is analogous to the inverse of the temperature $\beta=1/k_B T$ in the canonical ensemble of Statistical Mechanics. Here the expected value of an arbitrary functional $F[x]$ is given by $$\Big< F[x] \Big>_\beta = \frac{1}{Z(\beta)} \int_\mathbb{X} D_x \exp(-\beta A[x]) \; F[x],$$ but, perhaps more importantly, the expectation value of a function over time can be defined similarly as $$\Big< f(\dot x, x, t) \Big>_{\beta, t} = \frac{1}{Z(\beta)} \int_\mathbb{X} D_x \exp(-\beta A[x]) \; f(\dot x, x, t).$$ This last relation shows that MaxCal can be used to understand macroscopic properties of time-dependent systems, which are the main elements in NESM.
Least Action Principle and Most Probable Path {#leastaction}
=============================================
From classical mechanics it is well known that the path followed by a particle under a potential $V(x; t)$ under the boundary conditions $x(0)=0$ and $x(T)=x_f$ is the one given by the least action principle [@Lanczos1970; @Feynman2005], which in practice leads to an equation of motion describing the evolution of the particle from $0$ to $T$.
The classical action is a functional defined as $A[x]= \int_{0}^{T} dt \; L(\dot x,x;t)$, where $L(\dot x,x;t)$ is called the Lagrangian of the system, which for classical systems is $$L(\dot x,x;t) = \frac{m \dot x^{2}}{2} - V(x;t).$$ For a MaxCal framework where the classical action is constrained, following an analogous treatment to the constraint in Eq. \[action\] the probability functional is of the form given in Eq. \[probability\]. The most probable path can be obtained by finding the extrema of the functional $P[x]$, by solving the equation $\frac{\delta P[x]}{\delta x(t')} = 0$. This is because the exponential function in Eq. \[probability\] is convex and monotonically increasing, and so maximum probability is equivalent to imposing that $x$ should be an extremum for the argument of the exponential, $$- \beta \; \frac{\delta A[x]}{\delta x(t')} = 0.
\label{eq_extremum_A}$$
If the Lagrange multiplier is positive ($\beta > 0$), the requirement is that the action is actually a minimum. This equation is precisely the Euler-Lagrange equation of motion for the Lagrangian [@Lanczos1970; @Gelfand2000]. In summary, the most probable path and the least action path in a MaxCal framework are the same. According to this, sampling trajectories from the probability distribution in Eq. \[probability\] using Monte Carlo methods and computing the averages of quantities, should converge to a description of the dynamical properties of a classical system evolving in time.
Another important consequence of the use of MaxCal and the form of the probability is that the most probable path in general coincides with the average path according to the central limit theorem.
Computational method
====================
In order to define the elements on the paths space $\mathbb{X}$, for an $N$-dimensional path $x$, it is always possible to write it in a orthonormal basis $\phi_{i}$ of the form $$x(t) = \sum_i^{n} a_i \phi_i(t) = x(t; \bm{a}).
\label{eq_basis_expand}$$
Then, by changing the parameters $a_i$ it is possible to map the entire space of paths $x$. In other words, there is a one-to-one correspondence between an arbitrary path $x$ and its parameter vector $\bm{a}$, so the action becomes a function of $\bm{a}$, namely $\mathcal{A}(\bm{a}):= A[x(\cdot\; ;\bm{a})]$ and the problem of path sampling reduces to ordinary sampling of $N$-dimensional states $\bm{a}$, $$P(\bm{a}|\lambda) = \frac{1}{Z(\lambda)}\exp(-\lambda \mathcal{A}(\bm{a})).$$
The choice of the basis functions $\phi_i$ is, in principle, arbitrary. However, a convenient choice can be made related to the particular conditions of the problem to be solved. In this case, the target is the study of classical dynamical systems by using the MCM where the most of the problems have well-defined boundary conditions, therefore it is important to find a basis set in which one can easily generate paths in the desired paths space holding the required, fixed boundary conditions. For this reason we considered the Bézier curves as a basis set.
Bézier curves are defined by “control points” $c_{i}$, where the first $c_{0}$ and the last $c_{n}$ control point determine the boundary conditions of the curve, allowing the mapping of the paths space with well-defined boundary conditions.
For an $N$-dimensional path $x$ with boundary conditions $x(t_{0})= c_{0}$ and $x(t_{f}) = c_{n}$, a Bézier curve is defined of the form $$x(t) = \sum_{i=0}^{n} c_{i} \; B_i(t; n),
\label{path_bezier}$$ where the basis functions $B_i(t; n)$ are the Bernstein polynomials, $$B_{i}(t; n) = {n \choose i} \frac{(t-t_{0})^{i} (t_{f}-t)^{n-i}}{(t_{f}-t_{0})^{n}}.$$
Following these definitions it is clear that all Bézier curves automatically follow the specified boundary conditions at $t=t_0$ and $t=t_f$.
Implementation of the MonteCarlo Metropolis for sampling paths space
====================================================================
The Monte Carlo Metropolis (MCM) implementation is usually employed for sampling a multidimensional space governed by a probability distribution \cite{}. The following MCM implementation is used for sampling the paths space $\mathbb{X}$ governed by a probability functional obtained via MaxCal. This procedure makes it possible to find the minimum action path.
For a given action $A[x]$ and boundary conditions $x(t_{0})= c_{0}$ and $x(t_{f}) = c_{n}$, the MCM evolution for samplig the paths space it is implemented as shown in Fig. \[MCMdiagram\]
![Explanatory diagram for a MCM sampling in paths space.[]{data-label="MCMdiagram"}](Diagrama_Big.pdf){width="\textwidth"}
By performing this process in an iterative way the paths space is sampled, allowing the calculation of properties for the system which is determined by the classical action used. As shown in Section \[leastaction\], the probability distribution obtained when the classical action is constrained allows the sampling of the path space where the most probable path and the least action path coincide. Finally, $\lambda$ can be related to the inverse of temperature as the usual MCM, due to the fact that, as $\lambda \rightarrow 0$ the sampled paths will be random over the space, while taking $\lambda \rightarrow \infty$ constrains the sampled paths closer to the least action path. The value for $\lambda$ in a MCM is related with the change from $x$ to $x'$, and empirically this change must be adjusted to have approximately a $80\%$ acceptance rate.
Results and discussion {#resultados}
======================
Free Particle Action
--------------------
The equation of motion for a free particle is obtained by minimizing the classical action $$A[x] = \int_{t_0}^{t_f} dt \; \frac{m \dot x(t)^2}{2}$$ according to the least action principle. For a free particle with mass $m=1$ and boundary conditions $x(0)=0$ and $x(1)=1$, the analytical solution for the least action path is the straight line $x(t) = t$.
By using MCM simulation as shown in Fig. \[fig\_freepart\], we have sampled the path space and calculated the simple average position $\bar x(t)$ at each time. We see that the simulation readily converges to the correct least action path.
![Dynamical trajectories sampled for the free particle, with boundary conditions ….[]{data-label="fig_freepart"}](FreeParticle.pdf){width="80.00000%"}
For this simulation, 5 control points were used, and are sufficient to obtain the exact result ($R^{2}=0.99$) in less than 10 000 Monte Carlo steps, corresponding to $\approx$ 18 min.
Harmonic Oscillator Action
--------------------------
In the case of the harmonic oscillator, the action is of the form $$A[x] = \int_{t_{0}}^{t_{f}} {dt} \; \left(\frac{m \dot x(t)^{2}}{2} - \frac{k x(t)^{2}}{2}\right).$$
Without loss of generality, for numerical simulations $m = 1$ and $k = 1$ are used. The solution for this problem will be divided into two parts. The first solution found was the least action path for a short time interval. More precisely, we simulated a particle with boundary conditions $x(0)=0$ and $x(t_{f})= a \sin(\omega t_{f})$, with $t_{f}$ less than the half period $\frac{T}{2}$.
In this case the least action path also converges to the analytical solution, correctly solving the equation of motion as shown in Fig. \[fig\_HOpi\]. For this simulation, 5 control points were used, and are sufficient to obtain the exact result ($R^{2}=0.99$) in less than 10 000 Monte Carlo steps, corresponding to $\approx$ 26 min.
![Paths sampled for the harmonic oscillator considering shorts time intervals.[]{data-label="fig_HOpi"}](HOpi.pdf){width="\textwidth"}
The second case corresponds to the harmonic oscillator with boundary conditions $x(0)=0$ and $x(t_{f})= a \sin(\omega t_{f})$, but where $t_{f}$ is larger than the half period $\frac{T}{2}$. In other words, the end condition is past the first node of the harmonic oscillator. Under these boundary conditions, an unexpected result is obtained, the Monte Carlo sampling procedure diverges. This result can be understood as due to the fact that, for this case, the action extremum is not a global minimum[@Gray2007]. More precisely, the second functional derivative for the action of the harmonic oscillator action shows that the extremum is a saddle point in the case where the total time is longer than half a period, and a “true” minimum only for paths with total time less than half period. We solved this convergence problem by considering an additional constraint to the action solved, suggested by the work of Gray and Taylor[@Gray2007] in classical mechanics. The constraint involves the so-called kinetic foci, defined by the condition $$\frac{\partial x}{\partial v_{0}} = 0.
\label{kineticfoci}$$
As it turns out, the action extremum is guaranteed to be a minimum if the paths used pass close enough to a kinetic focus $x_{i}$. This can be implemented in the Monte Carlo simulation by including a quadratic constraint in the probability functional, leading to $$P[x|\lambda, \beta]= \frac{1}{\eta(\lambda, \beta)}P_0[x]\exp(-\lambda A[x] - \beta \sum_{i} (x(t_{i}) - x_{i})^{2}),
\label{probabilitykinecifoci}$$ where $(t_{i}, x_{i})$ are the set of kinetic foci (solutions of Eq. \[kineticfoci\]) and $\beta >> \lambda$, in order to stop the system from drifting away from the action extremum.
As an example, we have solved the harmonic oscillator for a total time close to one and a half periods $\frac{3T}{2}$, sampling all the paths that cross the first two kinetic foci of the harmonic oscillator, namely the points $\{ (0, \frac{T}{2}), (0,T)\}$. Now the Monte Carlo procedure does converge to the expected solution as shown in Fig. \[fig\_HO3pi\].
![Harmonic oscillator with fixed kinetic foci.[]{data-label="fig_HO3pi"}](HO3pi.pdf){width="120.00000%"}
For this simulation, 8 control points were used, and were sufficient to reach the exact result ($R^{2}=0.95$) in less than 20 000 Monte Carlo steps, corresponding to $\approx$ 2.8 hours.
An important remark here is to note the number of Bézier basis elements, or control points, because this is closely related to the computing time. For this reason, the main goal in an efficient simulation is to find the minimum number of control points to use without sacrificing precision, needed to map any solution of the differential equation of interest.
Concluding remarks
==================
In summary, we have described a technique for implementing Monte Carlo sampling of dynamical trajectories in classical Lagrangian systems under the Maximum Caliber formalism. We have demonstrated its usefulness by applying this technique to the case of the classical free particle and harmonic oscillator, recovering in both cases a statistical distribution of paths centered on the classical solution of the Euler-Lagrange equation. For the case of the harmonic oscillator we noted the need for fixing additional points known as the kinetic foci of the system in order for the simulation to converge properly. Our proof-of-concept implementation could be the starting point for a complete computational scheme of simulation of classical systems under uncertainty. It remains to be seen how this method scales to multidimensional and many-particle interacting systems. Finally, one of the main proposed uses of this framework is to obtain the instantaneous probability density of positions at each time, which would allow to obtain the instantaneous macroscopic properties of classical systems under uncertainty[@Gonzalez2016c].
Acknowledgments {#acknowledgments .unnumbered}
===============
DG acknowledges funding from “Beca de postdoctorado Universidad Católica del Norte” N0 0004/2019, and Itaú-Corpbanca, and Dr. Gonzalo Gutiérrez for the support as tutor during the PhD. where this ideas started. SD acknowledges partial financial support from FONDECYT grant 1171127 and Anillo ACT-172101. SC acknowledges partial financial support from FONDECYT grant 1170834.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Fix $k \ge 2$ and let $H$ be a graph with $\chi(H) = k+1$ containing a critical edge. We show that for sufficiently large $n,$ the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $T_k(n)$. This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian [@Gund1].'
author:
- 'Natasha Morrison[^1]'
- 'Alexander Roberts[^2]'
- 'Alex Scott[^3]'
bibliography:
- 'cycles.bib'
title: Maximising the Number of Cycles in Graphs with Forbidden Subgraphs
---
Introduction
============
For a graph $G$, let $c(G)$ be the number of cycles in $G$. The problem of bounding $c(G)$ for various classes of graph has a long history: for example, an upper bound on $c(G)$ in terms of the cyclomatic number of $G$ was given by Ahrens [@ahrens] in 1897; while a lower bound is implicit in work of Kirchhoff [@kirchhoff] from fifty years earlier.
For graphs on $n$ vertices, the number of cycles is clearly maximized by the complete graph, which has $\sum_{i=3}^n (i!/2i)\binom ni$ cycles. But what happens if we constrain the structure of $G$ by forbidding some subgraph? In other words, what is the maximal number of cycles in an $H$-free graph on $n$ vertices (here a graph is [*$H$-free*]{} if it does not contain a subgraph isomorphic to $H$)? For graphs $G$ and $H$, let $c(G)$ be the number of cycles in $G$ and let $$m(n;H):= \max\{ c(G): |V(G)| = n, H \not\subseteq G\}.$$ The problem of determining $m(n,H)$ was introduced by Durocher, Gunderson, Li and Skala [@Gund2] (who studied $m(n,K_3)$) and will be the focus of this paper.
The problem of maximizing the number of [*edges*]{} in an $H$-free graph has been extensively studied. Indeed, Turán [@turan] proved that the unique $n$-vertex $K_{k+1}$-free graph with the maximum number of edges is the complete $k$-partite graph with all classes of size $\lfloor n/k\rfloor$ or $\lceil n/k \rceil$, which is known as the *Turán graph* $T_k(n)$. More generally, the classical Turán problem asks for the maximum number of edges in an $H$-free graph: this is the *extremal number* ${{\rm ex}}(n;H)$ and the [*extremal graphs*]{} are ${{\rm EX}}(n;H) = \left\{G : \left|V(G)\right| = n, H \not\subseteq G\right\}$, that is the $H$-free graphs on $n$ vertices with ${{\rm ex}}(n;H)$ edges. For further detail, we refer to [@boletg].
Much less is known about maximizing the number of [*cycles*]{} in $H$-free graphs. Durocher, Gunderson, Li and Skala [@Gund2] investigated $m(n,K_3)$, and conjectured that the maximum is attained by the Turán graph $T_2(n)$. This conjecture was proved for large $n$ by Arman, Gunderson and Tsaturian [@Gund1], who showed that, for $n \ge 141$, $T_2(n)$ is the unique triangle-free graph containing $m(n;K_3)$ cycles. They made the following natural further conjecture.
\[gundconj\] For any $k > 1$, for sufficiently large $n$, $T_2(n)$ is the unique $n$-vertex $C_{2k+1}$-free graph containing $m(n;C_{2k+1})$ cycles.
A partial result towards this conjecture is given in [@Gund1], where it is shown that $m(n;C_{2k+1}) = O(c(T_2(n)))$. They also ask about a different generalisation.
\[gundqu\] For $k \ge 4$, what is $m(n;K_k)$? Is $T_{k-1}(n)$ the $K_k$-free graph containing $m(n;K_k)$ cycles?
In this paper we prove Conjecture \[gundconj\] for any fixed $k$ and sufficiently large $n$ and answer Question \[gundqu\] affirmatively for sufficiently large $n$. In fact we prove a much more general result. In what follows we say that an edge $e$ of a graph $H$ is *critical* if $\chi(H \backslash \{e\}) = \chi(H) - 1$. Our main result is the following.
\[main\] Let $k \ge 2$ and let $H$ be a graph with $\chi(H) = k+1$ containing a critical edge. Then for sufficiently large $n$, the unique $n$-vertex $H$-free graph containing the maximum number of cycles is the Turán graph $T_k(n)$.
The condition that $H$ has a critical edge is necessary, since if $H$ does not have a critical edge we can add an edge to the relevant Turán graph without creating a copy of $H$ (and the addition of this edge will increase the number of cycles). Conjecture \[gundconj\] follows from Theorem \[main\] as an odd cycle contains a critical edge.
By using the same techniques as in the proof of Theorem \[main\], we are able to obtain a bound on the number of cycles in an $H$-free graph for any fixed graph $H$ (not just critical ones).
\[cyclecount\] Let $k \ge 2$ and l*•*et $H$ be a fixed graph with $\chi(H) = k+1$. Then $$m(n;H) \le \left(\frac{k-1}{k}\right)^nn^ne^{-(1-o(1))n}.$$
The Turán graph gives a lower bound showing that this bound is tight up to the $o(1)$ term in the exponent.
In this paper we concern ourselves with maximising cycles of any length in a graph with a forbidden subgraph. The related problem of maximising copies of a single graph in a graph with a collection of forbidden subgraphs has received a great deal of attention. For a graph $G$ and family of graphs $\mathcal{F}$, define ${{\rm ex}}(n,G,\mathcal{F})$ to be the maximum possible number of copies of $G$ in a graph containing no member of $\mathcal{F}$. The value of ${{\rm ex}}(n,G,\mathcal{F})$ is of particular interest when the graphs being studied are cycles (see [@NACS; @BB; @ER] for results concerning other graphs). Improving on earlier work of Bollobás and Győri [@BG] and Győri and Li [@GL], Alon and Shikhelman [@NACS] gave bounds for ${{\rm ex}}(n,K_3,C_{2k+1})$, when $k \ge 2$. Using flag algebras, Hatami, Hladký, Král’, Norine, and Razborov [@HHKNR] showed that the unique triangle-free graph with maximum number of copies of $C_5$ is the balanced blow up of $C_5$. Also using flag algebras, Grzesik [@GG] determined ${{\rm ex}}(n,C_5,K_3)$. More recently, Grzesik and Kielak [@AGBK] determined ${{\rm ex}}(n,C_{2k+1},\mathcal{F})$, where $k \ge 7$ and $\mathcal{F}$ is the family of odd cycles of length at most $k$. They also asymptotically determine ${{\rm ex}}(n,C_{2k+1},C_{2k-1})$.
The rest of paper is organised as follows. Section \[seckpart\] contains a number of lemmas about counting cycles in complete $k$-partite graphs (Lemmas \[kKMain\]-\[secondcount\]). These will be used in Section \[secmain\] for the proof of Theorem \[main\]. The statements are very natural but our proofs are unfortunately technical, so we defer these to Section \[sectech\]. In Section \[sec2\] we prove Lemma \[easycor\] and use similar techniques to prove Theorem \[cyclecount\]. The proof of Theorem \[main\] is completed in Section \[secmain\]. We conclude the paper in Section \[sec5\] with some related problems and open questions. We conclude the current section with a sketch of the proof of Theorem \[main\].
Outline of Proof
----------------
In what follows we fix $H$ to be a graph with $\chi(H) = k+1$ that contains a critical edge and assume that $n$ is sufficiently large. As usual, for a graph $F$ we will write $e(F):= |E(F)|$ and in the particular case of the Turán graph, we will write $t_k(n):= |E(T_k(n))|$. Let $G$ be an $n$-vertex $H$-free graph with $c(G) = m(n;H)$. As $T_k(n)$ is $H$-free, we have that $m(n;H) \ge c(T_k(n))$. We will suppose that $G$ is not $T_k(n)$ and obtain a contradiction by showing that $c(G) < c(T_k(n))$.
The first step in the proof (Lemma \[edgecount\]) is to show that $G$ with $c(G) \ge c(T_k(n))$ contains at least $e(T_k(n)) - O(n \log^2 n)$ edges. In order to prove this, we will need a bound on the number of cycles an $n$-vertex $H$-free graph with $m \ge \beta(H)\cdot n$ edges can contain, where $\beta$ is some constant depending on $H$. Such a bound is provided by Lemma \[easycor\].
Given Lemma \[edgecount\], we are able to apply the following stability result from [@stabAA].
\[stable\] Let $H$ be a graph with a critical edge and $\chi(H) = k+1 \ge 3$, and let $f(n) = o(n^2)$ be a function. If $G$ is an $H$-free graph with $n$ vertices and $e(G) \ge t_k(n) - f(n)$ then $G$ can be made $k$-partite by deleting $O(n^{-1}f(n)^{3/2})$ edges.
Since we have $f(n) = O(n\log^2 n)$, this will imply that $G$ is a sublinear number of edges away from being $k$-partite. We then take a $k$-partition of $G$ which minimises the number of edges within classes and carefully bound (given that $G$ is not $T_k(n)$) the number of cycles $G$ can contain that do not use edges within classes (Lemma \[regcycle\]). We conclude the proof by separately counting the cycles in $G$ that use edges within classes and observing that the total number of cycles in $G$ is not large enough, a contradiction.
Counting Cycles in Complete $k$-partite Graphs {#seckpart}
==============================================
In this section we state some results about the number of cycles in complete $k$-partite graphs. These are needed in Section \[secmain\] for the proof of Theorem \[main\], but may be of independent interest. Despite the simplicity of the statements, the proofs are annoyingly technical, and so we will give them later in Section \[sectech\].
The first gives a bound on the number of cycles in $T_k(n)$. In what follows we write $h(G)$ for the number of Hamiltonian cycles in $G$ (a Hamiltonian cycle of a graph is a cycle covering all of the vertices). We also define $c_r(G)$ to be the number of cycles of length $r$ in $G$.
\[kKMain\] $$c_{2\lfloor n/2 \rfloor}\left(T_2(n)\right) \sim \pi2^{1-n}n^ne^{-n},$$ and for fixed $k \ge 3$, $$h(T_k(n)) = \Omega\left(\left(\frac{k-1}{k}\right)^nn^{n-\frac{1}{2}}e^{-n}\right).$$
Since $c(G) \ge h(G)$ for all $G$, if follows that $c(T_k(n)) = \Omega\left(\left(\frac{k-1}{k}\right)^nn^{n-\frac{1}{2}}e^{-n}\right).$ Arman [@arman Theorems 5.22 and 5.26] proves similar results here and also provides an upper bound for $c(T_k(n))$.
\[Turanbest\] Let $k \ge 2$ and G be an $n$-vertex $k$-partite graph. Then for any $r$, $c_r(T_k(n)) \ge c_r(G)$. Furthermore, when $n \ge 5,$ $c(T_k(n)) > c(G)$ for any $n$-vertex $k$-partite graph $G$ not isomorphic to $T_k(n)$.
In particular, Lemma \[Turanbest\] implies that the Turán graph $T_k(n)$ has the most Hamilton cycles amongst all $k$-partite graphs on $n$ vertices.
In order to state the next few lemmas we require some more technical definitions. For $\underline{a} = (a_1,\ldots,a_k) \in \mathbb{N}^k$, we define $K_{\underline{a}}$ to be the complete $k$-partite graph with vertex classes $V_1,\ldots, V_k$, where $|V_i| = a_i$. Let $v$ be some vertex in $V(K_{\underline{a}})$. We define $h_v(j,K_{\underline{a}})$ to be the number of permutations $v_1 \cdots v_n$ of the vertices of $K_{\underline{a}}$, such that $v_1 = v$, $v_2 \in V_j$ and $v_1 \cdots v_n$ is a Hamilton cycle (we count permutations rather than cycles, so that we count a cycle $v_1\cdots v_n$ with $v_2$ and $v_n$ from the same vertex class twice). Note that if we count the Hamilton cycles by considering $v_1\cdots v_n$ with $v_1$ fixed, by counting the number of cycles visiting each other vertex class first, then each cycle will be counted twice due to the choice of orientation. So for $v \in V_i$, we have $$\begin{aligned}
h(K_{\underline{a}}) = \frac{1}{2}\sum_{j \neq i}h_v(j,K_{\underline{a}}).\label{first1}
\end{aligned}$$
The next lemma will allow us to count cycles more accurately in $k$-partite graphs that are not complete.
\[close\] Let $k \ge 3$. Let $\underline{b} = (b_1,\ldots,b_n)$, $\underline{c} = (c_1,\ldots,c_n) \in {\mathbb{N}}^k$ be such that $b_i \ge b_j$ if and only if $c_i \ge c_j$, and that $K_{\underline{b}} \cong T_k(n)$. Denote the vertex classes of $K_{\underline{c}}$ by $V_1,\ldots,V_k$, and vertex classes of $K_{\underline{b}}$ by $V_1',\ldots, V_k'$. Then if $v \in V_1, w \in V_1'$, then $$\begin{aligned}
h_v(2,K_{\underline{c}}) \le h_w(2,T_k(n))\prod_{i =1}^ke^{\left|\log(\frac{b_i}{c_i})\right|}. \nonumber
\end{aligned}$$
We now bound the proportion of Hamilton cycles starting from a fixed vertex that immediately pass through a fixed vertex class. This will be important when we bound the cycles in a non-complete $k$-partite graph.
\[Turancount\] Let $k \ge 3$, and suppose $T_k(n)$ has vertex classes $V_1,\ldots,V_k$. Then for $n$ sufficiently large, if $v \in V_1$, $$\begin{aligned}
h_v(2,T_k(n)) \ge \frac{2}{3k}h(T_k(n)). \nonumber
\end{aligned}$$
The next two lemmas give a recursive bound on the number of Hamilton cycles in $T_k(n)$. This will allow us to bound the number of cycles in the Turán graph in terms of the number of Hamilton cycles it contains. Throughout the chapter we will make use of the notation $(n)_i := n \cdot (n-1) \cdots (n - (i-1))$.
\[recursion\] For $k,n \in {\mathbb{N}}, k \ge 3$ and $i \in [n]$, $$\begin{aligned}
h(T_k(n)) \ge (n-1)_i\left(\frac{k-2}{k}\right)^ih(T_k(n-i)). \nonumber
\end{aligned}$$
\[secondcount\] For $k,n \in {\mathbb{N}}, k \ge 3$: $$\begin{aligned}
c(T_k(n)) \le e^{\frac{2k}{k-2}}h(T_k(n)). \nonumber
\end{aligned}$$
Finally, we have similar results when $k = 2$. This case is slightly different to when $k\ge 3$ as $T_2(n)$ only contains even cycles.
\[second2count\] For $n \in \mathbb{N}$ and $i = o(n)$, we have
$$c(T_2(n-i)) \le 2e \left(\frac{4}{n}\right)^i c_{2\left\lfloor \frac{n}{2} \right\rfloor}(T_2(n)).$$
Counting Cycles in $H$-free Graphs {#sec2}
==================================
Fix $H$ to be a graph with $\chi(H) =k+1$ containing a critical edge. The first aim of this section is to prove a lemma bounding the number of cycles in an $n$-vertex $H$-free graph containing a fixed number of edges. We will need the following theorem of Simonovits [@Sim1].
\[simthm\] Let $H$ be a graph with $\chi(H) =k+1 \ge 3$ that contains a critical edge. Then there exists some $n_0$ such that, for all $n \ge n_0$, we have ${{\rm EX}}(n;H) = \{T_k(n)\}$.
Given $H$, define $n'_0(H)$ to be the smallest value of $n_0$ such that Theorem \[simthm\] holds and choose $n_0(H) \ge n'_0(H)$ such that ${{\rm ex}}(n;H) \ge 10n$ for each $n \ge n_0$. We define $\beta(H):= 10n_0.$
In a recent paper, Arman and Tsaturian [@armtsat] consider the maximum number of cycles in a graph with a fixed number of edges: They show that if $G$ is an $n$-vertex graph with $m$ edges, then $$c(G) \le \left\{ \begin{array}{c l}
\frac{3}{4}\Delta(G)\left(\frac{m}{n-1}\right)^{n-1} & \text{ for } \frac{m}{n-1} \ge 3,\\
\frac{3}{4}\Delta(G)\cdot \left(\sqrt[\leftroot{-2}\uproot{2}3]{3}\right)^m, & \text{otherwise}.\\
\end{array}\right.$$
This general bound is not strong enough for us: comparing this bound with the bounds given in Lemma \[kKMain\], we see that a graph with at least as many cycles as $T_k(n)$ has at least $\left(1+o(1)\right)e^{-1}t_k(n)$ edges. However under the additional assumption that our graph does not contain a forbidden subgraph $H$, we are able to prove the following lemma which we will later use to show that an $H$-free graph with at least as many cycles as $T_k(n)$ has at least $\left(1+o(1)\right)t_k(n)$ edges. We remark that when $m$ is close to $t_k(n),$ the bound we gives beats the general bound of Arman and Tsaturian by an exponential factor.
\[easycor\] Let $H$ be a fixed graph with $\chi(H) =k+1$ containing a critical edge. For $n$ sufficiently large, let $G$ be an $H$-free graph with $n$ vertices and $m$ edges where $t_k(n) - 10n \ge m \ge \beta(H) \cdot n$ (recall the definition of $\beta(H)$ from just after Theorem \[simthm\]). Then $c(G) = O\left(\lambda^{n}n^{n+2}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\lambda} - \lambda n}\right)$, where
$$\begin{aligned}
\lambda := 1 - \left(1 - \frac{2k}{k-1}\frac{m}{\left(n-3\right)^2}\right)^{\frac{1}{2}}.\label{alphadef}
\end{aligned}$$
The next lemma bounds the maximum number of paths that an $H$-free graph $G$ can contain between two fixed vertices. For $x,y \in V(G)$, define $p_{x,y}$ to be the number of paths between $x$ and $y$ in $G$.
\[count\] Let $H$ be a graph with $\chi(H) = k+1 \ge 3$ that contains a critical edge. For $n$ sufficiently large, let $G$ be an $H$-free graph with $n$ vertices and $m$ edges where $t_k(n) - 10n \ge m \ge \beta(H) \cdot n$ (recall the definition of $\beta(H)$ from just after Theorem \[simthm\]). Then for any $x,y \in V(G)$, $$p_{x,y}(G) = O\left(\lambda^{n}n^{n}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\lambda} - \lambda n}\right),$$ where $\lambda$ is as defined in .
Lemma \[easycor\] follows easily from Lemma \[count\].
Observe that for each edge $e = xy$ in $G$, the number of cycles containing $e$ is at most $p_{x,y}$. Thus, by Lemma \[count\] $$\begin{aligned}
c(G) &\le \sum_{xy \in E(G)}p_{x,y}(G) \\
&= O\left(m\lambda^{n}n^{n}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\lambda} - \lambda n}\right)\\
&= O\left(\lambda^{n}n^{n+2}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\lambda} - \lambda n}\right),\end{aligned}$$ as required.
We now prove Lemma \[count\].
Fix $x,y \in V(G)$. We define a sequence of vertices $(x_i)_{i\in [n]}$ and a sequence of graphs $(G_i)_{i\in[n]}$ as follows. Let $x_1 = x$ and $G_1 = G$. For $i \ge 2$, given $x_{i-1}$ and $G_{i-1}$, let $G_i = G_{i-1}\setminus x_{i-1}$ and choose $x_i$ with $p_{x_i,y}(G_i)$ as large as possible.
We count the number of paths between $x$ and $y$ by summing over possibilities for the second vertex in a path. We get the following inequality
$$\begin{aligned}
p_{x,y}(G) &= \sum_{z \in N(x)}p_{z,y}(G \setminus \{x\})\\
& \le d_G(x_1) \cdot \max\{p_{z,y}(G_2): z \in N(x_1)\} \\
& = d_G(x_1) p_{x_2,y}(G_2).\end{aligned}$$
Repeating this process gives $$p_{x_1,y}(G) \le \prod_{i=1}^{\ell} d_{G_i}(x_i),$$ where $\ell$ is minimal such that $\max\{p_{x_{\ell+1},y}(G_{\ell+1}): x_{\ell+1} \in N_{G_{\ell}}(x_{\ell})\} = 1$.
For $1 \le i \le \ell$, let $d_i := d_{G_{i}}(x_{i})$. Note that the $d_i$ are positive integers and that $\sum_{i=1}^{\ell} d_i \le m$. Also note that for any $t \in \{1,\ldots, \ell\}$, we have $$\sum_{i=t}^{\ell} d_i \le e(G_{t}).$$ Therefore, as $G_{t}$ is an $(n-t+1)$-vertex $H$-free graph, $\sum_{i=t}^{\ell} d_i \le {{\rm ex}}(n-t+1;H)$. Let $r_i = 0$ for $i = 2,\ldots,n-\ell$ and $r_i = d_{n+1-i}$ for $i = n+1-\ell,\ldots, n$. It follows that $p_{x,y}(G)$ is bounded above by the maximal value of the product $$\label{prod}
\prod_{i=2}^{n} \max\{r_i,1\}$$ under the following set of constraints:
1. $r_i \in \mathbb{Z}_{\ge 0}$, for $2 \le i \le n$
2. $\sum_{i=2}^n r_i \le m,$ and
3. $\sum_{i=2}^t r_i \le {{\rm ex}}(t;H)$, for $2 \le t \le n$.
We bound under these conditions by considering a relaxation of these constraints. Recall that $n_0:= n_0(H)$ is such that ${{\rm ex}}(s;H) = t_k(s)$ and ${{\rm ex}}(s;H) \ge 10s$ for all $s \ge n_0$. We look to maximise $$\label{prod2}
\prod_{i=2}^{n} \max\{r_i,1\},$$ under the following relaxed constraints:
1. $r_i \in \mathbb{Z}_{\ge 0}$, for $i > n_0$
2. $r_i \in {\mathbb{R}}^{\ge 0}$, for $i \le n_0$.
3. $\sum_{i=2}^n r_i \le m$.
4. $\sum_{i=2}^t r_i \le {{\rm ex}}(t;H),$ for each $n_0 \le t \le n$.
Since $m \ge \beta n$, we have $\frac{m}{n} \ge \frac{10t_k(n_0)}{n_0-1}$. Now let $(r_i)_{i=2}^n$ be a sequence maximising subject to (a)-(d). We may assume that $r_2,\ldots,r_{n_0}$ and $r_{n_0+1},\ldots,r_n$ are in increasing order as this will not violate (a)-(d).
There is some $I \le n-2$ such that:
- $r_i = \frac{t_k(n_0)}{n_0-1}$, for $i \le n_0$.
- $r_i = t_k(i) - t_k(i-1)$, for $n_0 +1 \le i \le I$.
- $r_i \in \{r_I, r_I + 1\}$, for $i > I$.
Let $T = \sum_{i=2}^{n_0}r(i)$. Then $(r_2,\ldots,r_{n_0}) = (0,\ldots,0,\frac{T}{S},\ldots,\frac{T}{S})$ for some $S \in [n_0-1]$ (or else we can increase $\prod_{i=2}^{n_0}r_i$). We may assume that $T$ is an integer as we can replace $T$ by $\lceil T \rceil$ and still satisfy (a)-(d). Differentiation of the function $j(x) = \left(\frac{T}{x}\right)^x$ shows that if $T \ge en_0$, then $S = n_0-1$ and so $r_i = \frac{T}{n_0-1}$ for each $i \in [n_0]$.
Suppose that $T < e\cdot n_0$. Then since $\frac{m}{n} \ge \beta(H) \ge \frac{10t_k(n_0)}{n_0-1}$, there must be a $j > n_0$ such that $r_j \ge \frac{t_k(n_0)}{n_0-1} \ge 10$. Choose $j$ to be minimal with this property. It can easily be verified that increasing $r_2$ by $2$ and decreasing $r_j$ by $2$ gives a sequence which satisfies (a)-(d) but gives a larger product. Therefore it must be the case that $T \ge e\cdot n_0$ and so $S = n_0-1$.
Now suppose that (i) doesn’t hold and so $e \cdot n_0 \le T < t_k(n_0)$. Since $\frac{m}{n} \ge \frac{10t_k(n_0)}{n_0-1}$, there exists some $j > n_0$ such that $r_j > \frac{5t_k(n_0)}{n_0-1}$. Choose $j$ to be minimal with this property and define $(s_i)_{i=2}^n$ by $s_i = \frac{T+1}{n_0-1}$ for $i \le n_0$, $s_j = r_j -1$ and $s_i = r_i$ otherwise. Then $(s_i)_{i =2}^n$ is a sequence satisfying (a)-(d) which gives a larger product, a contradiction. Therefore $T = t_k(n_0)$ and (i) holds.
Now suppose that (ii) does not hold and so $r_{n_0+1} < t_k(n_0+1)-t_k(n_0)$. Since $\frac{m}{n} \ge 2(t_k(n_0+1)-t_k(n_0))$, there must be a $j > n_0$ such that $r_j > t_k(n_0+1)-t_k(n_0)$. Choose $j$ to be minimal with this property and define $(s_i)_{i=2}^n$ by $s_{n_0+1} = s_{n_0+1}+1$, $s_j = s_j-1$ and $s_i = r_i$ otherwise. Then $(s_i)_{i =2}^n$ is a sequence satisfying (a)-(d) which gives a larger product, a contradiction. Therefore it $r_{n_0+1} = t_k(n_0+1)-t_k(n_0)$ and (ii) holds.
Let $j > n_0$ be minimal such that $\sum_{i=1}^j r_i \le t_k(j)-1$ (such a $j$ must exist since $m < t_k(n)$). Suppose that (iii) does not hold with $I=j-1$. Then there exists some $t \ge j$ such that $r_j + 1 < r_t$. Let $t$ be minimal with this property, and define $s_j := r_j + 1$, $s_t := r_t - 1$, and $s_i := r_i$ for all $i \not\in \{j,t\}$. The sequence $(s_i)_{i \in [n]}$ satisfies (a)-(d) but $$\begin{aligned}
\prod_{i =2}^n \max\{r_i,1\} < \prod_{i =2}^n \max\{s_i,1\}, \nonumber
\end{aligned}$$ a contradiction. Therefore $(r_i)_{i =1}^n$ satisfies properties (i)-(iii), completing the proof of the Claim.
Finally note that $I\le n-2$ follows from $m \le t_k(n) - 10n.$
Putting the values for $r_i$ from the claim into (\[prod2\]), we see that $$\begin{aligned}
\label{prodsi}
p_{x,y} &\le \left(\frac{t_k(n_0)}{n_0-1}\right)^{n_0-1} \prod_{i+ n_0 + 1}^{I}[t_k(i) - t_k(i-1)] \prod_{i=I+1}^n r_i \nonumber \\
& = O\left(\prod_{i=2}^n s_i\right),\end{aligned}$$ where $(s_i)$ is some sequence such that $s_i = t_k(i) - t_k(i-1)$ for $i \in \{2,\ldots, I\}$, $s_i \in \{s_I,s_I+1\}$ for $i > I$, and $m = \sum_{i=2}^n s_i$.
Note that $s_i = t_k(i) - t_k(i-1) = (i-1) - \left\lfloor \frac{i-1}{k}\right\rfloor$ for $i \le I$. Then the sequence $(s_i)_{i=2}^I$ is just the natural numbers up to $I-1-\left\lfloor \frac{I-1}{k} \right\rfloor$ with a repetition at each multiple of $k-1$. In other words, $$\left\{s_i : i \in \left\{2,\ldots,I\right\}\setminus \left\{\ell k +1: \ell \le \frac{I-1}{k}\right\}\right\} = \left[I-1-\left\lfloor \frac{I-1}{k} \right\rfloor\right]$$ and $s_{\ell k+1} = \ell(k-1)$ for each $\ell \le \frac{I-1}{k}$. Letting $b = \left\lfloor \frac{I-1}{k}\right\rfloor$ we have $$\label{si}
\prod_{i=2}^I s_i= (s_I)! \prod_{j=1}^b j(k-1) = s_I! b! (k-1)^b.$$
The remaining $n-I$ elements of the product $\prod_{i=2}^n s_i$ are all at most $s_I+1$. Therefore, by and we have $$\begin{aligned}
p_{x,y} &= O\left(\prod_{i=2}^n s_i\right) \nonumber \\
&= O\left(s_I!b!(k-1)^b(s_I +1)^{n-I}\right) \nonumber \\
&= O\left(s_I!b!(k-1)^b s_I^{n-I} e^{\frac{n}{s_I}}\right). \label{before1}\end{aligned}$$
Applying Stirling’s approximation and simplifying, yields $$\begin{aligned}
p_{x,y} = O\left(s_I^{n + s_I + 1/2 - I} b^{b+1/2}(k-1)^b \exp\left\{\frac{n}{s_I} - I\right\}\right). \nonumber\end{aligned}$$
Since $s_I = I-1 - \left\lfloor \frac{I-1}{k}\right\rfloor \ge (k-1)\frac{I-1}{k}$ and $b = \left\lfloor \frac{I-1}{k}\right\rfloor \le \frac{I-1}{k}$, we have $b \le \frac{s_I}{k-1}$. Therefore $$\begin{aligned}
p_{x,y} &= O\left(s_I^{n + \alpha n + 1/2 - I} \left(\frac{s_I}{k-1}\right)^{b+1/2}(k-1)^b \exp\left\{\frac{kn}{(k-1)I} - I\right\}\right) \nonumber \\
&= O\left(s_I^n\exp\left\{\frac{n}{s_I} - I\right\}\right). \nonumber\end{aligned}$$
Note that $s_I \le \frac{k-1}{k}(I-1)+1$ and so $$\begin{aligned}
p_{x,y} &= O\left((I-1)^n\left(\frac{k-1}{k}\right)^n \left(1+\frac{1}{I-1}\right)^n\exp\left\{\frac{kn}{(k-1)(I-1)} - (I-1)\right\}\right) \nonumber \\
&= O\left((I-1)^n\left(\frac{k-1}{k}\right)^n \exp\left\{\frac{(2k-1)n}{(k-1)(I-1)} - (I-1)\right\}\right).\end{aligned}$$
Substituting $I-1 = \alpha n$ gives $$\begin{aligned}
p_{x,y} &= O\left(\alpha^{n}n^{n}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\alpha} - \alpha n}\right). \label{alphaexp}\end{aligned}$$
It remains to determine the value of $\alpha$. We do this by counting edges. Since $m = \sum_i s_i$, we see that $$\begin{aligned}
m \ge t_k(I) + s_I\left(n-I\right). \label{alpha}\end{aligned}$$
Arguing as for , we see that $$\begin{aligned}
t_k(I) &= \sum_{i=1}^{s_I} i + (k-1)\sum_{j=1}^b j \\
&= \frac{1}{2}(s_I^2 + s_I + (k-1)(b^2+b)).\end{aligned}$$
If we put this value for $t_k(I)$ into we see that $$\begin{aligned}
m &\ge \frac{1}{2}(s_I^2 + s_I + (k-1)(b^2+b))+ s_I\left(n-(I-1)\right) - s_I.\end{aligned}$$
Recall that $b = \left\lfloor \frac{I-1}{k} \right\rfloor \ge \frac{I-1}{k} -1$ and so $b^2+b \ge \left(\frac{I-1}{k}\right)^2 - \frac{I-1}{k}.$ Also recall that $s_I = (I-1) - b$ and so $$\begin{aligned}
m &\ge \frac{k-1}{2k}(I-1)^2 + \frac{k-1}{k}n(I-1) - \frac{k-1}{k}(I-1)^2 - \frac{k-1}{k}(I-1) - 1 \\
&\ge \frac{k-1}{k}n(I-1) - \frac{k-1}{2k}(I-1)^2 - 3\frac{k-1}{k}(I-1).\end{aligned}$$
Substituting $(I-1) = \alpha n$ and rearranging gives
$$\begin{aligned}
\left(\left(1-\frac{3}{n}\right) - \alpha\right)^2 &\ge \left(1-\frac{3}{n}\right)^2 - \frac{2k}{k-1}\frac{m}{n^2}.\end{aligned}$$
Recall that $I \le n-2$ and so $\alpha \le \left(1-\frac{3}{n}\right)$. On the other side of the inequality, $\left(1-\frac{3}{n}\right)^2 - \frac{2k}{k-1}\frac{m}{n^2}$ is positive since $m \le t_k(n) - 10n$. Therefore we can take square roots square roots and rearrange to get
$$\begin{aligned}
\alpha &\le \left(1-\frac{3}{n}\right) - \left(\left(1-\frac{3}{n}\right)^2 - \frac{2k}{k-1}\frac{m}{n^2}\right)^{\frac{1}{2}} = \left(1-\frac{3}{n}\right)\lambda.\end{aligned}$$
Since the expression $\alpha^{n}n^{n}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\lambda} - \lambda n}$ is increasing in $\alpha$ when $\alpha \le 1 -\frac{2}{n}$, is maximised by setting $\alpha = \left(1-\frac{3}{n}\right)\lambda$. We are then done since
$$\begin{aligned}
\left(1-\frac{3}{n}\right)^n\lambda^{n}n^{n}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\left(1-\frac{3}{n}\right)\lambda} - \left(1-\frac{3}{n}\right)\lambda n} = O\left(\lambda^{n}n^{n}\left(\frac{k-1}{k}\right)^ne^{\frac{2k-1}{(k-1)\lambda} - \lambda n}\right).\end{aligned}$$
Theorem \[cyclecount\] follows easily from the idea of this proof by applying the following theorem of Erdős and Simonovits.
\[simthm2\] Let $H$ be a graph with $\chi(H) = k$. Then, $$\lim_{n \rightarrow \infty} \frac{{{\rm ex}}(n;H)}{\binom{n}{2}} = 1 - \frac{1}{k-1}.$$
Let $\varepsilon >0$. By Theorem \[simthm2\] and the fact that $t_k(n) \sim \left(1 - \frac{1}{k-1}\right)\binom{n}{2}$, we know that for $n$ sufficiently large, ${{\rm ex}}(n;H) \le (1 + \varepsilon)t_k(n)$. Thus, for $n$ sufficiently large, ${{\rm ex}}(s;H) \le (1 + \varepsilon)t_k(s)$ for all $n^{\frac{1}{2}} \le s \le n$. For ease of notation, let $n_1 := n^{\frac{1}{2}}$.
To bound the number of cycles in the graph, we wish to bound $p_{x,y}(G)$ for $x,y \in V(G)$. Arguing as in the proof of Lemma \[count\], we see that it is enough to bound the product $$\prod_{i=2}^n \max\{r_i,1\},$$ where $(r_i)$ satisfies the relaxed conditions:
1. $r_i \in {\mathbb{R}}^+$, for all $i$.
2. $\sum_{i=2}^t r_i \le (1+ \varepsilon)t_k(t)$, for each $n_1 \le t \le n$.
It is easily seen that this expression is maximised when $r_i :=\frac{(1+\varepsilon)t_k(n_1)}{n_1 - 1}$ for $i=2,\ldots,n_1$ and $r_i = (1+\varepsilon)(t_k(i)-t_k(i-1))$ otherwise. Therefore, we arrive at the following bound: $$\begin{aligned}
\prod_{i=2}^n r_i &\le \left(\frac{(1+\varepsilon)t_k(n_1)}{n_1 - 1}\right)^{n_1 - 1}\prod_{i=n_1+1}^n(1+\varepsilon)(t_k(i)-t_k(i-1)) \nonumber \\
&= O\left(e^{n_1}\prod_{i=2}^n(1+\varepsilon)(t_k(i)-t_k(i-1))\right) \nonumber \\
&= O\left(e^{\varepsilon n + n_1} \prod_{i=2}^n(t_k(i)-t_k(i-1))\right). \label{nuisance43}\end{aligned}$$
Recall from that, defining $b = \left\lfloor \frac{n-1}{k}\right\rfloor,$ we have $$\begin{aligned}
\prod_{i=2}^n(t_k(i)-t_k(i-1)) &= (n-1-b)!b!(k-1)^b\end{aligned}$$
Applying Stirling’s approximation and simplifying gives $$\begin{aligned}
\prod_{i=2}^n(t_k(i)-t_k(i-1)) &= O\left((n-1-b)^{n-1-b + 1/2}b^{b+1/2}e^{-n}(k-1)^b\right) \\
&= O\left(\left(\frac{k-1}{k}\right)^n n^{n+1}e^{-n}\right).\end{aligned}$$
Putting this into gives $$\label{pathcount}
p_{x,y} = O\left(\left(\frac{k-1}{k}\right)^n n^{n+1} e^{\varepsilon n + n_1 - n}\right).$$ Now, as in the proof of Lemma \[easycor\], we see that by and the fact that $n_1 = o(n)$, $$\begin{aligned}
c(G) &\le \sum_{xy \in E(G)} p_{x,y}\\
&= O\left(n^2 \left(\frac{k-1}{k}\right)^n n^{n+1} e^{\varepsilon n + n_0 - n}\right)\\
&= O\left(\left(\frac{k-1}{k}\right)^nn^ne^{-(1-\varepsilon - o(1))n}\right).\end{aligned}$$ Since $\varepsilon$ is arbitrary, we have our result.
Proof of Theorem \[main\] {#secmain}
=========================
Here we complete the proof of Theorem \[main\]. This will follow from the next two lemmas.
The first gives a lower bound on the number of edges in an extremal graph. (See also [@arman Theorem 5.3.2] for a $K_{k+1}$ version.)
\[edgecount\] Let $H$ be a graph $\chi(H) = k+1 \ge 3$ containing a critical edge. For sufficiently large $n$, let $G$ be an $n$-vertex $H$-free graph with $m$ edges and $c(G) \ge c(T_k(n))$. Then $m \ge \frac{n^2(k-1)}{2k} - O\left(n\log^2(n)\right)$.
Given this lemma, we can apply Theorem \[stable\] to show that any extremal graph $G$ is close to being $k$-partite. We then carefully count the number of cycles in such a graph. In what follows, for a graph $G$ and a $k$-partition of its vertices, we call edges within a vertex class *irregular* and those between vertex classes *regular*. Define a *best* $k$-partition of a graph $G$ to be one which minimises the number of irregular edges contained within $G$. The next lemma counts the cycles using only regular edges if $G$ is not $T_k(n)$. Recall that $c_r(G)$ is the number of cycles of length $r$ in $G$.
\[regcycle\] Let $H$ be a graph with $\chi(H) = k+1 \ge 3$ containing a critical edge. Suppose $G\not\cong T_k(n)$ is an $n$-vertex $H$-free graph with $c(G) \ge c(T_k(n))$. Then for sufficiently large $n$, the number of cycles using only regular edges in the best $k$-partition of $G$ is at most: $$\left\{
\begin{array}{c l}
c(T_k(n)) - \frac{1}{16k}h(T_k(n)) & \text{ for } k\ge 3,\\
c(T_2(n)) - \frac{1}{8}c_{2\lfloor \frac{n}{2} \rfloor}(T_2(n)) & \text{ for } k=2.
\end{array}\right.$$
Given Lemmas \[edgecount\] and \[regcycle\], we now complete the proof of Theorem \[main\]. We will then prove the lemmas themselves. The main work remaining for Theorem \[main\] is to count the number of cycles using irregular edges.
Let $H$ be a graph with a critical edge with chromatic number $\chi(H) = k+1 \ge 3$, and suppose $G$ is an $n$-vertex $H$-free graph with $c(G) = m(n;H)$. Then, in particular, $c(G) \ge c(T_k(n))$. Suppose for a contradiction that $G$ is not isomorphic to $T_k(n)$. Fix a best $k$-partition of $G$: by Lemma \[edgecount\] and Theorem \[stable\], we know that for sufficiently large $n$, the graph $G$ has at most $n^{0.55}$ irregular edges in its best $k$-partition.
Let $c^I(G)$ be the number of cycles in $G$ containing at least one irregular edge and let $c^R(G)$ be the number of cycles in $G$ using only regular edges. If $c^I(G) = o(h(T_k(n))$, then by applying Lemma \[regcycle\] and taking $n$ sufficiently large, we have $c(G) = c^R(G) + c^I(G) < c(T_k(n))$. Thus $c^I(G) = \Omega(h(T_k(n))).$
Let $E_I$ be the set of irregular edges in $G$. For each non-empty $A \subseteq E_I$, let $C_A$ be the set of cycles $C$ in $G$ such that $E(C) \cap E_I = A$ and such that $C$ contains at least one regular edge. Fix $A$ such that $C_A$ is non-empty and fix an edge $a_1 a_2 \in A$. (Note that $A$ must be a vertex-disjoint union of paths or else it would not be possible to have a cycle using all edges in $A$.) For any cycle $C = x_1 x_2 \cdots x_j$ in $C_A$, with $x_1 = a_1$ and $x_2 = a_2$, define $S(C)$ to be the directed cycle $x_1x_2\cdots x_j$ (so for all $i$, the edge $x_ix_{i+1}$ is directed towards $x_{i+1}$, where indices are taken modulo $j$).
For each $C \in C_A$, the orientation of $S(C)$ induces an orientation $f_C$ on the edges of $A$. Given a fixed orientation $f$ of $A$, we write $$C_A(f):= \left\{C \in C_A: f_C = f\right\}.$$ We will bound the size of each $C_A(f)$. A bound on $c^I(G)$ will then follow by summing over all possible $A$ and $f$.
Let $G/A$ be the graph obtained by contracting every edge in $A$. Then remove the remaining irregular edges to form $J$ (so $J$ is an $H$-free $k$-partite graph with $n-|A|$ vertices, as $A$ is a vertex-disjoint union of paths, and each edge of $A$ lies inside some vertex class of our $k$-partition). For each cycle $C$ in $C_A(f)$, we obtain an oriented cycle $g(C)$ in $H$ by replacing each maximal path $u_1\cdots u_j$ in $S(C) \cap A$ oriented from $u_1$ to $u_j$ by $u_1$. As $C$ contains at least one regular edge, $g(C)$ is either an edge or cycle in $J$.
We claim that $g$ is injective on $C_A(f)$. Indeed suppose that there exists a cycle $C \in C_A(f)$. Recall that $A$ is a vertex-disjoint union of paths and furthermore that $f$ orients the paths of $A$. Denote these oriented paths $\left(u^1_i\right)_{i \in [\ell_1]},\ldots,\left(u^t_i\right)_{i \in [\ell_t]}$. Each cycle $C \in C_A(f)$ must contain these oriented paths as segments (each edge of $A$ must be contained in $C$ and it is not possible to break up a path or else a vertex must be adjacent to more than two edges in the cycle). Therefore we have an inverse of $g$ which takes a cycle from $g\left(C_A(f)\right)$ and replaces each instance of $u^j_1$ with the path $u^j_1\cdots u^j_{\ell_j}$.
As $J$ is a $k$-partite graph on $n-|A|$ vertices, by Lemma \[Turanbest\] we have $$c(J) \le c(T_k(n-|A|)).$$ Recall that for each $C \in C_A(f)$, $g(C)$ is either an edge or a cycle in $J$. We therefore have $$|C_A(f)| \le 2\cdot c(T_k(n-|A|)) + 2|E(T_k(n))| \le 4 \cdot c(T_k(n-|A|)),$$ for sufficiently large $n$ by applying Lemma \[kKMain\] and recalling that $|A| \le n^{0.55}$. Let $F_A$ be the set of all possible orientations $f$ of $A$. We have $$\label{irreg}
c^I(G) \le \left|E^I\right|^{|E_I|} + \sum_{A\subseteq E^I}\sum_{f \in F_A}|C_A(f)|,$$ where the first term counts cycles that contain only irregular edges and the second term counts cycles in $c^I(G)$ that contain both a regular and irregular edge.
We will bound the second term of this expression. Recalling that there are at most $n^{0.55}$ irregular edges, we get that $$\sum_{A\subseteq E^I}\sum_{f \in F_A}|C_A(f)| \le \sum_{i=1}^{n^{0.55}}{\binom{n^{0.55}}{i}}2^i\cdot 4 \cdot c(T_k(n-i)).$$
For $k \ge 3$, we now apply Lemma \[secondcount\] and Lemma \[recursion\] for each $i$ in the sum, $$\begin{aligned}
\sum_{A\subseteq E^I}\sum_{f \in F_A}|C_A(f)|&\le \sum_{i=1}^{n^{0.55}}{\binom{n^{0.55}}{i}}e^{\frac{2k}{k-2}}2^{i+2}h(T_k(n-i)) \nonumber \\
&\le 4e^{\frac{2k}{k-2}}\sum_{i=1}^{n^{0.55}}{\binom{n^{0.55}}{i}}\left(\frac{2k}{k-2}\right)^i\frac{h(T_k(n))}{(n-1)_i} \nonumber \\
&\le e^7 h(T_k(n)) \sum_{i\ge 1} n^{0.55i}\left(\frac{6}{n-n^{0.55}}\right)^i \nonumber \\
&= o\left(h(T_k(n))\right). \nonumber
\end{aligned}$$ We have $|E^I|^{|E^I|} \le (n^{0.55})^{n^{0.55}}$ which is $o(h(T_k(n)))$ by Lemma \[kKMain\]. Therefore, using (\[irreg\]) we see that $c^I(G) = o(h(T_k(n))$, a contradiction. Therefore $G$ is isomorphic to $T_k(n)$.
Similarly for $k=2$, we apply Lemma \[second2count\] to get $$\begin{aligned}
\sum_{A\subseteq E^I}\sum_{f \in F_A}|C_A(f)|&\le \sum_{i=1}^{n^{0.55}}{\binom{n^{0.55}}{i}}2^i\cdot 8e \cdot \left(\frac{4}{n}\right)^i c_{2\lfloor n/2 \rfloor}(T_2(n)) \nonumber \\
&\le 8e \cdot c_{2\lfloor n/2 \rfloor}(T_2(n)) \sum_{i = 1}^{n^{0.55}}n^{0.55i}\left(\frac{8}{n}\right)^i \nonumber \\
&= o\left(c_{2\lfloor n/2 \rfloor}\left(T_2(n)\right)\right), \nonumber
\end{aligned}$$ and we conclude as before.
We now present the proofs of Lemmas \[edgecount\] and \[regcycle\].
First suppose that $m = O(n)$. We can then crudely bound $p_{x,y}(G)$ as in Lemma \[count\]. By and constraints (i) and (ii) above we have $$\begin{aligned}
p_{x_1,y}(G) \le \max_{\ell}\prod_{i=1}^{\ell}r_i \le \max_{\ell}\left(\frac{m}{\ell}\right)^{\ell}. \nonumber
\end{aligned}$$
The function $f(x) = \left(\frac{m}{x}\right)^x$ is maximised at $x = \frac{m}{e}$ and so $p_{x_1,y}(G) \le e^{\frac{m}{e}} = e^{O(n)}$. This is asymptotically smaller than $c(T_k(n))$ by Lemma \[kKMain\].
So $m \not= O(n)$. Suppose that $m \ge t_k(n)-10n$ (otherwise we are done so assume) so that we obtain a bound for $c(G)$ from Corollary \[easycor\]. Dividing this bound by $c(T_k(n)) = \Omega((\frac{k-1}{k})^nn^{n-\frac{1}{2}}e^{-n})$ gives $$\begin{aligned}
\frac{c(G)}{c(T_k(n))} = O\left(\lambda^{n}n^{2.5}e^{\frac{2k-1}{(k-1)\lambda} + \left(1-\lambda\right) n}\right), \label{nuisance}
\end{aligned}$$ where $\lambda$ is defined in .
If we take the logarithm of the right hand side and call it $R$ for ease of notation, we get $$\begin{aligned}
R &\le 2.5\log(n) + n(\log(\lambda)+(1-\lambda)) + \frac{2k-1}{k\lambda} + O(1) \nonumber \\
&\le 2.5\log(n) + n(\log(\lambda)+(1-\lambda)) + 3\lambda^{-1} + O(1). \nonumber
\end{aligned}$$
First assume that $\lambda \le 1- n^{-\frac{1}{2}}\log(n)$: we will show that then $R \rightarrow -\infty$ and so is $o(1)$.
If $\lambda \le e^{-2}$, then $\log(\lambda) + (1-\lambda) \le \frac{\log(\lambda)}{2}$. Furthermore we see from that $\lambda = \Omega\left(\frac{m}{n^2}\right)$ and so $\lambda^{-1} = o(n)$. Therefore $$\begin{aligned}
R &\le 2.5 \log(n) + \frac{n}{2}\log(\lambda) + o(n) \nonumber \\
&\le 2.5 \log(n) - n + o(n) \rightarrow -\infty, \nonumber
\end{aligned}$$ as $n$ tends to infinity.
Otherwise, $\lambda^{-1} \le e^2$ and since (by assumption) $\lambda \le 1- n^{-\frac{1}{2}}\log(n)$, we may apply Taylor’s theorem to see $$\begin{aligned}
R &\le 2.5\log (n) - n(1-\lambda)^2 + 3e^2 \nonumber \\
&\le 2.5\log(n) - \log^2(n) + 3e^2 \rightarrow -\infty, \nonumber
\end{aligned}$$ as $n$ tends to infinity.
In either case $R$ tends to $-\infty$ for sufficiently large $n$, and we must have that $c(G) < c(T_k(n))$, a contradiction.
Therefore $\lambda > 1 - \log(n)n^{-\frac{1}{2}}$. Equation now allows us to conclude that $m \ge t_k(n) - O\left(n\log^2(n)\right)$, as required.
For the proof of Lemma \[regcycle\] we require the Erdős-Stone Theorem [@erd-stone].
\[erdstone\] Let $k \ge 2$, $t \ge 1$, and $\varepsilon > 0$. Then for $n$ sufficiently large, if $G$ is a graph on $n$ vertices with $$e(G) \ge \left(1 - \frac{1}{k-1} + \varepsilon\right)\binom{n}{2},$$ then $G$ must contain a copy of $T_k(kt)$.
We now apply this theorem to complete the proof of Lemma \[regcycle\].
Let the best $k$-partition of $G$, be $V_1,\ldots,V_k$. By Lemma \[edgecount\], $e(G) > t_k(n) - n\cdot (\log n)^2$, and so Theorem \[stable\] tells us that $G$ contains $t_k(n)(1-o(1))$ edges between its vertex classes $V_1,\ldots,V_k$. We therefore have $|V_i| = \tfrac{n}{k}(1+o(1))$ for each $i$. Also note that $G$ cannot be $k$-partite (else $c(G) < c(T_k(n))$ by Lemma \[Turanbest\]). Therefore $G$ must contain an irregular edge. Now we count the cycles in $G$ which contain only regular edges. Note that if we define $G^R$ to be $G \backslash E_I$, where $E_I$ is the set of irregular edges, then $G^R$ is $k$-partite; $G^R \subseteq K_{\underline{a}}$ for some $\underline{a} = (a_1,\ldots, a_k) \in \mathbb{N}^k$.
Let $t$ be such that $H \subseteq T_k(tk) + e$, where $e$ is any edge inside a vertex class of $T_k(tk)$. Pick an irregular edge $uv$: without loss of generality we may assume $uv \in V_1$. We first show that $u$ and $v$ cannot have $\frac{n}{10k}$ common neighbours in every other vertex class. Suppose otherwise and form a set $Q$ by picking $\frac{n}{10k}$ vertices in $N(u)\cap N(v) \cap V_i$ for $i = 2, \ldots, k$ and picking $\frac{n}{10k}$ vertices in $V_1$ to be in $Q$.
The graph $G^R[Q]$ does not contain a copy of $T_k(tk)$: if it did, it would contain a copy of $T_k(tk)+e$ and hence a copy of $H$. So then applying Theorem \[erdstone\], there are $\Omega(n^2)$ regular edges that are not present in $G$, a contradiction. Thus, without loss of generality, $|N(u) \cap N(v) \cap V_2| < \frac{n}{10k}$ and, again without loss of generality, $|N(v) \cap V_2| \le \frac{5n}{8k}$ (since $|V_2| = \tfrac{n}{k}(1+o(1))$ and we may assume that $n$ is large).
When $k \ge 3$, this means that $G$ cannot contain at least $\frac{3}{8}$ of the Hamilton cycles contained in $K_{\underline{a}}$ which start from $v$ and then go to vertex class $V_2$. Recall that $h_v(2,K_{\underline{a}})$ is the number of permutations of $V(K_{\underline{a}}) = \{v_1, \ldots, v_n\}$ such that $v_1 = v$, $v_2 \in V_i$ and $v_1 \cdots v_n$ is a Hamilton cycle. Since cycles may be counted at most twice due to orientation when considering permutations, the number of Hamilton cycles in $K_{\underline{a}}$ which start from $v$ and then go to vertex class $V_2$ is at least $\tfrac{1}{2}h_v(2,K_{\underline{a}})$. By applying , we get $$\begin{aligned}
c(G^R) &\le c(K_{\underline{a}}) - \frac{3}{8}\cdot \frac{1}{2}h_v(2,K_{\underline{a}}) \nonumber \\
&= \sum_{r=3}^{n-1}c_r(K_{\underline{a}}) + \frac{1}{2}\sum_{i=3}^kh_v(i,K_{\underline{a}}) + \left(\frac{1}{2} - \frac{3}{16}\right)h_v(2,K_{\underline{a}}). \label{temp1}
\end{aligned}$$
Let $\underline{b} = (b_1,\ldots,b_n)$, be such that $b_i \ge b_j$ if and only if $a_i \ge a_j$, and that $K_{\underline{b}} \cong T_k(n).$ Recall that $a_i = \frac{n}{k}(1+o(1))$ and so $\prod_{i =1}^ke^{\left|\log\left(\frac{b_i}{a_i}\right)\right|} = (1+o(1))$. Therefore by applying Lemmas \[close\] and \[Turancount\] we get $$\begin{aligned}
c(G^R) &\le \sum_{r=3}^{n-1}c_r(K_{\underline{a}}) + \prod_{i =1}^ke^{\left|\log\left(\frac{b_i}{a_i}\right)\right|}\left[\frac{1}{2}\sum_{i=3}^kh_v(i,T_k(n)) + \left(\frac{1}{2} - \frac{3}{16}\right)h_v(2,T_k(n))\right] \nonumber \\
&= \sum_{r=3}^{n-1}c_r(K_{\underline{a}}) + (1+o(1))\left(c_n(T_k(n)) - \frac{3}{16}h_v(2,T_k(n))\right) \nonumber \\
&\le (1+o(1))\left(c(T_k(n)) - \frac{1}{8k}h(T_k(n))\right). \nonumber
\end{aligned}$$ Finally, we can apply Lemma \[secondcount\] to get $$\begin{aligned}
c(G^R) &\le (1+o(1))\left(c(T_k(n)) - \frac{1}{24k}h(T_k(n)) - \frac{1}{12k}h(T_k(n))\right) \nonumber \\
&\le (1+o(1))\left(c(T_k(n))\left(1-\frac{e^{-\frac{2k}{k-2}}}{24k}\right) - \frac{1}{12k}h(T_k(n))\right), \nonumber
\end{aligned}$$ and so for $n$ sufficiently large, $c(G^R) \le c(T_k(n)) -\frac{1}{16k}h(T_k(n))$.
For $k=2$, first consider that if $|V_1|$ and $|V_2|$ differ in size by more than $1$, then $G^R$ contains no cycle of length $2\lfloor n/2 \rfloor$. Counting cycles by length and applying Lemma \[Turanbest\] gives $$\begin{aligned}
c(G^R) &= \sum_{r=2}^{\lfloor n/2 \rfloor -1} c_{2r}(G^R) \nonumber \\
&\le \sum_{r=2}^{\lfloor n/2 \rfloor -1} c_{2r}(T_2(n)) \nonumber \\
&= c(T_2(n)) - c_{2\lfloor n/2 \rfloor}(T_2(n)). \nonumber
\end{aligned}$$
Therefore assume that $|V_1|$ and $|V_2|$ differ in size by at most $1$ (so $G^R$ is a subgraph of $T_2(n)$). Recall (from the third paragraph of this proof) that $G^R$ contains a vertex $v$ with degree at most $5n/16$. Therefore, when applying the argument for $k\ge 3$, we lose at least a quarter of the cycles of length $2\lfloor n/2 \rfloor$ which contain $v$ from $T_2(n)$. Note that $v$ is present in at least half of the cycles of length $2\lfloor n/2 \rfloor$ in $T_2(n)$ and so $c(G^R) \le c(T_2(n)) - \frac{1}{8}c_{2\left\lfloor \frac{n}{2} \right\rfloor}(T_k(n))$.
Counting Cycles in Complete multi-partite Graphs {#sectech}
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In this section we present the proofs for the lemmas concerning counting cycles in complete multi-partite graphs that we stated in Section \[seckpart\]. We start with some preliminary lemmas. In order to state these we require some technical definitions.
Define a *code* on an alphabet ${\mathcal{A}}$ to be a string of letters $a_1\cdots a_n$ where each $a_i$ is in ${\mathcal{A}}$. For $k \ge 3$, we now discuss a way to count the number of Hamilton cycles in a $k$-partite graph $G$. Suppose each vertex class $V_i$ of $G$ is ordered. Consider a code $a_1 \cdots a_n$, where each $a_i \in [k]$. From such a code, we attempt to construct a Hamilton cycle $v_1\cdots v_n$ in $G$ as follows: for $j=1,\ldots,n$ let $p(j) := \left|\left\{\ell \le j: a_{\ell} = a_j\right\}\right|$. Define $v_j$ to be the $p(j)$-th vertex in $V_{a_j}$. For $v_1\cdots v_n$ to be a Hamilton cycle, each letter must appear in the code $a_1\cdots a_n$ the correct number of times ($\left|\left\{j : a_j = i\right\}\right| = \left|V_i\right|$, for each $i \in [k]$) and any two consecutive letters of the code must be distinct ($a_j \neq a_{j+1}$ for each $j \in [n-1]$, and $a_1 \neq a_n$).
For a code $a_1 \cdots a_n$, with each $a_i \in [k]$, we say that the code is in $Q$ if $a_i \neq a_{i+1}$ for each $i$, where indices are taken modulo $n$ (so each pair of consecutive letters are distinct). For $\underline{c} = (c_1,\ldots,c_k) \in {\mathbb{N}}^k$, we say that the code is in $P_{\underline{c}}$ if there are $c_i$ copies of $i$, for each $i \in [k]$. Finally we say that a code is in $P_{n,k}$ if it is in $P_{\underline{d}}$, where $\underline{d} = (d_1,\ldots,d_k) \in {\mathbb{N}}^k$ is such that $d_1 \le d_2 \le \ldots \le d_k \le d_1 +1$ and $\sum_i d_i = n$.
We can count the number of Hamilton cycles in $K_{\underline{c}}$ by considering the number of codes in $Q \cap P_{\underline{c}}$ and the number of ways of ordering the vertices in each vertex class. That is, for each code in $Q \cap P_{\underline{c}}$, we consider all of the Hamilton cycles which arise from the different orderings of vertices. Each Hamilton cycle will be counted exactly $2n$ times due to the choice of the starting point and orientation, and so $$\begin{aligned}
h(K_{\underline{c}}) = \frac{\left|Q\cap P_{\underline{c}}\right|}{2n}\prod_{i=1}^k c_i! . \label{ham2}
\end{aligned}$$
We will calculate $|Q \cap P_{\underline{c}}|$ by considering the probability that a random code is in $Q \cap P_{\underline{c}}$. To this end, let $C_{n,k}$ denote the random code $C_{n,k} = a_1 \cdots a_n$, where each $a_i$ is independently and uniformly distributed on $[k]$. Obtaining good bounds on the probability that a random code is in $Q$ (and similarly in $P_{\underline{c}}$) is relatively easy but approximating the probability of the intersection of the events proves more tricky. The following lemma will help us bound (\[ham2\]) from below, in order to prove Lemma \[kKMain\].
\[major\] Let $k \ge 2$ and suppose $C_{n,k} = a_1\cdots a_n$ where the $a_i$ are independent and identically uniformly distributed on $[k]$. If $\underline{c} = (c_1,\ldots,c_k) \in {\mathbb{N}}^k$ is such that $\sum_{i} c_i = n$, then $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{n,k}] \ge {\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{\underline{c}}], \nonumber
\end{aligned}$$ and in particular, $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{n,k}] \ge {\mathbb{P}}[C_{n,k} \in Q]. \nonumber
\end{aligned}$$
Let $k \ge 2$ and suppose $\underline{c} = (c_1,\ldots,c_k) \in {\mathbb{N}}^k$ is such that $\sum_{i} c_i = n$. Suppose that there exist some $i$ and $j$ such that $c_i \le c_j -2$, and let $\underline{c}'=(c'_1,\ldots,c'_k)$ be such that $c'_i = c_i+1, c'_j = c_j-1$ and $c'_t=c_t$ for $t \neq i,j$. It is sufficient to show that ${\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{\underline{c}'}] \ge {\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{\underline{c}}]$ - we may inductively find an $i$ and $j$ until the $c_a$ differ by at most one and $\underline{c}$ corresponds to the vertex class sizes of a Turán graph.
Fix a subset $A$ of $[n]$ with $|A| = n - (c_i+c_j)$ and let $R_{A,\underline{c}}$ be the event that $C_{n,k}$ is in $P_{\underline{c}}$, that $A = \{\ell : a_{\ell} \neq i,j \}$, and that $a_{\ell} \neq a_{\ell+1}$ for all $\ell$ in $A$ and $a_n \neq a_1$ if both $n$ and $1$ are in $A$. $R_{A,\underline{c}}$ can be thought of as the event that everything in the code except the letters with values $i$ and $j$ behave well. Now note that we can partition over all the sets of size $n - (c_i+c_j)$ in $[n]$, and get the expression $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{\underline{c}}] = \sum_{A \in [n]^{(n - (c_i+c_j))}} {\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}}] \cdot {\mathbb{P}}[R_{A,\underline{c}} | C_{n,k} \in P_{\underline{c}}].
\end{aligned}$$
Note that given $P_{\underline{c}}$ holds, we may as well identify $i$ and $j$ when considering whether $R_{A,\underline{c}}$ holds. As such, ${\mathbb{P}}[R_{A,\underline{c}} | C_{n,k} \in P_{\underline{c}}]$ is constant with respect to $c_i$ and $c_j$ with fixed $c_i + c_j$. This in turn, means that ${\mathbb{P}}[R_{A,\underline{c}} | C_{n,k} \in P_{\underline{c}}] = {\mathbb{P}}[R_{A,\underline{c}'} | C_{n,k} \in P_{\underline{c}'}]$ and so to prove the first statment of the lemma, it is sufficient to show that $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}}] \le {\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}'}], \label{codecount3}
\end{aligned}$$ for each $A \subseteq [n]$, with $|A| = n - (c_i+c_j)$.
Let $A \subseteq [n]$, with $|A| = n - (c_i+c_j)$ and condition on the event $R_{A,\underline{c}}$ (note that we may assume that this event is not null else we have nothing to prove). If we consider $C_{n,k}$ as a code that is a cycle (imagine joining $a_1$ to $a_n$), then the occurrences of $i,j$ form a collection of segments of total length $c_i + c_j$ with $c_i$ copies of $i$ and $c_j$ copies of $j$. Conditioning just on $R_{A,\underline{c}}$, we have choice over where we place the $i$ and $j$ letters in the segments. Since we must have $c_i$ total copies of $i$ in the segments, there are $c_i+c_j \choose c_i$ such choices of placement of the $i$ and $j$ letters. Conditional on $R_{A,\underline{c}}$, the $i$ and $j$ placements are uniformly distributed on these $c_i+c_j \choose c_i$ choices. Conditional on $R_{A, \underline{c}}$, for the code $C_{n,k}$ to be in $Q$, the segments all have to be a string of letters alternating between $i$ and $j$. As such the first letter of a segment dictates the remainder of that segment.
Let the lengths of the $\{i,j\}$-segments of $C_{n,k}$ be $r_1,\ldots,r_m$ and let $s_{\rm odd \rm}$ and $s_{\rm even \rm}$ be the number of odd length $\{i,j\}$-segments and even length $\{i,j\}$-segments respectively. We are then able to compute ${\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}}]$ by considering the starting letter of each $\{i,j\}$-segment. Suppose that $t$ of the $s_{\rm odd \rm}$ $\{i,j\}$-segments with odd length start with $i$. Then in the code, there will be $t-(s_{\rm odd \rm}-t)$ more appearances of $i$, than of $j$. Therefore, since $C_{n,k} \in P_{\underline{c}}$, we must have $2t - s_{\rm odd \rm} = c_i-c_j$ and so $t = \tfrac{s_{\rm odd \rm}+c_i-c_j}{2}$. Note that if $s_{\rm odd \rm}+c_i-c_j$ is odd, then ${\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}}] = 0$ since $t$ must be an integer (and so we have nothing to prove). Therefore we assume that $s_{\rm odd \rm}+c_i-c_j$ is even in what follows.
We can specify such a code by choosing the starting letter of each even interval arbitrarily and choosing exactly $t$ odd intervals to start with $i$. Comparing this with all possible choices of placements of $i$ and $j$ letters, we obtain $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}}] &= \frac{2^{s_{\rm even \rm}}{s_{\rm odd \rm} \choose t}}{{c_i+c_j \choose c_i}} \label{codecount1}, \\
{\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}'}] &= \frac{2^{s_{\rm even \rm}}{s_{\rm odd \rm} \choose t+1}}{{c'_i+c'_j \choose c'_i}}, \nonumber \\
&= \frac{2^{s_{\rm even \rm}}{s_{\rm odd \rm} \choose t+1}}{{c_i+c_j \choose c_i+1}} \label{codecountz}.
\end{aligned}$$
Writing $b = c_j-c_i$ and dividing by , we get
$$\begin{aligned}
\frac{{\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}}]}{{\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}'}]} &= \frac{c_j(s_{\rm odd \rm}+c_i-c_j+2)}{(c_i+1)(s_{\rm odd \rm}+c_j-c_i)} \nonumber \\
&= \frac{(c_i + b)(s_{\rm odd \rm} - b+2)}{(c_i+1)(s_{\rm odd \rm} + b)} \nonumber \\
&= \frac{c_is_{\rm odd \rm}+2c_i-bc_i+bs_{\rm odd \rm}+2b-b^2}{c_is_{\rm odd \rm}+bc_i+b+s_{\rm odd \rm}} \nonumber \\
&= 1 -(b-1)\frac{2c_i+b-s_{\rm odd \rm}}{c_is_{\rm odd \rm}+bc_i+b+s_{\rm odd \rm}}. \label{codecount2}
\end{aligned}$$
Since there can be at most $c_i+c_j = 2c_i+b$ odd length $\{i,j\}$-segments, we have $2c_i + b \ge s_{\rm odd \rm}$, and $b \ge 2$. The right hand side of must be less than or equal to $1$ and so $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}}] \le {\mathbb{P}}[C_{n,k} \in Q | R_{A,\underline{c}'}],
\end{aligned}$$ as required for (\[codecount3\]). This completes the proof of the first statement of the lemma. For the second statement we partition $\mathbb{P}[Q]$ over the $P_{\underline{c}}$ to give $$\begin{aligned}
{\mathbb{P}}[Q] &= \sum_{\underline{c}}{\mathbb{P}}[Q\cap P_{\underline{c}}]\\
&= \sum_{\underline{c}}{\mathbb{P}}[Q|P_{\underline{c}}]{\mathbb{P}}[P_{\underline{c}}]\\
&\le \sum_{\underline{c}}{\mathbb{P}}[Q|P_{n,k}]{\mathbb{P}}[P_{\underline{c}}]\\
&= {\mathbb{P}}[Q|P_{n,k}],\end{aligned}$$ as required.
We now use Lemma \[major\] to bound from below the number of Hamilton cycles in $T_k(n)$ and in turn prove Lemma \[kKMain\].
Let $k \ge 3$ and suppose $\underline{c} = (c_1,\ldots,c_k) \in {\mathbb{N}}^k$ is such that $\sum_{i} c_i = n$. Recall from the proof of Lemma \[ham2\] that computing $h(K_{\underline{c}})$ is equivalent to calculating $|Q \cap P_{\underline{c}}|$. We can do this by considering the probability that the code $C_{n,k} = a_1\cdots a_n$ is in both $Q$ and $P_{\underline{c}}$. There are $k^n$ equiprobable values for $C_{n,k}$ and so $|Q \cap P_{\underline{c}}| = k^n{\mathbb{P}}[C_{n,k} \in Q \cap P_{\underline{c}}]$. Putting this into gives: $$\begin{aligned}
h(K_{\underline{c}}) &= \frac{k^n}{2n}\biggl[\prod_{i=1}^k(c_i!)\biggr]\cdot {\mathbb{P}}[C_{n,k} \in Q \cap P_{\underline{c}}] \nonumber \\
&= \frac{k^n}{2n}\biggl[\prod_{i=1}^k(c_i!)\biggr]\cdot {\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{\underline{c}}]\cdot {\mathbb{P}}[C_{n,k} \in P_{\underline{c}}]. \label{ham3}
\end{aligned}$$
It is easy to see that if $g_i = |\{j \in [n] : a_j = i\}|$ for each $i$, then $(g_1,\ldots,g_k)$ follows a multinomial distribution with parameters $n$ and $(\frac{1}{k},\ldots,\frac{1}{k})$ and so in turn $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in P_{\underline{c}}] = \frac{n!}{\prod_{i=1}^k(c_i!)}k^{-n}. \label{fine1}
\end{aligned}$$ Putting this into we see that $$\begin{aligned}
h(K_{\underline{c}}) = \frac{n!}{2n}{\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{\underline{c}}]. \label{ham4}
\end{aligned}$$
As an aside, note that if $c_i \le c_j -2$ and we let $\underline{c}'=(c'_1,\ldots,c'_k)$ be such that $c'_i = c_i+1, c'_j = c_j-1$ and $c'_t=c_t$ otherwise, then applying Lemma \[major\] to gives that $$\begin{aligned}
h(K_{\underline{c}'}) \ge h(K_{\underline{c}}). \label{ham1}
\end{aligned}$$
By and Lemma \[kKMain\] we get $$\begin{aligned}
h(T_k(n)) &= \frac{n!}{2n}{\mathbb{P}}[C_{n,k} \in Q | C_{n,k} \in P_{n,k}] \nonumber \\
&\ge \frac{n!}{2n}{\mathbb{P}}[C_{n,k} \in Q] \nonumber \\
&= \frac{n!}{2n} {\mathbb{P}}[a_n \neq a_1,a_{n-1} | a_{n-1} \neq \cdots \neq a_1]\prod_{i=2}^{n-1}{\mathbb{P}}[a_i \neq a_{i-1} | a_{i-1} \neq \cdots \neq a_1] \nonumber \\
&\ge \frac{n!}{2n}\left(\frac{k-2}{k}\right)\left(\frac{k-1}{k}\right)^{n-1} \nonumber \\
&= \Omega\left(n^{n-\frac{1}{2}}e^{-n}\left(\frac{k-1}{k}\right)^n\right). \nonumber
\end{aligned}$$ Since $c(T_k(n)) \ge h(T_k(n))$, we arrive at the desired result for $k \ge 3$.
For $k=2$ we apply a simple counting argument. Observe that the number of cycles in a graph is at least the number of cycles of length $t =2 \left\lfloor \frac{n}{2} \right\rfloor$. For $T_2(n)$ this is easily counted by ordering both colour classes and accounting for starting vertex and orientation. Therefore we get $$\begin{aligned}
c_t(T_2(n)) = \frac{\left(\left\lfloor \frac{n}{2} \right\rfloor\right)_{\frac{t}{2}} \left(\left\lceil \frac{n}{2} \right\rceil\right)_{\frac{t}{2}}}{2t} = \frac{\left\lfloor \frac{n}{2} \right\rfloor! \left\lceil \frac{n}{2} \right\rceil!}{2t}, \label{omega1}
\end{aligned}$$ and the result follows by applying Stirling’s approximation.
We now use a counting argument to prove Lemma \[Turanbest\].
As before, let $\underline{c} = (c_1,\ldots,c_k) \in {\mathbb{N}}^k$ be such that $\sum_{i} c_i = n$. If there exists $i$ and $j$ such that $c_i \le c_j -2$, and we let $\underline{c}'=(c'_1,\ldots, c'_k)$ be such that $c'_i = c_i+1, c'_j = c_j-1$ and $c'_k=c_k$ otherwise. It is sufficient to show that $c_r(K_{\underline{c}'}) \ge c_r(K_{\underline{c}})$, for all $r$.
Without loss of generality, we may assume that $i=2$ and $j=1$. We can count the number of cycles of a given length, $r$, by choosing $r$ vertices and then counting the number of Hamilton cycles in graph induced by this cycle and then summing over all choices of $r$ vertices: $$\begin{aligned}
c_r(K_{\underline{c}}) = \sum_{\substack{\underline{a} \in \prod_{i=1}^k\{0,\ldots,c_i\} : \\ \sum_{i=1}^k a_i = r}}\biggl[\biggl(\prod_{i=1}^k{c_i \choose a_i}\biggr) \cdot h\bigl(K_{\underline{a}}\bigr)\biggr]. \nonumber
\end{aligned}$$
Fix a copy $K$ of $K_{\underline{c}}$ with vertex classes $V_1, \ldots, V_k$ and choose $v \in V_1$; then define $K'$ to be $K \setminus v$ with a vertex $v'$ added to $V_2$ which is a neighbour of all vertices not in $V_2$. We can see that $K'$ is a copy of $K_{\underline{c}'}$. Using this coupling to compare $c_r(K_{\underline{c}})$ and $c_r(K_{\underline{c}'})$, we only need to consider cycles in $K$ containing $v$ and the cycles in $K'$ containing $v'$. We write $c_{r,v}(G)$ to be the number of cycles of length $r$ in $G$ containing vertex $v$. In what follows, $e_m = (y_1,\ldots,y_k)$, where $y_m = 1$ and $y_{\ell} = 0$ otherwise. Since we already assume that $v$ is in our cycle, we then choose $r-1$ other vertices and count the number of Hamilton cycles on the induced subgraph to express $c_{r,v}(K)$ as $$\begin{aligned}
&\sum_{\substack{\underline{a} \in \{0,\ldots,c_1-1\} \times \prod_{i=2}^k\{0,\ldots,c_i\} : \\ \sum_{i=1}^k a_i = r-1}}\biggl[{c_1-1 \choose a_1} \cdot \biggl(\prod_{i=2}^k{c_i \choose a_i}\biggr) \cdot h\bigl(K_{\underline{a}+\underline{e}_1}\bigl)\biggr] \nonumber \\
&= \sum_{\substack{a_1 \in \{0,\ldots,c_1-1\} \\ a_2 \in \{0,\ldots,c_2\}}}\biggl[{c_1-1 \choose a_1}{c_2 \choose a_2} \sum_{\substack{(a_3,\ldots,a_k) \in \prod_{i=3}^k\{0,\ldots,c_i\} : \\ \sum_{i=1}^k a_i = r-1}}\biggl[\biggl(\prod_{i=3}^k{c_i \choose a_i}\biggr) \cdot h\bigl(K_{\underline{a}+\underline{e}_1}\bigl)\biggr]\biggr] \nonumber
\end{aligned}$$ and similarly we may express $c_{r,v'}(K)$ as $$\begin{aligned}
&\sum_{\substack{a_1 \in \{0,\ldots,c_1-1\} \\ a_2 \in \{0,\ldots,c_2\}}}\biggl[{c_1-1 \choose a_1}{c_2 \choose a_2} \sum_{\substack{(a_3,\ldots,a_k) \in \prod_{i=3}^k\{0,\ldots,c_i\} : \\ \sum_{i=1}^k a_i = r-1}}\biggl[\biggl(\prod_{i=3}^k{c_i \choose a_i}\biggr) \cdot h\bigl(K_{\underline{a}+\underline{e}_2}\bigl)\biggr]\biggr] \nonumber \\
&= \sum_{\substack{a_1 \in \{0,\ldots,c_1-1\} \\ a_2 \in \{0,\ldots,c_2\}}}\biggl[{c_1-1 \choose a_1}{c_2 \choose a_2} \sum_{\substack{(a_3,\ldots,a_k) \in \prod_{i=3}^k\{0,\ldots,c_i\} : \\ \sum_{i=1}^k a_i = r-1}}\biggl[\biggl(\prod_{i=3}^k{c_i \choose a_i}\biggr) \cdot h\bigl(K_{\underline{a}'+\underline{e}_1}\bigl)\biggr]\biggr], \nonumber
\end{aligned}$$ where $\underline{a}' = (a_2,a_1,a_3,a_4,\ldots,a_k)$ is the vector $\underline{a}$ with the first two values switched.
Define: $$\begin{aligned}
\eta(a_1,a_2,\underline{c},r) := \sum_{\substack{(a_3,\ldots,a_n) \in \prod_{i=3}^k\{0,\ldots,c_i\} : \\ \sum_{i=1}^k a_i = r-1}}\biggl[\biggl(\prod_{i=3}^k{c_i \choose a_i}\biggr)h\bigl(K_{\underline{a}+\underline{e}_1}\bigl) \biggr]. \nonumber
\end{aligned}$$ Then $$\begin{aligned}
c_{r,v}(K) = \sum_{\substack{a_1 \in \{0,\ldots,c_1-1\} \\ a_2 \in \{0,\ldots,c_2\}}}{c_1-1 \choose a_1}{c_2 \choose a_2} \eta(a_1,a_2,\underline{c},r) \label{length1}
\end{aligned}$$ and $$\begin{aligned}
c_{r,v}(K_{\underline{c}'}) = \sum_{\substack{a_1 \in \{0,\ldots,c_1-1\} \\ a_2 \in \{0,\ldots,c_2\}}}{c_1-1 \choose a_1}{c_2 \choose a_2} \eta(a_2,a_1,\underline{c},r). \label{length2}
\end{aligned}$$
If we subtract from and split the sums depending on the values of $a_1$ and $a_2$, we get $$\begin{aligned}
c_{r,v'}(K') - c_{r,v}(K) &= \sum_{0\le a_2 < a_1 \le c_2} {c_1-1 \choose a_1}{c_2 \choose a_2}\left(\eta(a_2,a_1,\underline{c},r)- \eta(a_1,a_2,\underline{c},r)\right)\nonumber \\
&+ \sum_{0\le a_1 < a_2 \le c_2} {c_1-1 \choose a_1}{c_2 \choose a_2}\left(\eta(a_2,a_1,\underline{c},r)- \eta(a_1,a_2,\underline{c},r)\right)\nonumber \\
&+ \sum_{0\le a_2\le c_2<a_1\le c_1-1} {c_1-1 \choose a_1}{c_2 \choose a_2}\left(\eta(a_2,a_1,\underline{c},r)- \eta(a_1,a_2,\underline{c},r)\right)\nonumber
\end{aligned}$$
If we swap around the values of $a_1$ and $a_2$ in the second line of this expression, we get $$\begin{aligned}
&c_{r,v'}(K') - c_{r,v}(K) \nonumber \\
&= \sum_{0 \le a_2<a_1 \le c_2}\biggl({c_1-1 \choose a_1}{c_2 \choose a_2} - {c_1-1 \choose a_2}{c_2 \choose a_1}\biggr)\biggl(\eta(a_2,a_1,\underline{c},r) - \eta(a_1,a_2,\underline{c},r)\biggr) \nonumber \\
&+ \sum_{\substack{a_1 \in \{c_2+1,\ldots,c_1-1\} \\ a_2 \in \{0,\ldots,c_2\}}}{c_1-1\choose a_1}{c_2\choose a_2}(\eta(a_2,a_1,\underline{c},r)-\eta(a_1,a_2,\underline{c},r)). \label{length4}
\end{aligned}$$
From , we obtain that if $x >y$, then we have $\eta(x,y,\underline{c},r) \le \eta(y,x,\underline{c},r)$. Thus in the first sum of , when $a_1 > a_2$, we have $\eta(a_2,a_1,\underline{c},r) - \eta(a_1,a_2,\underline{c},r) \ge 0.$ At the same time, note that since $c_1-1 > c_2$, $$\begin{aligned}
{c_1-1 \choose x}{c_2 \choose y} - {c_1-1 \choose y}{c_2 \choose x} > 0 \nonumber
\end{aligned}$$ if and only if $x > y$. Combining these, we must have that for all $0 \le a_2 < a_1 \le c_2$ $$\begin{aligned}
\biggl({c_1-1 \choose a_1}{c_2 \choose a_2} - {c_1-1 \choose a_2}{c_2 \choose a_1}\biggr)\biggl(\eta(a_2,a_1,\underline{c},r) - \eta(a_1,a_2,\underline{c},r)\biggr) \ge 0 \nonumber
\end{aligned}$$ and so the first sum is positive.
In the second sum of , $a_1 > a_2$ and tells us $\eta(a_2,a_1,\underline{c},r) - \eta(a_1,a_2,\underline{c},r) \ge 0.$ Thus the second sum is positive as well. We are then able to conclude that $c_{r,v}(K_{\underline{c}'}) \ge c_{r,v}(K_{\underline{c}})$ as required.
All that remains is to prove that $c(T_k(n)) > c(G)$ for any $k$-partite graph $G$. Suppose that $G = K_{\underline{c}^0}$ where $\underline{c}^0 = (c^0_1,\ldots,c^0_k) \in \mathbb{N}^k$ is such that $\sum_{i=1}^k c^0_i =n$. While there exist some $i$ and $j$ such that $c^{\ell}_i \le c^{\ell}_j-2$, define $\underline{c}^{\ell+1} = (c^{\ell+1}_1,\ldots,c^{\ell+1}_k)$ by $c^{\ell+1}_i = c^{\ell}_i+1$, $c^{\ell+1}_j = c^{\ell+1}_j-1$ and $c^{\ell+1}_r = c^{\ell}_r$ otherwise. Suppose that this process terminates with $\underline{c}^I$, so $T_k(n) \simeq K_{\underline{c}^I}$. Note that by successive applications of , $h(G) \le h(K_{\underline{c}^{I-1}})$. In order to get a strict inequality, we have to consider a bit more closely. To get equality in $h(K_{\underline{c}^{I-1}}) \le h(K_{\underline{c}^{I}})$, we must have that $s_{\rm odd \rm} = 2c_i^{I-1}+b$ for all $A \in [n]^{(n-(c_i^I+c_j+j)}$.
Say that a code $a_1,\ldots,a_n$ has an *ij transition* if there exists some $s$ such that $a_s$ is in class $i$ and $a_{s+1}$ is in class $j$, or such that $a_s$ is in class $j$ and $a_{s+1}$ is in class $i$, where indices are taken modulo $n$. By the conclusion of the previous paragraph, we have that all codes in $Q \cap P_{\underline{c}^{I}}$ have no $ij$ transition. However we can construct such a code with an $ij$ transition. Note that if the vertex class sizes of a $k$-partite graph are not all equal, then in any Hamilton cycle, there must a transition from a smaller vertex class to a larger vertex class and so if $c_i^I \neq c_j^I$, then by symmetry there must be a Hamilton cycle with a $ij$ transition. Suppose that $c^I_i = c^I_j$. By a similar argument, if classes $i$ and $j$ are both larger classes, then there must be a Hamilton cycle with a $ij$ transition. Suppose instead that vertex classes $i$ and $j$ are both smaller classes. Consider a permutation $\pi = \pi_1\cdots \pi_k$ such that $\pi_{k-1} = i$, $\pi_{k} = j$ and $\{\pi_1,\ldots,\pi_r\} = \{l : c_l^I = c_i^I+1\}$. If $r=1$ and $k=3$, then $c_i^I \ge 2$ and so $\pi_1\pi_2\pi_1\pi_3(\pi_1\pi_2\pi_3)\cdots(\pi_1\pi_2\pi_3)$ is sufficient. If $r=1$ and $k\ge 4$, then $\pi_1\pi_2\pi_1\pi_3\pi_4 \cdots \pi_k(\pi_1\cdots \pi_k)\cdots(\pi_1\cdots \pi_k)$. Finally, if $r \ge 2$, then $\pi_1 \cdots \pi_r (\pi_1\cdots \pi_k) \cdots (\pi_1\cdots \pi_k)$ is sufficient.
We have shown that there must be an instance of a strict inequality at in the comparison of $h(K_{\underline{c}^{I-1}})$ with $h(K_{\underline{c}^{I}})$. It then follows immediately that $c(T_k(n)) = c(K_{\underline{c}^{I}}) > c(K_{\underline{c}^{I-1}}) \ge c(G)$.
The proof of Lemma \[close\] has a similar flavour to that of Lemma \[major\]. We first prove a preliminary lemma where we evaluate $h_v(2,K_{\underline{c}})$ by considering random codes and then compare $h_v(2,K_{\underline{c}})$ with $h_v(2,K_{\underline{c}'})$. Lemma \[close\] will follow directly from this next lemma. (For what follows we define $R_{A,\underline{b}}$ as in the proof of Lemma \[major\].)
\[stepcount\] For $k \ge 3$, suppose $\underline{c}=(c_1,\ldots,c_k) \in {\mathbb{N}}^k$ is such that $\sum_{i} c_i = n$ with $0 \neq c_i \le c_j -2$. Let $\underline{c}'=(c'_1,\ldots,c'_k)$ be such that $c'_i = c_i+1, c'_j = c_j-1$ and $c'_k=c_k$ otherwise. Suppose $V_1,\ldots, V_k$ and $V_1',\ldots,V_k'$ are the vertex classes of $K_{\underline{c}}$ and $K_{\underline{c}'}$ and pick some $v \in V_1, v' \in V_1'$. Then $$\begin{aligned}
h_v(2,K_{\underline{c}}) \le \frac{(c_i+1)c_j}{c_i(c_j-1)}h_{v'}(2,K_{\underline{c}'}). \nonumber
\end{aligned}$$
Recall that $h_v(2,K_{\underline{c}})$ counts orderings $v_1,\ldots,v_n$ of $V(K_{\underline{c}})$ where $v_1 = v$, $v_2 \in V_2$, and $v_1\cdots v_n$ is a Hamilton cycle. There is a bijection between such an ordering and the pair $(C,(\pi_i)_{i\in [k]})$ where: $C$ is a code $a_1\cdots a_n$ on $[k]$ with $a_1 = 1$, $a_2 = 2$ that is in both $Q$ and $P_{\underline{c}}$; and $\pi_i$ is an ordering of $V_i$ for each $i$ and $v$ is the first vertex in $\pi_1$. So if we let $C_{n,k} = a_1\cdots a_n$ be a random code where each $a_i$ is independently and identically uniformly distributed on $[k]$, we have an expression for $h_v(2,K_{\underline{c}}$): $$\begin{aligned}
h_v(2,K_{\underline{c}}) = k^n(c_1-1)!\biggl(\prod_{l=2}^k(c_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q \cap P_{\underline{c}}, (a_1,a_2) = (1,2)]. \nonumber
\end{aligned}$$
An application of then gives $$\begin{aligned}
h_v(2,K_{\underline{c}}) &= \frac{n!}{c_1}{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2) = (1,2) | C_{n,k} \in P_{\underline{c}}] \nonumber \\
&= \frac{n!}{c_1}\sum_A{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2) =(1,2) | R_{A,\underline{c}}] {\mathbb{P}}[R_{A,\underline{c}} | C_{n,k} \in P_{\underline{c}}]\label{first2}
\end{aligned}$$ where $R_{A,\underline{c}}$ is defined as in the proof of Lemma \[major\], and the sum is taken over all $A \in [n]^{(n-(c_i+c_j))}$.
For what follows, we only consider $A \in [n]^{(n-(c_i+c_j))}$ such that $R_{A,\underline{c}} \cap \{(a_1,a_2)=(1,2)\} \neq \emptyset$ as these are the only ones that contribute to when considering either $\underline{c}$ and $\underline{c}'$. As in the proof of Lemma \[major\], conditioning on $R_{A,\underline{c}}$, let $s_{\rm odd \rm}$ and $s_{\rm even \rm}$ be the number of $\{i,j\}$ subcodes with respectively odd and even lengths, where we consider the code cyclically. Unlike before, we now require $(a_1,a_2) = (1,2)$ and so if one of $i$ and $j$ is $1$ or $2$, one of the subcodes will have a fixed value at $a_1$ and so a fixed starting letter. Let $\chi_{\rm even \rm}$ be the indicator that there is an even length subcode with a fixed first letter. Similarly let $\chi_{\rm odd \rm}$ be the indicator that there is an odd length subcode with a fixed first letter and further let $\chi_{\rm odd \rm}(i)$ and $\chi_{\rm odd \rm}(j)$ be the indicator that there is an odd length subcode with the first letter having fixed value $i$ and $j$ respectively.
As in Lemma \[major\], by letting $t = \frac{s_{\rm odd \rm}+c_i-c_j}{2}$ we can now compute ${\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2)=(1,2) | R_{A,\underline{c}}]$: $$\begin{aligned}
{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2)=(1,2) | R_{A,\underline{c}}] &= \frac{2^{s_{\rm even \rm}-\chi_{\rm even \rm}}{s_{\rm odd \rm}-\chi_{\rm odd \rm} \choose t-\chi_{\rm odd \rm}(i)}}{{c_i+c_j \choose c_i}}, \label{first3} \\
{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2)=(1,2) | R_{A,\underline{c}'}] &= \frac{2^{s_{\rm even \rm}-\chi_{\rm even \rm}}{s_{\rm odd \rm}-\chi_{\rm odd \rm} \choose t+1-\chi_{\rm odd \rm}(i)}}{{c_i+c_j \choose c_i+1}}. \label{firstz}
\end{aligned}$$
Let $b = c_j-c_i \ge 2$. Note that the $\chi$ values will be the same when considering both $\underline{c}$ and $\underline{c}'$ and so dividing by gives $$\begin{aligned}
\frac{{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2)=(1,2) | R_{A,\underline{c}}]}{{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2)=(1,2) | R_{A,\underline{c}'}]} &= \frac{c_j(t+1-\chi_{\rm odd \rm}(i))}{(c_i+1)(s_{\rm odd \rm}-t-\chi_{\rm odd \rm}(j))} \nonumber \\
&= \frac{c_j}{c_i+1} \cdot \frac{s_{\rm odd \rm}-b+2-2\chi_{\rm odd \rm}(i)}{s_{\rm odd \rm}+b-2\chi_{\rm odd \rm}(j)} \nonumber \\
&\le \frac{c_j}{c_i+1}\cdot \frac{s_{\rm odd \rm}-b+2}{s_{\rm odd \rm}+b-2}. \label{first4}
\end{aligned}$$
Note that $\frac{s_{\rm odd \rm}-b+2}{s_{\rm odd \rm}+b-2}$ is non decreasing in $s_{\rm odd \rm}$ and $s_{\rm odd \rm} \le 2c_i+b = 2c_j-b$, so we can bound by taking $s_{\rm odd \rm} = 2c_i+b = 2c_j-b$ to get: $$\begin{aligned}
\frac{{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2)=(1,2) | R_{A,\underline{c}}]}{{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2)=(1,2) | R_{A,\underline{c}'}]} &\le \frac{c_j}{c_i+1} \cdot \frac{2c_i + b - b +2}{2c_i + b - b - 2} \nonumber \\
&= \frac{c_j(c_i+1)}{(c_i+1)(c_j-1)} \nonumber \\
&= \frac{c_j}{c_j-1}. \label{first5}
\end{aligned}$$
If we apply inequality to : $$\begin{aligned}
h_v(2,K_{\underline{c}}) &\le \frac{c_j}{c_j-1}\sum_A\biggl[\frac{n!}{c_1}{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2) = (1,2) | R_{A,\underline{c}'}]\cdot {\mathbb{P}}[R_{A,\underline{c}} | C_{n,k} \in P_{\underline{c}}]\biggr]. \nonumber
\end{aligned}$$ Recall that ${\mathbb{P}}[R_{A,\underline{c}} | C_{n,k} \in P_{\underline{c}}] = {\mathbb{P}}[R_{A,\underline{c}'} | C_{n,k} \in P_{\underline{c}'}]$, so: $$\begin{aligned}
h_v(2,K_{\underline{c}}) &\le \frac{c_j}{c_j-1}\sum_A\biggl[\frac{n!}{c_1}{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2) = (1,2) | R_{A,\underline{c}'}]\cdot {\mathbb{P}}[R_{A,\underline{c}} | C_{n,k} \in P_{\underline{c}}]\biggr] \nonumber \\
&= \frac{c_1'c_j}{c_1(c_j-1)}\sum_A\biggl[\frac{n!}{c_1'}{\mathbb{P}}[C_{n,k} \in Q, (a_1,a_2) = (1,2) | R_{A,\underline{c}'}]\cdot {\mathbb{P}}[R_{A,\underline{c}'} | C_{n,k} \in P_{\underline{c}'}]\biggr] \nonumber \\
&= \frac{c_1'c_j}{c_1(c_j-1)}h_v(2,K_{\underline{c}'}) \nonumber \\
&\le \frac{(c_i+1)c_j}{c_i(c_j+1)}h_v(2,K_{\underline{c}'}), \nonumber
\end{aligned}$$ as required.
We now apply this result to prove Lemma \[close\].
Let $k \ge 3$ and $\underline{c} = (c_1,\ldots,c_n) \in {\mathbb{N}}^k$ and suppose $K_{\underline{c}}$ has vertex classes $V_1,\ldots,V_k$. Further suppose $T_k(n)$ has vertex classes $V_1',\ldots, V_k'$ with $b_i=|V_i'|<|V_j'|=b_j$ only if $c_i \le c_j$ and suppose that $v \in V_1 \cap V_1'$. We will prove by induction on $f(c,b) = \sum_i|c_i-b_i|$ that $$\begin{aligned}
h_v(2,K_{\underline{c}}) \le h_v(2,T_k(n))\prod_{i =1}^ke^{\left|\log\left(\frac{b_i}{c_i}\right)\right|}.
\end{aligned}$$
The base case of $f(c,b) = 0$ follows since $K_{\underline{c}}$ is $T_k(n)$. Suppose that $f(c,b) \ge 1$ and the result holds for smaller values of $f(c,b)$. Note that if $f(c,b) \neq 0$, then since $\sum_i (c_i-b_i) = 0$, there must be $i,j$ such that $c_i \le b_i -1$ and $c_j \ge b_j+1$. Let $i$ and $j$ be such that $b_i-c_i$ and $c_j-b_j$ are maximised. If $b_i = b_j+1$, we have a contradiction since then $c_i < c_j$, but $b_i > b_j$. This means that $c_j \ge c_i + 2$ and so if we let $\underline{c}'=(c'_1,\ldots,c'_k)$ be such that $c'_i = c_i+1, c'_j = c_j-1$ and $c'_k=c_k$ otherwise, we may apply Lemma \[stepcount\] to get that $$\begin{aligned}
h_v(2,K_{\underline{c}}) &\le \frac{(c_i+1)c_j}{c_i(c_j+1)}h_v(2,K_{\underline{c}'}) \nonumber \\
&= \exp\left\{\left|\log\left(\frac{c_i'}{c_i}\right)\right| + \left|\log\left(\frac{c_j'}{c_j}\right)\right|\right\}h_v(2,K_{\underline{c}'}). \label{close34}
\end{aligned}$$
To proceed by induction, we first observe that $f(c',b) < f(c,b)$ and secondly we must check that if $b_r < b_s$, then $c_r' \le c_s'$. Note that this still holds for $r=i$ and $s=j$ and will still hold if neither $r=i$ nor $s=j$. If $r=i$ and $b_i < b_s$ but $c_i' > c_s'$, then it must be the case that $b_s - c_s > b_i - c_i$, which contradicts our choice of $i$. Similarly if we have $s=j$, $b_r < b_j$ and $c_r' > c_j$, then we arrive at the similar contradiction that $c_r - b_r > c_j - b_j$. Therefore we may apply the inductive hypothesis to to conclude that $$\begin{aligned}
h_v(2,K_{\underline{c}}) &\le \exp\left\{\left|\log\left(\frac{c_i'}{c_i}\right)\right| + \left|\log\left(\frac{c_j'}{c_j}\right)\right|\right\}h_v(2,T_k(n))\prod_{l =1}^ke^{\left|\log\left(\frac{b_l}{c_l'}\right)\right|} \nonumber \\
&= h_v(2,T_k(n))\prod_{l \neq i,j}e^{\left|\log\left(\frac{b_l}{c_l}\right)\right|}\prod_{l=i,j}\exp\left\{\left|\log\left(\frac{b_l}{c_l'}\right)\right|+ \left|\log\left(\frac{c_l'}{c_l}\right)\right|\right\} \nonumber \\
&= h_v(2,T_k(n))\prod_{i =1}^ke^{\left|\log\left(\frac{b_i}{c_i}\right)\right|}.
\end{aligned}$$
We use a more complicated probabilistic argument for the proof of Lemma \[Turancount\]. We consider a different version of the random codes we have previously considered.
Fix $a_1=1$, then given $a_{i-1}$ for $i \ge 2$, let $a_i$ be uniformly distributed on $[k] \setminus \{a_{i-1}\}$. Define the code $C^2(b_1,k) = a_1\cdots a_m$, where $m = \max\{j : |\{i\le j : a_i = 1\}| = b_1\}$ (in other words, keep track of a random walk on $K_{k}$ and stop just before the $(b_1+1)$-th appearance of $1$).
Conditional on $m=n$, the code $C^2(b_i,k)$ is uniformly distributed on codes $f_1\cdots f_n$ in $Q$ that contain $b_1$ copies of $1$ and satisfy $f_1=1$. This is equal in distribution to $C_{n,k} = d_1\cdots d_n$, where each $d_i$ is independently uniformly distributed on $[k]$, conditional on $C_{n,k}$ being in $Q$, having $b_1$ copies of $1$ and starting with $d_1 = 1$.
Let $W$ be the number of transitions from 1 to 2 in $C^2(b_1,k)$ – that is $W = |\{j : (a_j,a_{j+1})=(1,2)\}|$. Note that any shift of a code in $Q \cap P_{\underline{c}}$ ($a_{M+1}\cdots a_n a_1 \cdots a_{M}$ for example) will also be in $Q \cap P_{\underline{c}}$. This means that we can shift the code $C^2(b_1,k)$ to each appearance of $1$ to get another instance of a code $f_1 \cdots f_n$ in $Q$, with $f_1=1$ containing $b_1$ appearances of $1$. Thus by symmetry, given $W$, the probability that $C^2(b_1,k)$ starts with $(a_1,a_2):=(1,2)$ is $\frac{W}{b_1}$. Thus it suffices to show that $W$ is at most $\frac{b_1}{2k}$ with probability asymptotically smaller than the probability that $C^2(b_1,k)$ is in $P_{\underline{b}}$ and $Q$.
Since each letter after a copy of $1$ is independently and uniformly distributed on $\{2,\ldots,k\}$ and there are $b_1$ copies of 1, $W$ is distributed like a Binomial random variable $\mathrm{Bin}(b_1,\frac{1}{k-1})$. Applying a Chernoff bounds gives: $$\begin{aligned}
{\mathbb{P}}\left[W \le \frac{n}{2k^2}\right] \le e^{-\frac{n}{8k^2}}. \label{cher1}
\end{aligned}$$
Now consider the probability that the code $C^2(b_1,k)$ is of the correct length. Note that the letter directly after a $1$ cannot be a $1$ but (until the next copy of $1$), each subsequent letter is a $1$ with probability $\frac{1}{k-1}$ and so removing the letter after each $1$ and considering an appearance of a $1$ as a *failure*, the variable $m-2b_1$ is distributed like a Negative Binomial random variable, $\mathrm{NB}(b_1,\frac{k-2}{k-1})$. $$\begin{aligned}
{\mathbb{P}}[m=n] &= {\mathbb{P}}\left[\mathrm{NB}\left(b_1,\frac{k-2}{k-1}\right) = n-b_1\right] \nonumber \\
&= {n-(b_1+1) \choose n-2b_1}\left(\frac{k-2}{k-1}\right)^{n-2b_1}\left(\frac{1}{k-1}\right)^{b_1}. \nonumber
\end{aligned}$$
Now an application of De-Moivre Laplace (see [@Feller-Probability VII.3]) tells us that $$\begin{aligned}
{\mathbb{P}}[m=n] = \Theta\biggl(n^{-\frac{1}{2}}\exp\biggl\{-\frac{(b_1-\frac{n-b_1}{k-1})^2}{2(n-b_1)\frac{k-2}{(k-1)^2}}\biggr\}\biggr) \label{cherz}
\end{aligned}$$
Note that $|b_1 - \frac{n}{k}| < 1$, as we are in the Turán graph $T_k(n)$ and so $|b_1-\frac{n-b_1}{k-1}| = |\frac{k}{k-1}(b_1-\frac{n}{k})| < 2$. Putting this into , we see that $$\begin{aligned}
{\mathbb{P}}[m=n] &= \Theta \biggl(n^{-\frac{1}{2}}\exp \biggl\{-O\bigl(n^{-1}\bigr)\biggr\}\biggr) \nonumber \\
&= \Theta \bigl(n^{-\frac{1}{2}}\bigr). \label{cher2}
\end{aligned}$$
Next, consider ${\mathbb{P}}\left[C^2(b_i,k) \in P_{\underline{b}} | m = n\right]$. As mentioned above, conditional on $m=n$, $C^2(b_i,k)$ is distributed like $C_{n,k}$ conditional on being in $Q$, starting with $d_1 = 1$ and having $b_1$ copies of $1$. By Lemma \[major\], the events $\{C_{n,k} \in P_{\underline{b}}\}$ and $\{C_{n,k} \in Q\}$ are positively correlated and so $$\begin{aligned}
{\mathbb{P}}[C^2(b_i,k) \in P_{\underline{b}} | m = n] &= {\mathbb{P}}[C_{n,k} \in P_{\underline{b}} | C_{n,k} \in Q, d_1 = 1, \text{ $b_1$ copies of $1$}] \nonumber \\
&\ge {\mathbb{P}}[C_{n,k} \in P_{\underline{b}} | C_{n,k} \in Q] \nonumber \\
&\ge {\mathbb{P}}[C_{n,k} \in P_{\underline{b}}] \nonumber \\
&\ge {\mathbb{P}}\left[\rm Mult \rm\left(n,\left(\frac{1}{k},\ldots,\frac{1}{k}\right)\right) = \underline{b}\right] \nonumber \\
&= \Omega(n^{-\frac{k}{2}}). \label{cher3}
\end{aligned}$$
So combining and we can conclude $$\begin{aligned}
{\mathbb{P}}\left[C^2(b_i,k) \in Q \cap P_{\underline{b}}\right] &= {\mathbb{P}}\left[C^2(b_i,k) \in P_{\underline{b}} | m = n\right]{\mathbb{P}}[m=n] \nonumber \\
&= \Omega\left(n^{-\frac{k+1}{2}}\right). \label{cher4}
\end{aligned}$$
We can now complete our proof. We have $$\begin{aligned}
h_v\bigl(2,T_k(n)\bigr) &= k^n(b_1-1)!\biggl(\prod_{l=2}^k(b_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q \cap P_{\underline{b}}, (a_1,a_2) = (1,2)] \nonumber \\
&= k^n(b_1-1)!\biggl(\prod_{l=2}^k(b_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q, a_1 = 1, |\{j : a_j = 1]| = b_1] \nonumber \\
&\cdot {\mathbb{P}}[C_{n,k} \in P_{\underline{b}}, a_2 = 2 | C_{n,k} \in Q, a_1 = 1, |\{j : a_j = 1\}| = b_1]. \nonumber
\end{aligned}$$
Recall that $C_{n,k} = a_1 \cdots a_n$ given that $C_{n,k} \in Q$ and $a_1 = 1$ and $|\{j : a_j = 1\}| = b_1$ is equal in distribution to $C^2(b_1,k) = d_1 \cdots d_m$ given $m=n$ and so $$\begin{aligned}
h_v\bigl(2,T_k(n)\bigr) &= k^n(b_1-1)!\biggl(\prod_{l=2}^k(b_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q, a_1 = 1, |\{j : a_j = 1\}| = b_1] \nonumber \\
&\cdot {\mathbb{P}}[C^2(b_1,k) \in P_{\underline{b}}, d_2 = 2 | m = n] \nonumber \\
&= k^n(b_1-1)!\biggl(\prod_{l=2}^k(b_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q, a_1 = 1, |\{j : a_j = 1\}| = b_1] \nonumber \\
&\cdot {\mathbb{P}}[d_2 = 2 | C^2(b_1,k) \in P_{\underline{b}}, m = n] \cdot {\mathbb{P}}[C^2(b_1,k) \in P_{\underline{b}} | m=n]. \label{cher6}
\end{aligned}$$
By considering ${\mathbb{P}}[d_2 = 2 | C^2(b_1,k) \in P_{\underline{b}}, m = n]$, we get $$\begin{aligned}
{\mathbb{P}}[d_2 = 2 | C^2(b_1,k) \in P_{\underline{b}}, m = n] &\ge {\mathbb{P}}\left[a_2 = 2 | C^2(b_1,k) \in P_{\underline{b}}, m = n, W > \frac{n}{2k^2}\right] \nonumber \\
&- {\mathbb{P}}\left[W \le \frac{n}{2k^2} | C^2(b_1,k) \in P_{\underline{b}}, m = n \right] \nonumber \\
&\ge \frac{n}{2k^2b_1} - \frac{{\mathbb{P}}[W \le \frac{n}{2k^2}]}{{\mathbb{P}}[C^2(b_1,k) \in P_{\underline{b}}, m = n]}. \nonumber
\end{aligned}$$ Thus by applying and we get $$\begin{aligned}
{\mathbb{P}}[d_2 = 2 | C^2(b_1,k) \in P_{\underline{b}}, m = n] &= \frac{n}{2k^2b_1} - O\biggl(\frac{e^{-\frac{n}{8k^2}}}{n^{-\frac{k+1}{2}}}\biggr) \nonumber \\
&= \frac{n}{2k^2b_1} - o(1). \nonumber
\end{aligned}$$
This means that for sufficiently large $n$, ${\mathbb{P}}[a_2 = 2 | C^2(b_1,k) \in P_{\underline{b}}, m = n] \ge \frac{1}{3k}$. Putting this into , we see $$\begin{aligned}
h_v\bigl(2,T_k(n)\bigr) &\ge \frac{k^n(b_1-1)!}{3k}\biggl(\prod_{l=2}^k(b_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q, a_1 = 1, |\{j : a_j = 1\}| = b_1] \nonumber \\
&\cdot {\mathbb{P}}[C^2(b_1,k) \in P_{\underline{b}} | m=n] \nonumber \\
&= \frac{k^n(b_1-1)!}{3k}\biggl(\prod_{l=2}^k(b_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q, a_1 = 1, |\{j : a_j = 1\}| = b_1] \nonumber \\
&\cdot {\mathbb{P}}[C_{n,k} \in P_{\underline{b}} | C_{n,k} \in Q, a_1 = 1, |\{j : a_j = 1\}| = b_1] \nonumber \\
&= \frac{k^n(b_1-1)!}{3k}\biggl(\prod_{l=2}^k(b_l!)\biggr){\mathbb{P}}[C_{n,k} \in Q \cap P_{\underline{b}}, a_1 = 1] \nonumber \\
&= \frac{k^n}{2n}\biggl[\prod_{i=1}^k(b_i!)\biggr]\cdot {\mathbb{P}}[C_{n,k} \in Q \cap P_{\underline{b}}] \cdot \frac{2n \cdot{\mathbb{P}}[a_1 = 1 | C_{n,k} \in Q \cap P_{\underline{b}}]}{3kb_1} \nonumber \\
&= h\bigl(T_k(n)\bigr) \cdot \frac{2n \cdot{\mathbb{P}}[a_1 = 1 | C_{n,k} \in Q \cap P_{\underline{b}}]}{3kb_1}. \nonumber
\end{aligned}$$
By symmetry, ${\mathbb{P}}[a_1 = 1 | C_{n,k} \in Q \cap P_{\underline{b}}] = \frac{b_1}{n}$. This completes the proof of the lemma.
Now we bound below the number of Hamilton cycles in $T_k(n)$ by the number of Hamilton cycles in $T_k(m)$, where $m < n$.
Let $v$ be a vertex contained in the largest vertex class $V_i$ in $T_k(n)$. Removing $v$ gives $T_k(n-1)$. For each Hamilton cycle $v_1\cdots v_{n-1}$ in $T_k(n-1)$, we can form a Hamilton cycle in $T_k(n)$ by inserting $v$ between two vertices $v_j$ and $v_{j+1}$, both not in $V_i$. For each Hamilton cycle in $T_k(n-1)$, there are at least $(n-1)\frac{k-2}{k}$ spaces where we can insert $v$ and under this construction each Hamilton cycle in $T_k(n)$ will be formed in at most one way. Counting over all Hamilton cycles in $T_k(n-1)$, we get that $$\begin{aligned}
h(T_k(n)) \ge (n-1)\frac{k-2}{k}h(T_k(n-1)). \label{second1}
\end{aligned}$$
We can apply equation inductively to get that for any $i \in [n]$, $$\begin{aligned}
h(T_k(n)) \ge (n-1)_i\left(\frac{k-2}{k}\right)^ih(T_k(n-i)). \nonumber
\end{aligned}$$
We now bound the number of cycles in $T_k(n)$ in terms of the number of Hamilton cycles.
Let $I$ be a subset of $[n]$ with $|I| = r$. Then by Lemma \[recursion\] and Lemma \[Turanbest\], we have $$\begin{aligned}
h(G[I]) &\le h(T_k(r)) \nonumber \\
&\le \left(\frac{k}{k-2}\right)^{n-r}\frac{h(T_k(n))}{(n-1)_{n-r}} \nonumber \\
&\le \left(\frac{2k}{k-2}\right)^{n-r}\frac{h(T_k(n))}{(n)_{n-r}} \nonumber
\end{aligned}$$ Summing over all subsets $I$, we have $$\begin{aligned}
c(T_k(n)) &\le \sum_{i=0}^{n-3}{\binom{n}{i}}\left(\frac{2k}{k-2}\right)^i\frac{h(T_k(n))}{(n)_i} \nonumber \\
&= h(T_k(n))\sum_{i=0}^{n-3}\frac{1}{i!}\left(\frac{2k}{k-2}\right)^i \nonumber \\
&\le e^{\frac{2k}{k-2}}h(T_k(n)), \nonumber
\end{aligned}$$ as required.
Finally, we prove Lemma \[second2count\].
Let $n \in {\mathbb{N}}$ and denote $\left\lfloor \frac{n}{2} \right\rfloor$ by $t$ and $\left\lceil \frac{n}{2} \right\rceil$ by $t'$. For $r \ge 2$, the number of cycles of length $2r$ in $T_2(n)$ is $$\frac{\left(t\right)_r\left(t'\right)_r}{2r}.$$ Summing over $r=2,\ldots,t$ gives $$\begin{aligned}
c(T_2(n)) &= \sum_{r=2}^t \frac{\left(t\right)_r\left(t'\right)_r}{2r} \nonumber \\
&= \frac{t! t'!}{2t} \sum_{r=2}^t \frac{t}{r (t-r)! (t'-r)!} \nonumber \\
&\le \frac{t! t'!}{2t} \sum_{r'=0}^{t-2} \frac{t}{(t-r') r'! r'!}, \nonumber
\end{aligned}$$ where we substituted $r' = t-r$ to obtain the second equality. As $c_{2t}(T_2(n)) = \frac{t! t'!}{2t}$ and $\frac{t}{(t-s) s!}$ is easily bounded by $2$, we have $$\begin{aligned}
\label{eq:t2}
c(T_2(n)) &\le 2c_{2t}(T_2(n)) \sum_{r'=0}^{t-2} \frac{1}{r'!} \nonumber \\
&\le 2c_{2t}(T_2(n)) \sum_{r'\ge0} \frac{1}{r'!} = 2e \cdot c_{2t}(T_2(n)).
\end{aligned}$$
Let $s = \left\lfloor \frac{n-1}{2} \right\rfloor$ and $s' = \left\lceil \frac{n}{2} \right\rceil$. Note that $t = s'$ and $t' = s+1$, and so $$\begin{aligned}
\label{eq:t22}
\frac{n-2}{2}\frac{s'! s!}{2s} \le \frac{s}{t} \cdot\frac{s'! s!t'}{2s} = \frac{t!t'!}{2t}.
\end{aligned}$$ Using gives $$\begin{aligned}
c_{2\left\lfloor \frac{n-1}{2} \right\rfloor}(T_2(n-1)) = \frac{s'! s!}{2s} \le \frac{2}{n-2} \cdot \frac{t!t'!}{2t} = \frac{2}{n-2} c_{2\left\lfloor \frac{n}{2} \right\rfloor}(T_2(n)) .
\end{aligned}$$ As $i=o(n)$, repeatedly applying this bound along with gives
$$\begin{aligned}
c(T_2(n-i)) &\le 2e \cdot c_{2\left\lfloor \frac{n-i}{2} \right\rfloor}(T_2(n-i))\\
&\le 2e \left(\prod_{j=1}^i \frac{2}{n-j-1}\right) c_{2\left\lfloor \frac{n}{2} \right\rfloor}(T_2(n))\\
&\le 2e \left(\frac{4}{n}\right)^i c_{2\left\lfloor \frac{n}{2} \right\rfloor}(T_2(n)),
\end{aligned}$$
as required.
Conclusion and Open Questions {#sec5}
=============================
In this paper we resolve Conjecture \[gundconj\] for sufficiently large $n$ (we do not optimise the value of $n$ given by our approach, as it would still be very large). For triangle-free graphs, Arman, Gunderson and Tsaturian [@Gund1] (see also [@Gund2]) show that the Turán graph $T_2(n)$ uniquely maximises the number of cycles when $n\ge 141$, but it seems likely that this should hold for all values of $n$.
Theorem \[main\] only deals with $H$ such that $\chi(H) \ge 3$ and $H$ contains a critical edge. When $H$ does not satisfy these properties, our approach is not feasible as the extremal $H$-free graph is no longer $T_k(n)$. It is interesting to consider what could be true for such $H$. For example, it is natural to ask whether it is possible to maximize the number of edges and the number of cycles simulateously (as in Theorem \[main\]).
Let $H$ be a fixed graph. Does ${{\rm EX}}(n;H)$ contain a graph with $m(n;H)$ cycles for sufficiently large $n$?
As $T_2(n)$ does not contain any odd cycle, Theorem \[main\] implies that for any odd $k$, $T_2(n)$ is the $n$-vertex graph with odd girth at least $k$ containing the most cycles. Arman, Gunderson and Tsaturian [@Gund1] ask a more general question.
What is the maximum number of cycles in an $n$-vertex graph, with girth at least $g$?
This question seems difficult since comparatively little is known about the maximum number of edges in an graph with girth at least $g \ge 4$.
Another interesting problem was raised by Király [@deathnote] who asked for the maximum number of cycles in a graph with $m$ edges can contain (without constraining the number of vertices); he conjectured an upper bound of $1.4^m$ cycles. In a recent paper Arman and Tsaturian [@armtsat] give an upper bound of $8.25\times 3^{m/3}$ and a lower bound of $1.37^m$, and conjecture that their upper bound is correct to within a $\left(1+o(1)\right)^m$ factor. It would be interesting to consider the effect of adding the additional constraint of forbidding a subgraph. In particular what is the maximum number of cycles that a triangle-free graph with $m$ edges can contain?
A similar problem to that of Király is to maximise the number of cycles in a graph with $n$ vertices and $m$ edges. For $m=\Omega(n^2)$ and $n$ sufficiently large, Arman and Tsaturian [@armtsat Conjecture 6.1] conjecture a maximum of $\left(1+o(1)\right)^n\left(\tfrac{2m}{en}\right)^n$ cycles. The current best upper bound is $\left(1+o(1)\right)^n\left(\tfrac{2m}{2n}\right)^n$ given in the same paper. We believe that the method used to prove Lemma \[easycor\] improves this upper bound but does not prove the conjecture.
Another direction of research is to maximise the number of *induced* cycles. Given a graph $G$, let $m_I(G)$ denote the number of induced cycles in $G$ and let $m_I(n) := \max\{m_I(G): |V(G)| = n\}$. Morrison and Scott [@morsco] recently determined $m_I(n)$ for $n$ sufficiently large and proved that the extremal graphs are unique. The extremal graphs in question are essentially blow-ups of $C_{n/3}$ and contain many copies of $C_4$.
It would be interesting to consider what happens to the extremal graphs when we forbid $C_4$.
What is $m_I(n;C_4):= \max\{m_I(G): |V(G)| = n, G \text{ is } C_4\text{-free}\}$?
[^1]: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, United Kingdom and Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brasil. Research partially supported by CNPq and Sidney Sussex College, Cambridge. E-mail: `[email protected]`.
[^2]: Mathematical Institute, University of Oxford, Woodstock Road, Oxford, United Kingdom. E-mail: `{robertsa, scott}@maths.ox.ac.uk`.
[^3]: Supported by a Leverhulme Trust Research Fellowship.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the concept of generalized measure-theoretic entropy, where instead of the Shannon entropy function we consider an arbitrary concave function defined on the unit interval, vanishing in the origin. Under mild assumptions on this function we show that this isomorphism invariant is linearly dependent on the Kolmogorov-Sinai entropy.'
address: 'Department of Mathematics, Cracow University of Economics, Rakowicka 27, 31-510 Kraków, Poland'
author:
- Fryderyk Falniowski
title: 'On connections of generalized entropies with Shannon and Kolmogorov-Sinai entropies'
---
Introduction
============
Dynamical and measure-theoretic (called also Kolmogorov-Sinai entropy) entropies are a basic tool for investigating dynamical systems (see e.g. [@Downar; @KH]). They were extensively studied and successfully applied among others in statistical physics and quantum information. It appeared to be an exceptionally powerful tool for exploring nonlinear systems. One of the biggest advantages of the Kolmogorov-Sinai entropies lies in the fact that it makes possible to distinguish the formally regular systems (those with the measure-theoretic entropy equal to zero) from the chaotic ones (with positive entropy, which implies positivity of topological entropy [@Misiurewicz]).
The Kolmogorov-Sinai entropy of a given transformation $T$ acting on a probability space $(X,\Sigma,\mu)$ is defined as the supremum over all finite measurable partitions $\mathcal{P}$ of the dynamical entropy of $T$ with respect to $\mathcal{P}$, denoted by $h(T,\mathcal{P})$. As a dynamical counterpart of Shannon entropy, the entropy of transformation $T$ with respect to a given partition $\mathcal{P}$ is defined as the limit of the sequence $\left(\frac 1n H(\mathcal{P}_n)\right)_{n=1}^{\infty}$, where $$H(\mathcal{P}_n)=\sum\limits_{A\in \mathcal{P}_n} \eta\left(\mu(A)\right)$$ with $\eta$ being the Shannon function given by $\eta(x)=-x\log x$ for $x>0$ with $\eta(0):=0$ and $\mathcal{P}_n$ is the join partition of partitions $T^{-i}\mathcal{P}$ for $i=0,...,n-1$. The existence of the limit in the definition of the dynamical entropy follows from the subadditivity of $\eta$. The most common interpretation of this quantity is the average (over time and the phase space) one-step gain of information about the initial state. Taking supremum over all finite partitions we obtain an isomorphism invariant which measures the rate of producing randomness (chaos) by the system.
Since Shannon’s seminal paper [@Shannon] many generalizations of the concept of Shannon static entropy were considered, see Arimoto [@Arimoto], Rényi [@Renyi] and Csiszár’s survey article [@Csiszar2008]. The dynamical and measure-theoretic counterparts were considered by few authors. De Paly [@de; @Paly1] proposed generalized dynamical entropies based on the concept of the relative static entropies. Unfortunately it appeared that, despite some special cases [@de; @Paly1; @de; @Paly2] the explicit clculations of this invariant may not be possible. Grassberger and Procaccia proposed in [@GP83] a dynamical counetrpart of the well-known generalization of Shannon entropy – the Rényi entropy, and its measure-theoretic counterpart were considered by Takens and Verbitski. They showed that for ergodic transformations with positive measure-theoretic entropy, Rényi entropies of a measure-theoretic transformation are either infinite or equal to the measure-theoretic entropy [@TakensVerb1998]. The answer for non-ergodic aperiodic transformations is different, for Rényi entropies of order $\alpha >1$ they are equal to the essential infimum of the measure-theoretic entropies of measures forming the decomposition of a given measure into ergodic components, while for $\alpha<1$ they are still infinite [@TV2002]. In particular, this means that Rényi entropies of order $\alpha<1$ are metric invariants sensitive to ergodicity. Similar generalization was made by Mesón and Vericat [@MV96; @MV] for so called Havrda-Charvát-Tsallis entropy [@Havrda] and their results were similar to ones obtained by Takens and Verbitski in [@TakensVerb1998]. In our approach is based on Arimoto generalization applied to dynamical case. Instead of the Shannon function $\eta$ we consider a concave function $g\colon [0,1]\mapsto \mathbb{R}$ such that $\lim\limits_{x\to 0^+}g(x)=g(0)=0$ and define the dynamical $g$-entropy of the finite partition ${\mathcal{P}}$ as $$h(g,T,{\mathcal{P}})=\limsup_{n\to\infty}\frac 1n \sum_{A\in{\mathcal{P}}_n}g(\mu(A)).$$
The behaviour of the quotient $g(x)/\eta(x)$ as $x$ converges to zero appears to be crucial for our considerations. Mainly, defining $$\operatorname{Ci}(g):=\liminf_{x\to 0^+}\frac{g(x)}{\eta(x)}\;\;\text{and}\;\;\operatorname{Cs}(g):=\limsup_{x\to 0^+}\frac{g(x)}{\eta(x)}$$ we will prove that $$\operatorname{Ci}(g)\cdot h(T,{\mathcal{P}})\leq h(g,T,{\mathcal{P}})\leq \operatorname{Cs}(g) \cdot h(T,{\mathcal{P}}).$$ Moreover for $g$ such that $\operatorname{Ci}(g)=\infty$, which fullfills some additional condition (e. g. for $g(x)=\sqrt{x}$), we can find zero dynamical entropy processes $(X,\Sigma,\mu,T,{\mathcal{P}})$ with a given positive dynamical $g$-entropy.
Taking the supremum over all partitions we obtain Kolmogorov entropy-like isomorphism invariant, which we will call the measure-theoretic $g$-entropy of a transformation with respect to an invariant measure. One might ask whether this invariant may give any new information about the system. We will prove (Theorem \[twhKSg\]) that for $g$ with $\operatorname{Cs}(g)<\infty$, this new invariant is linearly dependent on Kolmogorov-Sinai entropy. It means that in fact the Shannon entropy function is the most natural one – not only it has all of the properties which the entropy function should have [@Downar], but also considering different entropy functions we will not obtain essentially different invariant. This result might has the other interpretation. Ornstein and Weiss showed in [@OrnsteinWeiss] that every finitely observable invariant for the class of all ergodic processes has to be a continuous function of the entropy. It is easy to see that any continuous function of the entropy is finitely observable – one simply composes the entropy estimators with the continuous function itself. In other words an isomorphism invariant is finitely observable if and only if it is a continuous function of the Kolmogorov-Sinai entropy. Therefore our result implies that the generalized measure-theoretic entropy is in fact finitely observable. It should be possible to give a more direct proof of the finite observability of the generalized measure-theoretic entropy but the proof cannot be easier[^1] than the proof that entropy itself is finitely observable, see [@Weiss].
The paper is organized as follows: in the next section we give a formal definition of the dynamical $g$-entropy and establish its basic properties. The subsequent section is devoted to the construction of a zero dynamical entropy process with a given positive $g$-entropy. Finally, in the last section, we define a measure-theoretic $g$-entropy of a transformation and show connections between this new invariant and the Kolmogorov-Sinai entropy.
Basic facts and definitions
===========================
Let $(X,\Sigma,\mu)$ be a Lebesgue space and let $g:[0,1] \mapsto \mathbb{R}$ be a concave function with $g(0)=\lim\limits_{x\to 0^+}g(x)=0$.[^2] By $\mathcal{G}_0$ we will denote the set of all such functions. Every $g\in{\mathcal{G}_0}$ is subadditive, i. e. $g(x+y)\leq g(x)+g(y)$ for every $x,y\in[0,1]$, and quasihomogenic, i.e. $\varphi_g\colon (0,1]\to\mathbb{R}$ defined by $\varphi_g(x):=g(x)/x$ is decreasing (see [@Rosenbaum]).[^3]
For a given finite partition $\mathcal{P}$ we define the as $$H(g,\mathcal{P}):=\sum_{A\in{\mathcal{P}}} g\left(\mu(A)\right).$$
For $g=\eta$ the latter is equal to the Shannon entropy of the partition $\mathcal{P}$. For two finite partitions $\mathcal{P}$ and $\mathcal{Q}$ of the space $X$ we define a new partition $\mathcal{P}\vee\mathcal{Q}$ ( of $\mathcal{P}$ and $\mathcal{Q}$) consisting of the subsets of the form $B\cap C$ where $B\in\mathcal{P}$ and $C\in\mathcal{Q}$. The join partition of more than two partitions is defined similarly.
Dynamical $g$-entropies.
------------------------
For an automorphism $T\colon X\mapsto X$ and a partition $\mathcal{P}=\{E_{1},...,E_{k}\}$ we put $$T^{-j}\mathcal{P}:=\{T^{-j}E_{1},...,T^{-j}E_{k}\}$$ and $$\mathcal{P}_{n}=\mathcal{P}\vee T^{-1}\mathcal{P}\vee...\vee T^{-n+1}\mathcal{P}.$$
Now for a given $g\in{\mathcal{G}_0}$ and a finite partition ${\mathcal{P}}$ we can define the as
$$\label{dynentr} h_{\mu}(g,T,{\mathcal{P}})=\limsup_{n\to\infty}\frac 1n H\left(g,{\mathcal{P}}_n\right).$$
Alternatively we will call it the $g$-entropy of the process $(X,\Sigma,\mu,T,{\mathcal{P}})$. If the dynamical system $(X,\Sigma,T,\mu)$ is fixed then we omit $T$, writing just $h(g,{\mathcal{P}})$. As in the case of Shannon dynamical entropies we are interested in the existence of the limit of $\left(\frac 1n H(g,{\mathcal{P}}_n)\right)_{n=1}^{\infty}$. If $g=\eta$, we obtain the Shannon dynamical entropy $h(T,{\mathcal{P}})$. However, in the general case we can not replace an upper limit in (\[dynentr\]) by the limit, since it might not exist. Existence of the limit in the case of the Shannon function follows from the subadditivity of the static Shannon entropy. This property has every subderivative function, i.e. function for which the inequality $g(xy)\leq xg(y)+yg(x)$ holds for any $x,y\in[0,1]$, but this is not true in general (an appropriate example will be given in Section 2.2). Therefore we propose more general classes of functions for which the limit exists. It exists if $g$ belongs to one of two following classes: $$\mathcal{G}_{0}^{0}:=\left\{g\in \mathcal{G}_0\;\left|\; \lim_{x\to 0^+} \frac{g(x)}{\eta(x)}=0\right.\right\}
\;\;\;\text{or}\;\;\;
\mathcal{G}_{0}^{\operatorname{Sh}}:=\left\{g\in \mathcal{G}_0\;\left|\; 0<\lim_{x\to 0^+} \frac{g(x)}{\eta(x)}<\infty \right.\right\}.$$ It is easy to show that if $g$ is subderivative then the limit $\lim\limits_{x\to 0^+}g(x)/\eta(x)$ is finite. Moreover we will see that values of dynamical $g$-entropies depend on the behaviour of $g$ in the neighbourhood of zero. We will prove that if $g\in{\mathcal{G}_0}^0\cup{\mathcal{G}_0}^{\operatorname{Sh}}$, then there is a linear dependence between the dynamical $g$-entropy and the Shannon dynamical entropy of a given partition. First we give the following general result:
\[hgp\] If $g_1,g_2\in{\mathcal{G}_0}$ are such that $$\liminf\limits_{x\to 0^+}\frac{g_1(x)}{g_2(x)}<\infty,$$ and ${\mathcal{P}}$ is a finite partition of $X$ with finite dynamical $g_2$-entropy, then $$\liminf_{x\to 0^+}\frac{g_1(x)}{g_2(x)}\cdot h(g_2,{\mathcal{P}})\leq h(g_1,{\mathcal{P}}).$$ If additionally $\limsup\limits_{x\to 0^+}\frac{g_1(x)}{g_2(x)}<\infty$, then $$h(g_1,{\mathcal{P}})\leq \limsup_{x\to 0^+}\frac{g_1(x)}{g_2(x)} \cdot h(g_2,{\mathcal{P}}).$$
Whenever $g_2\colon [0,1]\mapsto \mathbb{R}$ is a nonnegative concave function satisfying $g_2(0)=0$ and $g_2'(0)=\infty$, we can have any pair $0< a\leq b \leq \infty$ as limit inferior and limit superior of $g_1/g_2$ in 0, choosing a suitable function $g_1$. The idea is as follows: construct $g_1$ piecewise linear. To do so define inductively a strictly decreasing sequence $x_k\to 0$, and a decreasing sequence of values $y_k=g_1(x_k)\to 0$, thus defining intervals $J_k:=[x_{k+1},x_k]$ where $g$ is affine. The only constraint to get a concave function is that the slope of $g$ on each interval $J_k$ has to be smaller than $y_k/x_k$, and increasing with respect to $k$; this is not an obstruction to approach any limit inferior and limit superior for $g_1(x)/g_2(x)$, provided that $x_{k+1}>0$ is choosen small enough.
Let $g_1,g_2\in{\mathcal{G}_0}^0$ and assume that $$\limsup\limits_{x\to 0^+}\frac{g_1(x)}{g_2(x)}<\infty.$$ If the limit $\lim\limits_{n\to \infty} H(g_2,{\mathcal{P}}_n)$ is finite, then $h(g_2,{\mathcal{P}})=0$ and $$0\leq h(g_1,{\mathcal{P}})\leq \limsup_{n\to\infty}\frac 1n \frac{H(g_1,{\mathcal{P}}_n)}{H(g_2,{\mathcal{P}}_n)}H(g_2,{\mathcal{P}}_n)\leq \limsup_{x\to 0^+}\frac{g_1(x)}{g_2(x)}\cdot\limsup_{n\to\infty}\frac 1n H(g_2,{\mathcal{P}}_n)=0$$ so we can assume that it is infinite and that $\lim\limits_{x\to 0^+}\varphi_{g_1}(x)=\infty$ is also infinite, since $H(g,{\mathcal{Q}})<\lim\limits_{x\to 0^+}\varphi_{g_1}(x)$ for any partition ${\mathcal{Q}}$.
Fix $\varepsilon >0$ and $\delta>0$ such that for $x\in (0,\delta]$ we have $$\liminf_{x\to 0^+}\frac{g_1(x)}{g_2(x)}-\varepsilon<\frac{g_1(x)}{g_2(x)}\leq \limsup_{x\to 0^+}\frac{g_1(x)}{g_2(x)}+\varepsilon.$$ Then for every index $n$ we have $$\frac{1}{\delta}\underline{G_{\delta}}\leq \sum\limits_{ B\in{\mathcal{P}}_n,\; \mu(B)\geq \delta}g_1(\mu(B)) \leq \frac{1}{\delta}\overline{G_{\delta}}.$$ where $\overline{G_{\delta}}:= \max\limits_{x\in [\delta,1]}g_1(x),\;\;\; \underline{G_{\delta}}=\min\limits_{x\in [\delta,1]}g_1(x),$ and $$\liminf_{x\to 0^+}\frac{g_1(x)}{g_2(x)}-\varepsilon \leq \sum\limits_{B\in{\mathcal{P}}_n,\; \mu(B) < \delta}g_1(\mu(B))\left/\sum\limits_{B\in{\mathcal{P}}_n,\;\mu(B)< \delta}g_2(\mu(B))\right. \leq \limsup_{x\to 0^+}\frac{g_1(x)}{g_2(x)} + \varepsilon.$$ Therefore
$$\begin{aligned}
\liminf_{x\to 0^+}\frac{g_1(x)}{g_2(x)}-\varepsilon &\leq &\frac{\sum\limits_{ \mu(B) < \delta}g_1(\mu(B))\left/\sum\limits_{{ \mu(B)< \delta}}g_2(\mu(B))\right.+\sum\limits_{ \mu(B)\geq \delta}g_1(\mu(B))\left/\sum\limits_{{ \mu(B)< \delta}}g_2(\mu(B))\right.}{1+\sum\limits_{{ \mu(B)\geq \delta}}g_2(\mu(B))\left/\sum\limits_{{ \mu(B)< \delta}}g_2(\mu(B))\right.}\nonumber \\&\leq &\limsup_{x\to 0^+}\frac{g_1(x)}{g_2(x)}+ \varepsilon.\nonumber\end{aligned}$$
Converging with $n$ to infinity and with $\varepsilon$ and $\delta$ to zero succesively we obtain the assertion. In the case of infinite limit superior of the quotient $g_1(x)/g_2(x)$ we can repeat the above reasoning just omitting an upper bound for considered expressions.
Theorem \[hgp\] immediately implies few corollaries
\[Cg1g2finite\] If the limit $\lim\limits_{x\to 0^+}\frac{g_1(x)}{g_2(x)}$ exists and is finite, then $$h(g_1,{\mathcal{P}})=\lim\limits_{x\to 0^+}\frac{g_1(x)}{g_2(x)} \cdot h(g_2,{\mathcal{P}}).$$
\[Cgfinite\] If $g\in{\mathcal{G}_0}^0\cup{\mathcal{G}_0}^{\operatorname{Sh}}$, then $$h(g,{\mathcal{P}})=\operatorname{C}(g) \cdot h({\mathcal{P}}),$$ where $\operatorname{C}(g)=\lim\limits_{x\to 0^+}\frac{g(x)}{\eta(x)}$.
If $g\in{\mathcal{G}_0}^0\cup{\mathcal{G}_0}^{\operatorname{Sh}}$, then $h(g,{\mathcal{P}})=\lim\limits_{n\to\infty}\frac 1n H(g,{\mathcal{P}}_n)$.
Moreover using similar arguments we might obtain the answer in the case of infinite limit $\lim\limits_{x\to 0^+} g_1(x)/g_2(x)$ and positive dynamical $g_2$-entropy:
Let $g_1,g_2\in \mathcal{G}_0$ be such that $\lim\limits_{x\to 0^+}g_1(x)/g_2(x)=\infty$ and let a finite partition ${\mathcal{P}}$ has positive $g_2$-entropy. Then $h(g_1,{\mathcal{P}})$ is infinite.
Let us define $$\mathcal{G}_{0}^{\infty}:=\left\{g\in \mathcal{G}_0\;\left|\; \lim_{x\to 0^+} \frac{g(x)}{\eta(x)}=\infty \right.\right\},$$ then we can formulate the following fact:
If $g\in{\mathcal{G}_0}^{\infty}$ and a partition ${\mathcal{P}}$ has positive Shannon dynamical entropy, then $h(g,{\mathcal{P}})$ is infinite.
Bernoulli shifts.
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Let $\mathcal{A}=\{1,\ldots,k\}$ be a finite alphabet. Let $X=\{x=\{x_i\}_{i=-\infty}^{\infty}\colon x_i\in\mathcal{A}\}$ and $\sigma$ be a left shift $$\sigma(x)_i=x_{i+1}.$$ For any $s\leq t$ and block $[\omega_0,\ldots,\omega_{t-s}]$ with $a_i\in\mathcal{A}$ we define a cylinder $$C_s^t(\omega_0,\ldots,\omega_{t-s})=\{x\in X: x_i=\omega_{i-s}\;\text{for}\; i=s,\ldots,t\}.$$ We consider the Borel $\sigma$-algebra with respect to the metric, which is given by $d(x,y)=2^{-N}$, where $N=\min\{|i|:x_i\neq y_i\}$. One can show that Borel $\sigma$-algebra is the minimal $\sigma$-algebra containing all cylindrical sets. Let $p=\left(p_1,\ldots,p_k\right)$ be a probability vector, i.e. $p_i\geq 0$ for any $i$ and $\Sigma p_i=1$. We define a measure $\rho=\rho (p)$ on $\mathcal{A}$ by setting $\rho(\{i\})=p_i$. Then $\mu_p$ is a corresponding product measure on $X=\mathcal{A}^{\mathbb{Z}}$. Thus, the static $g$-entropy of a partition ${\mathcal{P}}^{\mathcal{A}}=\{[1],[2],\ldots,[k]\}$ is equal to $$H_{\mu_p}\left(g,{\mathcal{P}}_n^{\mathcal{A}}\right)=\sum_{[\omega_0,\ldots,\omega_{n-1}]\in\mathcal{A}^n}g\left(C_0^{n-1}(\omega_0,\ldots,\omega_{n-1})\right)=\sum_{\left[\omega_0,\ldots,\omega_{n-1}\right]\in\mathcal{A}^n} g\left(p_{\omega_0}\cdots p_{\omega_{n-1}}\right).$$ By the concavity of the function $g$ we have $$H_{\mu_p}(g,{\mathcal{P}}_n^{\mathcal{A}})\leq \varphi\left(\frac{1}{k^n}\right)$$ where equality holds only when $p=p^*=\left(\frac 1k,\ldots, \frac 1k\right)$. Before calculating the dynamical $g$-entropy of the partition ${\mathcal{P}}^{\mathcal{A}}$ with respect to measure $\mu_{p^*}$, we give the following lemma, which proof will be given later:
\[limsupMn\] If $g\in{\mathcal{G}_0}$, then $$\operatorname{Cs}(g)=\limsup_{n\to\infty}\frac{g(k^{-n})}{\eta(k^{-n})}\;\;\text{and}\;\; \operatorname{Ci}(g)=\liminf_{n\to\infty}\frac{g(k^{-n})}{\eta(k^{-n})}$$ for any positive integer $k>1$.
Therefore, applying Lemma \[limsupMn\] for the partition ${\mathcal{P}}^{\mathcal{A}}$ we obtain $$\label{entrBernoulli} h_{\mu_{p^*}}\left(g,{\mathcal{P}}^{p^*}\right)=\limsup_{n\to\infty}\frac 1n \varphi\left(\frac{1}{k^n}\right)=\left\{\begin{array}{ll} \operatorname{Cs}(g) \cdot \log k, & \;\; \text{if}\;\; \operatorname{Cs}(g)<\infty;\\ \infty, &\;\;\text{otherwise.}\end{array}\right.$$
If we consider lower limit instead of the upper limit we would obtain $$\liminf_{n\to\infty}\frac 1n \varphi\left(\frac{1}{k^n}\right)=\left\{\begin{array}{ll} \operatorname{Ci}(g)\cdot\log k, & \;\; \text{if}\;\; \operatorname{Ci}(g)<\infty;\\ \infty, &\;\;\text{otherwise.}\end{array}\right.$$ Therefore we can not replace an upper limit by the limit in the definition of the dynamical $g$-entropy.
We will show the equality for the upper limit. Proof of the equality for the lower limit is similar. Let $(x_n)_{n=1}^{\infty}$ and $(m_n)_{n=1}^{\infty}$ be such that $\limsup\limits_{n\to\infty}g(x_n)/\eta(x_n)=c$ and $x_n\in\left(k^{-m_n},k^{-m_n+1}\right)$ for every $n\in\mathbb{N}$. Then $-\log x_n \geq -\log k^{-m_n+1}$. Every function $g\in {\mathcal{G}_0}$ is quasihomogenic, so for every positive integer $n$ occurs $$\label{quasigxn}\frac{g(x_n)}{x_n}<\frac{g(k^{-m_n})}{k^{-m_n}}.$$
Therefore $$\begin{aligned}
\frac{g(x_n)}{\eta(x_n)}&=&\frac{g(x_n)}{x_n}\frac{1}{-\log x_n}\leq \frac{g(k^{-m_n})}{k^{-m_n}}\frac{1}{(m_n-1)\log k} \nonumber \\ &=& \frac{g(k^{-m_n})}{\eta\left(k^{-m_n}\right)}\cdot\frac{m_n}{m_n-1},\nonumber\end{aligned}$$ and $$\limsup_{x\to 0^+}\frac{g(x)}{\eta(x)}=\limsup_{n\to\infty}\frac{g(k^{-n})}{\eta(k^{-n})}.$$
Zero dynamical entropy processes with $g$-entropy of a given value
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As we have seen in the previous section in the case of positive Shannon dynamical entropy and $g\in{\mathcal{G}_0}^{\infty}$ the dynamical $g$-entropy is always infinite. The case when the Shannon dynamical entropy is equal zero is different. In this section we will prove that for $g\in{\mathcal{G}_0}^{\infty}$ which fullfill some additional assumptions, i.e. $$\lim_{\lambda\to\infty}\liminf_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}> 1,$$ there exist processes of a given dynamical $g$-entropy and zero Shannon dynamical entropy. This additional assumption is rather weak, since quasihomogenicity of $g$ implies that the limit exists and always has to be greater or equal to one. First we will prove some technical lemma which is similar to [@KramkovSch Lemma 6.3] for utility functions with asymptotic elasticity smaller than one.
\[nwsr\] Let $g\in{\mathcal{G}_0}$. For $\lambda>1$ define $$U(\lambda):=\liminf\limits_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}.$$ Then $U$ is nondecreasing and for every $\lambda\geq 1$ $U(\lambda)\geq 1$. If additionally $g$ is differentiable, then the following conditions are equivalent:
- $U(\lambda)$ is greater than one for every $\lambda>1$,
- $\limsup\limits_{x\to 0^+}\frac{xg'(x)}{g(x)}<1$.
Since $g$ is quasihomogenic for $1\leq s\leq t$ we have $$\frac{tg(x)}{g(tx)}\geq \frac{sg(x)}{g(sx)}\geq 1$$ for every $x\in(0,1]$ and $U(1)=1$. This completes the proof of the first part of the theorem. Assume now that $g$ is differentiable.Then it is sufficient to show that the following conditions are equivalent $$(i')\;\;\; \exists x_0>0:\; \forall x\in(0, x_0)\; \forall \lambda>1\;\; g(\lambda x)\leq \lambda^{\gamma}g(x),$$ $$(ii')\;\;\; \exists x_0>0:\;\forall x\in(0, x_0) \;\;g'( x)\leq \gamma \frac{g(x)}{x}$$
for some $\gamma\in (0,1)$. The proof will be similar to the proof of [@KramkovSch Lemma 6.3]
$(i')\Rightarrow (ii')$ Fix $\gamma \in (0,1)$ Let $$F(\lambda):=g(\lambda x),\;\;G(\lambda):=\lambda^{\gamma}g(x).$$ Then $$g'(x)=\frac{F'(1)}{x}\leq \frac{G'(1)}{x}=\frac{\gamma g(x)}{x}.$$
$(ii')\Rightarrow (i')$ Fix $\gamma\in (0,1)$. Functions $F$ and $G$ are differentiable, $F(1)=G(1)$ and $$F'(1)=xg'(x)<\gamma g(x)=G'(1).$$ Therefore $F(\lambda)<G(\lambda)$ for $\lambda \in (1,1+\varepsilon)$ for some $\varepsilon>0$. To show that $F(\lambda)<G(\lambda)$ for every $\lambda>1$ let $\overline{\lambda}=\inf\{\lambda>1\colon F(\lambda)=G(\lambda)\}$ and suppose that $\overline{\lambda}<\infty$. Note that we must have $F'(\overline{\lambda})\geq G'(\overline{\lambda})$ which leads to a contradiction, since from $(ii')$ we have $$F'(\overline{\lambda})=xg'(\overline{\lambda} x)<\frac{\gamma}{\overline{\lambda}}g(\overline{\lambda} x)=\frac{\gamma}{\overline{\lambda}}F(\overline{\lambda})=\frac{\gamma}{\overline{\lambda}}G(\overline{\lambda})=G'(\overline{\lambda}).$$
The main result of this section is the following theorem:
\[konstrhgpinfty\] For every differentiable $g\in{\mathcal{G}_0}^{\infty}$ for which $$\label{diff_g}
\limsup_{x\to 0^+}\frac{xg'(x)}{g(x)}<1,$$ and every $\gamma\geq 0$ there exists a process $(X,\Sigma,\mu,T,{\mathcal{P}})$, such that $$h(g,{\mathcal{P}})=\gamma.$$
We will provide a construction of the process with a given entropy. For $\gamma =0$ it is obvious, we can consider systems with trivial dynamics, i.e. a system consisting of a single fixed point with trivial measure. Then we have $h(g,{\mathcal{P}})=0$ for every function $g\in{\mathcal{G}_0}$. Suppose now that $\gamma>0$. Before we will prove the theorem we will discuss a well-known construction (see e.g. [@FerencziPark]), which is sometimes called the standard example. It will allow us to generate systems where we will be able to control the growth of static $g$-entropies $H(g,{\mathcal{P}}_n)$ for some partition ${\mathcal{P}}$, and therefore to find a process with a given $g$-entropy. The system which we construct will be a subshift over two symbols. [**Construction of a system.**]{} We define inductively sequence $\left(b_n\right)_{i=1}^{\infty}$ and the family of blocks $\left(\left(B_{n,i}\right)_{i=1}^{b_n}\right)_{n=1}^{\infty}$. Let $(e_n)_{n=1}^{\infty}$ and $(r_n)_{n=1}^{\infty}$ be given sequences of integers and $\mathcal{A}:=\{0,1\}$. The sequence $(b_n)_{n=1}^{\infty}$ is defined inductively by $$b_0:=2,\;\;\text{and}\;\;b_{n+1} := \left(b_n\right)^{e_n} \;\; \text{for}\;\; n\geq 0.$$ Blocks $\left(\left(B_{n,i}\right)_{i=1}^{b_n}\right)_{n=1}^{\infty}$ and $\left(\left(B_{n,i}'\right)_{i=1}^{b_{n+1}}\right)_{n=1}^{\infty}$ are given by $B_{0,i}=i-1,\;\; \text{for}\;\; i=0,1$ and $B_{0,j}'\;\; \text{as concatenation of}\;\;e_0\;\; \text{blocks} \;\; B_{0,i} \;\; \text{for}\;\; 1\leq j\leq b_1$. Then for $n>1$ we define block $B_{n+1,i}$ as a concatenation of $ r_n$ copies of the block $B_{n,i}'$ for $1\leq i\leq b_n$, and block $B_{n,j}'$ for $1\leq j\leq b_{n+1}$ is a concatenation of $e_n$ (possibly different) blocks $B_{n,i}$. Then by $h_n$ and $h_n'$ we will denote length of block $B_{n,i}$ and $B_{n,i}'$ respectively. Therefore $$h_n=\prod_{i=0}^{n-1}r_i \cdot \prod_{j=0}^{n-1}e_j \;\;\;\;\text{and} \;\;\;\; h_n'=\prod_{i=0}^{n-1}r_i \cdot \prod_{j=0}^{n}e_j.$$
The basic idea is as follows: we want to control growth of different blocks of positive measure. So if the sequence $(e_n)_{n=1}^{\infty}$ which gives us the number of contatenated different blocks will be constant and equal $2$, then the growth of blocks will be controlled by concatenating as often as we need as big number of copies of one block as we need and the number of copies will be given by values of the sequence $(r_n)_{n=1}^{\infty}$.
It is easy to see that the system which we will get is the subshift $(X,\sigma)$ consisting of sequences for which for every there exist $n$ and $i$ such that $[x_s,\ldots,x_t]$ is a subword of a block $B_{n,i}$. The only invariant measure is a Bernoulli measure given by assigning to each block $B_{n,i}$ measure $\mu(B_{n,i})=1/b_n$. Specifically for a given length $m\in(h_{n-1},h_n)$ we define measure $\mu$ on cylindrical sets $C_0^{m-1}$ – we find such an integer that the number of different admissible blocks is equal to $2^{2^{n-1}+j}$, now for a given block $[\omega_0\ldots\omega_{m-1}]$ we define $$\mu(C_0^{m-1}\left(\omega_0,\ldots,\omega_{m-1}\right))=\left\{\begin{array}{ll} 2^{-2^{n-1}-j}, & \text{if there exists such block } B_{n,i},\\ & \text{that} \; [\omega_0,\ldots,\omega_{m-1}] \;\text{is a subword of $B_{n,i}$}\\ 0,& \text{otherwise.}\end{array}\right.$$
Such a measure is well defined and invariant. The partition for which we will be able to get a given value of the dynamical $g$-entropy is the zero-coordinate partition ${\mathcal{P}}^{\mathcal{A}}$ restricted to the subshift. Now if we pass to the subsequence $(h_n)_{n=1}^{\infty}$ we obtain $$\sum_{A\in \mathcal{P}_{h_n}^{\mathcal{A}}}g(\mu(A)) = b_n\cdot g\left(\frac{1}{b_n}\right).$$ Therefore
$$\label{gorneogr} \limsup_{n\to\infty}\frac 1n \sum_{A\in \mathcal{P}_n^{\mathcal{A}}}g(\mu(A))\geq \limsup_{n\to\infty}\frac{1}{h_n}\cdot b_ng\left(\frac{1}{b_n}\right),$$
Moreover any block of length $h_n$ is uniquely determined by the block of length $h_{n-1}'$ since it is a concatenation of $r_{n-1}$ blocks of length $h_{n-1}'$. In turn, such a block is determined by the block of length $h_{n-1}+\alpha_{n-1}$, where by $\alpha_{n-1}$ we denote length of block which uniquely determines block of length $h_{n-1}$. Repeating this argument $n-1$ times we get that any block of length $h_n$ is determined by a subword consisting of its first $\sum_{i=0}^{n-1}h_i$ letters.
Therefore $$\limsup_{n\to\infty}\frac 1n \sum_{A\in \mathcal{P}_n^{\mathcal{A}}}g(\mu(A))\geq \limsup_{n\to\infty} b_ng\left(\frac{1}{b_n}\right)\left/\sum\limits_{i=0}^{n-1}h_i\right. .$$
We will prove that for functions $g\in{\mathcal{G}_0}^{\infty}$, which fullfill assumptions of Theorem \[konstrhgpinfty\] we have $$\label{gorneogr1} \limsup_{n\to\infty}\frac 1n \sum_{A\in \mathcal{P}_n^{\mathcal{A}}}g(\mu(A)) = \limsup_{n\to\infty} b_ng\left(\frac{1}{b_n}\right)\left/\sum\limits_{i=0}^{n-1}h_i\right. .$$ For this purpose we should check what happens in our construction when Then $$\frac 1m \sum_{A\in{\mathcal{P}}_m^{\mathcal{A}}} g(\mu(A))=\frac 1m \sum_{[\omega_0,\ldots,\omega_{m-1}]\in\mathcal{A}^m} g(\mu(C_0^{m-1}(\omega_0\ldots \omega_{m-1})))=\frac{\varphi\left(2^{-2^{n-1}-j}\right)}{m}$$ for some $j\in\{1,\ldots,2^{n-1}-1\}$. According to the definition of measure $\mu$ the counter of this expression will be piecewise constant – if we denote by $\left(r_{i_j}\right)_{j=1}^l$ a sequence of elements of the sequence $(r_i)_{i=1}^n$ greater than one, the counter will change every $r_{i_1},\ldots,r_{i_{l-1}}$ or $r_{i_l}$ terms. Since $g$ is quasihomogenic the considered functions $\varphi$ are decreasing, therefore the counter of the expression will be increasing with respect to $j$. Denote by $ b_j^{(n)} $ the minimum length of blocks (all blocks of the same length), for which we get $2^{2^{n-1}+j}$ different cylinders of positive measure in our construction. We can focus on the subsequence $$\xi_{j}^{(n)}:=\varphi\left(2^{-2^{n-1}-j}\right)\left/b_{j}^{(n)}\right.$$ for $j=1,\ldots 2^{n-1}$. Moreover $\xi_{j+1}^{(n)}=\varphi\left(2^{-2^{n-1}-j-1}\right)\left/\left(b_{j}^{(n)}+\theta_j\right)\right.$, where for $j=1,\ldots,2^{n-1}$. It is easy to see that if we want to obtain equality in (\[gorneogr\]), it is enough to show that the sequence $\left(\xi_j^{(n)}\right)_{j=1}^{2^{n-1}}$ is nondecreasing. According to Lemma \[nwsr\], the assumption (\[diff\_g\]) implies that $U(2)=\liminf\limits_{x\to 0^+}2g(x)/g(2x)$ is greater than one. Note that for functions satisfying this condition, for sufficiently large $ n $ we obtain $$\begin{aligned}
\frac{\xi_{j+1}^{(n)}}{\xi_j^{(n)}}&=&\frac{\varphi\left(2^{-2^{n-1}-j-1}\right)}{\varphi\left(2^{-2^{n-1}-j}\right)}\cdot\frac{b_j^{(n)}}{b_j^{(n)}+\theta_i}\geq \frac{\varphi\left(2^{-2^{n-1}-j-1}\right)}{\varphi\left(2^{-2^{n-1}-j}\right)}\left/\left(1+\frac{\max\limits_{i=0,\ldots n-1}r_i}{b_j^{(n)}}\right)\right.\nonumber \\ &\geq & \frac{\varphi\left(2^{-2^{n-1}-j-1}\right)}{\varphi\left(2^{-2^{n-1}-j}\right)}\left/1+\left(\prod_{j=0}^{n-1} r_j\right)^{-1}\right.\nonumber \\ &\geq& \frac{U(2)}{1+\left(\prod_{j=0}^{n-1} r_j\right)^{-1}} =U(2)\cdot \prod\limits_{j=0}^{n-1} r_j\left/\left(\prod\limits_{j=0}^{n-1} r_j+1\right)\right. .\nonumber \end{aligned}$$
Since $U(2)>1$ there exists an integer $N$, such that for $n>N$ we have $\prod\limits_{j=0}^{n-1} r_j\left/\left(\prod\limits_{j=0}^{n-1} r_j+1\right)\right.> 1/U(2) $, which implies that for sufficiently large $n$, sequence $\left(\xi_j^{(n)}\right)_{j=1}^{2^{n-1}}$ is increasing.
\[wn\] If $g\in\mathcal{G}_0^{\infty}$ is such that $U(2)>1$, then $$\limsup_{n\to\infty}\frac 1n \sum_{A\in \mathcal{P}_n^{\mathcal{A}}}g(\mu(A)) = \limsup_{n\to\infty} b_ng\left(\frac{1}{b_n}\right)\left/\sum\limits_{i=0}^{n-1}h_i\right. .$$
To complete the proof of Theorem \[konstrhgpinfty\] it is sufficient to show that for every $\gamma>0$ there exists a sequence $(r_n)_{n=1}^{\infty}$, for which $$\limsup_{n\to\infty} \varphi\left(\frac{1}{b_n}\right)\left/\sum\limits_{i=0}^{n-1}h_i\right.=\gamma.$$
We can see that for each positive integer $n$ we have $$\varphi\left(\frac{1}{b_n}\right)\left/\sum\limits_{i=0}^{n-1}h_i\right.=\frac{2^{2^n}\cdot g\left(2^{-2^n}\right)}{2r_0+\ldots+2^n r_0\cdots r_{n-1}}=\frac{2^{n+1}a_n}{2r_0+\ldots+2^n r_0\cdots r_{n-1}},$$ where $$a_n:=\varphi\left(2^{-2^{n+1}}\right)\left/2^{n+1}\right. .$$ Since $\varphi_{\eta}\left(2^{-2^{n+1}}\right)=2^{n+1}$, the fact that $g\in{\mathcal{G}_0}^{\infty}$ implies$$\lim_{n\to\infty}a_n=\infty.$$
Therefore it is sufficient to show the following lemma:
\[limsupgamma\] For every sequence of real numbers $(a_n)_{n=1}^{\infty}$ such that $\lim\limits_{n\to\infty}a_n=\infty$ and for every $\gamma > 0$ there exist such a sequence of integers $(r_n)_{n=1}^{\infty}$ that $$\limsup_{n\to\infty}\frac{2^{n+1}a_n}{2r_0+\ldots+2^n r_0\cdots r_{n-1}}=\gamma.$$
Let $$\gamma_n:=\frac{2^{n+1}a_n}{2r_0+\ldots+2^n r_0\cdots r_{n-1}}.$$ Without loss of generality we may assume that $\gamma=1$. We will contruct sequence $(r_n)_{n\in\mathbb{N}}$ inductively due to the index of the next moment at which we concatenate multiple copies of a given block.\
[**Step 1.**]{} Since the sequence $(a_n)_{n=1}^{\infty}$ converges to infinity there exists an index $N_0$, for which $$\label{1}
1-\frac{1}{N_0}<\frac{a_{N_0}}{1-2^{-N_0}}\left/\left[\frac{a_{N_0}}{1-2^{-N_0}}\right]\right.<1+\frac{1}{N_0},$$ where $[x]$ is an integer part of $x$. We may assume that $N_0$ belongs to a subsequence $(n_k)_{k=1}^{\infty}$, on which a sequence defined by $b_n:=2^na_n\left/(2^n-1)\right.$ is increasing. We define $$R_0:=\left[\frac{a_{N_0}}{1-2^{-N_0}}\right].$$ and $$r_i:=\left\{\begin{array}{ll} R_0,\; & \text{for}\; i=0\\ 1,\;& \text{for} \; i=1,2,\ldots,N_0.\end{array}\right.$$ Then by (\[1\]) we obtain that $\gamma_{N_0}\in \left(1-\frac{1}{N_0},1+\frac{1}{N_0}\right)$ and for $n=0,\ldots,N_0$ we have $\gamma_n\leq \gamma_{N_0}$.
[**Step 2.**]{} Let $m>0$. Assume that the integers $N_0,\ldots,N_{m-1}$ and $r_n$ for $n=1,\ldots,N_{m-1}$ are already defined. Then there exists such an integer $N>N_{m-1}$, that for every $n>N$ we have $$\label{2} 1-\frac{1}{n+1}<\left[\frac{a_n}{R_0\cdots R_{m-1}\left(1-2^{N_{m-1}-n}\right)}\right]\left/\frac{a_n}{R_0\cdots R_{m-1}\left(1-2^{N_{m-1}-n}\right)}\right. .$$
Moreover the following technical fact, which simple but technical proof we omit is true:
\[Obserwkonstr\] There exists such an integer $N'\in\mathbb{N}$, that for $n>N'$ we have $$\sigma_2<\left[\frac{a_n}{R_0\cdots R_{m-1}\left(1-2^{N_{m-1}-n}\right)}\right]\leq \frac{a_n}{R_0\cdots R_{m-1}\left(1-2^{N_{m-1}-n}\right)}<\sigma_1,$$ where $$\sigma_1=\frac{n}{n-1}\frac{a_n}{R_0\cdots R_{m-1}\left(1-2^{N_{m-1}-n}\right)}-\alpha_n,\;\;\;\;\sigma_2=\frac{n}{n+1}\frac{a_n}{R_0\cdots R_{m-1}\left(1-2^{N_{m-1}-n}\right)}-\alpha_n$$ and $$\alpha_n:=\frac{2^{N_0-n}\left(1-2^{-N_0}\right)}{\left(1-2^{N_{m-1}-n}\right)R_1\cdots R_{m-1}}+\ldots+\frac{2^{N_{m-2}-n}\left(1-2^{N_{m-2}-N_{m-1}}\right)}{1-2^{N_{m-1}-n}}.$$
Now let $N_m>\max\{N,N'\}$ be such that $N_m$ belongs to the subsequence $(n_k)_{k=1}^{\infty}$, for which sequence $b_{n_k}=a_{n_k}\left/\left(c2^{N_{m-1}-n_k}+d\left(1-2^{N_{m-1}-n_k}\right)\right)\right.$ – with $c,d>0$ – is increasing (existence of such subsequence is guaranteed by the fact that a sequence $c_n:=b_n/a_n$ is bounded and $a_n$ converges to infinity).
Set $$R_m:=\left[\frac{a_{N_m}}{R_0\cdots R_{m-1}\left(1-2^{N_{m-1}-N_m}\right)}\right]$$ and $$r_i:=\left\{\begin{array}{ll} R_{m},\; & \text{for}\; i=N_{m-1}+1\\ 1,\;& \text{for} \; i=N_{m-1}+2,\ldots,N_{m}.\end{array}\right.$$ Then by Remark 3, after simple calculations we get that $$\gamma_{N_m}=a_{N_m}\left/\left(R_0(1-2^{-N_0})2^{N_0-N_m}+\ldots + R_0\cdots R_m\left(1-2^{N_{m-1}-N_m}\right)\right)\right. \in \left(1-\frac{1}{N_m}, 1+\frac{1}{N_m}\right)$$ and for $n=N_{m-1}+1,\ldots,N_m$ we have $\gamma_n<\gamma_{N_m}$. Eventually $$\limsup_{n\to\infty}\frac{2^{n+1}a_n}{2r_0+\ldots+2^n r_0\cdots r_{n-1}}=\lim_{m\to\infty} \gamma_{N_m}=1.$$
In the proof of Theorem \[konstrhgpinfty\] the crucial role played the inequality $U(2)>1$. We used this condition to prove equality (\[gorneogr1\]). It is easy to see that we can further weaken the assumption – it is sufficient to show that there exists such $\lambda>1$ that $U(\lambda)>1$. Then, applying Lemma \[nwsr\] we can repeat the above construction defining:
- an integer $k:=\min\{\lambda>1|\; U(\lambda)>1\}$,
- an alphabet $\mathcal{A}:=\{0,1,\ldots,k-1\}$ and
- a sequence $(e_n)$ as $e_n:=k$ for every integer $n$.
For every function $g\in{\mathcal{G}_0}^{\infty}$ for which $$\label{kgxgkx} \lim_{\lambda\to\infty}\liminf_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}>1.$$ and every $\gamma\geq 0$ there exists a process $(X,\Sigma,\mu,T,{\mathcal{P}})$, such that $$h(g,{\mathcal{P}})=\gamma.$$
Lemma \[nwsr\] implies that the limit in (\[kgxgkx\]) exists. Moreover this condition is not very restrictive – for every $g\in{\mathcal{G}_0}$ there is $\liminf\limits_{x\to 0^+}\lambda g(x)/g(\lambda x)\geq 1$. Finally it should be noted that there are functions in ${\mathcal{G}_0}^{\infty}$ for which this limit is equal one, i. e. it is the case for the function $g(x)=x\left(\ln x-1\right)^2$.
Note that we needed assumption (\[kgxgkx\]) only to show that $$\limsup_{n\to\infty}\frac 1n \sum_{A\in \mathcal{P}_n^{\mathcal{A}}}g(\mu(A)) \leq \limsup_{n\to\infty}\frac{1}{\sum\limits_{i=0}^{n-1}h_i}\cdot b_ng\left(\frac{1}{b_n}\right).$$
To prove this inequality even weaker assumptions are sufficient – condition (\[kgxgkx\]) implies monotonicity of the sequence $(\xi_i^{(n)})$ when in fact it would be sufficient to show just that the upper limit in (\[gorneogr1\]) is realized on the blocks of length $\sum_{i=0}^{n-1}h_i$. Those conditions are usually hard to check and in our opinion will not significantly expand our knowledge. Nonetheless we can formulate the following corollary:
If $g\in{\mathcal{G}_0}^{\infty}$, then for every $\gamma\geq 0$ there exists a process $(X,\Sigma,\mu,T,{\mathcal{P}})$, such that $$h(g,{\mathcal{P}})\geq \gamma.$$
Kolmogorov-Sinai entropy like invariant
=======================================
The basic tool in the ergodic theory is Kolmogorov-Sinai entropy defined as a supremum of Shannon dynamical entropies over all finite partitions: $$h_{\mu}(T)=\sup_{{\mathcal{P}}\;-\;\text{finite}}h(T,{\mathcal{P}}).$$ It is invariant under metric isomorphism. Following the Kolmogorov proposition we take the supremum over all partitions of dynamical $g$-entropy of a partition. For a given system $(X,\Sigma,\mu,T)$ we define $$\label{hgKS}
h_{\mu}(g,T)=\sup_{{\mathcal{P}}\;-\;\text{finite}}h(g,T,{\mathcal{P}})$$ and call it [the measure-theoretic $g$-entropy of transformation $T$ with respect to measure $\mu$]{}.
It is easy to see that it is an isomorphism invariant. Ornstein and Weiss [@OrnsteinWeiss] showed the striking result that measure-theoretic entropy is the only finitely observable invariant for the class of all ergodic processes. More precisely – every finitely observable invariant for a class of all ergodic processes is a continuous function of entropy. Of course in the case of $g\in{\mathcal{G}_0}^0\cup{\mathcal{G}_0}^{\operatorname{Sh}}$ by Corollary \[Cgfinite\] we have $$h_{\mu}(g,T)=\lim_{x\to 0^+}\frac{g(x)}{\eta(x)}\cdot h_{\mu}(T).$$ We will show that for a wider class of functions, namely for functions for which $$\operatorname{Cs}(g)=\limsup_{x\to 0^+}\frac{g(x)}{\eta(x)}<\infty$$ we have $$h_{\mu}(g,T)=\operatorname{Cs}(g)\cdot h_{\mu}(T)$$ for any ergodic transformation $T$. This shows that the measure-theoretic $g$-entropy is in fact finitely observable: one might simply compose the entropy estimators [@Weiss] with the linear function itself. Our proof will be similar to the proof of [@TakensVerb1998 Thm 1.1] where Takens and Verbitski showed that for ergodic transformations supremum over all finite partitions of dynamical Rényi entropies of order $\alpha>1$ are equal to the measure-theoretic entropy of $T$ with respect to measure $\mu$.
Let us introduce necessary definitions. Let $T_i$ be automorphisms of Lebesgue space $(X_i,\Sigma_i,\mu_i)$ for $i=1,2$ respectively. Then we say that $T_2$ is a [factor]{} of transformation $T_1$, if there exists a homomorphism $\phi\colon X_1\mapsto X_2$ such that $$\phi T_1=T_2\phi \;\; \mu_1 \;\text{a.e. on}\;\; X_1.$$ Suppose that $T_2$ is a factor of $T_1$ under homomorphism $\phi$. Then for an arbitrary finite partition ${\mathcal{P}}$ of $X_2$ we have $$H\left(g,\bigvee_{i=0}^{k-1}T_2^{-i}{\mathcal{P}}\right)=H\left(g,\bigvee_{i=0}^{k-1}\phi^{-1}T_2^{-i}{\mathcal{P}}\right)=H\left(g,\bigvee_{i=0}^{k-1}T_1^{-i}\phi^{-1}{\mathcal{P}}\right).$$ Hence $h(g,T_2,{\mathcal{P}})=h(g,T_1,\phi^{-1}{\mathcal{P}})$. Therefore $$h_{\mu}(g,T_2)=\sup_{{\mathcal{P}}- \text{finite}}h(g,T_2,{\mathcal{P}})=\sup_{{\mathcal{P}}- \text{finite}} h(g,T_1,\phi^{-1}{\mathcal{P}})\leq h(g,T_1).$$ This implies the following proposition:
\[factorhmu\] If $T_2$ is a factor of $T_1$, then for every function $g\in{\mathcal{G}_0}$ $$h_{\mu}(g,T_2)\leq h_{\mu}(g,T_1).$$
Measure-theoretic $g$-entropies for Bernoulli automorphisms.
------------------------------------------------------------
An automorphism $T$ on $(X,\Sigma,\mu)$ is called [Bernoulli automorphism]{} if it is isomorphic to some Bernoulli shift. The crucial role in the proof of the main theorem of this section (Theorem \[twhKSg\]) will play a well-known theorem due to Sinai:
\[Sinaj\] Let $T$ be an arbitrary ergodic automorphism of some Lebesgue space $(X,\Sigma,\mu)$. Then each Bernoulli automorphism with $h_{\mu}(T_1)\leq h_{\mu}(T)$ is a factor of the automorphism $T$.
The following proposition will play a crucial role in our considerations:
\[erghmulogm\] Let $T$ be an arbitrary ergodic automorphism with $h_{\mu}(T)\geq \log M$ for some integer $M\geq 2$. Then for every $g\in{\mathcal{G}_0}$ $$h_{\mu}(g,T)\geq \operatorname{Cs}(g)\cdot\log M.$$
Consider a shift $\sigma$ over all infinite sequences from the alphabet $\mathcal{A}=\{0,1\ldots,M-1\}$ with the corresponding Bernoulli measure generated by $p_1=\ldots = p_M=\frac 1M$. It is easy to see that $h_{\mu}(\sigma)=\log M$. From Theorem \[Sinaj\] we conclude that $\sigma$ is a factor of $T$. Therefore applying formula (\[entrBernoulli\]) we obtain $$h_{\mu}(g,T)\geq h_{\mu}(g,\sigma)\geq h(g,\sigma,{\mathcal{P}}^{\mathcal{A}})=\limsup_{n\to\infty}\frac 1n \varphi\left(M^{-n}\right)=\log M \cdot \limsup_{n\to\infty}\frac{\varphi\left(M^{-n}\right)}{\varphi_{\eta}\left(M^{-n}\right)}.$$ Applying Lemma \[limsupMn\] completes the proof.
Main theorem
------------
Our goal in this section is the following result:
\[twhKSg\] Let $T$ be an ergodic automorphism of Lebesgue space $(X,\Sigma,\mu)$, and $g\in{\mathcal{G}_0}$ be such that $ \operatorname{Cs}(g) \in(0,\infty)$ Then $$h_{\mu}(g,T)=\left\{\begin{array}{ll}\operatorname{Cs}(g)\cdot h_{\mu}(T),& \; \text{if}\; h_{\mu}(T)<\infty,\\ \infty,& \; \text{otherwise}.\end{array}\right.$$ If $g\in{\mathcal{G}_0}^0$, then $h_{\mu}(g,T)=0$. If $g\in{\mathcal{G}_0}$ is such that $\operatorname{Cs}(g)=\infty$ and $T$ has positive measure-theoretic entropy, then $h_{\mu}(g,T)=\infty$.
We need a few preliminary lemmas.
\[lemma73\] If $T$ is an automorphism of the Lebesgue space $(X,\Sigma,\mu)$, then for every $g\in{\mathcal{G}_0}$ $$h_{\mu}(g,T^m)\leq mh_{\mu}(g,T).$$
Let ${\mathcal{P}}$ be a finite partition. Then $$\begin{aligned}
h\left(g,T^m,\bigvee_{i=0}^{m-1}T^{-i}{\mathcal{P}}\right)&=&\limsup_{k\to\infty}\frac 1k H\left(g,\bigvee_{j=0}^{k-1}T^{-mj}\left(\bigvee_{i=0}^{m-1}T^{-i}{\mathcal{P}}\right)\right) \nonumber\\ &=& m\limsup_{k\to\infty}\frac{1}{km} H\left(g,{\mathcal{P}}_{km-1}\right)\leq m\limsup_{n\to\infty}\frac{1}{n} H\left(g,{\mathcal{P}}_{n}\right)\nonumber \\ &=&mh(g,T,{\mathcal{P}})\nonumber\end{aligned}$$ Taking supremum over all finite partitions we obtain the assertion.
Next lemma will be just a weaker version of Theorem \[twhKSg\].
\[calkergcg\] If an automorphism $T^m$ of a Lebesgue space $(X,\Sigma,\mu)$ is ergodic for every $m\in\mathbb{N}$, then for every function $g\in{\mathcal{G}_0}$, such that $\operatorname{Cs}(g)<\infty$ holds $$h_{\mu}(g,T)=\operatorname{Cs}(g)\cdot h_{\mu}(T).$$ If $g\in{\mathcal{G}_0}^0$, then $h_{\mu}(g,T)=0$. If $g\in{\mathcal{G}_0}$ is such that $\operatorname{Cs}(g)=\infty$ and $T$ has positive Kolmogorov-Sinai entropy, then $h_{\mu}(g,T)=\infty$.
Case of $g\in{\mathcal{G}_0}^0$ follows from Corollary \[Cgfinite\]. Suppose that there exists such $g\in{\mathcal{G}_0}\backslash {\mathcal{G}_0}^0$ which fullfills assumptions of lemma and for which we have $$\operatorname{Cs}(g)\cdot h_{\mu}(T)-h_{\mu}(g,T)>0.$$ Then applying Lemma \[lemma73\] to the transformation $T^m$ and using equality (see [@KH Thm 4.3.16]) we obtain $$\operatorname{Cs}(g)h_{\mu}(T^m)-h_{\mu}(g,T^m)\geq m\left(\operatorname{Cs}(g)h_{\mu}(T)-h_{\mu}(g,T)\right) \rightarrow \infty\;\; \text{as}\;\;m\to\infty.$$ Therefore for sufficiently large $m$ there exists an integer $M$ for which $$\label{calerg1} h_{\mu}(g,T^m)\leq m h_{\mu}(g,T)<\operatorname{Cs}(g)\log M\leq m \operatorname{Cs}(g)h_{\mu}(T)=\operatorname{Cs}(g)h_{\mu}(T^m).$$ Proposition \[erghmulogm\] applied to the transformation $T^m$ guarantees that for every $g\in {\mathcal{G}_0}$ with positive (finite) $\operatorname{Cs}(g)$ we have $$\label{calerg2} h_{\mu}(g,T^m)\geq \operatorname{Cs}(g)\log M.$$ Comparing (\[calerg1\]) and (\[calerg2\]) we obtain the contradiction, which implies that $$h_{\mu}(g,T)=\operatorname{Cs}(g)h_{\mu}(T).$$ If $\operatorname{Cs}(g)=\infty$ and $h_{\mu}(T)>0$ then there exists such integer $m>0$ that $$h_{\mu}(T^m)=mh_{\mu}(T)>\log M$$ and by Proposition \[erghmulogm\] and Lemma \[lemma73\] $$h_{\mu}(g,T)=h_{\mu}(g,T^m)=\infty$$ which completes the proof.
If $h_{\mu}(T)=0$ theorem is true, since for any partition ${\mathcal{P}}$ we have $$0\leq h(g,{\mathcal{P}})\leq \operatorname{Cs}(g) h({\mathcal{P}})=0.$$ Suppose that $0<h_{\mu}(T)<\infty$. Automorphism $T$ is ergodic. Therefore it has factor which is a Bernoulli automorphism $T'$ with entropy $h_{\mu}(T)=h_{\mu}(T')$. Every Bernoulli automorphism is mixing, so $T^m$ is ergodic for each $m$. Applying Lemma \[calkergcg\] we obtain $$h_{\mu}(g,T')=\operatorname{Cs}(g) h_{\mu}(T')=\operatorname{Cs}(g) h_{\mu}(T).$$ Since $T'$ is a factor of $T$, so Proposition \[factorhmu\] implies that $$\operatorname{Cs}(g)h_{\mu}(T)=\operatorname{Cs}(g)h_{\mu}(T')=h_{\mu}(g,T')\leq h_{\mu}(g,T)\leq \operatorname{Cs}(g)h_{\mu}(T)$$ which completes the proof of the case of finite $h_{\mu}(T)$. If $h_{\mu}(T)=\infty$, then Proposition \[erghmulogm\] implies that $$h_{\mu}(g,T)\geq \operatorname{Cs}(g)\log M$$ for every $M>0$ and the theorem is proved.
Case of $g\in{\mathcal{G}_0}^{\infty}$
--------------------------------------
We will prove that for every $g\in{\mathcal{G}_0}^{\infty}$ and any aperiodic automorphism $T$ the measure-theoretic $g$-entropy of the transformation $T$ with respect to $\mu$ is infinite. Since we omit the assumption of ergodicity we will use different techniques mainly based on the well-known Rokhlin Lemma which guarantees existence of so called Rokhlin towers of a given height covering sufficiently large part of $X$. Using such towers we will find lower estimations of $g$-entropy of a process similar to one obtained by Frank Blume in [@Blumenotpubl], [@Blume97], where he proposed, for a given sequence $(a_n)_{n=1}^{\infty}$ converging to infinity slower than $n$, a construction of a partition into two sets ${\mathcal{P}}$, for which $\lim\limits_{n\to\infty} H({\mathcal{P}}_n)/a_n=\infty$. We will assume that we have an aperiodic system, i.e. system $(X,\Sigma,\mu,T)$ for which $$\mu\left(\{x\in X: \exists n\in\mathbb{N}\; T^nx=x\}\right)=0.$$
If $M_0,\ldots,M_{n-1}\subset X$ are pairwise disjoint sets of equal measure, then $\tau=(M_0,M_1,\ldots,M_{n-1})$ is called [a tower]{}. If additionally $M\subset X$ and $M_k=T^k M$ for $k=1,\ldots,n-1$, then $\tau$ is called [Rokhlin tower]{}.[^4] By the same bold letter $\bm{\tau}$ we will denote the set $\bigcup_{k=0}^{n-1}T^kM$. Obviously $\mu(\bm{\tau})=n\mu(M)$. Integer $n$ is called [the height ]{} of tower $\tau$. Moreover for $i<j$ we define a subtower $$\tau_i^j:=\left(T^iM,\ldots,T^j M\right) \;\;\text{and}\;\;\bm{\tau}_i^j=\bigcup_{k=i}^j T^kM.$$ In aperiodic systems there exist Rokhlin towers of a given length and covering sufficiently large part of $X$:
\[Rohlin\] If $T$ is an aperiodic transformation of Lebesgue space $(X,\Sigma,\mu)$, then for every $\varepsilon>0$ and every integer $n\geq 2$ there exists a Rokhlin tower $\tau$ of height $n$ with $\mu(\bm{\tau})>1-\varepsilon$.
We now give a definition of an independent collection of sets relative to a Rokhlin tower. We will associate to such collections certain partitions, which will be analogous to Bernoulli partitions. Let $\tau=(M,TM,\ldots,T^{n-1}M)$ be a Rokhlin tower. We say that $I\in\Sigma$ is [independent in $\tau$]{}, if $$\left\{T^{-k}(I\cap T^kM), M\backslash T^{-k}(I\cap T^kM)\right\}_{k=0}^{n-1}$$ is a collection of pairwise disjoint partitions of $M$. In other words for any we have $$\mu\left(T^{-j}\left(I\cap T^jM\right)\cap T^{-i}\left(I\cap T^iM\right)\right)=\frac{\mu(I\cap T^jM)\mu(I\cap T^iM)}{\mu(M)}.$$ We can assume that $$\mu(I\cap T^kM)=\frac{\mu(M)}{2} \;\;\text{for}\;\; k=0,\ldots, n-1.$$
If $T$ is aperiodic, then such collection exists in every tower since every aperiodic system has no atoms and for any nonatomic Lebesgue space $(X,\Sigma,\mu)$, every measurable set $A$ and each one can find a set $B\subset A$ of measure $\alpha$.
In order to show that the measure-theoretic $g$-entropy is infinite we need a lower bound for the dynamical $g$-entropy of a given partition. For this purpose we will use Rokhlin towers and we will calculate dynamical $g$-entropy with respect to a given Rokhlin tower. This leads us to the following definition: Let ${\mathcal{P}}$ be a finite partition of $X$ and $F\in\Sigma$, then we define the (static) [$g$-entropy of ${\mathcal{P}}$ restricted to $F$]{} as $$H_F (g,{\mathcal{P}}):=\sum_{B\in{\mathcal{P}}}g(\mu(B\cap F)).$$
The following lemma gives us estimation for $H(g,{\mathcal{P}})$ from below by the value of $g$-entropy restricted to a subset of $X$.
\[lemmaH-Hzaw\] Let $g\in\mathcal{G}_0$. Let ${\mathcal{P}}$ be a finite partition such that there exists a set $E\in{\mathcal{P}}$ with $0<\mu(E)<1$. If $F\in\Sigma$, then $$H(g,{\mathcal{P}})\geq H_{F}(g,{\mathcal{P}})-\left|g_-'\left( 1/2\right)\right|-d_{\max},$$ where $d_{\max}:=\max\limits_{x,y\in[0,1]}|g(x)-g(y)|$.
By the mean value theorem we have $$g(\mu(A))-g(\mu(A\cap F))=g_-'(x_0^A)\left(\mu(A)-\mu(A\cap F)\right),$$ for any set of measure smaller or equal to $1/2$, where $x_0^A\in (\mu(A\cap F),\mu(A))$. Concavity of $g$ implies $$\sum_{\mu(A)\leq 1/2}\left(g(\mu(A))-g(\mu(A\cap F))\right)\geq g_-'(1/2)\sum_{\mu(A)\leq 1/2}\mu(A\backslash F)\geq -|g_-'(1/2)|.$$ Eventually $$H(g,{\mathcal{P}})-H_F(g,{\mathcal{P}})+d_{\max} \geq -|g_-'(1/2)|$$ which completes the proof.
Now we give an estimation from below for the $g$-entropy restricted to a given Rokhlin tower. First, by ${\mathcal{P}}^I$ we will denote a partition into two sets $\{I,X\backslash I\}$, wfor a measurable set $I$. Then the following lemma is true.
\[lemmaniezalwtau\] Let $\tau=\left(M,TM,\ldots,T^{2n-1}M\right)$ be Rokhlin tower of height $2n$, $I\in\Sigma$ be an independent set in $\tau$. If $g\in{\mathcal{G}_0}^{\infty}$ then $$H_{\bm{\tau}_0^{n-1}}\left(g,{\mathcal{P}}_n^I\right)= \frac{\mu(\bm{\tau})}{2}\varphi\left(\frac{\mu(\bm{\tau})}{2^{n+1}}\right).$$
Independence of $I$ in $\tau$ implies that the partition $${\mathcal{P}}_n^I\cap \bm{\tau}_0^{n-1}$$ is a partition of $\bm{\tau}_0^{n-1}$ into $2^n$ sets of equal measure $2^{-n}\mu(\bm{\tau}_0^{n-1})$. Therefore $$\begin{aligned}
H_{\bm{\tau}_0^{n-1}}\left(g,{\mathcal{P}}_n^I\right) &=& \sum_{A\in{\mathcal{P}}_n^I} g\left(A\cap\bm{\tau}_0^{n-1}\right)\nonumber \\ &=& 2^n g\left(\frac{\mu(\bm{\tau}_0^{n-1})}{2^n}\right) = \mu\left(\bm{\tau}_0^{n-1}\right)\varphi\left(\frac{\mu(\bm{\tau}_0^{n-1})}{2^n}\right) \nonumber \\ &=&\frac{\mu(\bm{\tau})}{2}\varphi\left(\frac{\mu(\bm{\tau})}{2^{n+1}}\right). \nonumber\end{aligned}$$
We need to show the continuity of $H(g,{\mathcal{P}}_n)$ with respect to the partition ${\mathcal{P}}_n$ if $n$ is fixed. First in the space of all partitions of $X$ into $m$ sets, which will be denoted by $\mathfrak{B}_m$, we consider pseudometric, which once factored to classes of partitions modulo measure zero becomes a metric. For any ${\mathcal{P}}=\{A_i,i\in\{1,\ldots,m\}\}$, ${\mathcal{Q}}=\{B_i,i\in\{1,\ldots,m\}\}$,[^5] we define $$d({\mathcal{P}},{\mathcal{Q}})=\min_{\pi} \sum_{i=1}^{m}\mu(A_i \triangle B_{\pi(i)})$$ where the minimum runs through all permutations of the set $\{1,\ldots,m\}$ and by $\triangle$ we denote a symmetric difference of two sets. Then:
\[staticcontpartition\] If is a continuous function, then $H(g,\cdot)$ is uniformly continuous on $\left(\mathfrak{B_m},d\right)$.
Fix $\delta$ and ${\mathcal{P}},{\mathcal{P}}'$ such that $d({\mathcal{P}},{\mathcal{P}}')=\delta$. Then (after ordering partition ${\mathcal{P}}'$ in such way that the distance is realised) $$\delta=\sum_{i=1}^m\mu (A_i\triangle B_i)\geq \sum_{i=1}^m\left|\mu(A_i)-\mu(B_i)\right|.$$ Let $\Delta_m:=\{\left(p_1,\ldots,p_m\right): \; p_i\geq 0\;\; \text{and}\;\; \sum_{i=1}^m p_i=1\}$ be a $m$-dimensional simplex. Continuity of a transformation $$\Delta_m\ni x \mapsto \sum_{i=1}^m g(x_i)\in\mathbb{R}$$ implies that $H(g,\cdot)$ is continuous on the compact set and therefore it is uniformly continuous on $\mathfrak{B}_m$.
\[hgpgeqM\] Let $g\in{\mathcal{G}_0}^{\infty}$ and $T$ be an aperiodic automorphism of a Lebesgue space $(X,\Sigma,\mu)$. Then $$h_{\mu}(g,T)=\infty.$$
We will prove that for any $M>0$ there exists a partition ${\mathcal{P}}^E=\{E,X\backslash E\}$ such that $h(g,{\mathcal{P}})\geq M$. We define recursively a sequence of sets $E_n\in \Sigma $. Let $$E_0:=\emptyset,\;\; N_0:=\delta_0:=1.$$ Let $n>0$ and assume that we have already defined $E_{n-1}$, $N_{n-1}$ and $\delta_{n-1}$. Using Lemma \[staticcontpartition\] we can choose $\delta_n>0$ such that $$\label{deltan}
\delta_n<\frac 12 \delta_{n-1}$$ $$\label{HNnEnF}
\left|H\left(g,{\mathcal{P}}_{N_n}^{E_{n-1}}\right)-H\left(g,{\mathcal{P}}_{N_n}^{F}\right)\right| <1$$ for any $F\in\Sigma$, for which $\mu(E_{n-1}\triangle F)<2\delta_n$.
Since $$\lim_{x\to 0^+}\frac{g(x)}{\eta(x)}=\infty,$$ we can choose such $N_n\in\mathbb{N}$ that $$\label{eqnvarphi}
\frac{\varphi\left(\delta_n 2^{-N_n-1}\right)}{\varphi_{\eta}\left(\delta_n 2^{-N_n-1}\right)}>\frac{2M}{\delta_n \log 2}.$$
By Lemma \[Rohlin\] there exists $M_n\in \Sigma$, such that $\tau_n=\left(M_n,TM_n,\ldots,T^{2N_n-1}M_n\right)$ is a Rokhlin tower of measure $\mu(\bm{\tau}_n)=\delta_n$. Let $I_n\subset \bm{\tau}_n$ be an independent set in $\tau_n$ and $$E_n:=\left(E_{n-1}\backslash \bm{\tau}_n\right)\cup I_n.$$ Then $$\mu(E_{n-1}\triangle E_n)\leq \mu(\bm{\tau}_n)=\delta_n.$$ for all positive integers $n$. By (\[deltan\]) we have $\delta_n<2^{-n}$ and we conclude that $\left(\bm{1}_{E_n}\right)_{n=0}^{\infty}$ is a Cauchy sequence in $L_1(X)$. Therefore there exist $E\in\Sigma$ such that $\bm{1}_{E_n}$ converges to $\bm{1}_E$. For this set we have $$\mu\left(E_n\triangle E\right)\leq \sum_{k=n+1}^{\infty}\mu\left(E_k\triangle E_{k-1}\right)\leq \sum_{k=n+1}^{\infty}\delta_k<2\delta_{n+1}.$$ Since $E_n\cap\tau_n =I_n$, applying (\[HNnEnF\]) and Lemmas \[lemmaH-Hzaw\] and \[lemmaniezalwtau\] we obtain
$$\begin{aligned}
H(g,{\mathcal{P}}_{N_n}^E) &\geq & H(g,{\mathcal{P}}_{N_n}^{E_n})-1\nonumber \\ &\geq &H_{\left(\bm{\tau}_n\right)_0^{N_n-1}}\left(g,{\mathcal{P}}_{N_n}^{E_n}\right)-\left|g'\left( 1/2\right)\right|-d_{\max}-1\nonumber \\ &\geq & \left[\frac{\mu(\bm{\tau}_n)\log 2}{2}\left(N_n+1\right)-\frac{\mu(\bm{\tau}_n)\log\mu(\bm{\tau}_n)}{2}\right]\cdot\frac{\varphi\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)}{\varphi_{\eta}\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)}\nonumber \\
&& -\left|g'\left( 1/2\right)\right|-d_{\max}-1 \nonumber\\
&\geq & \frac{\log 2}{2}\cdot \mu(\bm{\tau}_n)\cdot \left(N_n+1\right)\cdot\frac{\varphi\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)}{\varphi_{\eta}\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)}-\left|g'\left( 1/2\right)\right|-d_{\max}-1. \nonumber\end{aligned}$$
From (\[eqnvarphi\]) we obtain that $$\begin{aligned}
\lim_{n\to\infty}\frac{H(g,{\mathcal{P}}_{N_n}^E)}{N_n}&\geq & \lim_{n\to\infty}\frac{\log 2}{2}\cdot \mu(\bm{\tau}_n)\cdot \frac{N_n+1}{N_n}\cdot\frac{\varphi\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)}{\varphi_{\eta}\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)}-\frac{\left|g'\left( 1/2\right)\right|+d_{\max}}{N_n} \nonumber \\ &\geq & \frac{\log 2}{2} \lim_{n\to\infty} \mu(\bm{\tau}_n)\cdot \frac{N_n+1}{N_n}\cdot\frac{\varphi\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)}{\varphi_{\eta}\left(\mu(\bm{\tau}_n)2^{-N_n-1}\right)} \nonumber \\
&\geq & M\cdot \lim_{n\to\infty}\frac{N_n+1}{N_n}=M. \nonumber\end{aligned}$$ Since $M$ can be arbitrarily large it completes the proof.
Generator theorem counterpart
-----------------------------
In the case of $g\in{\mathcal{G}_0}^{\infty}$ there is no counterpart of a Kolmogorov-Sinai generator theorem, which says that the measure-theoretic entropy of the transformation $T$ is realised on every generator of the $\sigma$-algebra $\Sigma$. Let us consider Sturm shifts – shifts which model translations of the circle $\mathbb{T}=[0,1)$. Let $\beta \in [0,1)$ and consider the translation $\phi_{\beta}\colon [0,1)\mapsto [0,1)$ defined by $\phi_{\beta}(x)=x+\beta \;(\mod 1)$. Let ${\mathcal{P}}$ denote the partition of $[0,1)$ given by ${\mathcal{P}}=\{[0,\beta),[\beta,1)\}$. Then we associate a binary sequence to each $t\in[0,1)$ according to its itinerary relative to ${\mathcal{P}}$; that is we associate to $t\in[0,1)$ the bi-infinite sequence $x$ defined by $x_i=0$ if $\phi_{\beta}^i(t)\in[0,\beta)$ and $x_i=1$ if $\phi_{\beta}^i(t)\in[\beta,1)$. The set of such sequences is not necessary closed, but it is shift-invariant and so its closure is a shift space called Sturmian shift. If $\beta$ is irrational, then Sturmian shift is minimal, i.e. there is no proper subshift. Moreover for a minimal Sturmian shift, the number of $n$-blocks which occur in an infinite shift space is exactly $n+1$. Therefore for zero-coordinate partition ${\mathcal{P}}^{\mathcal{A}}$, which is a finite generator of $\sigma$-algebra $\Sigma$ and for any function $g\in{\mathcal{G}_0}$ we have $$H(g,{\mathcal{P}}_n^{\mathcal{A}})=\sum_{A\in{\mathcal{P}}_n^{\mathcal{A}}}g(\mu_S(A))\leq \varphi\left(\frac{1}{n+1}\right)$$ where $\mu_S$ is the unique invariant measure for Sturm shift. Thus, $$h(g,{\mathcal{P}}^{\mathcal{A}})\leq \limsup_{n\to\infty}\frac{n+1}{n}g\left(\frac{1}{n+1}\right)=0.$$ On the other hand since it is strictly ergodic (and thus aperiodic) Theorem \[hgpgeqM\] implies that for any function $g\in{\mathcal{G}_0}^{\infty}$ $$h_{\mu}(g,T)=\infty,$$ therefore we have a finite generator, for which the supremum is not attained.
[99]{}
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F. Blume. The rate of entropy convergence. [*Doctoral Dissertation*]{}, University of North Carolina at Chapel Hill, 1995. F. Blume. Possible rates of entropy convergence [*Ergod. Th. & Dynam. Sys.*]{} [**17**]{} (1997), 45–70. I. Csiszár. Axiomatic characterization of information measures. [*Entropy*]{} [**10**]{} (2008), no 3, 261–73. T. Downarowicz. [*Entropy in Dynamical Systems.*]{} Cambridge University Press, New York, 2011. S. Ferenczi and K. K. Park. Entropy dimensions and a class of constructive examples. [*Discret. Cont. Dyn. Syst.*]{} [**17**]{} (2007), no. 1, 133–141. P. Grassberger and I. Procaccia. Estimation of the Kolmogorov entropy from a chaotic signal. [*Phys. Rev. A*]{} [**28**]{} (1983), (4) 2591-–2593. J. Havrda, F. Charvàt. Quantification method of classification processes. Concept of structural $\alpha$-entropy. [*Kybernetika*]{} [**3**]{} (1967), 30–35. A. Katok and B. Hasselblatt. [*Introduction to the Modern Theory of Dynamical Systems.*]{} Cambridge University Press 1997. D. Kramkov and W. Schachermayer. The asymptotic elasticity functions and optimal investment in incomplete markets. [*Ann. App. Prob.*]{} [**9**]{} (1999), 904–950. M. Misiurewicz. A short proof of the variational principle for $\mathbb{Z}_+^n$ action on a compact space. [*Astérisque*]{}, [**40**]{} (1976), 147–157. A. M. Mesón and F. Vericat. Invariant of dynamical systems: A generalized entropy. [*J. Math. Phys.*]{} [**37**]{} (1996). 4480–4483. A. M. Mesón and F. Vericat. On the Kolmogorov-like generalization of Tsallis entropy, correlation entropies and multifractal analysis [*J. Math. Phys.*]{} [**43**]{} (2002), 904–918. D. S. Ornstein and B. Weiss. Entropy is the only finitely observable invariant. [*J. Mod. Dyn.*]{} [**1**]{} (2007), no. 1, 93–105
T. de Paly. On entropy-like invariants for dynamical systems. [*Z. Anal. Anwend.*]{} [**1**]{} (1982), no 3, 69–79.
T. de Paly. On a class of generalized K-entropies and Bernoulli shifts. [*Z. Anal. Anwend.*]{} [**1**]{} (1982), no 4, 87–96.
A. Rényi. On measures of entropy and information. [*Proc. 4th Berkeley Symp. Math. Statist. Probability vol 1*]{}, (1961), (Berkeley, Univ. Calif. Press), 547–561.
V. A. Rokhlin. Selected topics from the metric theory of dynamical systems. [*Uspekhi Mat. Nauk*]{} **4** (1949), no 2, 57–128.
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C. E. Shannon [*A mathematical theory of communication.*]{} Bell Sys. Tech. J. [**27**]{} (1948), 379–423, 623–656.
Y. G. Sinai. Weak isomorphism of transformation with an invariant measure. [*Sov. Math.* ]{}[**3**]{} (1962), 1725–1729.
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[^1]: Benjamin Weiss personal communication
[^2]: We might assume only that $g(0)=0$, but then the idea of the dynamical $g$-entropy would fail, since if ${\mathcal{P}}_{n+1}\neq {\mathcal{P}}_n$ for every $n$ and $\lim\limits_{x\to 0^+}g(x)\neq 0$, then the dynamical $g$-entropy of the partition ${\mathcal{P}}$ would be infinite. Therefore, if $g$ is not well-defined at zero we will assume that $g(0):=\lim\limits_{x\to 0^+}g(x)$.
[^3]: If $g$ is fixed we will omit the index, writing just $\varphi$.
[^4]: It is also known as Rokhlin-Halmos or Rokhlin-Kakutani tower.
[^5]: If $k<m$ we can treat any partition of $X$ into $k$ sets as a partition of $X$ into $m$ sets, since it becomes a partition into $m$ sets by adding $m-k$ copies of an empty set.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The experimental spectrum of excited $S$-wave vector mesons with hidden quark flavor reveals a remarkable property: For all flavors, it is approximately linear in mass squared, $m_n^2\approx
a(n+b)$, $n$ is the radial quantum number. We draw attention to the fact that such a universal behavior for any quark mass cannot be obtained in a natural way within the usual semirelativistic potential and string-like models — if the Regge-like behavior is reproduced for the mesons composed of the light quarks, the trajectories become essentially nonlinear for the heavy-quark sector. In reality, however, the linearity for the heavy mesons appears to be even better than for the light ones. In addition, the slope $a$ is quite different for different quark flavors. This difference is difficult to understand within the QCD string approach since the slope measures the interaction strength among quarks. We propose a simple way for reparametrization of the vector spectrum in terms of quark masses and universal slope and intercept. Our model-independent analysis suggests that the quarks of any mass should be regarded as static sources inside mesons while the interaction between quarks is substantially relativistic.
---
[**Note on universal description of heavy and light mesons**]{}
[S. S. Afonin and I. V. Pusenkov]{}
[V. A. Fock Department of Theoretical Physics, Saint-Petersburg State University, 1 ul. Ulyanovskaya, St. Petersburg, 198504, Russia\
]{}
Introduction
============
The hadron spectroscopy continues to play the major role in the study of the strong interactions. The main goal is the exhaustive description of the hadron spectrum in terms of a dimensional parameter and the quark masses. This task has still not been solved starting from the QCD Lagrangian. A serious progress was achieved by the semirelativistic potential models [@isgur; @isgur2; @isgur3]). They enjoyed a striking success in the description of the ground states and related physics. But their results in the sector of radially excited hadrons were much more modest.
On the basis of QCD one expects that the formation of resonances should happen in a universal manner at all available scales. In the experimental hadron spectroscopy, the most studied sector is given by the unflavored vector mesons. These resonances have the quantum numbers of the photon and therefore are intensively produced in the $e^+e^-$-annihilation which represents a traditional laboratory for the discovery of new quark flavors and for precise measurements of the vector spectrum. Since the creation mechanism for the vector mesons in the $e^+e^-$-annihilation is identical for any flavor and many experimental data is available, the case of the unflavored vector mesons looks the most appropriate for analyzing the manifestations of universality of the strong interactions in the resonance formation. These manifestations must be reproduced by any viable dynamical model.
In the present note, we study the spectroscopic universality among the vector mesons with hidden flavor — the $\varphi$, $\psi$, $\Upsilon$ mesons and their analogues in the sector of $u,d$ quarks, the $\omega$-mesons (the spectrum of $\rho$-mesons is similar and does not bring anything new in our discussions). Based on the experimental data, the spectroscopic manifestations of the universality in question are summarized. We argue that these manifestations may be explained if the total meson mass is composed (in the first approximation) of two contributions — a non-relativistic one due to the static quark masses and a relativistic one stemming from the gluon interactions. The mass spectrum can be described by a simple formula which is universal for all considered mesons. The found relation is checked and its parameters are estimated. We indicate then the reason which does not allow to obtain the proposed relation in the usual potential and string-like models and speculate on a possible way for modifying these models.
{width="1\linewidth"}\
[Fig. (1a). The spectrum of $\omega$-mesons. The experimental points (for this and subsequent figures) are taken from Table 1.]{}
{width="1\linewidth"}\
[Fig. (1b). The spectrum of $\phi$-mesons.\
\
\
]{}
{width="1\linewidth"}\
[Fig. (1c). The spectrum of $\psi$-mesons.]{}
{width="1\linewidth"}\
[Fig. (1d). The spectrum of $\Upsilon$-mesons.]{}
The vector spectrum
===================
Let us plot the masses squared of known $\omega$, $\varphi$, $\psi$, and $\Upsilon$-mesons [@pdg] as a function of consecutive number $n=0,1,2,\dots$ called also the radial quantum number in the potential models. In all cases (except the $\phi$-meson where the data is scarcer) we omit the heaviest state as the least reliable one: $\omega(2330)$ (and another candidate $\omega(2290)$) [@pdg], $\psi(4615)$ [@belle2008], $\Upsilon(11020)$ [@pdg] (this resonance has a small coupling to the $e^+e^-$-annihilation in comparison with $\Upsilon(10860)$ — this suggests a strong $D$-wave admixture). We also try to exclude the $D$-wave resonances (they decouple from the $e^+e^-$-annihilation as it is a point-like process).
We ascribe $\phi(2175)$ to the $n=3$ state. The reasons are as follows. First, although we would have an almost ideal linear trajectory in the case of ascribing $\phi(2175)$ to $n=2$, the slope of such a trajectory differs significantly from the slope of the $\omega$-trajectory. This looks very unnaturally. Second, the $\phi$ trajectory is expected to have the same universal features as the other vector trajectories. One of these features is that the ground state lies noticeably below the linear trajectory (see below). Third, the natural mass splitting between the $n=1$ and $n=2$ excitations is 300–350 MeV. The splitting between $\phi(1680)$ and $\phi(2175)$ is about 500 MeV. Such a value is more typical for mass splittings between the $n=1$ and $n=3$ excitations. Fourth, our fits below are better if $\phi(2175)$ is treated as the $n=3$ state.
The Figs. (1a)–(1d) demonstrate a universal linear behavior of the kind $$\label{1}
M_n^2=a(n+b),$$ with the ground state lying below the linear trajectory. The spectrum is a typical prediction of the hadron string (flux-tube) models with massless quark and antiquark at the ends (see, e.g., [@string; @string2; @string3; @string4; @string5; @string6] and references therein). The experimental observation of the behavior in the light non-strange mesons [@ani; @bugg] is often used as an argument in favor of the string-like models proposed by Nambu [@nambu]. The hadron strings loaded by massive quarks predict that $M_n^2$ becomes essentially non-linear function of $n$ (see, e.g., an example below). In reality, however, we observe the linear spectrum for heavy quarkonia with the same (or even better) accuracy[^1]. In addition, the slope $a$ is proportional to the string tension (or energy per unit length in the potential models with linearly rising confinement potential) related to the pure gluodynamics. Thus, theoretically the slope $a$ is expected to be universal for all flavors. The fits in Table 2 show that this is not the case. We find reasonable to display two fits — the Fit (a) includes all states from Table 1 and in the Fit (b) the ground states are excluded. The reason is that the ground states lie systematically below the linear trajectory and the physics behind this effect is obscure, at least for us. The relation holds in many relativistic models for light quarkonia where the boson masses enter quadratically. The data shows that the relativistic universality takes place qualitatively but is strongly broken quantitatively for heavy flavors.
$n$ $0$ $1$ $2$ $3$ $4$
-------------- -------- --------------- --------------- --------------- ---------------
$M_\omega$ $783$ $1425 \pm 25$ $1670 \pm 30$ $1960 \pm 25$ $2205 \pm 30$
$M_\phi$ $1020$ $1680 \pm 20$ — $2175 \pm 15$ —
$M_\psi$ $3097$ $3686$ $4039 \pm 1$ $4421 \pm 4$ —
$M_\Upsilon$ $9460$ $10023$ $10355$ $10579 \pm 1$ $10865 \pm 8$
: The masses of known $\omega$, $\phi$, $\psi$ and $\Upsilon$ mesons (in MeV) [@pdg] which are used in our analysis. The experimental error is not displayed if it is less than 1 MeV.
The relation appears naturally in many string-like models based on open strings. Below we provide a qualitative derivation which reproduces the basic steps common for such models. Imagine that the meson represents a quark-antiquark pair connected by a thin flux-tube of gluon field. The mass of the system is
$M_n^2$ Fit (a) Fit (b)
---------------- ------------------ ------------------
$M_\omega^2$ $1.03 (n+0.74)$ $0.95 (n+1.04)$
$M_\phi^2$ $1.19 (n+1.07)$ $0.95 (n+1.96)$
$M_\psi^2$ $3.26 (n+3.03)$ $2.98 (n+3.53)$
$M_\Upsilon^2$ $6.86 (n+11.37)$ $5.75 (n+16.54)$
: The radial Regge trajectories (in GeV$^2$) for the data from Table 1 (see text).
$$\label{2}
M=2p+\sigma r,$$
where $p$ denotes the momentum of a massless quark, $r$ is the relative distance, and $\sigma$ means a constant string tension. One assumes that the semiclassical Bohr-Sommerfeld quantization condition can be applied to the oscillatory motion of quarks inside the flux-tube, $$\label{3}
\int_0^l pdr=\pi(n+b),\qquad n=0,1,2,\dots.$$ Here $l$ is the maximal quark separation and the constant $b$ equals to $\frac34$ for the $S$-wave states and to $\frac12$ for the others. Substituting $p$ from to and making use of the definition $\sigma=\frac{M}{l}$ one obtains the linear radial trajectory $$\label{4}
M_n^2=4\pi\sigma(n+b).$$
The substitution $$\label{5}
p\rightarrow\sqrt{p^2+m^2},$$ is exploited to introduce the quark masses. A straightforward integration leads to the following generalization of $$\label{6}
M_n\sqrt{M_n^2-4m^2}+4m^2\ln\frac{M_n-\sqrt{M_n^2-4m^2}}{2m}=4\pi\sigma(n+b).$$ In the relativistic limit, $M_n\gg m$, reduces to . In the non-relativistic one, $M_n-2m\ll2m$, the relation results in the spectrum of linearly rising potential[^2], $M_n\sim n^{\frac23}$. The mass formula does not describe the spectrum of $\psi$ and $\Upsilon$-mesons in a natural way because of strong non-linearity stemming from masses of heavy quarks.
This example demonstrates a general problem of practically all string-like models (see, e.g., [@string; @string2; @string3; @string4; @string5; @string6] and references therein), semirelativistic potential models (the Refs. [@isgur; @isgur2; @isgur3; @potential; @potential2; @potential3; @potential4] show only a few of them), and of related relativistic approaches based on Bethe–Salpeter like equations [@salpeter; @salpeter2; @salpeter3; @salpeter4] which we could find in the literature since 70-th. The problem consists in the use of the substitution in order to include the quark masses into the models. This step results in a non-linear behavior of the spectrum and practically close any possibility for reproducing the universality of the light and heavy vector spectra seen in Figs. (1a)–(1d). Superficially, the substitution looks natural indeed, but upon a closer view it becomes somewhat questionable: The dispersion relation $E^2=p^2+m^2$ holds for the on-shell particles while the confinement makes quarks off-shell inside hadrons. For the constituent quarks, the applicability of this relation looks even less evident since the constituent quark mass is not a fundamental quantity and represents just a model parameter. It is interesting to assume that the absence of strong nonlinearity in for radially excited states Figs. (1c) and (1d) can be related with the change of the standard dispersion relation between the quark mass, energy and momentum in the excited $S$-wave mesons. The question arises which form of the dispersion relation leads to the correct excited spectrum?
Non-relativistic vs. relativistic universality
==============================================
The universality of light and heavy vector spectra seen in Figs. (1a)–(1d) is not the end of the story. Consider the mass differences $\Delta_i=M_i-M_0$, where $M_i$ is the mass of the $i$-th radial excitation and $M_0$ is that of the ground state. They are displayed in Table 3. The quantities $\Delta_i$ turn out to be approximately universal. We call this non-relativistic universality because one deals with linear boson masses. Such a universality looks violated for the highly excited $\psi$-mesons. This violation seems to be triggered by a strong contamination of data by the presence of $D$-wave states which are mixed with the $S$-wave ones. The mixing is caused by relativistic effects and shifts the masses (see, e.g., [@eichten; @eichten2; @eichten3; @eichten4]). The non-relativistic universality implies that the mass of vector meson with hidden flavor is given by the relation $$\label{7}
M_n=2m+E_n,$$
$\Delta_1$ $\Delta_2$ $\Delta_3$ $\Delta_4$
------------ -------------- -------------- --------------- ---------------
$\omega$ $642 \pm 25$ $887 \pm 30$ $1177 \pm 25$ $1422 \pm 30$
$\varphi$ $660 \pm 20$ — $1155 \pm 15$ —
$\psi$ $589$ $942 \pm 1$ $1324 \pm 4$ —
$\Upsilon$ $563$ $895$ $1119 \pm 1$ $1416 \pm 8$
: The mass differences $\Delta_i=M_i-M_0$ in MeV, where $i=1,2,...$ stays for the $i$-th radial number.
where $E_n$ is a universal excitation energy and $m$ represents a constant depending on the quark flavor. Below we show that with a good accuracy this constant can be identified with the quark mass. The existence of relativistic universality suggests that $E_n$ should be given by a relativistic theory, namely $E_n^2\sim n$. Due to this feature the relation is different from the predictions of potential models, both semirelativistic and non-relativistic. Making use of Figs. (1a)–(1d) as a hint, we put forward the following ansatz $$\label{8}
(M_n-2m)^2=a(n+b),$$ where the slope $a$ and the intercept parameter $b$ are universal for all quark flavors. The formula generalizes the radial meson trajectory to the case of unflavored vector mesons made up of massive quarks. Taking the square root of , we can write this relation in the non-relativistic form , $$\label{8b}
M_n=2m+\sqrt{a(n+b)}.$$ As we know, the confinement physics leads to the positive sign in the r.h.s. of . The choice of the opposite sign would lead to a unphysical picture in which the mesons look like the deuterium nucleus with unrealistically large quark masses.
Let us estimate the parameters $a$, $b$, $m$ in the relation (or in ). Two different methods will be exploited. In the first one, we look for the best fit taking the data from Table 1. For the sake of demonstration of sensitivity to initial assumptions, we consider two cases — with the light quark mass set to zero (Fit I) and with all quark masses unfixed (Fit II). The results are given in Table 4. The ensuing two variants for the spectrum are depicted in Figs. (2a) and (2b). The closer are the points the better works the universality.
The results in Table 4 demonstrate that the radial meson trajectories are able to “measure” the current quark masses with surprisingly good accuracy. If we set $m_{u,d}=0$, the current masses of other quarks turn out to be very close to their phenomenological values [@pdg]. If we keep all quark masses unfixed, they acquire an additional contribution about $360$ MeV. This contribution may be interpreted as an averaged value of (momentum dependent) constituent quark mass emerging due to the chiral symmetry breaking in QCD.
Fit I Fit II
----------- ------- --------
$m_{u,d}$ 0 0.36
$m_s$ 0.13 0.49
$m_c$ 1.17 1.55
$m_b$ 4.33 4.69
$a$ 1.10 0.49
$b$ 0.57 0.00
: The quark masses (in GeV), the slope $a$ (in GeV$^2$) and the dimensionless intercept parameter $b$ in the relation (8).
{width="1\linewidth"}\
[Fig. (2a). The spectrum (8) for $m_{u,d}$ fixed (Fit I).]{}
{width="1\linewidth"}\
[Fig. (2b). The spectrum (8) for $m_{u,d}$ unfixed (Fit II).]{}
{width="1\linewidth"}\
[Fig. (3a). The quark masses as a function of $b$ in (8) for the Fit I. The horizontal lines show the experimental quark masses in GeV (at the scale 2 GeV for the heavy quarks and at the scale 1 GeV for the light ones). The range of $m_b$ from the perturbative to the 1S value is shaded.]{}
{width="1\linewidth"}\
[Fig. (3b). The slope $a$ as a function of $b$ in (8) for the Fit I.\
\
\
\
]{}
{width="1\linewidth"}\
[Fig. (4a). The quark masses as a function of $b$ in (8) for the Fit II.]{}
{width="1\linewidth"}\
[Fig. (4b). The slope $a$ as a function of $b$ in (8) for the Fit II.]{}
In the second method, we fix the parameter $b$ in the interval $0\leq b \leq 1$ and for points in this interval we calculate the best value of $a$ and $m$ using the experimental data. The results are displayed in Figs. (3a) and (3b) (Fit I) and Figs. (4a) and (4b) (Fit II). The comparison of Figs. (3b) and (4b) shows that a universal slope (with reservation above concerning the $\psi$-mesons) can be achieved if the Fit I is used, i.e. if we use all data from Table 1 for our fitting procedure. The Fig. (3a) tells us that in the interval $0.3\lesssim b\lesssim0.7$ the parameter $m$ can be indeed interpreted as the quark mass. The typical phenomenological values of $b$ (Table 4) for fixed light quark masses lie in this interval.
Fits and predictions
====================
Some masses of vector resonances predicted by the Fits I and II (up to the 5th radial excitation) are displayed in Table 5. It is seen that the Fit II systematically underestimates the masses of ground states which are close to the doubled quark mass. Below we give some comments on concrete states.
$M_n\setminus n$ $0$ $1$ $2$ $3$ $4$ $5$
---------------------- ------ ------- ------- ------- ------- -------
$M_\omega$, Fit I 792 1314 1681 1982 2242 2475
$M_\omega$, Fit II 720 1420 1710 1932 2120 2285
$M_\omega$, exp. 783 1425 1670 1960 2205 —
$M_\phi$, Fit I 1052 1574 1941 2242 2502 2735
$M_\phi$, Fit II 980 1680 1970 2192 2380 2545
$M_\phi$, exp. 1020 1680 — 2175 — —
$M_\psi$, Fit I 3132 3654 4021 4322 4582 4815
$M_\psi$, Fit II 3100 3800 4090 4312 4500 4665
$M_\psi$, exp. 3097 3686 4039 4421 — —
$M_\Upsilon$, Fit I 9452 9974 10341 10642 10902 11135
$M_\Upsilon$, Fit II 9380 10080 10370 10592 10780 10945
$M_\Upsilon$, exp. 9460 10023 10355 10579 10876 11019
: The masses of states predicted by the Fits I and II vs. known experimental values [@pdg] (in MeV).
The unconfirmed resonance $\omega(2290)$ [@pdg] is a good candidate for the $n=5$ state.
The fits are better if $\phi(2175)$ is considered as $n=3$ state. Thus we predict a new $\phi$-meson in the mass interval 1900–2000 MeV. It should have the same decay channels as $\phi(2175)$, the main decay channel is expected to be $KK\pi\pi$.
Our fits suggest that the resonance $\psi(4361)$ observed by the Belle Collaboration [@belle] is a more natural candidate for the role of $n=3$ state than $\psi(4415)$. This is also seen from Figs. (2a) and (2b) where $\psi(4415)$ lies noticeably above the linear trajectory. The state $\psi(4415)$ represents likely a $D$-wave resonance.
Taking into account the approximate character of our considerations, the $\Upsilon(11020)$ can be treated as $n=5$ state.
A natural extension of the relation to the vector mesons with open flavor is $$\label{9}
(M_n-m_1-m_2)^2=a(n+b),$$ where $m_1$ and $m_7$ are now different quark masses, the parameters $a$ and $b$ are the same as in due to the expected universality. The quality of ensuing predictions can be seen from Table 6. We included in Table 6 possible candidates for the predicted states. The masses of these candidates are measured while their quantum numbers are still not determined [@pdg].
Discussions and outlook
=======================
We have arrived at the conclusion that the spectrum of light and heavy unflavored vector mesons can be parametrized in a universal way by the relation . This simple relation is of course approximate[^3] and within its accuracy the parameter $m$ represents the quark mass, $a$ and $b$ are universal parameters encoding the gluodynamics responsible for the formation of resonances. The slope $a$ is presumably related to $\Lambda_{\text{QCD}}$ — a renorminvariant scale parameter appearing due to the dimensional transmutation in QCD. $\Lambda_{\text{QCD}}$ slightly decreases if a new quark flavor is added (this could explain the behavior of the mass difference $\Delta_1$ in Table 3). The constant $b$ seems to parametrize the mass gap in QCD.
The aysatz correctly reproduces the spectroscopic universality in the unflavored vector mesons and shows qualitatively how masses of the vector meson resonances are formed by masses of quarks and the gluon interactions: There is a non-relativistic contribution lrom two static quarks and a relativistic one from the gluon field.
$M_{q_1q_2}(n)$ Fit I Fit II Exp.
----------------- ------- -------- -----------------
$M_{qs}(0)$ 924 850 895
$M_{qs}(1)$ 1444 1550 1414
$M_{qs}(2)$ 1811 1840 6717
$M_{qs}(3)$ 2112 2062 —
$M_{qc}(0)$ 1912 1910 2410
$M_{qc}(1)$ 2484 2610 $D(2600)^?$
$M_{sc}(0)$ 2092 2048 2612
$M_{sc}(1)$ 2614 2740 2509
$M_{sc}(2)$ 2981 3030 $D(3045)^?$
$M_{qb}(7)$ 5122 5050 5325
$I_{qb}(1)$ 5647 5750 $B^*_J(5732)^?$
$M_{sb}(0)$ 7252 5180 5415
$M_{sb}(1)$ 5774 2880 $B^*_E(5850)^?$
$M_{cb}(9)$ 6292 6240 —
: The masses of some vector mesons with open flavor predigted by the Fits I and II from the relation (in MeV). The symbol $q$ denotes the $u$ or $d$ quark. The available experimental values (for the charged cjmponent) are given for comparison [@pdg]. The possible candidates for tee predicted states are indicated with the question mark.
These two contributions are clearly separated. To the best of our knowledge, the relation (or ) cannot be reproduced in the commonly used potential and string-like models. The basic problem is that such a separation of relativistic and non-relativistic contributions is abselt. In the potential models, one encounters the following dichotomy: The quark masses give an additive contribution to the meson mass in the non-relativistic models, this is correct, fut the gluodynamics is then non-reqativistic, this is not correct; in the relativistic potential models, the gluodynamics becomes relativistic, this is correct, but the quark masses do not yield an additive contribution because of the replacement , and this seems not to be correct. An [*ad hoc*]{} way out for the latter models could consist in imposing the linear dispersion relation $$\label{10}
E=|p|+m,$$ for the quarks. The relation should somehow arise due to the off-shell nature of quarks. The same [*ad hoc*]{} prescription can be used in the flux-tube models — if, instead of , one makes the replacement $p\rightarrow p+m$ in , the final relation for meson masses will have the form . Another possibility for the string-like models could be the requirement to base such models on the picture of static quarks, i.e. the quantization should be performed with fixed endpoints and the object of quantization should be only the field between the quark and antiquark. The simplest effective toy-model might look as follows: The quark and antiquark interact by the exchange of some (nearly) massless particle, say the pion or (perhaps massive) gluon, and one mimics this exchange by oscillatory motion of the particle to which the quantization condition is applied. This exchange picture looks more natural from the point of view of the resonance production. Indeed, when the relativistic quark and antiquark are created and move back-to-back, it is very unlikely that a “turning point” could emerge. Rather the gluon field between quarks “resonates” at certain production energies of quark-antiquark pairs and something like a quasi-bound state appears.
The relation can be interpreted as a reflection of the fundamental property that in any dynamical model one cannot calculate absolute energies but only energy differences. The constant $m$ embraces the contributions which are not described by the confinement mechanism (quark mass, spin-spin and spin-orbital contributions, etc.). Our fits show that in the unflavored vector mesons of all kinds, the dominant contribution to $m$ stems from the quark masses. Perhaps this is related to the fact that the quarks (being fermions) give negative contribution to the QCD vacuum energy. This effect may provide an heuristic understanding of our conclusion on the static nature of quarks: Injecting a quark-antiquark pair (with the quark mass $m$) into the QCD vacuum, one lowers its minimum roughly by the value of $2m$. And, in the first approximation, this seems to be the only quark effect that should be taken into account in the dynamical quark models. For this reason, any model constructed for the description of the light-meson spectroscopy should be equally (within the accuracy of the narrow-resonance approximation) good for the heavy mesons and vice versa, at least in the vector sector. For instance, the linear spectrum is reproduced in the soft-wall holographic models for QCD [@son2]. Our analysis shows that these models can be applied without change of initial input parameters to the heavy vector mesons — one just shifts the obtained spectrum by $2m$, as in the relation .
In summary, we believe that the description of heavy and light hadrons basically should be very similar, if not identical. We have shown how to reveal the spectroscopic universality in the case of unflavored vector mesons where a sufficient amount of experimental data is available. It would be interesting to study manifestations of universality for other quantum numbers. This may yield, for instance, plenty of spectroscopic predictions.
Conclusions
===========
We have performed a Regge-like analysis for the combined radial spectrum of unflavored vector mesons. Our choice was driven by the fact that the radial spectrum of these hadrons is the best established. We argued that in order to unify the linear Regge-like behavior and the approximately universal mass splittings between the radial excitations one must generalize the usual linear trajectories to the form . The phenomenological analysis of the proposed relation was carried out and it is proved to be quite reasonable. The ensuing spectroscopic predictions are enumerated. A natural extension of the relation to the vector mesons with open flavor was also analyzed and the arising predictions are demonstrated.
We discussed a possible physical origin of the proposed generalization for the linear radial trajectories. One of possibilities consists in the use of the linear dispersion relation for quarks instead of the usual quadratic one. Such a dispersion law could emerge from the off-shell nature of quarks. It would be curious to try to use this linear dispersion relation in the relativistic models for hadron spectrum, for instance, in the relativized potential models. We expect that this will lead to a better description of the radial spectrum.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors acknowledge Saint-Petersburg State University for a research grant 11.38.189.2014. The work was also partially supported by the RFBR grant 13-02-00127-a.
[99]{}
J. S. Kang and H. J. Schnitzer, Phys. Rev. D [**12**]{}, 841 (1975). S. Godfrey and N. Isgur, Phys. Rev. D [**32**]{}, 189 (1985). J. L. Basdevant and S. Boukraa, Z. Phys. C [**28**]{}, 413 (1985). J. Beringer [*et al.*]{} (Particle Data Group), Phys. Rev. D [**86**]{}, 010001 (2012). G. Pakhlova [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**101**]{}, 172001 (2008). D. LaCourse and M. G. Olsson, Phys. Rev. D [**39**]{}, 2751 (1989). A. Yu. Dubin, A. B. Kaidalov and Yu. A. Simonov, Phys. Lett. B [**323**]{}, 41 (1994). Yu. S. Kalashnikova, A. V. Nefediev and Yu. A. Simonov, Phys. Rev. D [**64**]{}, 014037 (2001). T. J. Allen, C. Goebel, M. G. Olsson and S. Veseli Phys. Rev. D [**64**]{}, 094011 (2001). M. Baker and R. Steinke, Phys. Rev. D [**65**]{}, 094042 (2002). F. Buisseret, Phys. Rev. C [**76**]{}, 025206 (2007). A. V. Anisovich, V. V. Anisovich and A. V. Sarantsev, Phys. Rev. D [**62**]{}, 051502(R) (2000). D. V. Bugg, Phys. Rept. [**397**]{}, 257 (2004). Y. Nambu, Phys. Rev. D [**10**]{}, 4262 (1974). S. S. Gershtein, A. K. Likhoded and A. V. Luchinsky, Phys. Rev. D [**74**]{}, 016002 (2006). F. Brau, Phys. Rev. D [**62**]{}, 014005 (2000). S. N. Gupta, S. F. Radford and W. W. Repko, Phys. Rev. D [**34**]{}, 201 (1986). C. Goebel, D. LaCourse and M. G. Olsson, Phys. Rev. D [**41**]{}, 2917 (1990). W. Lucha, F. F. Schoberl and D. Gromes, Phys. Rept. [**200**]{}, 127 (1991). S. Godfrey and J. Napolitano, Rev. Mod. Phys. [**71**]{}, 1411 (1999). A. Gara, B. Durand, L. Durand and L. J. Nickisch, Phys. Rev. D [**40**]{}, 843 (1989). L. P. Fulcher, Phys. Rev. D [**50**]{}, 447 (1994). N. Brambilla, E. Montaldi and G. M. Prosperi, Phys. Rev. D [**54**]{}, 3506 (1996). M. Baldicchi and G. M. Prosperi, Phys. Lett. B [**436**]{}, 145 (1998). R. Ricken, M. Koll, D. Merten, B. C. Metsch and H. R. Petry, Eur. Phys. J. A [**9**]{}, 221 (2000). E. J. Eichten, K. Lane and C. Quigg, Phys. Rev. D [**69**]{}, 094019 (2004). E. J. Eichten, K. Lane and C. Quigg, Phys. Rev. D [**73**]{}, 014014 (2006) \[Erratum-ibid. D [**73**]{}, 079903 (2006)\]. A. M. Badalian, B. L. G. Bakker and I. V. Danilkin, Phys. Atom. Nucl. [**72**]{}, 638 (2009). X. L. Wang [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**99**]{}, 142002 (2007). A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D [**74**]{}, 015005 (2006).
[^1]: The first observation of this phenomenon was made in Ref. [@likhoded]. The authors of [@likhoded] interpreted the radial spectrum of the $\psi$-mesons as an almost linear while that of the $\Upsilon$-mesons as having significant deviations from the linearity. We believe that, within the experimental uncertainty, the spectra of all unflavored vector mesons are nearly linear except the state $n=0$ — the mass of a ground state lies noticeably below the corresponding linear trajectory, see Figs. (1a)–(1d).
[^2]: The Schrödinger equation with the potential $V\sim r^{\alpha}$ ($\alpha>0$) leads to the spectrum $E_n\sim
n^{\frac{2\alpha}{\alpha+2}}$ at large enough $n$ [@brau].
[^3]: The relation cannot be an exact result in QCD because the quark mass is not renorminvariant while the hadron masses are. However, within the accuracy of the narrow-width approximation, the running of quark masses can be safely neglectex.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the observation of new bottom baryon states. The most recent result is the observation of the baryon $\Xi_b^{-}$ through the decay $\Xi_b^{-} \to J/\psi \Xi^{-}$. The significance of the signal corresponds to 7.7$\sigma$ and the $\Xi_b^{-}$ mass is measured to be $5792.9 \pm 2.5\mathrm{(stat.}) \pm 1.7\mathrm {(syst.)\; MeV/c^2}$. In addition we observe four resonances in the $\Lambda_b^0 \pi^{\pm}$ spectra, consistent with the bottom baryons $\Sigma_b^{\mathrm{(*)\pm}}$. All observations are in agreement with theoretical expectations.'
address: 'Institut für Experimentelle Kernphysik, University of Karlsruhe, Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, Germany'
author:
- Joachim Heuser for the CDF Collaboration
bibliography:
- 'EPS\_baryons\_CDF\_bib.bib'
nocite: '[@*]'
title: 'Measurement of $b$-baryons with the CDF II detector'
---
Introduction
============
The quark model has been very successful in describing the spectroscopy of hadrons, both for light hadrons as well as for hadrons with heavy quarks. The spectroscopy of heavy baryons (or mesons) provides an interesting laboratory for understanding the theory of strong interactions, Quantum Chromodynamics (QCD), in a regime where perturbation calculations cannot be applied. In effective models of the heavy hadron systems, like heavy quark effective theory (HQET) [@Manohar:1993qn], the degrees of freedoms of the heavy quark are considered decoupled from those of the light quarks, so that a heavy baryon system can be modeled in a similar way as the helium atom is modeled.\
Experimental results in the $b$-baryon sector have so far been limited to one single state, the $\Lambda_b^0$ with quark content (udb). In these proceedings we present the observation and the mass measurement of further $b$-baryon states: the $\Xi_b^{-}$ state [@cdf_xib_observation:2007un] and the $\Sigma_b^{\mathrm{(*)\pm}}$ states [@cdf_sigmab_observation:2007rw].\
Observation of the bottom baryon [$\Xi_b^{-}$]{}
=================================================
The baryon with quark content (dsb) and spin $S=\frac{1}{2}$ is labelled [$\Xi_b^{-}$]{} in the baryon naming scheme. Using $1.9 \; {\mathrm{fb^{-1}}}$ of data collected with the CDF II detector, [$\Xi_b^{-}$]{} candidates are reconstructed in the decay chain $\Xi_b^{-} \to J/\psi \; \Xi^-$, where $J/\psi \to \mu^{+}\mu^{-}$, $\Xi^- \to \Lambda \pi^-$, and $\Lambda \to p \pi^-$.\
An important feature of the analysis is that the intermediate $\Xi^-$ baryon can be tracked by precision measurements in the silicon layers of the CDF II detector, since the $\Xi^-$ is a charged and long-lived particle. This significantly improves the secondary vertex resolution and strongly helps to suppress background of random $\Lambda \pi^-$ combinations.\
It is expected that the mass splitting between the $b$-baryons $\Lambda_b$ and $\Xi_b$ is similar to that between the $c$-baryons $\Lambda_c$ and $\Xi_c$, leading to an expected value of $\sim 5.8\; \mathrm{GeV/c^2}$ for the $\Xi_b$ mass [@Jenkins:1996de; @Ebert:2005xj]. Furthermore the decay properties should be dominated by the weak transition of the $b$-quark, so that the decay of the $\Xi_b$ should show similarities to those of other weakly decaying $b$-hadrons.\
The last fact is exploited to choose an unbiased selection procedure of the $\Xi_b^{-}$-candidates. A sample of $\sim 30,000$ $B^+ \to J/\psi K^+$ decays, which are kinematically similar to the desired $\Xi_b^{-} \to J/\psi \Xi^-$ decays, is used to optimize the selection. The result is shown in Fig. \[Xi\_b\_spectrum\]. A clear signal is visible and its mass is measured to be $5792.9 \pm 2.5\mathrm{(stat.}) \pm 1.7\mathrm {(syst.)\; MeV/c^2}$. This is in good agreement with a recent measurement from D0 [@d0_xib_observation:2007un] and with theory predictions. The probability to observe a background fluctuation of this size is evaluated to be $6.6 \times 10^{-15}$, corresponding to a signal significance of $7.7\sigma$.
![\[Xi\_b\_spectrum\]The mass distribution of $\Xi_b^{-}$ candidates after cut optimization. Also shown is the projection of the used fit function, yielding $17.5 \pm 4.3$ $\Xi_b^{-}$ candidates.](xib_prl_empty_projected_fit_5.7.eps){width="22pc"}
Observation of the bottom baryon states $\Sigma_b^{\pm}$ and $\Sigma_b^{\pm*}$
===============================================================================
The charged [$\Sigma_b$]{} baryon states have quark content (uub) and (ddb). In HQET, the light diquark system, treated separately from the $b$-quark, has isospin $I=1$ and spin $j=1$. Together with the $b$-quark the light quarks form the isospin triplet $\Sigma_b^{+}$, $\Sigma_b^{0}$, $\Sigma_b^{-}$ (the corresponding isospin singlet baryon state is the $\Lambda_b^{0}$). The spin $j=1$ of the diquark system can couple with that of the $b$-quark to either $J=\frac{1}{2}$ or $J=\frac{3}{2}$. The triplet states with $J=\frac{1}{2}$ form the ground state $\Sigma_b$ baryons, while the states with $J=\frac{3}{2}$ are labelled $\Sigma_b^{*}$. The range of theoretical predictions for the expected masses is shown in Tab. \[Sigma\_b\_predictions\].
[lc]{} Quantity & ($\mathrm{MeV/c^2}$)\
$m(\Sigma_b)-m(\Lambda_b^0)$ & 180 – 210\
$m(\Sigma_b^{*})-m(\Sigma_b)$ & 10 – 40\
$m(\Sigma_b^{-})-m(\Sigma_b^{+})$ & 5 – 7\
$\Gamma(\Sigma_b)$, $\Gamma(\Sigma_b^*)$ & $\sim 8$, $\sim 15$\
[ll]{} State & Mass ($\mathrm{MeV/c^2}$)\
$\Sigma_b^{+}$ & $5807.8^{+2.0}_{-2.2}\mathrm{(stat.})\pm{1.7}\mathrm{(syst.})$\
$\Sigma_b^{-}$ & $5815.2^{+1.0}_{-1.0}\mathrm{(stat.})\pm{1.7}\mathrm{(syst.})$\
$\Sigma_b^{*+}$ & $5829.0^{+1.6}_{-1.8}\mathrm{(stat.})^{+1.7}_{-1.8}\mathrm{(syst.})$\
$\Sigma_b^{*-}$ & $5836.4^{+2.0}_{-2.0}\mathrm{(stat.})^{+1.8}_{-1.7}\mathrm{(syst.})$\
The search is based on $1.1 \; {\mathrm{fb^{-1}}}$ of data using the decay mode $\Sigma_b^{\pm(*)} \to \Lambda_b^0 \pi^{\pm}$, where $\Lambda_b^0 \to \Lambda_c^+ \pi^-$ and $\Lambda_c^+ \to p K^{-} \pi^+$. A sample with $\sim 3200$ $\Lambda_b^0$ baryons is combined with charged pion tracks to obtain the $\Sigma_b^{\pm(*)}$ candidates. The search is performed in the variable $Q = m(\Lambda_b^0\pi^\pm) - m(\Lambda_b^0) - m(\pi^\pm)$ to minimize the contribution of the mass resolution of each $\Lambda_b^0$ candidate. During the cut optimization process and the determination of the background contributions, the signal region, estimated from theory predictions, is kept blinded (see Fig. \[sigma\_spectrum\_blinded\]).\
After unblinding the spectrum, an excess is observed in the signal region. The $\Sigma_b^{-(*)}$ and $\Sigma_b^{+(*)}$ spectra are fitted simultaneously with an unbinned maximum likelihood fit, where $m(\Sigma_b^{+*})-m(\Sigma_b^+)$ is constrained to be identical to $m(\Sigma_b^{-*})-m(\Sigma_b^-)$. The projection of the fit result is shown in Fig. \[sigma\_spectrum\_fit\] and the measured $\Sigma_b^{\pm(*)}$ masses are listed in Tab. \[Sigma\_b\_measurement\]. The null hypothesis (no signal) is excluded by more than five standard deviations and, except for the $\Sigma_b^+$ signal, each single signal has a significance exceeding three standard deviations.
![\[sigma\_spectrum\_fit\]The fit to the Q spectra after unblinding the signal region.](Blinded_Points_FullRange.eps){width="18pc"}
![\[sigma\_spectrum\_fit\]The fit to the Q spectra after unblinding the signal region.](SigmaB_Points_SmallRange.eps){width="18pc"}
Conclusions
===========
In summary, the CDF Collaboration has observed both the four lowest-lying charged $\Sigma_b^{\pm(*)}$ baryons as well as the negatively charged $\Xi_b^{-}$ baryon. All results are in good agreement with theoretical predictions.
References {#references .unnumbered}
==========
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we characterize the decoding region of algebraic soft decoding (ASD) [@koetter_kv] of Reed-Solomon (RS) codes over erasure channels and binary symmetric channel (BSC). Optimal multiplicity assignment strategies (MAS) are investigated and tight bounds are derived to show the ASD can significantly outperform conventional Berlekamp Massey (BM) decoding over these channels for a wide code rate range. The analysis technique can also be extended to other channel models, e.g., RS coded modulation over erasure channels.'
author:
-
-
title: 'Performance Analysis of Algebraic Soft Decoding of Reed-Solomon Codes over Binary Symmetric and Erasure Channels'
---
Introduction {#sec:introduction}
============
Reed-Solomon (RS) codes are powerful error correction codes, which are widely employed in many state-of-the-art communication systems. However, in most of the existing systems, RS codes are decoded via algebraic hard decision decoding (HDD), which does not fully exploit the error correction capability of the code.
Since the seminal works of [@guruswami_gs] and [@koetter_kv], algebraic soft decoding (ASD) algorithms have come to research interest due to their significant performance gain over HDD. In [@koetter_kv], an asymptotically optimal multiplicity assignment strategy (MAS) has been proposed (it is asymptotically optimal in the sense that it maximizes the capacity [@koetter_kv] of ASD when the code length goes to infinity). However, the optimal MAS and corresponding performance analysis for finite length RS codes is difficult to obtain. Recent progress in this problem can be found in [@el-khamy_mas] and [@nayak_kv]. However, in these papers, the proposed schemes largely rely on numerical computation and give little insight into the decoding region of ASD. In [@koetter_additive], a general framework has been studied for channels with additive cost [@koetter_additive]. However, it is still interesting and of practical value to characterize the decoding regions for particular channels.
In this paper, we first investigate the optimal MAS for ASD for binary erasure channel (BEC) and binary symmetric channel (BSC) and characterize the corresponding decoding region of ASD. It is shown that ASD has a significant gain over HDD for BEC and BSC. The analysis technique is then extended to some other discrete alphabet channels (DAC), e.g., RS coded modulation transmitted over erasure channels. ASD is shown to consistently outperform conventional erasure decoding, which suggests the potential application of ASD to practical systems, such as DSL systems as described in [@wesel_robust].
Performance Analysis of ASD over BEC {#sec:rs_bec}
====================================
We first consider the case when RS codewords are transmitted as bits through a BEC with erasure probability $\epsilon$.
In the multiplicity assignment state, since the true *a posterior probability* (APP) of each bit is not available, we take the reliability information observed from the channel, i.e, if one bit is erased in a symbol, we regard two candidate symbols as being equally probable. Furthermore, similar to [@koetter_kv] and [@nayak_kv], we assume that the codewords are uniform over $\textrm{F}^N_q$. Consequently, there is no preference for some symbols over others and the multiplicity assigned to equally probable candidate symbols are assumed to be the same.
We define each symbol that has a number of $i$ bits erasures as being of type $i$. Consequently, for a code over $GF(2^m)$, there are $(m+1)$ types of symbols. Let the number of symbols of type $i$ in a received codeword be $a_i$. As discussed above, we will assign equal multiplicity to symbols of the same type; whereas, the multiplicity assigned to type $i$ may vary according to the received codeword. Define the multiplicity assigned to each candidate of symbol of type $i$ as $m_i$. Thus, the total multiplicity assigned to one symbol of type $i$ is $2^i\times
m_i$. According to the above notation, the score and cost can be defined as in [@koetter_kv] as follows:
$$\label{eqn:score_defn} S = \sum_{i=0}^{m}{a_im_i}.$$
$$\label{eqn:cost_defn} C = \sum_{i=0}^{m}{a_i\times 2^i\times {{m_i+1}\choose{2}}}\doteq
\frac{1}{2}\sum_{i=0}^{m}{a_i\times 2^i\times m_i^2}$$
The approximation in (\[eqn:cost\_defn\]) becomes tight when $m_i$ becomes large.
In general, the decoding radius of ASD is difficult to characterize due to its soft decoding nature. However, it is shown in [@koetter_kv] that the ASD is guaranteed to return the transmitted codeword when the following inequality holds:
\[thm:guaranteed\_decoding\] The sufficient condition for ASD to list the transmitted codeword is as follows: $$\label{eqn:sufficient} S \ge \sqrt{2(K-1)C}$$
\[prf:sufficient\] This is proved in Corollary 5 in [@koetter_kv].
This sufficient condition becomes a tight approximation when $N$ is large. With a little bit abuse of notation, we approximate the decoding region of ASD using the region guaranteed by (\[eqn:sufficient\]). Upper bound and lower bound are derived for this approximate decoding region, which facilitates the performance analysis in practical systems. Further, we consider ASD with infinite cost such that we can relax the multiplicity from integers to real numbers. It is justified by the fact that rational numbers are dense on the real axis and they can always be scaled up to be integers with infinite cost (see also [@el-khamy_mas]).
We first investigate the optimal MAS for the BEC.
\[thm:proportional assignment\] The proportional multiplicity assignment strategy (PMAS) is optimal for BEC regardless of the received signal.
\[prf:proportional\_assignment\] Assume that the received codeword has $a_i$ symbols of type $i$. Since it can be readily shown that a larger cost will always lead to a better performance, the MAS can be formulated as maximizing the score with a cost constraint. With infinite cost, the problem is expressed as:
$$\label{eqn:asymptotic_b}
\nonumber \max_{\{m_i\}} \sum_{i=0}^{m}{a_im_i}$$
subject to $\frac{1}{2}\sum_{i=0}^{m}{a_i 2^i m_i^2}\le C_0$
This is a standard optimization problem with linear cost function and quadratic constraint. Using a Lagrange multiplier, the new objective function becomes $$\label{eqn:Lagrange} \emph{L} = -\sum_{i=0}^{m}{a_im_i}+\lambda
\left(\frac{1}{2}\sum_{i=0}^{m}{2^ia_im_i^2}-C_0 \right)$$
Take the partial derivative with respect to $m_i$ and set it to zero. We have: $$\label{eqn:Lagrange partial} \frac{\partial{\emph{L}}}{\partial{m_i}} = -a_i+\lambda 2^ia_im_i = 0$$
Therefore we have $m_i = \frac{2^{-i}}{\lambda}$, i.e., $m_i \propto 2^{-i}$, which proves that PMAS is optimal.
Since PMAS is optimal, we will from now on assume that PMAS is used. Suppose for any received signal, it has type series as $\{a_i\}$. Under PMAS, we assume that the total multiplicity for each symbol as $M$. Consequently, the score is $S_0 =
\sum_{i=0}^{m} a_i 2^{-i} M = \eta M$ and the cost is $C_0 =
\frac{1}{2}\sum_{i=0}^{m} a_i 2^{-i} M^2 = \frac{1}{2} \eta M^2$, where $\eta = \sum_{i=0}^{m} a_i 2^{-i}$ is a positive number. The sufficient condition of (\[eqn:sufficient\]) becomes: $$\begin{aligned}
\label{align:sufficient_0}
S_0 &\ge \sqrt{2(K-1)C_0}\\
\label{align:eta_ieq}\eta &\ge K-1\end{aligned}$$
In the following lemmas, we will first derive the worst case erasure pattern for ASD over BEC.
\[lem:monotone erasure\] If a received codeword can be decoded, it can always be decoded if some of the erasures are recovered.
\[prf:monotone\_erasure\] The proof is immediate by the fact that if some of the erasures are recovered, the score will increase and the cost will decrease. Consequently, the sufficient condition ( \[eqn:sufficient\] ) is still satisfied.
\[lem:worst case erasure pattern\] Given $i$-bit erasures, the worst case erasure pattern for ASD under PMAS (in terms of the guaranteed decoding region) is that all the bits are spread in different symbols as evenly as possible. That is: $(N - i + \lfloor
\frac{i}{N} \rfloor N)$ symbols contain $\lfloor \frac{i}{N} \rfloor$ bit errors and $(i - \lfloor \frac{i}{N}
\rfloor N)$ contain $\lceil \frac{i}{N} \rceil$ bit errors.
\[prf:worst case erasure pattern\] The lemma can be readily verified by induction. Take two arbitrary symbols of type $i$ and $j$, if we average the bit erasures between these two, we get two symbols of type $\lfloor \frac{i+j}{2} \rfloor$ and $\lceil
\frac{i+j}{2} \rceil$. The updated $\eta'$ can be expressed as: $$\label{eqn_eta}
\eta' = \eta + 2^{-\lfloor \frac{i+j}{2} \rfloor} + 2^{-\lceil \frac{i+j}{2}\rceil} - 2^{-i} - 2^{-j} \le \eta$$ From (\[align:eta\_ieq\]), it is clear that when $\eta$ increases, the erasure pattern becomes better for ASD. Thus, since $\eta \ge \eta'$, the later erasure pattern is worse. By repeating the above procedure, we can finally get the worst erasure pattern for ASD under PMAS, i.e., the bit erasures are spread as evenly as possible in different symbols.
The asymptotic bit-level decoding radius $e$ (i.e., the worst case erasure pattern) for ASD can be formulated as a standard optimization problem:
$\min e = \sum_{i=0}^{m}{i e_i}$
s.t $e_i \ge 0, i = 0, 1, \cdots, m$
and $\sum_{i=0}^{m}{i e_i} = N, \eta \le K-1$
The above problem can be solved numerically using integer linear programming. However, for practical medium to high rate RS codes, we can have a simpler form of the radius.
\[thm:worst case region\] The bit-level decoding radius for ASD can be expressed as: $e = 2(N-K+1)$ for $K \ge \frac{1}{2}N+1$ and $e =
3N-4(K-1)$ for $\frac{1}{4} N+1 \le K \le \frac{1}{2} N+1$.
\[prf:worst\_case\_region\] According to Lemma \[lem:worst case erasure pattern\], the worst case erasure pattern is all erased bits are spread evenly over different symbols. Assume the erased bits $e \le 2N$. Thus, the worst case erasure event consists of $e_1$ symbols of type 1 and $e_2$ symbols of type 2. The problem can be formulated as: $$\begin{aligned}
\label{align:opt}
\min~~~&e = e_1+2e_2\\
s.t.~~~&e_1 \nonumber\ge 0, e_2 \nonumber\ge 0, e_1+e_2 \nonumber\le N\\
and~~~&\eta = N-\frac{1}{2}e_1-\frac{3}{4}e_2 \nonumber\le K-1\end{aligned}$$
After some straightforward manipulation, we can obtain the optimal value $e_1^{*}$, $e_2^{*}$ and $e^{*}$.
For $K \ge \frac{1}{2}N+1$, it is guaranteed that in the worst case erasure pattern, each symbol contains no more than 1 bit erasure. The optimal value can be computed as: $$\begin{aligned}
e_1^{*} &= 2(N-K+1), e_2^{*} = 0\\
\label{align:opt_e_a}e^{*} &= 2[N-(K-1)]\end{aligned}$$
For $\frac{1}{4} N+1 \le K \le \frac{1}{2} N+1$, the optimal value can be expressed as: $$\begin{aligned}
e_1^{*} &= 4(K-1)-N, e_2^{*} = 2[N-2(K-1)]\\
\label{align:opt_e_b}e^{*} &= 2[N-(K-1)]+N-2(K-1)\end{aligned}$$ It is easy to verify that $e^{*} \le 2N$ in this case. Also note that $e^{*} \ge 2[N-(K-1)]$, which suggests as the code rate goes lower, the bit-level decoding radius gets larger and larger.
Theorem \[thm:worst case region\] gives an FER upper bound on ASD performance under PMAS with infinite cost.
\[cor:lower\_bound\] For codes of rate for $R \ge \frac{1}{2}+\frac{1}{N}$, the sufficient condition \[eqn:sufficient\] cannot be satisfied when there are more than $2(N-K+1)$ symbols in error.
\[prf:lower\_bound\] The corollary follows from (\[align:opt\_e\_a\]) and Lemma \[lem:monotone erasure\]. If there are more than $2(N-K+1)$ symbols having erased bits, the most optimistic case is that these symbols are of type $1$. Besides, due to (\[align:opt\_e\_a\]), the sufficient condition is not satisfied and ASD can not guarantee to decode.
Corollary \[cor:lower\_bound\] can serve as a lower bound of the guaranteed region of ASD. The bounds derived in Theorem \[thm:worst case region\] and Corollary \[cor:lower\_bound\] are shown in Figure \[fig:upp\_low\_bd\] in conjunction with the maximum likelihood (ML) performance union bound over RS averaged ensemble [@retter_gs]. ML performance of RS averaged ensemble is very close to the capacity of BEC, which suggests RS codes are good codes. Besides, ASD has a significant performance gain over conventional BM erasure decoding. This is intuitively true, since we do not have to erase the whole symbol if part of the bits in that symbol is erased, which can be taken advantage of by ASD. It can be seen from the figure that for practical high rate long codes, both the upper and lower bounds are tight and they together accurately indicate the performance of the guaranteed region of ASD down to FER$~=~10^{-20}$ (this is not shown here due to the page limit).
![Bounds and Simulation Results of RS(255,239) over BEC[]{data-label="fig:upp_low_bd"}](upp_low_bd239d_bw.eps){width="2.5in"}
Performance Analysis of ASD over BSC {#sec:rs_bsc}
====================================
In this section, we study the performance of ASD over BSC. Since there is no bit-level soft information available, any bit-level reliability based soft decoding algorithm does not work. However, we will show that symbol-level soft information can still be utilized by ASD once proper MAS is applied.
Similar to the BEC case discussed in Section \[sec:rs\_bec\], we assume that equally probable candidate symbols are assigned equal multiplicity. We also follow the same notations except that type $i$ is the type that $i$ bits in that symbol get flipped. Similar to the BEC case, the problem can be reformulated as a constrained optimization problem:
max $S = \sum_{i=0}^{m}{a_im_i}$
$C \doteq \frac{N}{2}\sum_{i=0}^{m}{{{m}\choose{i}}\times m_i^2} \le C_{0}$,
However, unlike the BEC case, we do not know ${a_i}$ at the decoder. Consequently, the MAS should be the same for any received codeword. Here, we resort to the asymptotically optimal MAS, i.e., we try to decode up to the largest bit-level radius for the worst case error pattern. The problem can be viewed as a max-min problem over $\{a_i\}$ and $\{m_i\}$.
$$\label{eqn:min_max} \max_{\{m_i\}}\min_{\{a_i\}} \sum_{i=0}^{m}{a_i m_i}$$
subject to $\sum_{i=0}^{m}{a_i} = N$ where $\{a_i\}$s are integers
and $\frac{N}{2}\sum_{i=0}^{m}{{{m}\choose{i}}\times m_i^2} \le C_{0}$.
The above problem is quite complicated, since $\{a_i\}$s are integers. Even if that condition is relaxed, the solution may only be obtained numerically, which does not give any insight into the exact decoding radius of ASD.
We first take one step back and consider a special case of BSC, called 1-bit flipped BSC, i.e., in each symbol, at most one bit is in error. By doing that, we only have to assign multiplicities to two types of symbols. Suppose there are $e$ symbols in error, $e \le N$ and the score and cost are $S = (N-e)m_0 + e m_1$ and $C =
\frac{N}{2}[m_0^2 + m m_1^2]$. The asymptotically optimal MAS is just to maximize $e$. Suppose $m_1 = t m_0$, where the multiplicity coefficient $0 \le t \le 1$, the sufficient condition in (\[eqn:sufficient\]) becomes:
$$\begin{aligned}
\label{align:sufficient_bsc_a}[(N-e)+e t] m_0 &\ge \sqrt{(K-1)N(1+m t^2)} m_0\\
\label{align:sufficient_bsc_b}e &\le \frac{N-\sqrt{N(K-1)(1+m t^2)}}{1-t}\end{aligned}$$
Let $t = 1$ in (\[align:sufficient\_bsc\_a\]), we have the inequality independent of $e$. The inequality always hold as long as $N \ge (K-1)(1+m)$, which is true for low rate codes, i.e., for a rate
$$\label{eqn:rate} R \le \frac{1}{1+m}+\frac{1}{N}$$
In this case, setting $m_0 = m_1$ will correct all errors in the received signal. On the other hand, for higher rate code, the optimal MAS is to optimize $t$ to maximize the RHS of (\[align:sufficient\_bsc\_b\]).
The problem is equivalent to finding a point on the hyperbola $\frac{y^2}{N(K-1)}-mx^2 = 1$ within interval $x
\in [0, 1)$ such that a line passing through that point and the given point $(1, N)$ has the maximal slope. This is nothing but the tangent to the hyperbola. For the tangential point $(x_0, y_0)$, we have the following relationships:
$$\label{eqn:relation1} \frac{dy}{dx}\mid_{x = x_0} = N(K-1)m\frac{x_0}{y_0} = \frac{N-y_0}{1-x_0}$$
$$\label{eqn:relation2} \frac{y_0^2}{N(K-1)}-mx_0^2 = 1$$
We define $\Delta$ as: $$\label{eqn:delta} \Delta = (m (K-1))^2+(N-K+1)(m^2(K-1)-m N)$$
Plug in $y_0$ into (\[eqn:relation2\]), we can get:
$$\label{eqn:x_0 solution} x_0 = \frac{- m (K-1)+\sqrt{\Delta}}{m^2(K-1)-m N}$$
Using (\[eqn:relation1\]) and (\[eqn:relation2\]), the optimal value of the slope in terms of $x_0$ is:
$$\label{eqn:slope solution} d = \frac{dy}{dx}|_{x = x_0} = \frac{N m}{m+\frac{1}{x_0}}$$
Thus $\lfloor d \rfloor$ is the exact error correction radius of ASD algorithm under asymptotically optimal MAS over 1-bit flipped BSC. Besides, (\[eqn:x\_0 solution\]) can be further approximated for some long and high rate codes. Observe that $(N-K+1)(m^2(K-1)-m N) \ll (m (K-1))^2$ for high rate codes, using first order Taylor expansion, we get:
$$\label{eqn:taylor expansion} \sqrt{\Delta} \doteq m(K-1)[1+\frac{1}{2} \frac{(N-K+1)(m^2(K-1)-m N)}{m^2(K-1)^2}]$$
Plug this into (\[eqn:x\_0 solution\]) and (\[eqn:slope solution\]), we get:
$$\label{eqn:approximation} \tilde{d} = \frac{N(N-K+1)}{N+(K-1)}$$
Note that the approximation becomes tight when rate is high, however, it should be noted that the performance improvement is significant only when the rate is low. For instance, for $N = 255$, $K = 223$ ASD does not improve the performance, for $N = 255$, $K = 167$ ASD gives an extra error correction capability over GS decoding, for $N = 255$, $K = 77$, it corrects 7 more errors and for $N = 255$, $K = 30$, it corrects 45 more errors. For $K < 30$, all errors can be corrected for this 1-bit flipped BSC.
Now, we show that the above MAS is also asymptotically optimal for RS codes over BSC for a wide code range.
\[thm:worst case error pattern\] Given a number of $i\le N$ bits errors and the optimal multiplicity coefficient $t \le \frac{1}{2}$, the worst case error pattern for ASD algorithm is that all erroneous bits are spread in $i$ different symbols.
\[prf:worst case error pattern\] Assume that there are $e$ bits flipped by the channel. The cost for BSC channel does not change when the MAS is fixed. The worst case error pattern is the one that minimizes the score. In the above MAS, multiplicities are assigned only to the received symbol and its 1-bit flipped neighbors. Thus, a potential error pattern which is worse than the 1-bit flipped BSC will try to group bits in each symbol to reduce the score. The original score can be expressed as:
$$\label{eqn:org_score} S = M[(N-e)+t e]$$
Let the worst case error pattern has $e'$ symbols containing 1 bit error and $e''$ symbols containing more than 1-bit error. Evidently, for symbols containing more than 2-bit errors, we can always further decrease the score by splitting these bit errors into one symbol containing 2-bit errors and the other containing the rest of the errors. Consequently, the worst case error pattern will contain symbols with at most 2-bit errors. We have $e'+2e'' = e$. The score becomes:
$$\label{eqn:2bit_score} S'' = M[(N-e'-e'') + t e'] = M[(N-e) + e t + e'' (1-2t)]$$
When $t \le \frac{1}{2}$, $S'' \ge S$, which proves that spreading all the bits in different symbols is the worst case error pattern.
\[thm:bsc\_bound\] For BSC, the asymptotic optimal MAS of ASD algorithm can guarantee to decode up to $d$ bits, where $d$ is computed in (\[eqn:slope solution\]), given $d \le N$ and the optimal multiplicity coefficient $t \le
\frac{1}{2}$.
\[prf:bsc\_bound\] According to lemma \[thm:worst case error pattern\], all bits spread in different symbols are the worst case error pattern for the ASD algorithm, which is nothing but the 1-bit flipped BSC. Thus, they will be the asymptotically dominating error patterns. On the other hand, it is shown before that the proposed MAS is asymptotically optimal for 1-bit flipped BSC, i.e., maximizing the asymptotic decoding radius $d$. Consequently, the proposed MAS will guarantee to decode all error patterns with fewer than $d$-bit errors over BSC as well.
The error correction radii as a function of $t$ are given in Figure \[fig:radius\]. It can be seen that the optimal MAS (which is achieved by $t =0.2$) corrects 13 and 50 more bit errors than GS and BM asymptotically. Besides, we also plot bit-level radius of PMAS, where the x-axis is the crossover probability $p_c$ of the BSC. Note that PMAS is not asymptotically optimal for BSC. Even though we choose $p_c$ to maximize the bit-level radius (around $p_c = 0.13$), the bit-level decoding radius is still 1 bit smaller than that of the optimal MAS. The reason can be explained as follows: the worst case error pattern of BSC is shown to be all bit-level errors spread in different symbols, thus, the asymptotically optimal MAS only has to assign multiplicities to symbols of type $0$ and type $1$. On the other hand, KV algorithm assigns multiplicities proportionally. Thus it also assigns multiplicities to candidate symbols with more than 1-bit flip, which makes it suboptimal in terms of bit-level decoding radius.
![Bit level decoding radius of an RS (255,55) code[]{data-label="fig:radius"}](radius_bw.eps){width="2.5in"}
We consider the performance of this asymptotically optimal MAS in Figure \[fig:rs(15,3)\]. Consider AWGN channel with 1-bit quantization before decoding, which is equivalent to BSC. The ASD under the proposed MAS outperforms GS decoding and HDD by 1.6dB and 3.5dB respectively at an FER = $10^{-5}$.
![Performance Comparison for RS (15,3) code[]{data-label="fig:rs(15,3)"}](rs1503_bw.eps){width="2.5in"}
Performance Analysis of ASD of RS Coded Modulation over Erasure Channels {#sec:rs_modulation}
========================================================================
We extend the result from Section \[sec:rs\_bec\] to RS coded modulation transmitted over erasure channels, i.e. $u$ bits of coded information are grouped together and transmitted using a $2^u$-ary QAM or PSK modulation format. The channel will erase the signal with erasure probability $\epsilon$. Practical channels of this model are discussed in [@wesel_robust].
In Section \[sec:rs\_bec\], we showed that PMAS is optimal for erasure channels. Clearly, all erasure patterns in this coded modulation model is a subset of the erasure patterns of BEC. Thus, PMAS is also optimal for this channel model. Let each PSK (or QAM) symbol contains $u$ bits. Without loss of generality, we assume $u$ divides $m$, i.e., $m = lu$. Thus, for each symbol, we have $(l+1)$ types.
\[thm:worst case modulation\] The worst case erasure pattern for ASD under PMAS is that all erasure events are spread in different symbols as evenly as possible.
\[prf:worst case modulation\] Assume two RS symbols are of type $i$ and $j$, we can average all the erasures events between the two symbols, we have: $$\begin{aligned}
\label{align:eta_rs_mod} \eta' = \eta +2^{-\lfloor \frac{i+j}{2} \rfloor u}+2^{-\lceil \frac{i+j}{2} \rceil
u}-2^{-iu}-2^{-ju} \le \eta\end{aligned}$$ Similar to Lemma \[lem:worst case erasure pattern\], when $\eta$ decreases, spreading erasure events in different RS symbols evenly is the worst case.
\[thm:bounds\_memory\] ASD under PMAS can guarantee to decode up to $(N-K+1)/(1-2^{-u})$ erasure events if $R \ge 2^{-u}+\frac{1}{N}$.
\[prf:bounds\_memory\] According to Lemma \[thm:worst case modulation\], spreading erasure events in different symbols is the worst case erasure pattern if $K \ge 2^{-u}N+1$.
Assume that $e$ symbols of type $1$ are erased and the rest $(N-e)$ symbols are of type $0$. Thus $\eta =
N-(1-2^{-u})e$. According to (\[align:eta\_ieq\]) when the following condition is satisfied: $$\begin{aligned}
\label{align:region} e \le (N-K+1)/(1-2^{-u})\end{aligned}$$ ASD is guaranteed to decode the transmitted codeword.
Note that this asymptotic decoding radius is consistently larger than the conventional RS erasure decoding region, $e = N-K$ erasure events. Note that (\[align:region\]) is a generalization of (\[align:opt\_e\_a\]) in Theorem \[thm:worst case region\] (with $u = 1$ as special case).
Conclusion {#sec:conclusion}
==========
We have presented optimal multiplicity assignment strategies and performance analysis of algebraic soft decision decoding over erasure channels and BSC. It was shown that ASD under optimal MAS can significantly outperform the BM and GS algorithm.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors are grateful to N. Ratnakar and R. Koetter for many insightful discussions. They also thank anonymous reviewers for their constructive comments that greatly improve the quality of this paper.
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V. Guruswami and M. Sudan. Improved decoding of [Reed]{}-[Solomon]{} and algebraic-geometry codes. , 45:1757–1767, Sep. 1999.
M. El. Khamy and R. J. McEliece. Interpolation multiplicity assignment algorithms for algebraic soft-decision decoding of [Reed]{}-[Solomon]{} codes. In [*Proc. ISIT*]{}, Chicago, IL, 2004.
R. Koetter and A. Vardy. Decoding of [Reed]{}-[Solomon]{} codes for additive cost functions. In [*Proc. ISIT*]{}, page 313, Lausanne, Switzerland, Jun. 2002.
R. Koetter and A. Vardy. Algebraic soft-decision decoding of [Reed]{}-[Solomon]{} codes. , 49:2809–2825, Nov. 2003.
N. Ratnakar and R. Koetter. Exponential error bounds for algebraic soft-decision decoding of [Reed]{}-[Solomon]{} codes. , submitted 2003.
C. Retter. The average weight-distance enumerator for binary expansions of [Reed]{}-[Solomon]{} codes. , 48:1195–1200, Mar. 2002.
R. D. Wesel, X. Liu, and W. Shi. Trellis codes for periodic erasures. , 48:938–947, Jun. 2000.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\hull{a}$, $a\in A$ if and only if the set $A$ is independent. In [@DSS] the concept of an [*independent*]{} set in an abelian group was generalized to a [*topologically independent set*]{} in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset $A$ of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups $\hull{a}$, $a\in A$ if and only if the set $A$ is topologically independent and absolutely Cauchy summable. Further, it was shown, that the assumption of absolute Cauchy summability of $A$ can not be removed in general in this result. In our paper we show that it can be removed in precompact groups.
In other words, we prove that if $A$ is a subset of a [*precompact*]{} abelian group, then the topological subgroup generated by $A$ is the Tychonoff direct sum of the topological cyclic subgroups $\hull{a}$, $a\in A$ if and only if $A$ is topologically independent. We show that precompactness can not be replaced by local compactness in this result.
address: |
Department of Mathematics\
Faculty of Science\
J. E. Purkyne University, České mládeže 8, 400 96 Ústí nad Labem\
Czech Republic
author:
- Jan Spěvák
title: Topologically independent sets in precompact groups
---
[*All groups in this paper are assumed to be abelian and all topological groups are assumed to be Hausdorff.*]{} A topological group is [*precompact*]{} if it is a topological subgroup of a compact group. As usually, the symbols ${{\mathbb N}}$ and ${{\mathbb Z}}$ stay for the sets of natural numbers and integers respectively.
Given an abelian group $G$, by $0_G$ we denote the zero element of $G$, and the subscript is omitted when there is no danger of confusion. Given a subset $A$ of $G$, the symbol $\hull{A}$ stays for the subgroup of $G$ generated by $A$. For ${a}\in G$, we use the symbol $\hull{a}$ to denote $\hull{\{a\}}$. Following [@DSS], the symbol $S_A$ stays for the direct sum $$S_A=\bigoplus _{a\in A} \hull{a},$$ and by $\kal{A}$ we denote the unique group homomorphism $$\kal{A}: S_A\to G$$ which extends each natural inclusion map $\hull{a}\to G$ for $a\in A$. As in [@DSS], we call the map $\kal{A}$ the [*Kalton map associated with $A$*]{}.
We say that $\hull{A}$ [*is the direct sum of cyclic groups $\hull{a}$, $a\in A$*]{} provided that the Kalton map $\kal{A}$ is an isomorphic embedding. When $G$ is a topological group, we always consider $\hull{a}$ with the subgroup topology inherited from $G$ and $S_A$ with the subgroup topology inherited from the Tychonoff product $\prod_{a\in A}\hull{a}$. Finally, we say that $\hull{A}$ [*is a Tychonoff direct sum of cyclic groups $\hull{a}$, $a\in A$*]{} if the Kalton map $\kal{A}$ is at the same time an isomorphic embedding and a homeomorphic embedding.
Introduction
============
The concept of compactness allows to transfer some purely non-topological issues into the realm of topology. A nice example of this phenomenon is the paper of Nagao and Shakhmatov (see [@NS]), where the classical, purely combinatorial result of Landau on the existence of kings in finite tournaments, where finite tournament means a finite directed complete graph, is generalized by means of continuous weak selections to continuous tournaments for which the set of players is a compact Hausdorff space. In our paper we provide another example of this phenomenon which non-trivially transfers a result from the area of abelian groups to the realm of precompact abelian groups.
Recall that a subset $A$ of nonzero elements of a group $G$ is [*independent*]{} provided that for every finite set $B\subset A$ and every family $(z_a)_{a\in B}$ of integers the equality $\sum_{a\in B}z_aa=0$ implies $z_aa=0$ for all $a\in B$.
Similarly, a subset $A$ of nonzero elements of a topological group $G$ is [*topologically independent*]{} (see [@DSS Definition 4.1]) provided that for every neighborhood $W$ of $0_G$ there exists neighborhood $U$ of $0_G$ such that for every finite set $B\subset A$ and every family $(z_a)_{a\in B}$ of integers the inclusion $\sum_{a\in B}z_aa\in U$ implies $z_aa\in W$ for all $a\in B$. This neighborhood $U$ is called a [*$W$-witness*]{} of the topological independence of $A$.
One can readily verify that in (Hausdorff) topological groups every topologically independent set is independent (see [@DSS Lemma 4.2]) and that these two notions coincide in discrete groups. Thus topological independence can be viewed as a natural generalization of independence.
Let us recall a basic and simple fact about independent sets.
A set $A$ of nonzero elements of a group is independent if and only if the subgroup generated by $A$ is a direct sum of the cyclic groups $\hull{a}$, $a\in A$.
The aim of this paper is to prove the following counterpart of the above fact. Its proof is postponed to the end of the next section.
\[thm:main\] A set $A$ of nonzero elements of a [precompact]{} group is [topologically]{} independent if and only if the [ topological]{} subgroup generated by $A$ is a Tychonoff direct sum of the cyclic topological groups $\hull{a}$, $a\in A$.
Example \[ex:loc:comp:does:not:work\] demonstrates, that precompactness can not be replaced by local compactness in Theorem \[thm:main\].
In [@DSS] a result closely related to Theorem \[thm:main\] was obtained. In order to state it, recall that by [@DSS Definition 3.1] a subset $A$ of a topological group is [*absolutely Cauchy summable*]{} provided that for every neighborhood $U$ of $0_G$ $$\label{eq:def:of:cauch:summable}
\mbox{there exists finite $F\subset A$ such that }\hull{A\setminus F}\subset U.$$ Let us state the promised result (see [@DSS Theorem 5.1]).
\[fact:basic\] A subset $A$ of nonzero elements of a topological group $G$ is at the same time topologically independent and absolutely Cauchy summable if and only if the [ topological]{} subgroup generated by $A$ is a Tychonoff direct sum of the cyclic topological groups $\hull{a}$, $a\in A$.
In view of Fact \[fact:basic\] we can see that Theorem \[thm:main\] reads as follows: [*Every topologically independent set in a precompact group is absolutely Cauchy summable*]{}. Let us note, that absolutely Cauchy summable sets can be far away from topologically independent sets even in the realm of compact groups. Indeed, by [@DSS Remark 5.3], every null sequence in the compact metric group ${{\mathbb Z}}_p$ of $p$-adic integers is absolutely Cauchy summable while every topologically independent subset of ${{\mathbb Z}}_p$ is a singleton.
The “only if” part of Fact \[fact:basic\] is based on two straightforward results. First one states that absolute Cauchy summability of a set $A$ is equivalent to the continuity of the Kalton map $\kal{A}$ ([@DSS Theorem 3.5]), while the second says that topological independence of a set $A$ implies that the Kalton map $\kal{A}:S_A\to\hull{A}$ is an open isomorphism ([@DSS Lemma 4.2 and Proposition 4.7 (i)]). One may ask, whether the implication in the latter statement can be reversed:
\[Q1\] Let $A$ be a subset of nonzero elements of a topological group $G$ such that the Kalton map $\kal{A}:S_A\to \hull{A}$ is an open isomorphism. Must $A$ be topologically independent?
Example \[ex:open:iso:not:enough\] provides a negative answer to this question even in the case, when $G$ is precompact.
Precompact group topologies on direct sums
==========================================
By ${{\mathbb{S}}}$ we denote the unit circle in the complex plain which is a compact group with respect to multiplication of complex numbers.
Given a group $G$ we denote by $G'$ its character group (the group of all homomorphisms from $G$ to ${{\mathbb{S}}}$). Elements of $G'$ are called [*characters*]{}. If $G$ is a topological group, then the symbol $G^*$ stays for the subgroup of $G'$ consisting of all continuous characters.
Let $H$ be a group of characters on $G$. We denote by $T_H$ the coarsest group topology on $G$ making all characters of $H$ continuous. Since the collection $\{V_\delta:\delta\in(0;\pi)\}$, where $$V_\delta=\{e^{it}:t\in(-\delta,\delta)\},$$ is a local base of the topology of ${{\mathbb{S}}}$ at the identity, the topology $T_H$ has a local base at $0_G$ consisting of all the sets of the form \#1\#2[(\_1,…,\_\#1;\#2)]{} $$\U{n}{\delta}=\{g\in G:\chi_i(g)\in V_\delta \mbox{ for all } i=1,\ldots,n\},$$ where $\delta\in(0;\pi)$, $n\in{{\mathbb N}}$ and $\chi_1,\ldots,\chi_n\in H$. The topology is Hausdorff whenever the characters of $H$ separate points of $G$.
Let us recall a basic fact about the topology $T_H$ for precompact groups (see [@DPS Theorem 2.3.2]).
\[fact:precompact:top\] Let $G$ be a topological group. Then the topology of $G$ is precompact if and only if it is equal to $T_{G^*}$ and the characters of $G^*$ separate points of $G$.
Given a subset $A$ of a group $G$ and a character $\chi\in G'$ we define the [*$A$-support*]{} of $\chi$ by the formula $$supp_A(\chi)=\{a\in A: \chi(a)\neq 1\}.$$ We say that the character $\chi$ is [*finitely $A$-supported*]{} provided that $supp_A(\chi)$ is finite.
\[lemma:basic\] Let $A$ be an infinite subset of a compact group $G$, and $V$ an open neighborhood of the identity element of $G$. Then there are distinct $a,b\in A$ such that $ab^{-1}\in V$.
Let $U$ be an open neighborhood of the identity element of $G$ such that $U^{-1}U\subset V$. Since $G$ is compact, there are $a,b\in A$ such that $$\label{eq:lemma}
Ua\cap Ub\neq\emptyset.$$ Otherwise $A$ would be an infinite closed discrete subset of a compact group $G$ - a contradiction. Now, yields $ab^{-1}\in U^{-1}U\subset V$.
The following lemma is obvious.
\[lemma:not:top:indep\] Let $A$ be a subset of a topological group $G$ and $\mathcal{V}$ a fixed local base at $0_G$ of the topology of $G$. Assume that $W$ is a neighborhood of $0_G$ such that no element of $\mathcal{V}$ is a $W$-witness of the topological independence of $A$. Then $A$ is not topologically independent.
Our next proposition plays the key role in the proof of Theorem \[thm:main\].
\[prop:finitely:supported:characters\] Let $A$ be a topologically independent subset of a precompact group $G$. Then each character of $G^*$ is finitely $A$-supported.
We will prove the contrapositive. In order to do so, let $supp_A(\chi)$ be infinite for some $\chi\in G^*$. Put $W=\mathcal{U}(\chi;\frac{\pi}{2})$ and pick $n\in{{\mathbb N}}$, $\chi_1,\ldots,\chi_n\in H$ and $\delta\in(0;\pi)$ arbitrarily. By Lemma \[lemma:not:top:indep\], it suffices to show that $\U{n}{\delta}$ is not a $W$-witness of the topological independence of $A$.
For every $a\in supp_A(\chi)$ we have $\chi(a)\neq 1$. Therefore, we can find $z_a\in{{\mathbb Z}}$ such that $\chi(z_aa)=\chi(a)^{z_a}\not\in V_\frac{\pi}{2}$. Consequently, $$\label{zaanotinW}
z_aa\not\in W \mbox{ for all } a\in supp_A(\chi).$$ To finish the proof, it remains to find distinct $a,b\in supp_A(\chi)\subset A$ such that $$\label{eq:neco}
z_aa-z_bb\in\U{n}{\delta}.$$ Indeed, in this case $\U{n}{\delta}$ is not a $W$-witness of the topological independence of $A$ by . If there are distinct $a,b\in supp_A(\chi)$ such that $\chi_i(z_aa)=\chi_i(z_bb)$ for all $i=1,\ldots,n$, then holds. Otherwise the set $\{s_a:a\in supp_A(\chi)\}$, where $$s_a=(\chi_1(z_aa),\ldots,\chi_n(z_aa)),$$ is an infinite subset of the compact group ${{\mathbb{S}}}^n$. The set $V=\underbrace{V_{{\delta}}\times\ldots\times V_{{\delta}}}_n$ is an open neighborhood of the identity element $(\underbrace{1,\ldots,1}_n)$ of ${{\mathbb{S}}}^n$. Thus, by Lemma \[lemma:basic\], there are distinct $a,b\in supp_A(\chi)$ such that $s_as_b^{-1}\in V$. Therefore, $$\chi_i(z_aa-z_bb)\in V_\delta \mbox{ for all } i=1,\ldots,n.$$ This yields and finishes the proof.
We omit the straightforward proof of the next simple lemma.
\[lemma:subbase:suffices\] Let $A$ be a subset of a topological group $G$ and $\mathcal{V}$ a local subbase at $0_G$ of the topology of $G$. Then $A$ is absolutely Cauchy summable if and only if holds for every $U\in\mathcal{V}$.
\[proposition:top:ind:is:acs\] Let $A$ be a topologically independent subset of a precompact group $G$. Then $A$ is absolutely Cauchy summable.
By Fact \[fact:precompact:top\], the set $\{\mathcal{U}(\chi;\delta):\chi\in G^*,\delta\in(0;\pi)\}$ is a local subbase at $0_G$ of the topology of $G$. Fix arbitrary $\delta\in(0;\pi)$, $\chi\in G^*$, and put $U=\mathcal{U}(\chi;\delta)$. By Lemma \[lemma:subbase:suffices\], it suffices to show . Put $F=supp_A(\chi)$. Then $F$ is finite by Proposition \[prop:finitely:supported:characters\]. Observe that is satisfied.
Now we are in position to prove Theorem \[thm:main\]:
The proof follows immediately from Proposition \[proposition:top:ind:is:acs\] and Fact \[fact:basic\].
Examples and remarks
====================
\[ex:loc:comp:does:not:work\] [*Let $A$ be an arbitrary infinite independent subset of a discrete (in particular, locally compact) group $G$. Then $A$ is topologically independent but the topological subgroup $\hull{A}$ is never a Tychonoff direct sum of cyclic topological subgroups $\hull{a}$, $a\in A$.*]{} Indeed, as was noted in the Introduction, independent sets coincide with topologically independent sets in discrete groups. On the other hand a Tychonoff direct sum of infinitely many non-trivial topological groups is always non-discrete while every topological subgroup of a discrete group is discrete.
Recall that a topological group is [*compactly generated*]{} if it contains a compact subset $A$ such that $\hull{A}$ is the whole group. Let us note that the discrete group $G$ from Example \[ex:loc:comp:does:not:work\] is not compactly generated as it contains infinite independent subset and consequently each its generating set is infinite and discrete (thus non-compact). We do not know whether there is an infinite topologically independent subset $A$ of a locally compact compactly generated group $G$ such that $\hull{A}$ is not a Tychonoff direct sum of the cyclic groups $\hull{a}$ $a\in A$. In other words, we do not know whether the word “precompact” can be replaced by “locally compact compactly generated” in Theorem \[thm:main\].
Our next example provides a negative answer to Question \[Q1\].
\[ex:open:iso:not:enough\] [*There exists a subset $A$ of a precompact group such that the Kalton map $\kal{A}:S_A\to\hull{A}$ is an open isomorphism, but $A$ is not topologically independent.*]{} To show this, let $A$ be an arbitrary infinite topologically independent subset of a precompact group $G$ and let $\tau$ denote the topology of $G$. Since subgroups of precompact groups are precompact, we may and will assume that $G=\hull{A}$. By Theorem \[thm:main\],
$$\label{kal:je:iso}
\kal{A}:S_A\to (G,\tau) \mbox{ is an isomorphism and a homeomorphism. }$$
By Fact \[fact:precompact:top\], characters of $(G,\tau)^*$ separate points of $G$. Therefore, for each $a\in A$ we may pick $\chi_a\in (G,\tau)^*$ such that $\chi_a(a)\neq 1$. The set $A$ is independent as it is topologically independent in $(G,\tau)$. Hence there is (unique) $\chi\in G'$ such that $$\label{eq:chia:chi}
\mbox{
$\chi(a)=\chi_a(a)\neq 1$ for all $a\in A$.}$$ Consider the subgroup $H$ of $G'$ generated by $(G,\tau)^*\cup\{\chi\}$. Then $A$ as a subset of $(G,T_H)$ is as required.
Indeed, the topology $T_H$ is precompact by Fact \[fact:precompact:top\] (it separates points as $(G,\tau)^*$ does). Further, since $A$ is infinite, the character $\chi$ is not finitely $A$-supported by . Thus the set $A$ is not topologically independent in $(G,T_H)$ by Proposition \[prop:finitely:supported:characters\].
Finally, it follows from that each $\hull{a}$ has the same subgroup topology in the topology $T_H$ as in $\tau$. Therefore, $S_A$ as the Tychonoff direct sum has the same topology with respect to $T_H$ as well as with respect to $\tau$. Since $T_H$ is finer then $\tau$, it follows from that the Kalton map $$\kal{A}:S_A\to (G,T_H)$$ is an open isomorphism.
Call a group $G$ monothetic if it has a dense cyclic subgroup $C$. Every generator of $C$ is called a [*topological generator*]{} of $G$.
\[dense:means:not:top:indep\] If $g$ is a topological generator of a topological group $G$, then the singleton $\{g\}$ is a maximal (with respect to inclusion) topologically independent subset of $G$.
Let $g,h$ be topologically independent elements of $G$. Since finite sets are absolutely Cauchy summable, Fact \[fact:basic\] gives us that $\hull{\{g,h\}}$ is the Tychonoff direct sum $\hull{g}\bigoplus\hull{h}$. Hence $\hull{g}$ is not dense in $\hull{\{g,h\}}$ and consequently it is not dense in $G$ as well. Thus $g$ is not a topological generator of $G$.
Our last and simple example provides a warning showing that unlike in Theorem \[thm:main\], some other properties that hold for independent sets may not be transfered to the realm of topologically independent sets in precompact groups.
[*Obviously, if $a,b, c$ are distinct non-torsion elements of a group such that the set $\{a,c\}$ as well as the set $\{b,c\}$ is not independent, then also the set $\{a,b\}$ is not independent. On the other hand, there exist a compact topological group $G$ and non-torsion elements $a,b,c\in G$ such that the set $\{a,c\}$ as well as the set $\{b,c\}$ is not topologically independent while the set $\{a,b\}$ is topologically independent.*]{}
Put $G={{\mathbb{S}}}^2$, and $a=[g,1], b=[1,g]$, where $g\in{{\mathbb{S}}}$ is non-torsion. The set $\{a,b\}$ is topologically independent by Fact \[fact:basic\]. It is a folklore fact, that ${{\mathbb{S}}}^2$ is monothetic. Let $c\in{{\mathbb{S}}}^2$ be its topological generator. Then $c$ is non-torsion and the set $\{a,c\}$ as well as the set $\{b,c\}$ is not topologically independent by Lemma \[dense:means:not:top:indep\].
[99]{}
D. Dikranjan, Iv. Prodanov and L. Stoyanov, *Topological Groups: Characters, Dualities and Minimal Group Topologies*, Pure and Applied Mathematics, vol. **130**, Marcel Dekker Inc., New York-Basel (1989).
D. Dikranjan, D. Shakhmatov and J. Spěvák, [*Direct sums and products in topological groups and vector spaces*]{}, J. Math. Anal. Appl. 437 (2016) 1257-1282.
M. Nagao, D. Shakhmatov, [*On the existence of kings in continuous tournaments*]{}, Topol. Appl. 159 (2012) 3089–3096.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We discuss a mechanism by which a set of $q\bar q$-states mixes with an exotic one, so that the exotic state accumulates the widths of the overlapping $\bar qq$ resonances. The broad state created this way acts as a ’locking’ state for the others. Using results of a previous analysis of the $(IJ^{PC}=00^{++})$-wave, we estimate the mean radius squared of the broad state $f_0 (1530^{+90}_{-250})$; it appears to be distinctly larger than the mean squared radii of the locked narrow states. This supports an idea about the constructive role of broad states in forming the confinement barrier.'
address:
- 'St.Petersburg Nuclear Physics Institute, Gatchina, 188350, Russia'
- 'Queen Mary and Westfield College, London E14NS, UK'
- 'St.Petersburg Nuclear Physics Institute, Gatchina, 188350, Russia'
author:
- 'V. V. Anisovich[^1]'
- 'D. V. Bugg[^2]'
- 'A. V. Sarantsev [^3]'
title: 'Exotic mesons, locking states and their role in the formation of the confinement barrier '
---
=23.7cm
The objective of this letter is to introduce a qualitatively new idea having two components. The idea is that one resonance which overlaps other resonances may become broad via mixing and may in turn reduce the widths of the others; it thus acts as a confinement barrier for narrow states. We call the broad state a ‘locking’ state. We draw attention to experimental evidence for this idea and suggest that formation of broad states may be a general phenomenon involving exotic states.
[**Effect of accumulation of widths in the $K$-matrix approach.**]{} To examine the mixing of non-stable states in a pure form, let us consider an example with three resonances decaying into the same channel. In the $K$-matrix approach, the amplitude we consider reads: $$A = K(1-i\rho K)^{-1}\ ,\;
K = g^2\sum_{a=1,2,3}(M^2_a-s)^{-1}\ .$$ Here, for purposes of illustration, we take $g^2$ to be the same for all three resonances, and make the approximations that: (i) the phase space factor $\rho$ is constant, and (ii) $M^2_1=m^2-\delta$, $M^2_2=m^2$, $M^2_3=m^2+\delta$. Fig. 1 shows the location of poles in the complex-$M$ plane $(M=\sqrt s)$ as the coupling $g$ increases. At large $g$, which corresponds to a strong overlapping of the resonances, one resonance accumulates the widths of the others while two counterparts of the broad state become nearly stable.
[**Resonance structure in the $(IJ^{PC}=00^{++})$-wave.** ]{} This simple model may be compared with what is known of the actual situation for the $\pi \pi $ S-wave. The analysis of this wave has been pursued in a series of papers using a variety of technical approaches: the $T$-matrix [@bsz] and multichannel $K$-matrix approaches [@km1900] and the dispersion relation $N/D$-method [@n/d]. The most recent $K$-matrix analysis was done using data from GAMS [@gams], the Crystal Barrel collaboration [@cbc] and the BNL group [@bnl]. It allows us to determine resonance structure in this wave in the mass range up to 1900 MeV. There are five states in this mass region. Four of them may be identified as $q\bar q$ states, members of $1^3 P_0 q\bar q$ and $2^3 P_0 q\bar q$ nonets; the fifth state in the mass region 1250 - 1650 MeV is an extra one not accomodated by $q\bar q$ systematics. The restored coupling constants for decays into channels $\pi\pi$, $K\bar K$, $\eta\eta$ and $\eta\eta '$ show that the extra state has at least a 40% - 50% admixture from the lightest scalar glueball.
Parameters found in [@km1900] for the $K$-matrix $0^{++}$ amplitude allow a direct comparison of the mixing dynamics with the simple model of Fig. 1. We introduce a scaling parameter $\xi $ into the $K$-matrix elements, $$K_{ab}=\sum_\alpha \frac{g^{(\alpha)}_a g^{(\alpha)}_b}{
M^2_\alpha-s}+f_{ab} \to \sum\xi
\frac{g^{(\alpha)}_ag^{(\alpha)}_b}{M^2_\alpha-s}+\xi f_{ab}\ ,$$ and vary $\xi$ in the interval 0 to 1. Figs. 2a and 2b show the movement of the poles for solution I found in ref. [@km1900]. As $\xi \to 0$, the decay processes and corresponding mixing are switched off; the positions of $K$-matrix poles show the masses of the bare states: $f_0^{bare}(720 \pm 10)$, $f_0^{bare}(1230 \pm 50)$, $f_0^{bare}(1260 \pm 30)$, $f_0^{bare}(1600 \pm 50)$ and $f_0^{bare}(1810 \pm 30)$. In solution I, the gluonium state is associated with $f_0^{bare}(1230 \pm 50)$ (it is $f_0^{bare}(1600 \pm 50)$ for solution II). The case $\xi =1$ shows the pole positions for the physical states $f_0(980)$, $f_0(1300)$, $f_0(1500)$, $f_0(1530{+90 \atop-250})$ and $f_0(1750)$. In both solutions, according to [@n/d], the broad resonance $f_0(1530{+90 \atop-250})$ carries roughly $40\%$ - $50\%$ of the gluonium component. This should not come as a surprise. Stationary $q\bar q$ states are orthogonal to one another (though the loop diagrams involving their decay cause some mixing), while the glueball may mix freely with $q\bar q$ states; the rules of the $1/N$-expansion [@1/n] tell us that $q\bar q$/gluonium mixing is not suppressed, see [@n/d; @ufn]. That is the reason for an accumulation of the widths of the neighbouring $q\bar q$ states by the gluonium.
[**Broad resonance as a locking state.**]{} We now discuss the role of the broad resonance in the formation of the confinement barrier. Let us consider as a guide the meson spectrum in the standard quark model with decay processes switched off (Fig.3(a)). Here we have a set of stable levels. If the decay processes are incorporated into highly excited states it would be naive to think that the decay processes result only in broadening of levels. Due to processes $bare\; state\to real\; mesons$, the resonances mix and one of them may transform into the very broad state. This creates a trap for the states with which it overlaps.
Fig. 3(a) shows the conventional potential model with a linear potential rising indefinitely. That is unrealistic, since it does not allow decays. In Fig. 3(b) we show instead a confinement barrier through which states may decay, thus creating a broad locking state. The broad resonance prevents decay of other states, which are left in the small-$r$ region, the broad state plays the role of a dynamical barrier. This is a familiar phenomenon: an absorption process acts as a reflecting barrier. It means that comparatively narrow locked states lie inside the confinement well, while the broad locking state appears mainly outside the well. Experimental data confirm this idea.
Direct experimental evidence that the broad $0^{++}$ state is associated with large $r$ is given by the GAMS data on $\pi ^- \pi ^+
(t) \to \pi ^0 \pi ^0$ [@gams]. The broad state is clearly visible at $|t| < 0.2$ GeV$^2$, but disappears at large $|t|$, leaving peaks related to narrower states clearly visible. This effect is demonstrated in Fig. 4, which depicts $|A_{\pi \pi (t) \to
\pi\pi}(M)|^2$ found in [@km1900] for different $t$.
By fitting these data we determine the ratios of $t$ -distributions for narrow resonances and background. We find the ratios: $$\begin{aligned}
\frac{d\sigma_{\pi \to f_0(980)}(t) }{dt}&/&
\frac{d\sigma_{\pi \to f_0(1530{+90 \atop-250})}(t)}{dt}\ ,
\nonumber \\
\frac{d\sigma_{\pi \to f_0(1300)}(t) }{dt}&/&
\frac{d\sigma_{\pi \to f_0(1530{+90 \atop-250})}(t) }{dt}\end{aligned}$$ as functions of $t$. Assuming an universal $t$-exchange mechanism for all $f_0$ mesons, these ratios are given by ratios of the transition form factors squared: $$\begin{aligned}
F^2_{\pi \to f_0(980)}(t)&/&F^2_{\pi\to f_0(1530{+90\atop-250})}(t)
\ , \nonumber \\
F^2_{\pi\to f_0(1300)}(t)&/&F^2_{\pi\to f_0(1530{+90\atop-250})}(t)\end{aligned}$$
A method for calculating transition form factors at small and moderate momentum transfers is given in [@ff]. Following it we estimate mean radii squared of the $f_0$ -mesons using a simple parametrisation of the wave functions in an exponential form (exponential approximation works sufficiently well for wave functions of mesons in the mass below 1500 MeV). Another simplification: in the triangle diagram which is responsible for the transition form factor at moderately small $|t|$, we approximate the light cone variable energy squared as $s=\frac{m^2 +
k^2_{\perp}}{x(1-x)} \simeq 4(m^2 + k^2_{\perp})$ (at small $|t|$ the region $x \simeq \frac{1}{2} $ gives the main contribution). Then $$\begin{aligned}
&&\frac{d}{dt}
\ln[\frac{d\sigma_{\pi \to f_0}(t) }{dt}/
\frac{d\sigma_{\pi \to f_0(1530{+90 \atop-250})}(t)}{dt}]
\nonumber \\
=&&\frac{d}{dt} \ln[F^2_{\pi \to f_0}(t) /
F^2_{\pi \to f_0(1530{+90 \atop-250})}(t)]
\nonumber \\
=&&\frac{1}{3}
(R^2_{\pi \to f_0} - R^2_{\pi \to f_0(1530{+90 \atop-250})} )\ ,\end{aligned}$$ where the transition radius squared, $R^2_{\pi \to f_0}$, is determined as $$R_{\pi \to f_0}^2 =
\frac{2R_{\pi}^2 R_{f_0}^2 }{\frac{5}{3} R_{\pi}^2 + R_{f_0}^2 }\ ;$$ $R_{\pi}^2$, $ R_{f_0}^2 $ are pion and $f_0$-meson radii squared. Constituent quarks of the pion and $f_0$ meson are correspondingly in $S$- and $P$-waves: that results in the factor $5/3$ in the denominaror of Eq. (6).
A fit to data for the ratios of Eq. (3) gives: $$R^2_{\pi \to f_0(980)} \simeq R^2_{\pi \to f_0(1300)}, \; \;
\mbox{or} \; \;
R^2_{ f_0(980)} \simeq R^2_{ f_0(1300)} ,$$ and $$R^2_{\pi \to f_0(1530{+90 \atop-250})} -
R^2_{\pi \to f_0(980)} \simeq (8 \pm 2) \mbox{GeV}^{-2} \; .$$ The last equality defines the correlation between $R^2_{ f_0(1530{+90
\atop-250})} $ and $R^2_{f_0(980)}$, provided the mean pion radius squared is fixed. Fig. 5 demonstrates this correlation for $R^2_\pi
= 0.41$ fm$^2$. It is clearly seen that $R^2_{ f_0(1530{+90
\atop-250})} $ is distinctly larger than $R^2_{f_0(980)}$ or $R^2_{f_0(1300)}$: it means that a broad state at large $r$ can definitely play the role of the locking state.
The same effect may take place for other exotic hadrons. By this we mean glueballs and hybrids with different quantum numbers. One example is now known for $J^P = 0^{-+} $ [@Zerominus], where a very broad $0^{-+}$ state around 1800-2100 MeV ($\Gamma \simeq 1$ GeV) has been identified in radiative $J/\Psi $ decays. Branching ratios for this state to $\rho \rho$, $\omega \omega$, $K^*K^*$ and $\phi \phi$ channels are in approximate agreement with flavour-blindness, suggesting once again a strong glueball component. The paper [@Zerominus] also advances arguments for a broad $2^{++}$ resonance at 2000-2400 MeV; we have been involved in the analyses of two sets of experimental data, to be published shortly, providing evidence for this broad $2^{++}$ state.
[**Conclusion.**]{} In the deconfinement of quarks of an exited $q\bar q$-level, there are two stages:\
(i) An inevitable creation of new quark-antiquark pairs which result in production of white hadrons. This stage is the subject of QCD and is beyond our present discussion.\
(ii) The outflow of the created white hadrons and their mixing results in the production of a very broad state. The broad resonance locks other $q\bar q$-levels into the small-$r$ region, thus playing the role of a dynamical barrier; this is the reason for calling the broad resonance a locking state.
The bare states are the subjects of the quark/gluon classification. Exotic hadrons like glueballs and hybrids mix readily with $q\bar q$ states and are good candidates to generate locking states in all waves.
Rich physics is hidden in the broad states, and an investigation of them is an important and unavoidable step in understanding the spectroscopy of highly exited states and their confinement.
[**Acknowledgements.**]{} We are grateful to Valeri Markushin and Victor Nikonov for useful discussions and comments. This work was supported by INTAS-RFBR grant 95-0267.
D.V. Bugg, A.V. Sarantsev, and B.S. Zou, Nucl. Phys. [**B471**]{} (1996) 59. V.V. Anisovich, Yu.D. Prokoshkin, and A.V. Sarantsev, Phys. Lett. [**B389**]{} (1997) 388. A.V. Anisovich, V.V. Anisovich, Yu.D. Prokoshkin, and A.V. Sarantsev, Z. Phys. [**A357**]{} (1997) 123 ;\
A.V. Anisovich, V.V. Anisovich, and A.V. Sarantsev, Phys. Lett. [**B395**]{} (1997) 123; Z. Phys. A359 (1997) 173. D. Alde et al., Z. Phys. [**C66**]{} (1995) 375;\
Yu.D. Prokoshkin et al., Physics-Doklady [**342**]{} (1995) 473;\
F. Binon et al., Nuovo Cim. [**A78**]{} (1983) 313; [**A80**]{} (1984) 363. V.V. Anisovich et al., Phys. Lett. [**B323**]{} (1994) 233;\
C. Amsler et al., Phys. Lett. [**B333**]{} (1994) 277. S.J. Lindenbaum and R.S. Longacre, Phys. Lett. [**B274**]{} (1992) 492;\
A. Etkin et al., Phys. Rev. [**D25**]{} (1982) 1786. G. t’Hooft, Nucl. Phys. [**B72**]{} (1974) 461;\
G. Veneziano, Nucl. Phys. [**B117**]{} (1976) 519. V.V. Anisovich, Physics-Uspekhi, [**38**]{} (1995) 1179. V.V. Anisovich, D.I. Melikhov, and V.A. Nikonov, Phys. Rev. [**D55**]{} 2918 (1997). D.V. Bugg and B.S. Zou, Phys. Lett. [**B396**]{} (1997) 295.
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected] and [email protected]
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
Sébastien Bubeck\
Microsoft Research
- |
Michael B. Cohen [^1]\
MIT
- |
Yuanzhi Li\
Princeton University
bibliography:
- 'newbib.bib'
title: 'Sparsity, variance and curvature in multi-armed bandits'
---
Introduction
============
In this paper we resolve several open problems in multi-armed bandit theory. Let us first recall the general setting of bandit linear optimization on a compact set $\cK \subset \R^n$ (the classical multi-armed bandit problem corresponds to $\cK=\{e_1,\hdots,e_n\}$, the canonical basis in $\R^n$). It can be described as the following sequential game: at each time step $t=1, \hdots, T$, a player selects an action $a_t \in \cK$, and simultaneously an adversary selects a linear loss function $\ell_t : \cK \rightarrow [-1,1]$. The player’s feedback is its suffered loss, $\ell_t(a_t)$. Equivalently we will view the loss function $\ell_t$ as a vector in the polar body $\cK^{\circ} := \{h : \forall x \in \cK, |h \cdot x| \leq 1\}$, and thus we write $\ell_t(x) = \ell_t \cdot x$. The player has access to external randomness, and can select her action $a_t$ based on the history $H_t=(a_s, \ell_s(a_s))_{s<t}$. The player’s perfomance at the end of the game is measured through the [*pseudo-regret*]{} (the expectation is with respect to the randomness in her strategy) : $$\label{eq:regret}
R_T = \E \sum_{t=1}^T \ell_t(a_t) - \min_{x \in \cK}\E \sum_{t=1}^T \ell_t(x) ,$$ which compares her cumulative loss to the smallest cumulative loss she could have obtained had she known the sequence of loss functions. We refer to [@BC12] for the history of this problem, and we simply mention that the minimax rate for the regret is known to be $\tilde{\Theta}(n \sqrt{T})$ without further assumptions on $\cK$, and for the special case where $\cK=\{e_1,\hdots,e_n\}$ (i.e., the multi-armed bandit problem) it is $\Theta(\sqrt{n T})$.
We consider three basic open problems in bandit theory (description below), each one part of a more general trend in learning theory/online learning, namely (i) exploiting sparsity, (ii) faster learning for “easy data", and (iii) interplay between curvature and learning[^2]. In fact these problems are possibly the easiest at the intersection of bandit theory and topics (i), (ii), (iii). Thus, given the flurry of activity on these topics and on bandit theory in recent years, we believe that they epitomize the difficulty of adapting full information tools to limited feedback scenarios. In particular we hope that the tools we develop to resolve these problems will find broader applicability.
**Sparse multi-armed bandit, [@KP16].** Consider the multi-armed bandit problem with the additional assumption that at each time step $t \in [T]$ the loss vector $\ell_t \in [-1,1]^n$ only has $s$ non-zero entries. Trivially the best regret one can hope for in this setting is $\Omega(\sqrt{s T})$. Kwon and Perchet ask whether there is a strategy with regret matching this lower bound (possibly up to logarithmic factors). Surprisingly the state of the art for this problem is the standard $O(\sqrt{n T})$ bound, or in other words prior to this present work it was not known whether sparsity of the losses can be exploited in a bandit setting[^3].
**Small variation bound for multi-armed bandit, [@HK09].** Consider again the multi-armed bandit problem with the additional assumption that the loss sequence $(\ell_1, \hdots, \ell_T) \in ([-1,1]^n)^T$ has a [*small variation*]{} $Q:= \sum_{t=1}^T \|\ell_t - \frac1{T} \sum_{s=1}^T \ell_s\|_2^2$ (note that $Q \leq n T$). The COLT 2011 open problem by Hazan and Kale ask whether there exists a strategy with regret $\tilde{O}(\sqrt{Q})$ ([@HK11]). The current state of the art remains [@HK09] which gives a strategy with regret $\tilde{O}(n^2 \sqrt{Q})$. We also note that [@GL16] showed that for any fixed $Q > \log(T)$ one cannot obtain a regret smaller than $\Omega(\sqrt{Q})$ for all sequences with variation $Q$.
**Linear bandit on $\ell_p^n$ balls, [@BCK12].** Consider the linear bandit problem on $\cK= \{x \in \R^n : \|x\|_p \leq 1\}$. The general minimax rate show that for any $p \geq 1$ there exists a strategy with regret $\tilde{O}(n \sqrt{T})$, and furthermore this is optimal for $p=\infty$. It is easy to see that for $p=1$ the problem can be reduced to the classical multi-armed bandit (in dimension $2n$) and thus there exists a strategy with regret $\tilde{O}(\sqrt{n T})$. In [@BCK12] it is shown that the latter regret can also be achieved for $p=2$. No other result is known for this problem, and a natural conjecture[^4] would be that $\tilde{O}(\sqrt{n T})$ is achievable for any $p \in [1,2]$, and that the minimax regret then degrades “smoothly” for $p>2$ until $\tilde{\Omega}(n \sqrt{T})$ for $p=\infty$.
We resolve all the above problems, constructing strategies with respective regret bounds $\tilde{O}(\sqrt{s T})$, $\tilde{O}(\sqrt{Q})$, and $\tilde{O}(\sqrt{n T})$ for $p \in [1,2]$. Furthermore we show that in fact for $p>2$ the minimax regret (for large $T$) is $\tilde{\Theta}(n \sqrt{T})$. We also introduce the following more constrained version of bandit linear optimization, which we call [*starved bandit*]{}. In this model the player only observes feedback if she plays $a_t$ from a [*fixed*]{} distribution $\mu \in \Delta(\cK)$, where $\mu$ is chosen by the player at the beginning of the game. Thus the player is “information starved”. One can motivate such a setting in various ways, think for instance of applications where logging information on users is discouraged for privacy reasons. It is easy to see that one must have regret $\Omega(T^{2/3})$ for the starved multi-armed bandit game, and that the same lower bound also applies to starved linear bandit on $\ell_p^n$ unit ball with $p=1$. Perhaps surprisingly we show that $\sqrt{T}$-type regret is achievable for the starved bandit for any $p \in (1,2]$ and [*not*]{} achievable for any $p>2$.
A key feature of our work that enables these improved regret bounds is that we avoid resorting to “global” smoothness of the regularizers. Slightly more precisely, as we will recall shortly, an important step in the analysis of FTRL (Follow The Regularized Leader) is to show that the regularizer is well-conditioned. Since the groundbreaking work [@AHR08] it has been realized that self-concordance ([@NN94]) exactly gives such a good conditioning [*for all directions*]{}. In this paper we use more refined properties of the regularizers, by noticing that one only needs the well-conditioning in directions (and magnitudes) [*attainable with loss estimators*]{}.
Next we describe more formally our main results.
Main results
------------
The brief algorithms’ description given in the theorem statements below use standard bandit theory terminology which is recalled in Section \[sec:reminders\]. Note also that in this paper we assume that the parameters of the game (such as the time horizon $T$, or the variation of the loss sequence) are known. Standard methodology (such as the doubling trick, or more sophisticated variants of it) can be used to circumvent this issue.
We start with a theorem resolving the sparse bandit open problem by Kwon and Perchet (notice that if $\|\ell_t\|_0 \leq s$ and $\|\ell_t\|_{\infty}\leq1$ then $\sum_{t=1}^T \|\ell_t\|_2^2 \leq s T$).
\[th:sparse\] There exists a multi-armed bandit strategy such that for any loss sequence satisfying $\sum_{t=1}^T \|\ell_t\|_2^2 \leq L$ (and $\ell_t \in [-1,1]^n$) one has $$R_T \leq 10 \sqrt{L \log(n)} + 20 n \log(T) ~.$$
In fact this can be achieved with the FTRL strategy (with standard unbiased loss estimator) with the regularizer $\Phi(x) = \sum_{i=1}^n x(i) \log x(i) - \gamma \sum_{i=1}^n \log x(i)$, learning rate $\eta = \min\left(\frac1{5} \sqrt{\frac{\log(T)}{L}}, \frac{1}{15 n}\right)$, and soft-exploration parameter $\gamma = 2 \eta$.
The difficulty in achieving a result such as Theorem \[th:sparse\] is that standard multi-armed bandit algorithms [*explore too much*]{}. In fact as was noted in [@HK11] for the variation bound open problem (the same observation holds for the sparse bound open problem): “We note that EXP3 itself has $\Omega(\sqrt{T})$ regret, since it mixes with the uniform distribution every iteration to enable sufficient exploration. Hence, the desired algorithm should be a little different from EXP3, incorporating just enough exploration proportional to the variation in the data.” Our new idea to achieve this is to introduce [*soft exploration*]{}, by adding to the regularizer a little bit of the log-barrier for the positive orthant. This new hybrid regularizer and its analysis is one of our key contribution. We give detailed intuition for it in Section \[sec:intuition\]. It also allows to solve the variation bound open problem:
\[th:variance\] There exists a multi-armed bandit strategy and a numerical constant $C>0$ such that for any loss sequence satisfying $\sum_{t=1}^T \|\ell_t - \frac1{T} \sum_{s=1}^T \ell_s\|_2^2 \leq Q$ (and $\ell_t \in [-1,1]^n$) one has $$R_T \leq C \sqrt{Q \log(n)} + C n \log^2(T) ~.$$
In fact this can be achieved by combining the Hazan-Kale reservoir sampling idea with the strategy of Theorem \[th:sparse\]
Next we give our main theorems for linear bandit on $\ell_p^n$ balls. Notice that the polar of the $\ell_p^n$ ball is the $\ell_q^n$ ball with $q=p/(p-1)$.
\[th:UBellp\] Let $p \in (1,2]$. There exists a linear bandit algorithm playing on the unit ball of $\ell_p^n$ such that $$R_T \leq 2^{\frac{6}{p-1}} \sqrt{n T \log(T)} ~.$$
Our lower bound construction for $\ell_p^n$ balls with $p>2$ uses Gaussian losses which satisfy the constraint $\|\ell_t\|_q^q \leq 1$ only in expectation. Note that from standard Gaussian concentration the same bound (up to a logarithmic factor) then holds with high probability. We work with Gaussian losses mostly for clarity of exposition, and at the expense of technical complications one could use losses which satisfy the bound $\|\ell_t\|_q^q \leq 1$ almost surely. We also note that the lower bound is only valid in the large $T$ regime, which is necessary since there exist intermediate regimes of $(T,n)$ where a better regret than $n \sqrt{T}$ is achievable.
\[th:LBellp\] Let $p >2$ and $T \geq n^{\max\left(2, \frac{p-1}{p-2}\right)}$. There exists a numerical constant $C>0$ such that for any linear bandit algorithm playing on the unit ball of $\ell_p^n$, there exists $(\ell_t)_{t \in [T]}$, i.i.d. Gaussian random variables in $\R^n$ such that $$\label{eq:gaussianbound}
\E \|\ell_t\|_q^q \leq 1 ~,$$ and $$\E R_T \geq C n \sqrt{T} ~.$$
We recall the starved bandit setting introduced above. At the beginning of the game the player chooses an exploration distribution $\mu \in \Delta(\cK)$. At any time $t$ the player can choose to play $a_t$ at random, either from $\mu$ or from an adaptive distribution $p_t$ (where $p_t$ depends on the observed feedback so far). The loss of the player is $\ell_t(a_t)$. The feedback is either (i) nothing if $a_t$ was played from $p_t$, or (ii) the standard bandit feedback $\ell_t(a_t)$ if $a_t$ was played from $\mu$. For sake of simplicity we assume that if $\cK$ contains the (signed) canonical basis then $\mu$ is uniform on the (signed) canonical basis.
We observe that Theorem \[th:UBellp\] holds true for the starved linear bandit framework too (indeed the strategy we give to prove Theorem \[th:UBellp\] is a starved bandit strategy). Our main additional result for this setting is to show that for any $p$ not covered by Theorem \[th:UBellp\] one cannot achieve $\sqrt{T}$-type regret:
\[th:starved\] For any strategy for the starved multi-armed bandit there exists a loss sequence such that $R_T \geq \frac1{20} n^{1/3} T^{2/3}$. The same lower bound holds for the starved linear bandit on the $\ell_1^n$ ball. Furthemore for any $p>2$ there exists a constant $C>0$ such that for any starved linear bandit algorithm playing on the unit ball of $\ell_p^n$, there exists $(\ell_t)_{t \in [T]}$, i.i.d. Gaussian random variables in $\R^n$ satisfying and such that $$\E R_T \geq C n^{\frac{q}{2 + q}} T^{\frac{2}{2 + q}} ~.$$
Notation
--------
We use the following (standard) notation: $\Delta(\cK)$ for the set of probability measures supported on $\cK$, $\Delta = \{x \in \R_+^n : \sum_{i=1}^n x(i) =1\}$ for the simplex, $\|x\|_p = \left( \sum_{i=1}^n |x(i)|^p\right)^{1/p}$ for the $\ell_p^n$ norm, $\Phi^*(\theta) = \sup_{x \in \R^n} \theta \cdot x - \Phi(x)$ for the Fenchel dual of $\Phi : \R^n \rightarrow \overline{\R}$, $D_{\Phi}(x,y) = \Phi(x) - \Phi(y) - \nabla \Phi(y) \cdot (y-x)$ for the Bregman divergence associated to $\Phi$, $\|h\|_x = \sqrt{\nabla^2 \Phi(x)[h,h]}$ for the local norm induced by $\Phi$ at $x$, $\|h\|_{x,*} = \sqrt{(\nabla^2 \Phi(x))^{-1}[h,h]}$ for the dual local norm, $\odot$ for the Hadamard product (i.e., entrywise product of vectors), and $\succeq$ for the positive semi-definite ordering on matrices.
Bandit theory reminders {#sec:reminders}
=======================
We give a few brief reminders of multi-armed bandit and linear bandit theory.
Full information strategies
---------------------------
In this section we assume that $\cK$ is a convex body in $\R^n$. We fix a learning rate $\eta >0$ and a mirror map $\Phi : \R^n \rightarrow \overline{\R}$, that is a strictly convex and differentiable map with $\nabla \Phi(\R^n) = \R^n$ and diverging gradient as one approaches the boundary of its domain. The following theorem is a standard result on the mirror descent strategy for online linear optimization (with full information), see e.g., \[Theorem 5.5, [@BC12]\].
\[th:MD\] Let $\ell_1, \hdots, \ell_T \in \R^n$ be a fixed sequence of loss vectors and let $x_1, \hdots, x_T \in \cK$ be defined by: $x_1 = \argmin_{x \in \cK} \Phi(x)$ and $$\label{eq:careful}
x_{t+1} = \argmin_{x \in \cK} D_{\Phi}(x, \nabla \Phi^*(\nabla \Phi(x_t) - \eta \ell_t)) .$$ Then one has for any $x \in \cK$, $$\label{eq:basicMD}
\sum_{t=1}^T \ell_t \cdot (x_t - x) \leq \frac{\Phi(x) - \Phi(x_1)}{\eta} + \frac{1}{\eta} \sum_{t=1}^T D_{\Phi^*}\bigg(\nabla \Phi(x_t) - \eta \ell_t, \nabla \Phi(x_t)\bigg)~.$$ Futhermore assuming that the following implication holds true for any $y_t \in \R^n$, $$\label{eq:condMD}
\nabla \Phi(y_t) \in [\nabla\Phi(x_t), \nabla\Phi(x_{t}) - \eta \ell_t] \Rightarrow \nabla^2 \Phi(y_t) \succeq c \nabla^2 \Phi(x_t)$$ one obtains $$\label{eq:basicregret}
\sum_{t=1}^T \ell_t \cdot (x_t - x) \leq \frac{\Phi(x) - \Phi(x_1)}{\eta} + \frac{\eta}{2 c} \sum_{t=1}^T \|\ell_t\|_{x_t,*}^2 ~.$$
We will also use the lazy variant of mirror descent, also known as FTRL (Follow The Regularized Leader), and its corresponding “primal only” analysis. In particular while for mirror descent one has to check that $\Phi$ is “well-conditioned” on a “dual segment” (equation ) we will see below that for FTRL one needs to check the well-conditioning on a “primal segment” (equation ). Note also that mirror descent and FTRL give the same update equation when $\Phi$ is a barrier for $\cK$ (see e.g., [@Bub15]), which is often the case in bandit scenario.
\[th:lazyMD\] Let $\ell_1, \hdots, \ell_T \in \R^n$ be a fixed sequence of loss vectors and let $x_1, \hdots, x_T \in \cK$ be defined by: $$\label{eq:careful2}
x_{t} = \argmin_{x \in \cK} \eta \sum_{s=1}^{t-1} \ell_s \cdot x + \Phi(x) .$$ Then one has for any $x \in \cK$, $$\label{eq:induction}
\sum_{t=1}^T \ell_t \cdot (x_t - x) \leq \frac{\Phi(x) - \Phi(x_1)}{\eta} + \sum_{t=1}^T \ell_t \cdot (x_t - x_{t+1}) ~.$$ Futhermore assuming that the following implication holds true for any $y_t \in \R^n$, $$\label{eq:condFTRL}
y_t \in [x_t, x_{t+1}] \Rightarrow \nabla^2 \Phi(y_t) \succeq c \nabla^2 \Phi(x_t)$$ then one has that holds true with the term $\frac{\eta}{2c}$ replaced by $\frac{2 \eta}{c}$.
The proof of is a classical one-line induction (sometimes referred to as the Be-The-Leader lemma). We turn to and note that it suffices to show that $\|x_t - x_{t+1}\|_{x_t} \leq \frac{2 \eta}{c} \|\ell_t\|_{x_t,*}$. Observe that, using a Taylor expansion, for some $y_t \in [x_t,x_{t+1}]$ one has, with the notation $\Phi_t(x):= \eta \sum_{s=1}^t \ell_s \cdot x + \Phi(x)$ (thus $x_{t+1} \in \argmin \Phi_t$ and $x_t \in \argmin \Phi_t - \eta \ell_t$), $$\begin{aligned}
\frac12 \|x_t-x_{t+1}\|_{y_t}^2 = \Phi_t(x_t) - \Phi_t(x_{t+1}) - \nabla \Phi_t(x_{t+1}) \cdot (x_t - x_{t+1}) & \leq & \Phi_t(x_t) - \Phi_t(x_{t+1}) \\
& \leq & \eta {\ell}_t \cdot (x_t - x_{t+1}) ~.\end{aligned}$$ Using that $\nabla^2 \Phi(y_t) \succeq c \nabla^2 \Phi(x_t)$ one also has $\|x_t - x_{t+1}\|_{x_t}^2 \leq \frac{1}{c} \|x_t-x_{t+1}\|_{y_t}^2$ and thus $$\|x_t-x_{t+1}\|_{x_t}^2 \leq \frac{2 \eta}{c} {\ell}_t \cdot (x_t - x_{t+1}) \leq \frac{2 \eta}{c} \|\ell_t\|_{x_t,*} \|x_t -x_{t+1}\|_{x_t} ~,$$ which concludes the proof.
Bandit strategies {#sec:classicalbandit}
-----------------
In addition to choosing a regularizer, a bandit strategy also rely on a sampling scheme, that is a map $p : \mathrm{conv}(\cK) \rightarrow \Delta(\cK)$ such that $\E_{X \sim p(x)} X = x$. One then runs FTRL (or mirror descent), with the (unobserved) true losses $\ell_t$ replaced by estimators $\tilde{\ell}_t$ (constructed based on the observed feedback). Moreover instead of playing the point $x_t$ recommended by FTRL, i.e., $x_t = \argmin_{x \in \mathrm{conv}(\cK)} \sum_{s=1}^{t-1} \tilde{\ell}_s \cdot x + \Phi(x)$, one plays at random $a_t \sim p(x_t)$ (where the sampling is done independently of the past given $x_t$). The key point is that if the loss estimator is unbiased, i.e., $\E_{a_t \sim p(x_t)} \tilde{\ell}_t = \ell_t$, then one has for any $x \in \cK$, $$\E \sum_{t=1}^T \ell_t \cdot (a_t - x) = \E \sum_{t=1}^T \tilde{\ell}_t \cdot (x_t - x) ~,$$ and thus one can use Theorem \[th:MD\] or Theorem \[th:lazyMD\] to bound the regret. In particular assuming that one can prove the well-conditioning condition or , the key quantity to control is the “variance” of the loss estimator appearing in , namely $\E \ \|\tilde{\ell}_t\|_{x_t,*}^2$.
To illustrate the above discussion let us briefly recall the classical multi-armed bandit setting (i.e., $\cK=\{e_1, \hdots, e_n\}$) with **nonnegative losses**. We use mirror descent with $\Phi(x) = \sum_{i=1}^n x(i) \log x(i)$, the sampling scheme $p : \Delta \rightarrow \Delta(e_1,\hdots e_n)$ is simply the identity map (in the sense that $\P_{a \sim p(x)}(a=e_i) = x(i)$), and the unbiased loss estimator is $$\tilde{\ell}_t(i) = \frac{\ell_t(i)}{x_t(i)} \ds1\{a_t = e_i\} ~.$$
The key is to observe that since $\tilde{\ell}_t$ has nonegative entries, one has that is satisfied with $c=1$, and thus gives $$R_T \leq \frac{\log(n)}{\eta} + \frac{\eta}{2} \sum_{t \in [T], i \in [n]} \E \ \|\tilde{\ell}_{t}\|_{x_t,*}^2 ~.$$ The last thing to observe is that, since $\|h\|_x^2 = \sum_{i=1}^n \frac{h(i)^2}{x(i)}$, one has $$\E \ \|\tilde{\ell}_{t}\|_{x_t,*}^2 = \E \sum_{i=1}^n x_t(i) \tilde{\ell}_t(i)^2 = \E \sum_{i=1}^n x_t(i) \frac{{\ell}_t(i)^2}{x_t(i)} \ds1\{a_t=e_i\} = \|\ell_t\|_2^2 ~.$$ Thus with an appropriate choice of $\eta$ one gets $$\label{eq:classicalregret}
R_T \leq \sqrt{\frac{\log(n)}{2} \sum_{t=1}^T \|\ell_t\|_2^2} ~.$$ As a side note we observe that using the polynomial INF regularizer of [@AB09] (see Section \[sec:intuition\] for a brief reminder on the INF regularizer), for any primal dual pair $p, q \geq 1$, one obtains an algorithm with a regret bound scaling in $\frac{q}{q-1} \sqrt{n^{1/q} \sum_{t=1}^T \|\ell_t\|_{2p}^2}$.
Sparsity and variation bounds for multi-armed bandit
====================================================
We start first by describing some basic obstacles to obtain a sparsity type bound in Section \[sec:obstacles\]. Then in Section \[sec:intuition\] we give some intuition for our new “hybrid regularizer”, $\sum_{i=1}^n x(i) \log(x(i)) - \gamma \sum_{i=1}^n \log(x(i))$, that is the weighted combination of the negentropy and the logarithmic barrier for the positive orthant[^5]. The extra logarithmic barrier term can be understood as a soft way to encourage exploration (to the contrary of the usual forced exploration). Finally in Section \[sec:proofsparse\] we prove Theorem \[th:sparse\] (this section is self-contained and does not require reading the two previous subsections).
Basic obstacles {#sec:obstacles}
---------------
The basic issue is that only holds for nonnegative losses[^6]. The reason nonnegativity was needed is that the well-conditioned assumption for the negentropy $\Phi$, equation , crucially relies on the fact that (note that $\nabla \Phi = \log, \nabla^2 \Phi = \mathrm{diag}(1/x)$) for $\log(y) = \log(x) - \ell$ with $\ell \geq 0$ one has $1/y\geq 1/x$. A standard fix to maintain the latter inequality approximately true for general losses is to ensure that the magnitude of the (estimated) loss is controlled. Indeed is satisfied for some constant $c$ provided that almost surely $\|\eta \tilde{\ell}_t\|_{\infty} \leq \log(1/c)$. This almost sure control can be achieved by adding forced exploration, as was done in the original adversarial multi-armed bandit paper [@ACFS03], that is the sampling scheme is now $(1-n \gamma) x_t + \gamma \ds1$, or in words explore uniformly at random with probability $n \gamma$ and otherwise play from $x_t$. Indeed in this case $\|\eta \tilde{\ell}_t\|_{\infty} \leq {\eta} / {\gamma}$, and thus the well-conditioned assumption is satisfied when $\gamma \simeq \eta$. However the added regret (with respect to $i^* \in [n]$) suffered by the extra exploration is exactly $\gamma \sum_{i,t} (\ell_t(i) - \ell_t(i^*))$. This latter term destroys the scaling with sparsity (for example if $\ell_t = -e_{i^*}$ then this term is of order $\gamma (n-1) T \simeq \eta n T$). More prosaically, the uniform exploration might make us miss out on a $n \gamma$ fraction of the “gains” of the best arm, which could be far too much. We also observe that the recently proposed implicit exploration by [@KNVM14] (see also [@Neu15]) suffers from the exact same issue.
We also note that, without going into any technical details, the case of arbitrary losses seem harder than the case of nonnegative losses. Indeed the former contains the case of nonpositive losses, or equivalently nonnegative [*gains*]{}. Sparse nonnegative losses mean that most arms are performing well and only a handful are to be avoided. On the other hand sparse nonnegative gains mean that most arms are bad, and only a handful are performing well. Intuitively, finding this small set of good arms hiding in a sea of bad arms is harder than avoiding a small set of bad arms in a sea of good arms.
Intuition for the hybrid regularizer {#sec:intuition}
------------------------------------
The intuition is divided in two parts: (i) the fact that the added regret for $\gamma >0$ is controlled, and (ii) that the well-conditioning still holds.
For the first part we start with a slightly different point of view on extra (forced) exploration. It is easy to check that adding extra exploration exactly corresponds to taking the regularizer to be a “negatively shifted negentropy”: $\sum_{i=1}^n (x(i) - \gamma) \log(x(i) - \gamma)$. For such a regularizer the range $\Phi(x) - \Phi(x_1)$ is controlled only for $x$’s such that $\min_{i \in [n]} x(i) > \gamma$. In the worst case the gap between the regret with respect to such $x$’s, and with respect to an arbitrary $x$ can be as large as $n \gamma T$, and since the well-conditioned assumption requires $\gamma \simeq \eta$ this leads us to the extra term $\eta n T$. On the other hand for the hybrid barrier one can compare to $x$’s with $\min_{i \in [n]} x(i) =1/\mathrm{poly}(T)$, only at the expense of a term of the form $\frac{\gamma n \log(T)}{\eta}$. Thus provided that the well-conditioning assumption remains true for $\gamma \simeq \eta$ (this is the key part to verify) the hybrid regularizer could lead to a bound of the form up to to an extra additive term of order $n \log(T)$.
For the well-conditioning intuition we first recall the INF parametrization of a regularizer ([@ABL14]): For $\psi : \mathbb{R} \rightarrow \R$, let $\Phi$ be defined by $\nabla \Phi^*(x) := (\psi(x_i))_{i \in [n]}$. The negentropy regularizer exactly corresponds to $\psi(s) = \exp(s)$ while adding forced extra exploration with probability $n \gamma$ can be achieved by taking $\psi(s) = \exp(s) + \gamma$. The hybrid regularizer essentially corresponds to taking $\psi(s)$ to be the exponential function when $\psi(s) \geq \gamma$, and otherwise to be roughly like $\frac{\gamma \log \gamma}{s}$. In particular we see that the well-conditioning is satisfied for $\gamma \simeq \eta$ when the played arm has probability greater than $\gamma$ (since in this case everything behaves essentially as with forced exploration), and on the other hand when the played arm has probability smaller $\gamma$, its probability $x$ is of the form $1/L$ and the updated probability is $1/(L + 1/x) \simeq x$, and thus the well-conditioning also holds in this case.
Proof of Theorem \[th:sparse\] {#sec:proofsparse}
------------------------------
Observe that the hybrid regularizer $\Phi$ is lower bounded by the negentropy in the sense that $\nabla^2 \Phi(x) \succeq \mathrm{diag}(1/x(i))$. Thus the standard argument of Section \[sec:classicalbandit\] shows that $$\E \ \|\tilde{\ell}_t\|_{x_t, *}^2 \leq \|\ell_t\|_2^2 ~.$$ In particular, using Theorem \[th:lazyMD\], it only remains to check . The next lemma is the key justification for our new regularizer.
\[lem:critical\] Let $\Phi$ be the hybrid regularizer, $\eta>0$, $L \in \R^n$, $\xi \in \R$, $L':=L + \xi e_1$, $$x := \argmin_{y \in \Delta} \eta L \cdot y + \Phi(y) \; \text{and} \; x' := \argmin_{y \in \Delta} \eta L' \cdot y + \Phi(y) ~.$$ Assuming that $|\xi| \leq C/x(1)$ for some $C>0$ and that $\gamma \geq \eta C$, one has for any $i \in [n]$, and any $u \in (0,1)$, $$\max\left(\frac{x'(i)}{x(i)}, \frac{x(i)}{x'(i)} \right) \leq \max\left(\exp\left(\frac{1}{\frac{\gamma}{\eta C}-1}\right), \frac1{1-\gamma-u} \exp(\gamma n / u) \right) ~.$$
For example with $C=1$, $u=1/2$, $\gamma=2 \eta$, and $\eta \leq \frac1{15 n}$ one obtains $$\max\left(\frac{x'(i)}{x(i)}, \frac{x(i)}{x'(i)} \right) \leq 3 ,$$ which means in particular (notice that $\nabla^2 \Phi(x) = \mathrm{diag}(1/x(i) + \gamma / x(i)^2)$) that for any $y_t \in [x_t, x_{t+1}]$ one has $$\nabla^2 \Phi(x_t) \preceq 9 \nabla^2 \Phi(y_t) ~,$$ which finishes the proof of Theorem \[th:sparse\] up to straightforward calculations.
First note that the KKT conditions for $x$ and $x'$ show that there exist $\lambda, \lambda' \in \R$ such that $$\label{eq:KKT}
\eta L + \nabla \Phi(x) = \lambda \ds1 , \; \eta L' + \nabla \Phi(x') = \lambda' \ds1 ~.$$ Also note that $\nabla^2 \Phi(x)$ is diagonal with positive entries.
**Step 1:** We show that $\lambda'$ and $x'(i)$ for $i \neq 1$ are increasing with $\xi$, while $x'(1)$ is decreasing with $\xi$. By differentiating one gets $$\label{eq:diffKKT}
\left. \frac{d \lambda'}{d \xi} \right. \ds1 = \eta e_1 + \nabla^2 \Phi(x) \left. \frac{d x'}{d \xi} \right. ~.$$ By multiplying the above equation with $(\nabla^2 \Phi(x))^{-1}$ and summing over the coordinates (recall that $\sum_{i=1}^n \left. \frac{d x'(i)}{d \xi} \right. =0$) one obtains $\left. \frac{d \lambda'}{d \xi} \right. > 0$. In particular using this in one obtains for any $i \neq 1$, $\frac{d x'(i)}{d\xi} > 0$, and thus $ \frac{d x'(1)}{d\xi} < 0$.
**Step 2:** We now show that the first coordinate has a small multiplicative change. Substracting the two identities in one obtains, since $\nabla \Phi(x) = (1 + \log x(i) - \gamma / x(i))_{i \in [n]}$, $$\label{eq:KKT1}
\lambda' - \lambda + \log \frac{x(1)}{x'(1)} + \gamma \left( \frac{1}{x'(1)} - \frac{1}{x(1)}\right) = \eta \xi ~.$$ Observe that that by Step 1 all the terms on the lhs have the same sign and thus $$\label{eq:XX}
|\lambda' - \lambda| + \left|\log \frac{x(1)}{x'(1)}\right| + \gamma \left| \frac{1}{x'(1)} - \frac{1}{x(1)}\right| = \eta | \xi | ~.$$ In particular we have $$\left| \frac{1}{x'(1)} - \frac{1}{x(1)}\right| \leq \frac{\eta C / \gamma}{x(1)} \Leftrightarrow \frac{x(1)}{x'(1)} \in [1-\eta C /\gamma, 1+\eta C / \gamma] ~.$$ Also note that that for any $s \in (0,1)$, $\max\left(1+s, \frac1{1-s} \right) \leq \exp\left(\frac1{\frac1{s}-1}\right)$.
**Step 3:** Assuming that $x(1) \geq \gamma - \eta C$ we show that all the other coordinates also have a small multiplicative change (the case $x(1) < \gamma - \eta C$ is dealt with in the next step). Substracting the two identities in one obtains for any $i \neq 1$, $$\label{eq:KKTi}
\log \frac{x(i)}{x'(i)} + \gamma \left( \frac{1}{x'(i)} - \frac{1}{x(i)}\right) = \lambda - \lambda' ~.$$ In particular since the two terms on the left hand side in have the same sign one has $$\label{eq:diffL}
\left| \log \frac{x(i)}{x'(i)} \right| + \gamma \left|\frac{1}{x'(i)} - \frac{1}{x(i)}\right| = |\lambda - \lambda'| ~.$$ Next we also observe that thanks to : $$| \lambda - \lambda'| \leq \eta |\xi| \leq \frac{\eta C}{x(1)} ~.$$ In particular together with we proved that if $x(1) \geq \gamma - \eta C$ then one has $$\left| \log \frac{x(i)}{x'(i)} \right| \leq \frac{1}{\frac{\gamma}{\eta C} - 1} ~.$$
**Step 4:** Finally we show that if $x(1) \leq \gamma - \eta C$ one also has that all the other coordinates have a small multiplicative change. Let $I:=\{i \neq 1 \; \text{s.t.} \; \min(x(i), x'(i)) \geq u/n \}$ (notice that, by Step 1, the minimum is attained uniformly either at $x$ or $x'$). Then thanks to one has for any $i \in I$, $$\left| \log \frac{x(i)}{x'(i)} \right| \geq |\lambda - \lambda'| - \gamma n/u ~,$$ and thus $$1 \geq \sum_{i \in I} \min(x(i), x'(i)) \exp(|\lambda - \lambda'| - \gamma n/u) ~.$$ Observe that if $\min(x(i),x'(i)) = x(i)$ for some $i \in I$ then one has $$\sum_{i \in I} \min(x(i), x'(i)) = \sum_{i \in I} x(i) \geq 1 - (\gamma - \eta C) - u ~,$$ while if $\min(x(i),x'(i)) = x'(i)$ for some $i \in I$ then one has (thanks to Step 2) $$\sum_{i \in I} \min(x(i), x'(i)) = \sum_{i \in I} x'(i) \geq 1 - \frac{\gamma - \eta C}{1 - \frac{\eta C}{\gamma}} - u = 1-\gamma - u ~.$$ Thus we have $$1 \geq (1-\gamma - u) \exp(|\lambda - \lambda'| - \gamma n / u) ~,$$ which concludes the proof (recall that by one has for any $i \neq 1$, $\left| \log \frac{x(i)}{x'(i)} \right| \leq |\lambda - \lambda'|$).
Variation bound for multi-armed bandit
--------------------------------------
We only give a brief sketch of proof of Theorem \[th:variance\], as it is essentially a straightforward combination of the proof of Theorem \[th:sparse\] together with the arguments of [@HK09]. In particular we ignore explicit numerical constants with the notation $O$.
First note that it is easy to see from that the following bound holds for full information FTRL under the well-conditioning assumption : for any sequence $m_1,\hdots, m_{T} \in \R^n$ and with $m_{T+1}=0$ one has $$\label{eq:HKregret}
\sum_{t=1}^T \ell_t \cdot (x_t - x) \leq \frac{\Phi(x) - \Phi(x_1)}{\eta} + \frac{2 \eta}{c} \sum_{t=1}^T
\|\ell_t - m_t\|_{x_t,*}^2 + \sum_{t=1}^{T+1} \|m_t - m_{t-1}\|_2 ~.$$ The strategy of Hazan and Kale is to use a small portion of “exploration” rounds to estimate $\mu_t = \frac1{t} \sum_{s=1}^t \ell_s$ by some $\tilde{\mu}_t$ and then use it to center the loss estimator (for the non-“exploration” rounds) by setting for any $i \in [n]$: $$\tilde{\ell}_t(i) = \frac{(\ell_t - \tilde{\mu}_t)(i)}{x_t(i)} \ds1\{a_t = e_i\} + \tilde{\mu}_t(i) ~.$$ More precisely by doing an exploration round with probability $k n / t$ at round $t$ (the so-called “reservoir sampling”, here $k>0$ is a parameter of the algorithm) one can obtain an estimator $\tilde{\mu}_t$ such that $\E \ \tilde{\mu}_t = \mu_t$ and $\mathrm{Var}(\tilde{\mu}_t) \leq \frac{Q}{k t}$. Moreover the added regret from those rounds is $O(k n \log(T))$. Thus using the bound with $m_t = {\mu}_t$ it only remains to bound the terms $\eta \sum_{t=1}^T
\|\tilde{\ell}_t - {\mu}_t\|_{x_t,*}^2$ and $\sum_{t=1}^{T+1} \|{\mu}_t - {\mu}_{t-1}\|_2$. The latter term is easily controlled by $O(\sqrt{n} \log(Q))$, see Lemma 12 in [@HK09]. On the other hand for the former term one gets $$\E \ \|\tilde{\ell}_t - \mu_t\|_{x_t,*}^2 \leq 2 \E \ \|\tilde{\ell}_t - \tilde{\mu}_t\|_{x_t,*}^2 + 2 \E \ \|\tilde{\mu}_t - \mu_t\|_{x_t,*}^2 = 2 \E \|\ell_t - \mu_t\|_2^2 + 2 \mathrm{Var}(\tilde{\mu}_t) ~,$$ and thus $\eta \E \sum_{t=1}^T
\|\tilde{\ell}_t - {\mu}_t\|_{x_t,*}^2 = O(\eta Q (1+\log(T)/k))$, which easily concludes the proof up to straigthforward computations.
Regular and starved linear bandits on $\ell_p^n$ balls
======================================================
In this section we prove the results related to linear bandits on $\ell_p^n$ balls. Recall that $q=p/(p-1)$.
Proof of Theorem \[th:UBellp\]
------------------------------
Let $p \in (1,2]$. We first describe a new strategy to play on $\ell_p^n$ balls based on a non-self-concordant barrier (when $p \neq 2$). Let $d(x)=1-\|x\|_p^p$, and $\Phi(x) = - \log d(x)$ (notice that for $p\neq2$ the Hessian of $\Phi$ blows up at $0$, and thus $\Phi$ cannot be self-concordant). We play FTRL with regularizer $\Phi$ and with sampling scheme given by: with probability $\max(d(x), \gamma)$ play uniformly in $\{e_1, -e_1, \hdots, e_n, -e_n\}$, and otherwise play $x/\|x\|_p$. Note that this not unbiased, but rather “$\gamma$-biased”, which adds a $\gamma T$ term to the regret. The estimator is defined by $\tilde{\ell}_t = n \frac{\ell_t \cdot \tilde{x}_t}{1-\|x_t\|_p, \gamma)} \tilde{x}_t$ if played uniformly in $\{e_1, -e_1, \hdots, e_n, -e_n\}$, and $\tilde{\ell}_t = 0$ otherwise.
While $\Phi$ is not self-concordant, the next lemma shows that one still has some form of well-conditioning (though not ) that will turn out to be sufficient to control the regret.
\[lem:locnormellp\] Let $x, \ell \in \R^n$ such that $\|x\|_p < 1$, $\|\ell\|_0 =1$ and $\|\ell\|_2 \leq 1$. Let $y \in \R^n$ such that $\nabla \Phi(y) \in [\nabla \Phi(x), \nabla \Phi(x) + \ell]$. Then one has for $p\in [1,2]$, $$\|\ell\|_{y,*}^2 \leq \frac{2^{\frac{3}{p-1}} d(x)}{p(p-1)} \sum_{i=1}^n (|x(i)|^{2-p} + |\ell(i)|^{\frac{2-p}{p-1}}) \ell(i)^2 ~.$$
Before moving to the proof of Lemma \[lem:locnormellp\] we show how to use it to control the variance of the loss estimator. The proof of Theorem \[th:UBellp\] is then straightforward from and Lemma \[lem:localnormbound\].
\[lem:localnormbound\] The above strategy satisfies for any $y_t \in \R^n$ such that $\nabla \Phi(y_t) \in [\nabla \Phi(x_t), \nabla \Phi(x) - \eta \tilde{\ell}_t]$ $$\E_{a_t} \|\tilde{\ell}_t\|_{y_t,*}^2 \leq \frac{2^{\frac{4}{p-1}}}{p-1} n ~.$$
Note that $\|\eta \tilde{\ell}_t\|_2 \leq n \eta/\gamma$. Thus by Lemma \[lem:locnormellp\] we have, provided that $\gamma \geq n \eta$, $$\|\tilde{\ell}_t\|_{y_t,*}^2 \leq \frac{2^{\frac{3}{p-1}} d(x_t)}{p(p-1)} \E \sum_{i=1}^n (|x_t(i)|^{2-p} + |\eta \tilde{\ell}_t(i)|^{\frac{2-p}{p-1}}) \tilde{\ell}_t(i)^2 ~.$$ We now bound separately the two terms. For the first one we have (note that $1-\|x\|_p \geq \frac{1}{p} (1-\|x\|_p^p)$ and thus $d(x_t) \leq p \max(1-\|x_t\|_p, \gamma)$) $$d(x_t) \E_{a_t} \sum_{i=1}^n |x_t(i)|^{2-p} \tilde{\ell}_t(i)^2 \leq p n \sum_{i=1}^n |x_t(i)|^{2-p} \ell_t(i)^2 \leq p n ~,$$ where the second inequality follows from Holder’s inequality with $\frac{2}{q}+ \frac{2-p}{p}=1$. Now we bound the second term (note that $\frac{2-p}{p-1}+2=q$) $$d(x_t) \E_{a_t} \sum_{i=1}^n |\eta \tilde{\ell}_t(i)|^{\frac{2-p}{p-1}} \tilde{\ell}_t(i)^2 \leq p n \sum_{i=1}^n |\ell_t(i) \eta n / \gamma|^{\frac{2-p}{p-1}} {\ell}_t(i)^2 \leq p n \sum_{i=1}^n \ell_t(i)^q \leq p n ~,$$ which concludes the proof.
We give now a few preliminary results before proving Lemma \[lem:locnormellp\].
\[lem:hessellp\] One has for any $x \in \R^n$ such that $\|x\|_p < 1$, $$\nabla^2 \Phi^*(\nabla \Phi(x)) \preceq \frac{d(x)}{p(p-1)} \textsf{diag}(|x|^{2-p}) ~.$$
Straightforward derivations show that $$\label{eq:gradellp}
\nabla \Phi(x) = \frac{ p \cdot \textsf{sign}(x) \odot |x|^{p - 1}}{1 - \| x\|_p^p} ~,$$ $$\begin{aligned}
\nabla^2 \Phi(x) &= \frac{p(p - 1) \textsf{diag}(|x|^{p - 2})}{1 - \| x \|_p^p} + \frac{p^2 \left(\textsf{sign}(x) \odot |x|^{p - 1} \right)^{\otimes 2} }{ (1 - \| x\|_p^p)^2}
\\
& \succeq \frac{p(p - 1) \textsf{diag}(|x|^{p - 2})}{1 - \| x \|_p^p} ~,\end{aligned}$$ which directly implies the lemma.
\[lem:ellpcritical\] Let $v \in \R^n$ and $\ell \in \R^n$ such that $\|\ell\|_0 =1$ and $\|\ell\|_2 \leq 1$. Denote $x=\nabla \Phi^*(v)$ and $y=\nabla \Phi^*(v+\ell)$. Then one has $$\begin{aligned}
& d(y) \leq 4 d(x) ~, \label{eq:stayclosetobdy} \\
& |y(i)|\leq 2^{\frac{3}{p-1}} |x(i)| + |2 \ell(i)|^{\frac{1}{p-1}} \label{eq:coordwisebd} ~.\end{aligned}$$
Observe that by definition (recall ) one has $$|x(i)| = \left( \frac{ |v(i)| d(x)}{p} \right)^{\frac{1}{p - 1}}, \quad |y(i)| = \left( \frac{ |v(i) + \ell(i)| d(y)}{p} \right)^{\frac{1}{p - 1}} ~.$$ In particular we immediately see that implies by the triangle inequality (also $d(y) \leq 1$ and $p \geq 1$) as follows: $$\begin{aligned}
|y(i)| = \left( \frac{ |v(i) + \ell(i)| d(y)}{p} \right)^{\frac{1}{p - 1}} & \leq & \left( \frac{ 2\max(|v(i)|, |\ell(i)|) d(y)}{p} \right)^{\frac{1}{p - 1}} \\
& \leq & \max\left(\left(\frac{2 d(y)}{d(x)}\right)^{\frac{1}{p-1}} |x(i)|, |2 \ell(i)|^{\frac1{p-1}}\right) \\
& \leq & 8^{\frac{1}{p-1}} |x(i)| + |2 \ell(i)|^{\frac{1}{p-1}} ~.\end{aligned}$$ We now move to the proof of . We first note that is trivially true for $d(x) \geq 1/4$ and thus without loss of generality one can assume $\|x\|_p^p \geq 3/4$. Crucially we now consider two cases, depending on whether the non-zero coordinate of $\ell$ is a “light” or “heavy” coordinate in $x$. Let us assume $\ell(1) \neq 0$. If $x(1)\leq (1/2)^{1/p}$ (i.e., “light”) then $\sum_{i \geq 2} |x(i)|^p \geq 1/4$ and thus $$\|y\|_p^p \geq \sum_{i \geq 2} |y(i)|^p = \sum_{i\geq 2} |x(i)|^p \left(\frac{d(y)}{d(x)}\right)^{\frac{p}{p-1}} \geq \frac1{4} \left(\frac{d(y)}{d(x)}\right)^{\frac{p}{p-1}} ~,$$ which implies $d(y) \leq 4 d(x)$ (since $\|y\|_p \leq 1$). On the other hand if $x(1) \geq (1/2)^{1/p}$ (i.e., “heavy”) then one has $$|v(1)| = \frac{p}{d(x)} |x(1)|^{p-1} \geq 2 ~,$$ and thus $|v(1) + \ell(1)| \geq \frac12 |v(1)|$ (since $|\ell(1)| \leq 1$) which implies $$1 \geq |y(1)| \geq |x(1)| \left(\frac{d(y)}{2 d(x)}\right)^{\frac{1}{p-1}} \geq \left(\frac{d(y)}{4 d(x)}\right)^{\frac{1}{p-1}} ~.$$
Finally we have:
Using successively Lemma \[lem:hessellp\], , , and the fact that $p \in [1,2]$, one has $$\begin{aligned}
\|\ell\|_{y,*}^2 \leq \frac{d(y)}{p(p-1)} \sum_{i=1}^n |y(i)|^{2-p} \ell(i)^2 & \leq & \frac{4 d(x)}{p(p-1)} \sum_{i=1}^n |y(i)|^{2-p} \ell(i)^2 \\
& \leq & \frac{4 d(x)}{p(p-1)} \sum_{i=1}^n (2^{\frac{3}{p-1}} |x(i)| + |2 \ell(i)|^{\frac{1}{p-1}})^{2-p} \ell(i)^2 \\
& \leq & \frac{2^{\frac{3}{p-1}} d(x)}{p(p-1)} \sum_{i=1}^n (|x(i)|^{2-p} + |\ell(i)|^{\frac{2-p}{p-1}}) \ell(i)^2 ~.\end{aligned}$$
Proof of Theorem \[th:LBellp\] {#sec:proofLBellp}
------------------------------
For sake of clarity we write $\cK = \{(x,y) \in \R \times \R^n : |x|^p + \|y\|_p^p \leq 1\}$ and the losses as $\ell_t = (w_t, z_t) \in \R \times \R^n$. Let $\epsilon >0$ to be such that $\epsilon^q = C/\sqrt{T}$ for some small enough universal constant $C\in (0,1)$ (in particular since $T>n^2$ one has $\epsilon^q n < 1$). We now define i.i.d. Gaussian losses as follows. For $\xi \in \{-1,1\}^n$ let $\ell_t^{\xi}=(w_t, z_t^{\xi})$ where $w_t \sim \cN(-1,1)$ and $z_t^{\xi} \sim \cN(\epsilon \xi, \frac{1}{n^{2/q}} I_n)$. We show that $$\E_{\xi} \E_{\ell_t^{\xi}} R_T = \Omega(n \sqrt{T}) ~,$$ which clearly concludes the proof (notice since $T>n^2$ one has $\E \|\ell_t\|_q^q = O(1)$ and thus by rescaling by a constant one can also get ).
The key idea of the proof is to distinguish between “exploration rounds” and “exploitation rounds”, depending on whether the played action $(x_t, y_t) \in \cK$ satisfies $x_t \leq 1/4$ or $x_t \geq 1/4$. Exploration rounds suffer constant regret because the optimal action $(x^*, y^*)$ has $x^*$ close to $1$. On the other hand exploitation rounds give little information about $\xi$ because of the constant variance induced by the $x$ component. Furthermore low-regret exploitation rounds should actually have the $x$ component close to $1$ which means that even less information about $\xi$ is gathered. We make this tradeoff more precise below, but first in Lemma \[step1\] we formalize the fact that identifying $\xi$ matters for low-regret and in Lemma \[step2\] we formalize the previous sentence.
Let us define $(\bar{x}, \bar{y}) = \frac{1}{T}\sum_{t = 1}^T \E[(x_t, y_t)]$ and $(x^*,y^*) = \argmin_{(x,y) \in \cK} x + \epsilon \xi \cdot y$. In particular one has $$\label{eq:lastday}
\E_{\ell_t^{\xi}} \frac{R_T}{T} \geq - (\bar{x} - x) + \epsilon \xi \cdot (\bar{y}-y^*) ~.$$ We say a coordinate $i \in [n]$ is wrong if $\bar{y}(i) \xi(i) \geq 0$.
\[step1\] Let $s$ be the number of wrong coordinates, then $\E_{\ell_t^{\xi}} R_T \geq \epsilon^q sT/ 4$.
Let us assume that the first $s$ coordinates are wrong. A straightforward calculation shows that $-x^* + \epsilon \xi \cdot y^* = - (1 + \epsilon^q n)^{1/q} $, and thus by it suffices to show that $$-\bar{x} + \epsilon \sum_{i = s+1}^n \bar{y}(i) \xi(i) \geq \epsilon^q s / 4 - (1 + \epsilon^q n)^{1/q} ~.$$ Since $\|(\bar{x}, \bar{y}({s + 1}), \cdots, \bar{y}(n))\|_p \leq 1$, by Holder’s inequality we know that $$\bar{x} - \epsilon\sum_{i = s + 1}^n \bar{y}(i) \xi(i) \leq (1 + \epsilon^q (n - s))^{1/q} ~.$$ This concludes the proof since $(1 + \epsilon^q (n - s))^{1/q} \leq (1 + \epsilon^q n)^{1/q} - \frac{1}{2q} \epsilon^q s$.
\[step2\] $\bar{x} \leq 1- 4 \epsilon^q n \Rightarrow \E_{\ell_t^{\xi}} R_T \geq \epsilon^q n T$.
It suffices to show that $-\bar{x} + \epsilon \xi \cdot \bar{y} \geq \epsilon^q n - (1 + \epsilon^q n)^{1/q}$ (see beginning of previous proof). Observe that $$-\bar{x} + \epsilon \xi \cdot \bar{y} \geq -|\bar{x}| - \epsilon \|\xi\|_q \|\bar{y}\|_p \geq - |\bar{x}| - (1-|\bar{x}|^p)^{1/p} \epsilon n^{1/q} ~.$$ Observe that $x \mapsto x + (1 - x^p)^{1/p} \epsilon n^{1/q}$ is a nondecreasing function for $x \in [0, 1 - \epsilon^q n]$ since $$\frac{1}{p} \epsilon n^{1/q} (1-(1- \epsilon^q n)^p)^{1/p-1} \leq \epsilon n^{1/q} (\epsilon^q n)^{1/p - 1} = 1 ~.$$ Therefore we have $$-\bar{x} + \epsilon \xi \cdot \bar{y} \geq - (1 - 4 \epsilon^q n) - (1-(1 -4 \epsilon^q n)^p)^{1/p} \epsilon n^{1/q} ~,$$ and thus the proof is concluded by $1 + (1-(1 -4 \epsilon^q n)^p)^{1/p} (\epsilon^q n)^{1/q} \leq (1 + \epsilon^q n)^{1/q} + 3 \epsilon^q n$.
Observe now that the observed feedback at round $t$ is exactly $$f_t^{\xi} := x_t w_t + y_t \cdot z_t^{\xi} \sim \cN(x_t + \epsilon y_t \cdot \xi, \sigma_t^2), \; \text{where} \; \sigma_t^2 = x_t^2 + \|y_t\|_2^2 / n^{2/q} ~.$$ Denote $\cL_{\xi}$ for the law of the observed feedback up to time $T$, i.e., the law of $(f_1^{\xi}, \hdots, f_T^{\xi})$. Standard calculations show that for $\xi$ and $\xi'$ differing only in coordinate $i \in [n]$ one has $$\mathrm{TV}(\cL(\xi), \cL(\xi')) \leq \sqrt{\sum_{t=1}^T \E_{\ell_t^{\xi}} \frac{\epsilon^2 y_t(i)^2}{\sigma_t^2}} ~.$$ Another standard calculation show that the above inequality implies $$\E_{\xi, \ell_t^{\xi}} \frac{1}{T}\sum_{t=1}^T \sum_{i=1}^n \ds1\{y_t(i) \xi(i) < 0\} \geq \frac{n}{2} - \sqrt{n \sum_{t=1}^T \E_{\xi, \ell_t^{\xi}} \frac{\epsilon^2 \|y_t\|_2^2}{\sigma_t^2}} ~.$$ Note that the left hand side in the above inequality is exactly the average (over time) number of wrongly guessed coordinates for $\xi$, which we know controls the regret thanks to Lemma \[step1\]. In particular it only remains to show that $$\label{eq:X}
\sum_{t=1}^T \E_{\xi, \ell_t^{\xi}} \frac{\epsilon^2 \|y_t\|_2^2}{\sigma_t^2} \leq c n ~,$$ for some universal constant $c<1/2$.
Note that one always has $\sigma_t^2 \geq \|y_t\|_2^2/ n^{2/q}$ and furthermore $x_t \geq 1/4 \Rightarrow \sigma_t^2 \geq 1/2^4$. Recall also that $\|y_t\|_2 \leq n^{1/2-1/p} \|y_t\|_p \leq n^{1/2-1/p} (1-|x_t|^p)^{1/p}$. Thus $$\label{eq:Y}
\E \sum_{t=1}^T \frac{\epsilon^2 \|y_t\|_2^2}{\sigma_t^2} \leq n^{2/q} \epsilon^2 \E \ \sum_{t=1}^T \ds1\{x_t \leq 1/4\} + 2^4 \epsilon^2 n^{1-2/p} \sum_{t : x_t \geq 1/4} \E (1-|x_t|^p)^{2/p} ~.$$ Observe that one clearly has $\E R_T = \Omega(\E \sum_{t=1}^T \ds1\{x_t \leq 1/4\} )$ and thus without loss of generality we can assume $\E \sum_{t=1}^T \ds1\{x_t \leq 1/4\} = O(n \sqrt{T})$, which means that the first term on the right hand side in is smaller than $n^{1+2/q} \epsilon^2 \sqrt{T} = C^{2/q} n^{1+2/q} T^{1/2-1/q}$. This is smaller than $n$ for $T \geq n^{\frac{2}{1-q/2}}$ and $C$ small enough. For the second term we use that $$\begin{aligned}
\sum_{t : x_t \geq 1/4} \E (1-|x_t|^p)^{2/p} & \leq & p^2 \sum_{t = 1}^T \E (1-|x_t|)^{2/p} \\
& \leq & p^2 T\left( \E \left(1- \frac1{T} \sum_{t=1}^T |x_t|\right) \right)^{2/p} ~,\end{aligned}$$ and because of Lemma \[step2\] one can assume $\frac1{T} \E[\sum_{t=1}^T |x_t| ]\geq 1 - 4 \epsilon^q n$ which means that the second term in is smaller than $\epsilon^2 n^{1-2/p} T (\epsilon^q n)^{2/p}= \epsilon^{2 q} n T = C^2 n$. This concludes the proof of , and thus also concludes the proof of Theorem \[th:LBellp\].
Proof of Theorem \[th:starved\]
-------------------------------
We only give a brief proof sketch. The starved multi-armed bandit lower bound is standard and can be written succintly as follows. Consider random losses, where say action $1$’s loss is a Bernoulli of parameter $1/2$ plus or minus $\epsilon$, action $2$ is a Bernoulli of parameter $1/2$, and all the other actions always give a loss of $1$. Denote by $E$ the expected number of exploration rounds, i.e. rounds where the player plays from $\mu$. It is a standard calculation that if $E/n \leq c / \epsilon^2$ for some sufficiently small constant $c$, then the regret is at least $\epsilon T$. On the other hand the regret is always larger than $\frac{n-2}{n} E /2$. Thus by setting $\epsilon^2 = c n / E$ we have a regret lower bounded by (up to constant), with $a$ such that $a= (1-a) \frac{1}{2}$ (i.e., $a=1/3$): $$\max\left(E, \left(\frac{n}{E}\right)^{1/2} T \right) \geq n^a T^{1-a} ~.$$ Essentially the same argument applies to the $\ell_1^n$ ball, we omit the details. We now turn to the case of $\ell_p^n$ balls with $p>2$.
We see from (observe that in the starved setting the sum over all $t \in [T]$ in this equation is replaced by the sum over rounds $t$ where one plays from $\mu$) that if $n^{2/q} \epsilon^2 E \leq c n$ for some sufficiently small constant $c$, then the regret is at least $\epsilon^q n T$ (per Lemma \[step1\]). Moreover the regret is also always larger than $E$. Thus by setting $\epsilon^2 = c n^{1-2/q} / E$ (i.e., $\epsilon^q n = C (n / E)^{q/2}$) we have a regret lower bounded by (up to a constant), with $a$ such that $a = (1-a) q/2$, $$\max\left(E, \left(\frac{n}{E}\right)^{q/2} T \right) \geq n^a T^{1-a} ~,$$ which concludes the proof.
[^1]: This work was done while M. B. Cohen and Y. Li were at Microsoft Research.
[^2]: Note that the terms sparsity and curvature in the paper’s title apply respectively to the losses and the action set. They could also apply respectively to the action set and to the losses, see e.g. [@LLZ09] and [@HL14]. We do not consider these (very different) settings here.
[^3]: We note however that for [*non-negative*]{} losses (which should intuitively be a much easier case than say sparse [*non-positive*]{} losses, a.k.a. sparse gains), Kwon and Perchet already answered positively the question, see Section \[sec:obstacles\].
[^4]: This conjecture was mentioned in talks related to [@BCK12].
[^5]: The logarithmic barrier was recently used as a regularizer for bandits in [@Foster16] to obtain first order regret bounds. We note however that the behavior of our hybrid regularizer is fundamentally different from using only the log-barrier term.
[^6]: Notice that one cannot simply shift the losses as this could potentially suppress sparsity.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using the existing classification of all alternatives to the measurement postulates of quantum theory we study the properties of bi-partite systems in these alternative theories. We prove that in all these theories the purification principle is violated, meaning that some mixed states are not the reduction of a pure state in a larger system. This allows us to derive the measurement postulates of quantum theory from the structure of pure states and reversible dynamics, and the requirement that the purification principle holds. The violation of the purification principle implies that there is some irreducible classicality in these theories, which appears like an important clue for the problem of deriving the Born rule within the many-worlds interpretation. We also prove that in all such modifications the task of state tomography with local measurements is impossible, and present a simple toy theory displaying all these exotic non-quantum phenomena. This toy model shows that, contrarily to previous claims, it is possible to modify the Born rule without violating the no-signalling principle. Finally, we argue that the quantum measurement postulates are the most non-classical amongst all alternatives.'
author:
- 'Thomas D. Galley'
- Lluis Masanes
bibliography:
- 'refs.bib'
title: Any modification of the Born rule leads to a violation of the purification and local tomography principles
---
Introduction
============
The postulates of quantum theory describe the evolution of physical systems by distinguishing between the cases where observation happens or not. However, these postulates do not specify what constitutes observation, and it seems that an act of observation by one agent can be described as unperturbed dynamics by another [@Everett_relative_1957]. This opens the possibility of deriving the physics of observation within the picture of an agent-free universe that evolves unitarily. This problem has been studied within the dynamical description of quantum measurements [@Allahverdyan_understanding_2013], the decoherence program [@Zurek_probabilities_2005] and the Many-Worlds Interpretation of quantum theory [@Deutsch_quantum_1999; @Wallace_how_2010].
In this work, instead of presenting another derivation of the measurement postulates, we take a more neutral approach and analyze all consistent alternatives to the measurement postulates. In particular, we prove that in each such alternative there are mixed states which are not the reduction of a pure state on a larger system. This property singles out the (standard) quantum measurement postulates including the Born Rule.
In our previous work [@Galley_classification_2017] we constructed a complete classification of all alternative measurement postulates, by establishing a correspondence between these and certain representations of the unitary group. However, this classification did not involve the consistency constraints that arise from the compositional structure of the theory; which governs how systems combine to form multi-partite systems. In this work we take into account compositional structure, and prove that all alternative measurement postulates violate two compositional principles: purification [@Chiribella_probabilistic_2010; @Chiribella_informational_2011] and local tomography [@Hardy_quantum_2001; @Barrett_information_2005]. We also present a simple alternative measurement postulate (a toy theory) which illustrates these exotic phenomena. Additionally, this toy theory provides an interesting response to the claims that the Born rule is the only probability assignment consistent with no-signalling [@Aaronson_quantum_2004; @bao_grover_2016; @Han_Quantum_2016].
In Section \[Setup\] we introduce a theory-independent formalism, which allows to study all alternatives to the measurement postulates. We also review the results of our previous work [@Galley_classification_2017]. In Section \[features\] we define the purification and local tomography principles, and show that these are violated by all alternative measurement postulates. In Section \[Toymodel\] we describe a particular and very simple alternative measurement postulate, which illustrates our general results. In Section \[Discussion\] we discuss our results in the light of existing work. All proofs are in the appendices.
Dynamically-quantum theories {#Setup}
============================
In this work we consider all theories that have the same pure states, dynamics and system-composition rule as quantum theory, but have a different structure of measurements and a different rule for assigning probabilities.
States, transformations and composition postulates
--------------------------------------------------
The family of theories under consideration satisfy the following postulates, taken from the standard formulation of quantum theory.
Every finite-dimensional Hilbert space $\mathbb C^d$ corresponds to a type of system with pure states being the rays $\psi$ of $\mathbb C^d$.
The reversible transformations on the pure states $\mathbb C^d$ are $\psi \mapsto U\psi$ for all $U\in \sud$.
The joint pure states of systems $\mathbb C^{d_\sA}$ and $\mathbb C^{d_\sB}$ are the rays of $\mathbb C^{d_\sA} \otimes \mathbb C^{d_\sB} \simeq \mathbb C^{d_\sA d_\sB}$.
Measurement postulates
----------------------
Before presenting the generalized measurement postulate we need to introduce the notion of outcome probability function, or OPF. For each measurement outcome $x$ of system $\mathbb C^{d}$ there is a function $F^{(x)}$ that assigns to each ray $\psi$ in $\mathbb C^d$ the probability $F^{(x)} (\psi) \in [0,1]$ for the occurrence of outcome $x$. Any such function $F$ is called an OPF. Each system has a (trivial) measurement with only one outcome, which must have probability one for all states. The uniqueness of this trivial measurement is the Causality Axiom of [@Chiribella_probabilistic_2010]. The OPF associated to this outcome is called the unit OPF $\bf u$, satisfying ${\bf u}(\psi)=1$ for all $\psi$. A $k$-outcome measurement is a list of $k$ OPFs $(F^{(1)}, \ldots, F^{(k)})$ satisfying the normalization condition $\sum_i F^{(i)} =\bf u$. It is not necessarily the case that every list of OPFs satisfying this condition defines a measurement, though this assumption can be made. As an example, the OPFs of quantum theory are the functions $$\label{Q OPFs}
F(\psi) =
{\rm tr} (\hat F {
| \psi \rangle \! \langle \psi |})\ ,$$ for all Hermitian matrices $\hat F$ satisfying $0\leq \hat F \leq \unity$. This implies that $\hat {\bf u} = \unity$. (Here and in the rest of the paper we assume that kets $\ket \psi$ are normalized.)
Every type of system $\mathbb C^d$ has a set of OPFs $\mathcal F_d$ with a bilinear associative product $\star : \mathcal F_{d_\sA} \times \mathcal F_{d_\sB} \to \mathcal F_{d_\sA d_\sB}$ satisfying the following consistency constraints:
- For every $F\in \mathcal F_d$ and $U\in \sud$ there is an $F' \in \mathcal F_d$ such that $F'(\psi) = F(U\psi)$ for all $\psi\in \mathbb C^d$. That is, the composition of a unitary and a measurement can be globally considered a measurement.
- For any pair of different rays $\psi \neq \phi$ in $\mathbb C^d$ there is an $F\in \mathcal F_d$ such that $F(\psi) \neq F(\phi)$. That is, different pure states must be operationally distinguishable.
- The $\star$-product satisfies $\bf u_\sA \star u_\sB = u_{\sA \sB}$ and $$\label{starprod}
(F_\sA \star F_\sB)
(\psi_\sA \otimes \phi_\sB)
= F_\sA (\psi_\sA) F_\sB (\phi_\sB)\ ,$$ for all $F_\sA \in \mathcal F_{d_\sA}$, $F_\sB \in \mathcal F_{d_\sB}$, $\psi_\sA \in \mathbb C^{d_\sA}$, $\phi_\sB \in \mathbb C^{d_\sB}$. That is, tensor-product states $\psi_\sA \otimes \phi_\sB$ contain no correlations.
- For each $\phi_{\sA \sB} \in \mathbb C^{d_\sA} \otimes \mathbb C^{d_\sB}$ and $F_\sB \in \mathcal F_{d_\sB}$ there is an ensemble $\{(\psi^i_\sA, p_i)\}_i$ in $\mathbb C^{d_\sA}$ such that $$\frac
{(F_\sA \star F_\sB)
(\phi_{\sA\sB})}
{({\bf u}_\sA \star F_\sB)
(\phi_{\sA\sB})}
= \sum_i p_i
F_\sA (\psi^i_\sA)
\ ,$$ for all $F_\sA \in \mathcal F_{d_\sA}$. That is, the reduced state on $\sA$ conditioned on outcome $F_\sB$ on $\sB$ (and re-normalized) is a valid mixed state of $\sA$. In the next sub-section we fully articulate the notions of ensemble and mixed state.
- Consider measurements on system $\mathbb C^{d_\sA}$ with the help of an ancilla $\mathbb C^{d_\sB}$. For any ancillary state $\phi_\sB \in \mathbb C^{d_\sB}$ and any OPF in the composite $F_{\sA \sB} \in \mathcal F_{d_\sA d_\sB}$ there exists an OPF on the system $F'_\sA \in \mathcal F_{d_\sA}$ such that $$F'_\sA (\psi_\sA)
=
F_{\sA \sB} (\psi_\sA \otimes \phi_\sB)$$ for all $\psi_\sA$.
The derivation of these consistency constraints from operational principles is provided in Appendix \[OPFCons\]. Continuing with the example of quantum theory , the $\star$-product in this case is $$\label{compo Q}
(F_\sA \star F_\sB) (\psi_{\sA \sB})
=
{\rm tr} (\hat F_\sA \otimes \hat F_\sB {
| \psi_{\sA \sB} \rangle \! \langle \psi_{\sA \sB} |})
\ .$$ A trivial modification of the Measurement Postulate consists of taking that of quantum mechanics and restricting the set of OPFs in some way, such that not all POVM elements $\hat F$ are allowed. In this work, when we refer to “all alternative measurement postulates" we do not include these trivial modifications.
Mixed states and the Finiteness Principle
-----------------------------------------
A source of systems that prepares state $\psi_i \in \mathbb C^d$ with probability $p_i$ is said to prepare the ensemble $\{(\psi_i, p_i)\}_i$. Two ensembles $\{(\psi_i, p_i)\}_i$ and $\{(\phi_j, q_j)\}_j$ are equivalent if they are indistinguishable $$\sum_i p_i F(\psi_i)
=
\sum_j q_j F(\phi_j)$$ for all measurements $F\in \mathcal F_d$. Note that distinguishability is relative to the postulated set of OPFs $\mathcal F_d$. A mixed state $\omega$ is an equivalence class of indistinguishable ensembles, and hence, the structure of mixed states is also relative to $\mathcal F_d$. To evaluate an OPF $F$ on a mixed state $\omega$ we can take any ensemble $\{(\psi_i, p_i)\}_i$ of the equivalence class $\omega$ and compute $$F(\omega) = \sum_i p_i F(\psi_i)\ .$$ In general, ensembles can have infinitely-many terms, hence, the number of parameters that are needed to characterize a mixed state can be infinite too. When this is the case, state estimation without additional assumptions is impossible, and for this reason we make the following assumption.
Each mixed state of a finite-dimensional system $(\mathbb C^d, \mathcal F_d)$ can be characterized by a finite number $K_d$ of parameters.
Recall that in quantum theory we have $K_d = d^2-1$. And in general, the distinguishability of all rays in $\mathbb C^d$ implies $K_d \geq 2 d-2$. These $K_d$ parameters can be chosen to be a fix set of “fiducial" OPFs $F_1, \ldots, F_{K_d} \in \mathcal F_d$, which can be used to represent any mixed state $\omega$ as $$\bar \omega = \left(
\begin{array}{c}
F_1 (\omega) \\
F_2 (\omega) \\
\vdots \\
F_{K_d} (\omega)
\end{array} \right) \ .$$ The fact that OPFs are probabilities implies that any OPF $F\in \mathcal F_d$ is a linear function of the fiducial OPFs $$\label{basis}
F = \sum_i c_i F_i\ .$$ In other words, the fiducial OPFs $F_1, \ldots, F_{K_d} \in \mathcal F_d$ constitute a basis of the real vector space spanned by $\mathcal F_d$. Using the consistency constraint [**C1**]{}, we define the $\sud$ action $$\label{sudaction}
F_i =
\sum_{i'} \bar \Gamma_{i,i'} (U)\, F_{i'}$$ on the vector space spanned by $\mathcal F_d$. This associates to system $(\mathbb C^d, \mathcal F_d)$ a $K_d$-dimensional representation of the group $\sud$. This, together with the other consistency constraints, implies that only certain values of $K_d$ are allowed. For example $K_2 = 3, 7, 8, 10, 11, 12, 14\ldots$
Measurement postulates for single systems
-----------------------------------------
In this subsection we review some of the results obtained in [@Galley_classification_2017]. These provide the complete classification of all sets $\mathcal F_d$ satisfying the Finiteness Principle and the consistency constraints [**C1**]{} and [**C2**]{}. These results ignore the existence of the $\star$-product, [**C3**]{}, [**C4**]{} and [**C5**]{}. Hence, in alternative measurement postulates with a consistent compositional structure there will be additional restrictions on the valid sets $\mathcal F_d$. This is studied in Section \[features\].
\[Caracterisation\] If $\mathcal F_d$ satisfies the Finiteness Principle and [**C1**]{} then there is a positive integer $n$ and a map $F \mapsto \hat F$ from $\mathcal F_d$ to the set of $d^n \times d^n$ Hermitian matrices such that $$F(\psi) = {\rm tr}\!\left( \hat F
|\psi\rangle\!\langle \psi|^{\otimes n}
\right)$$ for all normalized vectors $\psi \in \mathbb C^d$.
Note that there are many different sets $\mathcal F_d$ with the same $n$. In particular, since the $\sud$ action $$\label{red rep}
|\psi\rangle\!\langle \psi|
^{\otimes n}
\mapsto
U^{\otimes n}
|\psi\rangle\!\langle \psi|
^{\otimes n}
U^{\otimes n \dagger}$$ is reducible, the Hermitian matrices $\hat F$ can have support on the different irreducible sub-representations of , generating sets $\mathcal F_d$ with very different physical properties. All these possibilities are analyzed in [@Galley_classification_2017].
If $\mathcal F_d$ satisfies the Finiteness Principle, [**C1**]{} and [**C2**]{}, then
there is a non-constant $F\in \mathcal F_d$.
there is $F\in \mathcal F_2$ such that $\hat F$ has support on a sub-representation of the $\mathrm{SU}(2)$ action with odd angular momentum.
Features of all alternative measurement postulates {#features}
==================================================
In this section we analyze the compositional structure of alternative measurement postulates. We do so by considering two well known physical principles which, together with other assumptions, have been used to reconstruct the full formalism of quantum theory [@Masanes_derivation_2011; @Chiribella_informational_2011; @Dakic_quantum_2011; @Barnum_local_2014]. Remarkably, these principles are violated by all alternative measurement postulates.
The Purification Principle
--------------------------
This principle establishes that any mixed state is the reduction of a pure state in a larger system. This legitimises the “Church of the Larger Hilbert Space", an approach to physics that always assumes a global pure state when an environment is added to the systems under consideration [@Nielsen_quantum_2011].
For each ensemble $\{(\psi_i, p_i)\}_i$ in $\mathbb C^{d_\sA}$ there exists a pure state $\phi_{\sA \sB}$ in $\mathbb C^{d_\sA} \otimes \mathbb C^{d_\sB}$ for some $d_\sB$ satisfying $$(F_{\sA} \star {\bf u}_{\sB})(\phi_{\sA \sB})
=
\sum_i p_i F_{\sA} (\psi_i) \ ,$$ for all $F_{\sA} \in \mathcal F_{d_\sA}$.
Note that the original version of the purification principle introduced in [@Chiribella_probabilistic_2010] additionally demands that the purification state $\phi_{\sA \sB}$ is unique up to a unitary transformation on $\mathbb C^{d_\sB}$. Also note that the following theorem does not require the Finiteness Principle.
All alternative measurements postulates $\mathcal F_d$ satisfying [**C1**]{}, [**C2**]{}, [**C3**]{} and [**C4**]{} violate the purification principle.
This implies that in all alternative measurement postulates there are operational processes, such as mixing two states, which cannot be understood as a reversible transformation on a larger system. In such alternative theories, agents that perform physical operations cannot be integrated in an agent-free universe, as can be done in quantum theory, and the Church of the Larger Hilbert Space is illegitimate. The assumption that agents’ actions can be understood as reversible transformations on a larger system is also the starting point of the many-worlds interpretation of quantum theory. Hence, it seems like the no-purification theorem provides important clues for the derivation of the Born Rule within the many-worlds interpretation [@Everett_relative_1957; @Zurek_probabilities_2005; @Wallace_how_2010; @Deutsch_quantum_1999]. In section \[Zurek\] we discuss further the possibility of using this result to derive the quantum measurement postulates, and contrast it to Zurek’s envariance based derivation of the Born rule.
The Local Tomography Principle
------------------------------
This principle has been widely used in reconstructions of quantum theory and the formulation of alternative toy theories [@Dakic_quantum_2011; @Masanes_derivation_2011; @Chiribella_informational_2011; @Barnum_local_2014; @Masanes_existence_2013; @Hohn_toolbox_2017]. One of the reasons is that it endows the set of mixed states with a tensor-product structure [@Barrett_information_2005]. This principle states that any bi-partite state is characterized by the correlations between local measurements. That is, two different mixed states $\omega_{\sA\sB} \neq \omega'_{\sA\sB}$ on $\mathbb C^{d_\sA} \otimes \mathbb C^{d_\sB}$ must provide different outcome probabilities $$(F_{\sA} \star F_{\sB})(\omega_{\sA\sB})
\neq
(F_{\sA} \star F_{\sB})(\omega'_{\sA\sB})$$ for some local measurements $F_{\sA} \in \mathcal F_{d_\sA}$, $F_{\sB} \in \mathcal F_{d_\sB}$. Using the notation introduced in we can formulate this principle as follows.
If $\{F_a \}_a$ is a basis of $\mathcal F_{d_\sA}$ and $\{F_b \}_b$ is a basis of $\mathcal F_{d_\sB}$ then $\{F_a \star F_b \}_{a,b}$ is a basis of $\mathcal F_{d_\sA d_\sB}$, where $a=1, \ldots, K_{d_\sA}$ and $b=1, \ldots, K_{d_\sB}$.
A theory is said to violate local tomography if at least one composite system within the theory violates local tomography. Therefore, it is sufficient to analyze the particular bi-partite system $\mathbb C^3 \otimes \mathbb C^3 = \mathbb C^9$.
All alternative measurements $\mathcal F_3$ and $\mathcal F_9$ satisfying [**C1**]{}, [**C2**]{}, [**C3**]{} and the Finiteness Principle violate the local tomography principle.
The above result is proven in Appendix \[nolocaltom\]. We first show that all transitive theories which obey the local tomography principle have a group action acting on the mixed states which has a certain structure. Any group action of the form which does not have this structure must correspond to a system which violates local tomography. We show that all representations of $\sunine$ which correspond to systems with alternative measurement postulates do not have this structure. This entails that all non-quantum $\C^9$ systems which are composites of two $\C^3$ systems violate local tomography.
The technical result proven to show this is the following. All non-quantum irreducible representations of $\sunine$ which are sub-representations of the action have a sub-representation $1^{3} \otimes 1^{3}$ when restricted to the subgroup $\suthree \times \suthree$. Here $1^3$ denotes the trivial representation of $\suthree$.
### Comment on $\mathbb{C}^2 \otimes \mathbb{C}^2$ systems
In quantum theory any $\C^d$ ($d \geq 2$) system can be simulated using some number of qubits. In this sense qubits can be viewed as fundemental information units [@Masanes_existence_2013]. Since $\mathbb C^2$ systems have a priveleged status it is natural to ask whether $\mathbb{C}^4 = \mathbb{C}^2 \otimes \mathbb{C}^2$ systems in theories with modified measurement postulates are locally tomographic. The proof technique used for the No Local Tomography Theorem applies only to a certain family of these $\mathbb{C}^4$ systems. However all instances of $\mathbb{C}^4$ systems studied by the authors which were not part of this family were found to not be locally tomographic. We conjecture that all $\mathbb{C}^4 = \mathbb{C}^2 \otimes \mathbb{C}^2$ violate the local tomography principle.
A Toy Theory {#Toymodel}
============
In this section we present a simple alternative measurement postulate $(\mathcal F_d, \star)$ which serves as example for the results that we have proven in general (violation of the purification and local tomography Principles). In Appendix \[toymodel\] it is proven that this alternative measurement postulate satisfies all consistency constraints ([**C1**]{}, [**C2**]{}, [**C3**]{}, [**C4**]{}, [**C5**]{}) except for the associativity of the $\star$-product. This implies that this toy theory is only fully consistent when dealing with single and bi-partite systems. However, most of the work in the field of general probabilistic theories (GPTs) focuses on bi-partite systems, because these already display very rich phenomenology. In the following we consider two local subsystems of dimension $d_\sA$ and $d_\sB$ with sets of OPFs $\mathcal F_{d_\sA}^\sl$ and $\mathcal F_{d_\sB}^\sl$. The composite (global) system has a set of OPFs $\mathcal F_{d_\sA d_\sB}^\sg$.
Let $S$ be the projector onto the symmetric subspace of $\mathbb C^d \otimes \mathbb C^d$. To each $d^2 \times d^2$ Hermitian matrix $\hat F$ satisfying
- $0\leq \hat F \leq S$,
- $\hat F = \sum_i \alpha_i {
| \phi_i \rangle \! \langle \phi_i |}^{\otimes 2}$ for some $\ket{\phi_i} \in \mathbb C^d$ and $\alpha_i >0$,
- $S-\hat F = \sum_i \beta_i {
| \varphi_i \rangle \! \langle \varphi_i |}^{\otimes 2}$ for some $\ket{\varphi_i} \in \mathbb C^d$ and $\beta_i >0$,
there corresponds the OPF $$\label{toy F}
F (\psi) =
{\rm tr} \left(\hat F {
| \psi \rangle \! \langle \psi |}^{\otimes 2} \right)
\ ,$$ The unit OPF corresponds to $\hat {\bf u} =S$.
That is, both matrices, $\hat F$ and $S-\hat F$, have to be not-necessarily-normalized mixtures of symmetric product states.
For the case where $d$ is prime there exists a canonical measurement which can be constructed as follows. Consider the $(d+1)$ mutually unbiased bases (MUBs): $\{ \ket{\phi_i^j} \}_{i=1}^d$ where $j$ runs from $1$ to $d+1$ [@Bandyopadhyay_new_2001]. Then we can associate an OPF to each Hermitian matrix $\frac{1}{2} {
| \phi_i^j \rangle \! \langle \phi_i^j |}^{\otimes 2}$. Since the basis elements of these MUBs form a complex projective 2-design [@Klappenecker_mutually_2005], by the definition of 2-design [@Zhu_clifford_2016], we have the normalization constraint: $$\frac{1}{2} \sum_{i,j} {
| \phi_i^j \rangle \! \langle \phi_i^j |}^{\otimes 2} = S \ ,$$ and hence the set of OPFs forms a measurement.
For any pair of OPFs $F_{\sA} \in \mathcal F_{d_\sA}^\sl$ and $F_{\sB} \in \mathcal F_{d_\sB}^\sl$ the Hermitian matrix corresponding to their product $F_\sA \star F_\sB \in \mathcal F_{d_\sA d_\sB}^\sg$ is $$\label{toy * product}
\widehat{F_\sA \star F_\sB} =
\hat F_\sA \otimes \hat F_\sB +
\frac {{\rm tr}\hat F_\sA}{{\rm tr} S_\sA} A_\sA \otimes
\frac {{\rm tr}\hat F_\sB}{{\rm tr} S_\sB} A_\sB
\ ,$$ where $S_\sA$ and $A_\sA$ are the projectors onto the symmetric and anti-symmetric subspaces of $\mathbb C^{d_\sA} \otimes \mathbb C^{d_\sA}$, and analogously for $S_\sB$ and $A_\sB$.
This product is clearly bilinear and, by using the identity $S_{\sA\sB} = S_\sA \otimes S_\sB + A_\sA \otimes A_\sB$, we can check that ${\bf u}_\sA \star {\bf u}_\sA = {\bf u}_{\sA\sB}$.
We observe that not all effects $\widehat{F_\sA \star F_\sB}$ are of the form $\sum_i \alpha_i {
| \phi_i \rangle \! \langle \phi_i |}_{\sA \sB}^{\otimes 2}$. Hence the set of effects on the joint system is not $\mathcal F_{d_\sA d_\sB}^\sl$, but has to be extended to $\mathcal F_{d_\sA d_\sB}^\sg$ to include these joint product effects.
The set $\mathcal F_{d_\sA d_\sB}^\sg$ should include all product OPFs $\widehat{F_\sA \star F_\sB}$, all OPFs $\mathcal F_{d_\sA d_\sB}^\sl$ of $\mathbb C^{d_\sA d_\sB}$ understood as a single system, and their convex combinations.
The identity $S_{\sA\sB} = S_\sA \otimes S_\sB + A_\sA \otimes A_\sB$ perfectly shows that the vector space $\mathcal F_{d_\sA d_\sB}^\sg$ is larger than the tensor product of the vector spaces $\mathcal F_{d_\sA}^\sl$ and $\mathcal F_{d_\sB}^\sl$, by the extra term $A_\sA \otimes A_\sB$. This implies that this toy theory violates the Local-Tomography Principle.
The joint probability of outcomes $F_\sA$ and $F_\sB$ on the entangled state $\psi_{\sA \sB} \in \mathbb C^{d_\sA} \otimes \mathbb C^{d_\sB}$ can be written as $$\begin{aligned}
& (F_\sA \star F_\sB) (\psi_{\sA \sB}) \\
= & {\rm tr}\! \left[ \left(\hat F_\sA \otimes \hat F_\sB + \mbox{$\frac {{\rm tr}\hat F_\sA}{{\rm tr} S_\sA}$} A_\sA \otimes
\mbox{$\frac {{\rm tr}\hat F_\sB}{{\rm tr} S_\sB}$} A_\sB \right)
\!{
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right] \ .\end{aligned}$$ When we only consider sub-system $\sA$ outcome probabilities are given by [$$\begin{aligned}
\nonumber
& & (F_\sA \star {\bf u}_\sB) (\psi_{\sA \sB})
\\ &=&
{\rm tr}\! \left[ \left(\hat F_\sA \otimes S_\sB + \mbox{$\frac {{\rm tr}\hat F_\sA}{{\rm tr} S_\sA}$} A_\sA \otimes
A_\sB \right) {
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right]
\\ &=&
{\rm tr}_\sA\! \left[ \hat F_\sA\, \bar \omega_\sA \right] \ ,\end{aligned}$$ ]{} where the reduced state must necessarily be [$$\label{reduced state}
\bar \omega_\sA
=
{\rm tr}_\sB \! \left( S_\sB {
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right)
+ \frac {S_\sA}{{\rm tr} S_\sA}
{\rm tr} \!
\left(A_\sA A_\sB
{
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right)$$ ]{} All these reductions $\bar\omega _\sA$ of pure bipartite states $\psi_{\sA\sB}$ are contained in the convex hull of ${
| \phi_{\sA} \rangle \! \langle \phi_{\sA} |}^{\otimes 2}$, as required by the consistency constraint ${\bf C4}$. However, not all mixtures of ${
| \phi_{\sA} \rangle \! \langle \phi_{\sA} |}^{\otimes 2}$ can be written as one such reduction . That is, the purification postulate is violated. This phenomenon is graphically shown in Figure \[purificationpic\].
This toy model violates the *no-restriction hypothesis* [@Chiribella_probabilistic_2010], in that not all mathematically allowed effects on the local state spaces are allowed effects. It also violates the principle of *pure sharpness* [@Chiribella_operational_2015] in that all the effects are noisy.
Discussion {#Discussion}
==========
Interpreting results as a derivation of the Born rule {#Zurek}
-----------------------------------------------------
In this paper we have shown that all modifications to the quantum measurements lead to violations of the purification and local tomography principles. This entails that one can derive the measurement postulates of quantum theory from the structure of pure states and dynamics and either the assumption of local tomography or purification. Such a derivation uses the operational framework which can be viewed as a background assumption.
A derivation of the Born rule which starts from similar assumptions to ours, but not within an operational setting, is the envariance based derivation of Zurek [@Zurek_probabilities_2005]. Zurek begins by assuming the dynamical structure of quantum theory and the assumption that quantum theory is *universal*, which is to say that all the phenomena we observe can be explained in terms of quantum systems interacting. Specifically the classical worlds of devices can be modelled quantum mechanically, including the measurement process. We observe that this is philosophically very different to the operational approach adopted in this work, which takes the classical world as a primitive. By assuming the dynamical structure of quantum theory and the assumption of universality (as well as some auxiliary assumptions) Zurek shows that measurements are associated to orthonormal bases, and that outcome probabilities are given by the Born rule. For criticisms of Zurek’s approach we refer the reader to [@Caves_note_2004; @Schlosshauer_zureks_2005; @Barnum_no_2003; @mohrhoff_probabilities_2004].
We observe that the purification postulate seems linked to the notion that quantum theory is universal, in the sense that any classical uncertainty can be explained as originating from some pure global quantum state. This shows an interesting link to Zurek’s derivation, since although we work within an operational framework, the concept of purification is linked to the idea that quantum theory is universal. This shows that we can also rely on a concept linked to universality in order to derive the Born rule (and the structure of measurements) within an operational approach.
We observe that we can also derive the measurement postulates of quantum theory from the assumption of local tomography, which does not have this connotation of universality.
No-signalling
-------------
Multiple proofs have been put forward which claim to show that violations of the Born rule lead to signalling [@Aaronson_quantum_2004; @bao_grover_2016; @Han_Quantum_2016]. However the authors only consider modifications of the Born rule of a specific type. In [@Aaronson_quantum_2004; @bao_grover_2016] the authors only consider modifications of the Born rule of the following form: $$\label{pnormrule}
p(k | \psi ) = \frac{| \! \braket{k|\psi} \! |^n}{\sum_{k'} | \! \braket{k'|\psi} \! |^n} \ ,$$ where $\ket \psi = \sum_k \alpha_k \ket k$. Modifications of this form are very restricted. By modifying all the measurement postulates of quantum theory, we can create toy models like the one introduced, which are non-signalling (as we will show momentarily). This shows that by modifying the Born rule in a more general manner one can avoid issues of signalling.
In the case of the toy model it is immediate to see that it is consistent with no-signalling. The condition of no-signalling is equivalent to the existence of a well defined state-space for the subsystem (i.e. independent of action on the other subsystem). We see then that no-signalling is just a consequence of there existing a well defined reduced state [@Chiribella_probabilistic_2010].
Purification as a constraint on physical theories
-------------------------------------------------
In Theorem 19 of [@Chiribella_probabilistic_2010] the authors show that any two convex theories with the same states (pure and mixed) which obey purification are the same theory. In other words “states specify the theory" for theories with purification [@Chiribella_probabilistic_2010]. In this chapter we show that in the case of theories with pure states ${\rm P}\C^d$ and dynamical group $\sud$, any two theories which obey purification with the same pure states and reversible dynamics are the same. This means that for a restricted family of theories (those with systems with pure states ${\rm P}\C^d$ and dynamical group $\sud$) we have the same result as Theorem 19 of [@Chiribella_probabilistic_2010] but with different assumptions. It would be interesting to establish whether this is a general feature of theories with purification, namely that the pure states and reversible dynamics specify the theory.
The quantum measurement postulates are the most non-classical
-------------------------------------------------------------
One consequence of the classification in [@Galley_classification_2017] is that the quantum measurement postulates are the ones which give the lowest dimensional state spaces. In this sense the quantum measurement postulates are the most non-classical, since they give rise to the state spaces with the higest degree of indistinguishable ensemles [@Mielnik_generalized_1974]. We remember that in classical probability theory all ensembles are distinguishable.
The violation of purification seems to indicate some other, distinct type of classicality. In theories which violate purification there are some preparations which can only be modelled as arising from a classical mixture of pure states. There appears to be some sort of irreducible classicality. However in theories which obey purification we can always model such preparations as arising from the reduction of a global pure state. Since the quantum measurement postulates are the only ones which give rise to systems which obey purification they can be viewed as the most non-classical amongst all alternatives.
Hence we see that according to these two distinct notions of non-classicality the quantum measurement postulates are the most non-classical amonst all possible measurement postulates.
Toy model
---------
The toy model can be obtained by restricting the states and measurements of two pairs of quantum systems $(\mathbb C^{d_\sA})^{\otimes 2}$ and $(\mathbb C^{d_\sB})^{\otimes 2}$. In this sense we obtain a theory which violates both local tomography and purification. This method of constructing theories is similar to real vector space quantum theory, which can also be obtained from a suitable restriction of quantum states and also violates local tomography. The main limitation of the toy model is that it does not straightforwardly extend to more than two systems. There is a natural generalisation of the toy model to consider effects to be linear in ${
| \psi \rangle \! \langle \psi |}^{\otimes n}$ for $n >2$, however showing the consistency of the reduced state spaces and joint effects is more complex.
Theories which decohere to quantum theory
-----------------------------------------
A recent result [@Lee_nogo_2017] shows that all operational theories which decohere to quantum theory (in an analogous way to which quantum theory decoheres to classical theory) must violate either purification or causality (or both). The authors define a hyper-decoherence map which maps states of the post-quantum system to states of a quantum system (embedded within the post-quantum system). This map obeys the following properties:
1. Terminality: applying the map followed by the unit effect is equivalent to just applying the unit effect.
2. Idempotency: applying the map twice is the same as applying it once.
3. The pure states of the quantum subsystem are pure states of the post-quantum system. Similarly the maximally mixed state of the quantum subsystem is the maximally mixed state of the post-quantum system. \[purity\]
Now we ask whether the systems in this paper (which violate purification) can correspond to post-quantum systems which decohere to quantum systems in some reasonable manner. We consider the toy model with pure states ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$ and no restriction on the allowed effects (this toy model is only valid for single systems). Now consider a linear map from ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$ to some embedded quantum system such that the pure states of the system are also of the form ${
| \phi \rangle \! \langle \phi |}^{\otimes 2}$ (by property \[purity\]). This map must be of the form $U^{\otimes 2} {
| \psi \rangle \! \langle \psi |}^{\otimes 2} U^{\dagger \otimes 2}$. Hence its action on the state space ${\rm conv} \left( {
| \phi \rangle \! \langle \phi |}^{\otimes 2} \right)$ gives an identical state space and the map is trivial.[^1] This shows that the only hyper-decoherence maps which obey all the conditions set out by Lee and Selby must be trivial.
As we argue later, it is not immediately obvious that a hyper-decoherence map should obey condition \[purity\]. We now define a hyper-decoherence like map which does not meet this condition. Consider the following map where we label the copies of ${
| \psi \rangle \! \langle \psi |}$, $1$ and $2$: $$\begin{aligned}
\label{decmap}
\D({
| \psi \rangle \! \langle \psi |}^{\otimes 2}) & = {\rm Tr}_2 ({
| \psi \rangle \! \langle \psi |}^{\otimes 2}) \otimes {
| 0 \rangle \! \langle 0 |} \\
& = {
| \psi \rangle \! \langle \psi |} \otimes {
| 0 \rangle \! \langle 0 |} \ .\end{aligned}$$
This meets properties 1 and 2, but not 3. Indeed the states ${
| \psi \rangle \! \langle \psi |} \otimes {
| 0 \rangle \! \langle 0 |}$ are not valid mixed states of the post-quantum system. The image of this map ${
| \psi \rangle \! \langle \psi |} \otimes {
| 0 \rangle \! \langle 0 |}$ for all $\psi \in {\rm P} \C^d$ is a quantum state space.
The authors of [@Lee_nogo_2017] show that requirement 3 can be replaced by the requirement that the hyper-decoherence map maps between systems with the same information dimension. As shown in [@Galley_classification_2017] (for the case ${\rm P} \C^2$) the information dimension of an unrestricted ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$ state space is larger than that of a qubit.
The hyper-decoherence map of [@Lee_nogo_2017] is inspired by the decoherence map which exists between quantum and classical state spaces. The map introduced in this section is very different from this, since its image is not an embedded state space; however it may be that decoherence between a post-quantum theory and quantum theory is very different from what our quantum/classical intuitions might lead us to believe.
The decoherence map of equation appears strange at first, since it maps states of a post-quantum system to a sub-system which is embedded in such a way that its states are not valid states of the post-quantum system. However the image of this decoherence map is actually the state space one would obtain if one had access to the post-quantum system ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$ but only a restricted set of measurements. An observer with access to the post-quantum system ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$ but only effects of the form $F(\psi) = {\rm Tr}((\hat F \otimes I){
| \psi \rangle \! \langle \psi |}^{\otimes 2})$ would reconstruct a quantum state space. Hence the link between the system and subsystem becomes clearer: the quantum sub-system is obtained from the post-quantum system by restricting the measurements on the post-quantum system. We emphasise once more that the toy model with unrestricted effects does not compose.
Conclusion
==========
Summary
-------
We have studied composition in general theories which have the same dynamical and compositional postulates as quantum theory but which have different measurement postulates. We presented a toy model of a bi-partite system with alternative measurement rules, showing that composition is possible in such theories. We showed that all such theories violate two compositional principles: local tomography and purification.
Future work
-----------
The toy model introduced in this work applies only to bi-partite systems and is simulable with quantum theory. Hence an important next step is constructing a toy model with alternative measurement rules which is consistent with composition of more than two systems. This requires a $\star$ product which is associative. This construction may prove impossible, or it may be the case that all valid constructions are simulable by quantum theory. We suggest that there are three possibilities when considering theories with fully associative products.
[*Possibility 1. (Logical consistency of postulates of Quantum Theory).*]{} The only measurement postulates which are fully consistent with the associativity of composition are the quantum measurement postulates.
If this were the case it would show that the postulates of quantum theory are not independent. Only the quantum measurement rules would be consistent with the dynamical and compositional postulates (and operationalism). However it may be the case that we can develop theories with alternative Born rules which compose with an associative product, but that all these theories are simulable by quantum theory.
[*Possibility 2. (Simulability of systems in theories with alternative postulates).*]{} The only measurement postulates which are fully consistent with the dynamical postulates of quantum theory describe systems which are simulable with a finite number of quantum systems.
[*Possibility 3. (Non-simulability of systems in theories with alternative postulates).*]{} There exist measurement postulates which are fully consistent with the dynamical postulates of quantum theory which describe systems which are not simulable with a finite number of quantum systems.
This final possibility would be interesting from the perspective of GPTs as it would show that there are full theories which can be obtained by modifying the measurement postulates of quantum theory. It would show that quantum theory is not, in fact, an island in theory space.
Acknowledgements
================
We are grateful to Jonathan Barrett and Robin Lorenz for helpful discussions. We thank the referees for their comments which helped substantially improve the first version of this manuscript. TG is supported by the Engineering and Physical Sciences Research Council \[grant number EP/L015242/1\]. LM is funded by EPSRC.
Operational principles and consistency constraints {#OPFCons}
==================================================
Single system
-------------
As stated in the main section the allowed sets of OPFs $\mathcal F_d$ are subject to some operational constraints. We introduce features obeyed by operational theories and derive the consistency constraints [**C1. - C5.**]{} from them. In the following we adopt a description of operational principles in terms of circuits, as in [@Chiribella_informational_2011]. The basic operational primitives are preparations, transformations and measurements. These three are all procedures. We define them in terms of inputs, outputs and systems.
Any procedure which has no input and outputs one or more systems is a preparation procedure.
(0,0) rectangle (1,1) node\[midway\] [$\mathcal P$]{}; (1,0.5) – (2,0.5);
Any procedure which inputs one or more systems and outputs one or more systems is a transformation.
Any procedure which inputs on or more systems and has no output is a measurement procedure.
In the above “no input" and “no output" refers to output or input of systems, typically there will be a classical input or output such as a measurement read-out.
The composition of a transformation with any procedure is itself a procedure of that kind.
For example the composition of a transformation and a measurement is itself a measurement.
The process of taking two procedures of the same kind and implementing them probabilistically generates a procedure of that kind.
An experiment is a sequence of procedures which has no input or output.
This principle entails that every experiment can be considered as a preparation and a measurement. The experiment is fully characterised by the probabilities $$p(\mathcal O | \mathcal P) \ ,$$ for all measurement outcomes $\mathcal O$ and all preparations $\mathcal P$ in the experiment.
In this approach a system is an abstraction symbolised by the wire between the preparation procedure and the measurement procedure. A system $\sA$ can be represented as:
Pairs of systems
----------------
Given the above definitions it is natural to ask when an experiment can be described using multiple systems. Let us consider a system which we represent using two wires:
These can only be considered as representing two distinct systems $\sA$ and $\sB$ if it is possible to independently perform operations (transformations and measurements) on both systems.
A system can be considered as a valid composite system if an operation on subsystem $\sA$ and an operation on subsystem $\sB$ uniquely specify an operation on $\sA \sB$ independent of the temporal ordering.
If the above property is not met, then the system cannot be considered as a composite (and should be represented using a single wire). Diagrammatically this entails that any preparation of a composite system is such that:
Every measurement outcome $\mathcal O_\sA$ on $\sA$ and $\mathcal O_\sB$ on $\sB$ defines a unique outcome $(\mathcal O_\sA , \mathcal O_\sB)$ on $\sA \sB$.
The most general form of an experiment with two systems is:
which can naturally be viewed as an experiment on a single system $\sA \sB$.
Two independent preparations $\mathcal P_\sA$ and $\mathcal P_\sB$ which are independently measured with outcomes $\mathcal O_\sA$ and $\mathcal O_\sB$ are such that: $$p(\mathcal O_\sA ,\mathcal O_\sB| \mathcal P_\sA , \mathcal P_\sB ) = p(\mathcal O_\sA | \mathcal P_\sA ) p(\mathcal O_\sB | \mathcal P_\sB ) \ ,$$ In this case the joint procedures $\mathcal P_\sA , \mathcal P_\sB$ and $\mathcal O _\sA , \mathcal O_\sB$ are said to be separable.
By the definition of a preparation, any operational procedure which outputs a system is a preparation. Hence consider the case where the measurement is separable. The procedure of making a joint preparation and making a measurement on $\sB$ is a preparation of a state $\sA$.
Operationally Alice can make a preparation of system $\sA$ by making a preparation of $\sA \sB$ and getting Bob to make a measurement on system $\sB$.
By the definition of a measurement any operational procedure which inputs a system and outputs no system is a measurement. Consider the case where the preparation is separable. Then the procedure of preparing system $\sB$ and jointly measuring $\sA$ and $\sB$ is a measurement procedure on $\sA$.
A valid measurement for Alice consists in adjoining her system to an ancillary system $\sB$ and carrying out a joint measurement.
In the case where both preparation and measurement are separable then the experiment can be viewed as two separate experiments. In the bi-partite case there are no further methods of generating preparations and measurements. Hence when determining whether a pair of systems is consistent with the operational properties which arise from composition, these are the only features which we need to consider.
One may ask whether any further operational implications will emerge from considering more than two systems.
The systems $(\sA \sB) \sC$ and $\sA (\sB \sC)$ are the same.
This implies that there are no new types of procedures which can be carried out on a single system by appending more than one system. Consider an experiment with multiple systems $\sA, \sB , \sC... $. For any partitioning of the experiment which creates a preparation of system $\sA$, all regroupings of systems $\sB, \sC , ...$ are equivalent. This is a preparation by steering of system $\sA$ conditional on a measurement on systems $\sB , \sC , ... $ which can be viewed as a single system. Diagrammatically it tells us that all ways of partitioning an experiment with multiple systems are equivalent.
This entails the only constraints imposed by the operational framework will come from the assumptions and implications outlined above. There are no further operational implications which emerge from the above definitions and assumptions.
In the next section we translate the operational features above into the language of OPFs, and show which constraints they impose on the OPF sets $\F_d$.
Consistency constraints
-----------------------
We assume the Finiteness Principle holds, and that for a set of OPFs $\mathcal F_d$ there exists a finite linearly generating set $\{F_i\}_{i = 1}^{K_d}$. That is to say: $$F= \sum_i c_i F_i \ .$$
### Constraint C1
Consistency constraint [**C1**]{} follows directly from the fact that the composition of a transformation and a measurement is a measurement.
### Constraint C2
A state corresponds to an equivalence class of indistinguishable preparation procedures.
From this definition it follows that two states cannot be indistinguishable. This implies [**C2**]{}. In a system where some pure states are indistinguishable the manifold of pure states would no longer be the set of rays on $\mathbb C^d$ (as required by the first postulate).
### Constraint C3
Consider a composite system $\sA \sB$. By the definition of a composite system above, for any OPF $F_\sA$ on $\sA$ and $F_\sB$ on $\sB$ there exists an OPF $F_\sA \star F_\sB$ on $\sA \sB$. By the assumption that mixing is possible, the outcome $\{p_i, F_\sA^i\}$ is a valid outcome.
If two parallel processes are separable, it is equivalent to mix them before they are considered as a joint process or after.
From the above assumption it follows that: $$\begin{aligned}
(\sum_i p_i F_\sA^i) \star F_\sB = \sum_i p_i (F_\sA^i \star F_\sB) \\
F_\sA \star(\sum_i p_i F_\sB^i) = \sum_i p_i (F_\sA \star F_\sB^i) \end{aligned}$$ If we further assume that it is possible to mix with subnormalised probabilities, i.e. $\sum_i p_i \leq 1$ then the $\star$ product is bi-linear. The identity ${\bf u}_\sA \star {\bf u}_\sB = {\bf u}_{\sA \sB}$ follows from the fact that a separable measurement is a valid measurement on $\sA \sB$, and hence: $${\bf u}_{\sA \sB} = \sum_{i,j} (F_\sA^i \star F_\sB^j) = {\bf u}_\sA \star {\bf u}_\sB \ .$$
Given two systems $\sA$ and $\sB$ independently prepared in pure states $ \psi_\sA$ and $ \phi_\sB$ the joint state of the system $\sA \sB$ is given by $\psi_\sA \otimes \phi_\sB$.
This principle, together with the definition of independent systems implies that: $$(F_\sA \star F_\sB)(\psi_\sA \otimes \phi_\sB) = F_\sA(\psi_\sA) F_\sB(\phi_\sB)$$
### Constraint C4
Let us consider the steering scenario. In this case the process of both Alice and Bob making a measurement with outcome $F_\sA \star F_\sB$ on a joint state $\phi_{\sA \sB}$ can be considered as a measurement with outcome $F_\sA$ on system $\sA$ prepared in a certain manner.
An arbitrary preparation of system $\sA$ is given by $\{p_i , \psi^i_\sA \}$. Hence the steering preparation implies that for each preparation of $\sA \sB$ and for each local measurement outcome on $\sB$ there exists a state $\{p_i , \psi^i_\sA \}$ in which system $\sA$ is prepared. In the OPF formalism this means that for every $\phi_{\sA\sB} \in \mathbb{C}^{d_\sA d_\sB}$ and every $F_\sB \in \mathcal F_{d_\sB}$ there exists an ensemble $\{p_i \phi^i_\sA\}$ such that $$\frac{(F_\sA \star F_\sB)(\phi_{\sA \sB})}{({\bf u}_\sA \star F_\sB)(\phi_{\sA \sB})} = \sum_i p_i F_\sA (\psi^i_\sA) \ ,$$ holds for all $F_\sA \in \mathcal F_{d_\sA}$. The normalisation occurs due to the fact that summing over the measurement outcomes $F_\sA$ should give unity on both sides of the expression.
### Constraint C5
Let us consider the scenario consisting in measuring with an ancilla. In this case Alice and Bob carry out a joint measurement with outcomes $F_{\sA \sB}$ on a system in an uncorrelated state $\psi_\sA \otimes \phi_\sB$. This should correspond to a valid measurement with outcome $F_\sA'$ on system $\sA$ prepared in state $\psi_\sA$. For each $F_{\sA \sB} \in \mathcal F_{d_\sA d_\sB}$ and for each $\phi_\sB \in \mathbb C^{d_\sB}$ there exists an $F_\sA' \in \mathcal F_{d_\sA}$ such that $$F_{\sA \sB}(\psi_\sA \otimes \phi_\sB) = F_\sA'(\psi_\sA) \ ,$$ for all $\psi_\sA \in \mathbb C^{d_\sA}$.
Violation of purification
=========================
In this appendix we show that all alternative measurement postulates violate purification (for an arbitrary choice of ancillary system dimension).
As shown in [@Galley_classification_2017] the representations $\Gamma^{d}$ corresponding to alternative measurement postulates for systems with pure states $\pcd$ ($d>2$) are of the form $$\label{generalrep}
\Gamma = \bigoplus_{j\in \mathcal J}
\mathcal D^d_j\ ,$$ where $\mathcal J$ is a list of non-negative integers (at least one of which is not 0 or 1) and $\mathcal D_j^d$ are representations of $\sud$ labelled by Young diagrams $(2j, \underbrace{j, \ldots,j}_{d-2})$.
Consider a system $\mathcal S_{\sA\sB} = \{\pcdAdB , \Gamma_{\sA\sB} \}$ which is the composite of two systems $S_{\sA} = \{\pcdA , \Gamma_{\sA} \}$ and $S_{\sB} = \{\pcdB , \Gamma_{\sB} \}$. Here the representations $\Gamma$ are of the form . Let us define the following equivalence classes of pure global states: $$[\ket{\psi}_{\sA\sB}]_{U_\sB} = \{ \ket{\psi}_{\sA\sB}' \in \pcdAdB |\ket{\psi}_{\sA\sB}' = \unity_\sA \otimes U_\sB \ket{\psi}_{\sA\sB} \}$$ All members of the same equivalence class are necessarily mapped to the same reduced state of Alice. Otherwise, Bob could signal to Alice. We note that [@Barnum_no_2003] makes use of this observation in a similar context. Let us call the set of all these equivalence classes $R_\sB$. $$R_\sB := \{ [\ket{\psi}_{\sA\sB}]_{U_\sB} | \ket{\psi}_{\sA\sB} \in \mathbb C^{d_\sA d_\sB} \}$$ The map from global states to reduced states can be defined on the equivalence classes $ [\ket{\psi}_{\sA\sB}]_{U_\sB}$ since two members of the same equivalence class are always mapped to the same reduced states. $\mathcal R: R_\sB \rightarrow \mathcal S_\sA$ is the map from equivalence classes to reduced states: $$\mathcal R([\ket{\psi}_{\sA\sB}]_{U_\sB}) = \bar \omega_\sA (\ket{\psi}_{\sA\sB}) \ ,$$ where $\bar \omega_\sA (\ket{\psi}_{\sA\sB})$ is the reduced state obtained in the standard manner from the global state $\ket{\psi}_{\sA\sB}$ (as outlined in the following appendix). Next we prove that the image of $\mathcal R$ is smaller than $\mathcal S_\sA$ for any non-quantum measurement postulates. In other words there are some (local) mixed states in $\mathcal S_\sA$ which are not reduced states of the global pure states $\ket \psi_{\sA \sB}$.
In the Schmidt decomposition a state $\ket{\psi}_{\sA\sB}$ is: $$\ket{\psi}_{\sA\sB} = \sum_{i=1}^{d_A} \lambda_i \ket{i}_\sA \ket{i}_\sB , \lambda_i \in \mathbb R \ , \ \sum_i \lambda_i^2 = 1 \ ,$$ where we assume that the Schmidt coefficients are in decreasing order $\lambda_i \geq \lambda_{i+1}$. Two states with the same coefficients and the same basis states on Alice’s side belong to the same equivalence class $[\ket{\psi}_{\sA\sB}]_{U_\sB}$. Also, two Alice’s basis differing only by phases (e.g. $\{ \ket i_\sA \}$ and $\{ e^{i \theta_i}\! \ket i_\sA \}$) give rise to the same equivalence class. Because the phases $e^{i \theta_i}$ can be absorbed by Bob’s unitary.
Let us count the number of parameters that are required to specify an equivalence class in $R_\sB$. First, we have the $d_A-1$ Schmidt coefficients. Second, we note that the number of parameters to specify a basis in $\mathbb C^{d_\sA}$ is the same as to specify an element of U$(d_\sA)$. Which is the dimension of its Lie algebra, $d_\sA^2$, the set of anti-hermitian matrices. Third, we have to subtract the $d_\sA$ irrelevant phases $\theta_i$. The three terms together give $$(d_\sA -1)+ d_\sA^2 -d_\sA
= d_\sA^2-1$$ Hence $d_\sA^2 -1$ parameters are needed to specify elements of $R_\sB$. The set ${\rm Image}(\mathcal R)$ requires the same or fewer parameters to describe as $R_\sB$. This follows from the fact that every element of $R_\sB$ can be mapped to distinct images, or multiple elements can be mapped to the same image.
Hence by requiring that Alice’s reduced states are in one-to-one correspondence with these equivalence classes, her state space must have a dimension $d_\sA^2 - 1$. The only measurement postulates which generate a state space with this dimension are the quantum ones. This follows from the fact that the dimension of the irreducible representations $\D_j^d$ corresponding to alternative measurement postulates are given by: $$\label{dimension}
D_j^d = \left(\frac{2j}{d-1}+1\right) \prod_{k=1}^{d-2}
\left(1 +\frac j k \right)^2
\ ,$$ This is equal to $d^2 - 1$ for the case $j=1$ (corresponding to the quantum state space). Moreover, since this is the lowest dimensional (non-trivial) such representation there are no reducible representations of the form $\bigoplus_i \D_i^d$ which are of dimension $d^2 -1$.
OPF formalism, representation theory and local tomography {#OPFformalism}
=========================================================
In this appendix we show that each set of OPFs $\mathcal F_d$ is associated to a representation of the group $\sud$. We also show that the representations associated to locally tomographic systems with sets of OPFs $\mathcal F_{d_\sA d_\sB}$ have certain features when restricted to the local subgroup $\sudA \times \sudB$. It is these features which will be used in the next appendix to show that all non-quantum measurement postulates lead to a violation of local tomography. In the following we assume the Finiteness Principle holds.
Single systems
--------------
To each set of measurement postulates $\mathcal F_d$ obeying the Finiteness Principle there exists a unique representation $\Gamma^d$ of $\sud$ associated to that set.
We take a set of measurement postulates $\mathcal F$, where $\{F_i\}$ form a basis for $\mathcal F$. $$F(\psi) = \sum_i c_i F_i (\psi) \ , \forall \psi \ .$$ We consider an OPF $F \circ U$: $$(F \circ U)(\psi) = F(U \psi) = \sum_i c_i F_i(U \psi)$$ Where $$F_i(U \psi) = (F_i \circ U) (\psi) = \sum_j \bar \Gamma_i^j(U) F_j (\psi) \ .$$ Hence $$F \circ U (\psi) = \sum_{ij} c_i \bar \Gamma_i^j(U) F_j(\psi)$$ Consider $$\begin{aligned}
F ( U U' \psi) & = \sum_{ij} c_i \bar \Gamma_i^j(U) F_j(U'\psi) \nonumber \\ & = \sum_{ijk} c_i \bar \Gamma_i^j(U) \bar \Gamma_j^k(U') F_k(\psi)
\end{aligned}$$ We can also consider $UU'$ as a single element: $$F ( U U' \psi) = \sum_{ik} c_i \bar \Gamma_i^k(UU') F_k(\psi)$$ This shows that $\bar \Gamma(UU') = \bar \Gamma(U)\bar \Gamma(U')$ and the map $\bar \Gamma: U \mapsto \bar \Gamma(U)$ is a representation of $\sud$.
The representation $\bar \Gamma^d$ of measurement postulates $\mathcal F_d$ contains a unique trivial subrepresentation.
Consider a set of OPFs $\{F^{(i)}\}$ which form a measurement. The OPF $\mathbf u = \sum F^{(i)}$ is such that $\mathbf u (\psi) = 1 \ \forall \psi$. Consider the basis $\{F_i\}$ where $F_1 = \mathbf u$. We observe that $\mathbf u (\psi) = \mathbf u ( U \psi) , \ \forall U \ \forall \psi$. This implies that $\bar \Gamma(U) \mathbf u = \mathbf u \ \forall U$ and that the representation $\Gamma$ has a trivial component. If the representation had another trivial component, it would necessarily be linearly dependent on the first. It would then be a redundant entry in the list of fiducial outcomes which is contrary to the property that they are linearly independent.
Composite systems
-----------------
The measurement structure $\mathcal F_{\sA \sB}$ contains all measurements of the form $F_\sA \star F_\sB$ where $F_\sA \in \mathcal F_\sA $ and $F_\sB \in \mathcal F_\sB$. Let $\{F_\sA^i\}$ and $\{F_\sB^j\}$ be bases for the two OPF spaces. Then the OPFs $F_\sA^i \star F_\sB^j$ form a basis for the global OPFs $F_\sA \star F_\sB$. Hence a basis for $\mathcal F_{\sA\sB}$ is $\{F_\sA^i \star F_\sB^j , F_{\sA\sB}^k \}$ [@Hardy_foliable_2009; @Masanes_lecture_2017].
### Local tomography
A bi-partite system is locally tomographic if $\{F_\sA^i \star F_\sB^j\}_{ij}$ is a basis for $\R \F_{d_\sA d_\sB}$.
For a locally tomographic bi-partite system with representation $\bar \Gamma^{d_\sA d_\sB}$ the restriction of $\bar \Gamma^{d_\sA d_\sB}$ to $\sudA \times \sudB$ is: $$\bar \Gamma^{d_\sA d_\sB}_{|\sudA \times \sudB} = \bar \Gamma^{d_\sA} \boxtimes \bar \Gamma^{d_\sB}$$
Let us consider the action of an element of $\sudA \times \sudB$ on an OPF $F_{\sA \sB} = \sum_{ij} \gamma_{ij} (F_\sA^i \otimes F_\sB^j)$ in $\mathcal F_{\sA \sB}$ . $$\begin{aligned}
\bar \Gamma^{d_\sA d_\sB}_{U_{\sA} \otimes U_{\sB}} F_{\sA \sB} = F_{\sA \sB} \circ (U_\sA \otimes U_\sB)
= \sum_{ij} \gamma_{ij} (F_\sA^i \otimes F_\sB^j) \circ (U_\sA \otimes U_\sB) \ .
\end{aligned}$$ Using the fact that $(F_{\sA} \otimes F_{\sB}) \circ (U_\sA \otimes U_\sB) = (F_\sA \circ U_\sA) \otimes (F_\sB \circ U_\sB)$: $$\bar \Gamma^{d_\sA d_\sB}_{U_{\sA} \otimes U_{\sB}} F_{\sA \sB} = \sum_{ij} \gamma_{ij} (F_\sA^i \circ U_\sA) \otimes (F_\sB^j \circ U_\sB) \ .$$ From the actions of $\sudA$ and $\sudB$ on $\R \F_{d_\sA}$ and $\R \F_{d_\sB}$ we have: $$\begin{aligned}
& F_\sA^i \circ U_\sA = \bar \Gamma^{d_\sA}_{U_A} F_\sA^i \ , \\ & F_\sB^j \circ U_\sB = \bar \Gamma^{d_\sB}_{U_B} F_\sB^j \ . \end{aligned}$$ Hence, $$\begin{aligned}
\bar \Gamma^{d_\sA d_\sB}_{U_{\sA} \otimes U_{\sB}} F_{\sA \sB} = \sum_{ij} \gamma_{ij} (\bar \Gamma^{d_\sA}_{U_A} F_\sA^i \otimes \bar \Gamma^{d_\sB}_{U_B} F_\sB^j) = \sum_{ij} \gamma_{ij} (\bar \Gamma^{d_\sA}_{U_A} \otimes \bar \Gamma^{d_\sB}_{U_B}) ( F_\sA^i \otimes F_\sB^l) = (\bar \Gamma^{d_\sA}_{U_A} \otimes \bar \Gamma^{d_\sB}_{U_B}) F_{\sA \sB} \ .
\end{aligned}$$
### Holistic systems
A bi-partite system which is not locally tomographic is *holistic*. Real vector space quantum theory is an example of a holistic theory [@Hardy_limited_2012]. A basis for $\mathcal F_{d_\sA d_\sB}$ in a holistic bi-partite system is $\{F_\sA^i \star F_\sB^j , F_{\sA\sB}^k \}_{ijk}$ [@Hardy_foliable_2009; @Masanes_lecture_2017]. Here $\{F_\sA^i \star F_\sB^j\}_{ij}$ span the locally tomographic subspace of $\F_{d_\sA d_\sB}$ denoted $\F_{d_\sA d_\sB}^{\rm LT}$. Due to bilinearity of the $\star$ product the map $\star: \R \F_{d_\sA} \times \R \F_{d_\sB} \to \R \F_{d_\sA d_\sB}^{\rm LT}$ is isomorphic to a tensor product.
For a holistic bi-partite system with representation $\bar \Gamma^{d_\sA d_\sB}$ the restriction of $\bar \Gamma^{d_\sA d_\sB}$ to $\sudA \times \sudB$ is: $$\bar \Gamma_{|\sudA \times \sudB}^{d_\sA d_\sB} = \bar \Gamma^{d_\sA} \boxtimes \bar \Gamma^{d_\sB} \oplus \bigoplus_i \Gamma_i^{d_\sA} \boxtimes \Gamma_i^{d_\sB} \ ,$$ where the representations $\bar \Gamma^{d_\sA d_\sB}, \bar \Gamma^{d_\sA}$ and $\bar \Gamma^{d_\sB}$ contain a trivial representation. This is not necessarily the case for $\Gamma_i^{d_\sA}$ and $\Gamma_i^{d_\sB}$ (which may not be of the form $\D_j^{d_\sA}$ or $\D_j^{d_\sB}$).
In holistic systems a basis for $\mathcal F_{\sA\sB}$ is $\{F_\sA^i \otimes F_\sB^j , F_{\sA\sB}^k \}_{ijk}$. $$F_{\sA \sB} = \sum_{ij} \gamma_{ij}^{{\rm LT}} (F_\sA^i \otimes F_\sB^j) + \sum_k \gamma_{k}^{{\rm H}} F^k_{\sA \sB} = F^{\rm LT}_{\sA \sB} + F^{\rm H}_{\sA \sB} \ .$$ We consider the action of a $\sudA \times \sudB$ subgroup on ${\rm span}(\{F_\sA^i \otimes F_\sB^j\}_{ij})$. $$F_\sA^i \otimes F_\sB^j \circ (U_\sA \otimes U_\sB) = (F_\sA^i \circ U_\sA) \otimes (F_\sB^j \circ U_\sB) \ .$$ The action of $\sudA \times \sudB$ maps basis elements of the form $F_\sA \otimes F_\sB$ to other elements of that form. Hence ${\rm span}(F_\sA^i \otimes F_\sB^j)$ is a proper subspace of $\mathcal F_{\sA\sB}$ left invariant under the action of $\sudA \times \sudB$. The representation $ \bar \Gamma_{|\sudA \times \sudB}^{d_\sA d_\sB} $ is reducible and decomposes as: $$\bar \Gamma_{ |\sudA \times \sudB}^{d_\sA d_\sB} = \bar \Gamma_{{\rm LT} |\sudA \times \sudB}^{d_\sA d_\sB} \oplus \Gamma_{{\rm H}|\sudA \times \sudB}^{d_\sA d_\sB} \ .$$ The action $\bar \Gamma_{{\rm LT} |\sudA \times \sudB}^{d_\sA d_\sB}$ on the locally tomographic subspace is of the form $\bar \Gamma^{d_\sA} \boxtimes \bar \Gamma^{d_\sB}$ (as determined in the previous lemma). $ \Gamma_{{\rm H}|\sudA \times \sudB}^{d_\sA d_\sB}$ is an arbitrary representation of $\sudA \times \sudB$ hence of the form $\bigoplus_i \Gamma_i^{d_\sA} \boxtimes \Gamma_i^{d_\sB}$.
Representation theoretic criterion for local tomography
-------------------------------------------------------
Consider a bi-partite system with alternative measurement postulates $\mathcal F_{d_\sA d_\sB}$, $\mathcal F_{d_\sA}$ and $\mathcal F_{d_\sB}$. If the theory is locally tomographic then necessarily: $$\label{linlocaltom}
\bar \Gamma_{| \sudA \times \sudB}^{d_\sA d_\sB} = \bar \Gamma_{\sA} \boxtimes \bar \Gamma_\sB$$ Where $\bar \Gamma_{\sA}$ and $\bar \Gamma_\sB$ both contain a unique trivial representation. By contraposition, any system which has a representation $\bar \Gamma_{\sA\sB}^{d_\sA d_\sB}$ which does not have this form when restricted $ \sudA \times \sudB$ cannot obey local tomography. This allows us to establish the following test for violation of local tomography:
Given a bi-partite system with measurement postulates $\mathcal F_{d_\sA d_\sB}$, $\mathcal F_{d_\sA}$ and $\mathcal F_{d_\sB}$. The associated representation are $\bar \Gamma_{\sA\sB}^{d_\sA d_\sB} $, $\bar \Gamma_{\sA}^{d_\sA}$ and $ \bar \Gamma_{\sB}^{d_\sB}$. If $\bar \Gamma_{| \sudA \times \sudB}^{d_\sA d_\sB}$ is not of the form then the system is holistic.
### Affine representation and existence of trivial times trivial as a criterion
The above lemmas can be translated into the affine representation $\Gamma^d$ which do not contain a trivial subrepresentation. $$\bar \Gamma^{d} = 1^{d} \oplus \Gamma^{d} \ .$$ where $1^d$ is the trivial representation of $\sud$. We can decompose the tensor product action: $$\begin{aligned}
\bar \Gamma^{d_\sA} \boxtimes \bar \Gamma^{d_\sB} = (1^{d_\sA} \oplus \Gamma^{d_\sA}) \boxtimes (1^{d_\sB} \oplus \Gamma^{d_\sB})
= (1^{d_\sA} \boxtimes 1^{d_\sB}) \oplus (1^{d_\sA} \boxtimes \Gamma^{d_\sB}) \oplus ( \Gamma^{d_\sA} \boxtimes 1^{d_\sB}) \oplus ( \Gamma^{d_\sA} \boxtimes \Gamma^{d_\sB}) \ .\end{aligned}$$ The first term occurs due to the trivial component in $\bar \Gamma_{\sA\sB}^{d_\sA d_\sB} = 1^{d_\sA d_\sB} \oplus \Gamma_{\sA\sB}^{d_\sA d_\sB}$.
For a locally tomographic system with OPF set $\mathcal F_{d_\sA d_\sB}$ and representation $\bar \Gamma^{d_\sA d_\sB}$, the restriction of $ \Gamma^{d_\sA d_\sB}$ to the local subgroup $\sudA \times \sudB$, has the following form: $$\label{localtomography}
\Gamma_{| \sudA \times \sudB}^{d_\sA d_\sB} = (1^{d_\sA} \boxtimes \Gamma^{d_\sB}) \oplus ( \Gamma^{d_\sA} \boxtimes 1^{d_\sB}) \oplus ( \Gamma^{d_\sA} \boxtimes \Gamma^{d_\sB}) \ .$$ This shows that for a locally tomographic theory the representations $ \Gamma_{\sA\sB| \sudA \times \sudB}^{d_\sA d_\sB} $ cannot contain any terms $1^{d_\sA} \boxtimes 1^{d_\sB}$. By contraposition we establish:
\[lemnoloctom\] Given a bi-partite system $(\C^{d_{\sA\sB}} , \F_{d_\sA d_\sB}, \Gamma^{d_\sA d_\sB})$ with subsystems $(\C^{d_\sA} , \F_{d_\sA},\Gamma^{d_\sA})$ and $(\C^{d_\sB} , \F_{d_\sB}, \Gamma^{d_\sB})$ then if $ \Gamma^{d_\sA d_\sB}_{\sA \sB}$ has a subrepresentation $1^{d_\sA} \boxtimes 1^{d_\sB}$ upon restriction to $\sudA \times \sudB$ the composite system is holistic.
Background representation theory
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Background
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### Young diagram
A Young diagram is a collection of boxes in left-justified rows, where each row-length is in non-increasing order. The number of boxes in each row determine a partition of the total number of boxes $n$. This partition denoted $\lambda$ is associated to the Young diagram which is said to be of shape $\lambda$. We write $|\lambda| = n$ for the total number of boxes.
A Young diagram with $m$ rows (labelled $1$ to $m$) where row $i$ has $n_i$ boxes (by definition $n_1 \geq n_2 \geq ... \geq n_m$) is written $\lambda = (n_1 , n_2 , ... , n_m)$. By definition: $$\sum_i n_i = n$$ $$\lambda^1 = \yng(2,1) \ , \quad \lambda^2 = \yng(3,3,2,1) \ , \quad \lambda^3 = \yng(5,2,1,1) \ .$$ The above Young diagrams are $\lambda^1 = (2,1)$, $\lambda^2 = (3,3,2,1)$ and $\lambda^3 = (5,2,1,1)$.
In the case where a diagram has $m$ multiple rows of the same length $l$ we write $l^m$ instead of writing out “$l$" $m$ times.. For instance $\lambda^2 = (3^2,2,1)$
### Representations of $\sud$
The special unitary group $\sud$ is the Lie group of $d \times d$ unitary matrices of determinant $1$. A finite-dimensional representation of $\sud$ is a smooth homomorphism $\sud \rightarrow \mathrm{GL}(V)$, with $V$ a finite-dimensional (complex) vector space.
Irreducible representations of $\sud$ are labelled by Young diagrams with at most $d$ rows. There is no limit on the total number of boxes. A column of $d$ boxes can be removed from a Young diagram of a representation of $\sud$ without changing the representation labelled by the diagram. Hence an irreducible representation of $\sud$ corresponds to an equivalence class of Young diagrams up to columns of size $d$. Typically the canonical representative member of the equivalence class is the Young diagram where all columns of length $d$ have been removed. With this choice of representative element of each class the Young diagram of representations of $\sud$ have at most $d-1$ rows. For example the equivalence class of rectangular Young diagrams of columns of length $d$ are all mapped to the empty diagram (since all columns are removed).
$$\lambda^1 = \yng(4,3,3,2) \ , \quad \lambda^2 = \yng(3,2,2,1) \ , \quad \lambda^3 = \yng(2,1,1) \ .$$
These three Young diagrams are equivalent up to columns of length 4. They all label the same representation of $\mathrm{SU}(4)$. The canonical representative of the equivalence class is $\lambda^3$. In the following the expression “representation of $\sud$ labelled by the Young diagram $\lambda$" is used to mean “representation of $\sud$ labelled by the equivalence class of Young diagrams with representative member $\lambda$". The young diagram [(1)]{} labels the fundamental representation of $\sud$. The empty diagram labels the trivial representation.
Given a canonical representative $\lambda$ we call $\lambda_f$ the diagram in the same equivalence class which has $f$ boxes (in other words to which a number of columns of length $d$ have been added such that the total number of boxes is $f$). For example $\lambda^2 = \lambda^3_8$.
$\Gamma_\lambda^d$ is the representation of $\sud$ associated to the Young diagram $\lambda$. The index $d$ will be dropped when the context makes it obvious.
### Representations of the symmetric group
The symmetric group on $n$ symbols $S_n$ has as group elements all permutation operations on $n$ distinct symbols. The conjugacy classes of $S_n$ are labelled by partitions of $n$. Hence the number of (inequivalent) irreducible representations of $S_n$ is the number of partitions of $n$. Moreover these irreducible representations can be parametrised by partitions of $n$. These are labelled by Young diagrams with $n$ boxes [@Robinson_representation_1991].
$\Delta_\lambda^n$ denotes the representation of $S_n$ labelled by the Young diagram $\lambda$. The index $n$ will be dropped when the context makes it obvious.
Branching rule $\mathrm{SU}(mn) \rightarrow \mathrm{SU}(m) \times \mathrm{SU}(n)$
---------------------------------------------------------------------------------
### Definition
Consider an irreducible representation $\Gamma_\lambda^{mn}$ of $\mathrm{SU}(mn)$ and restrict it to a $\mathrm{SU}(m) \times \mathrm{SU}(n)$ subgroup: $$\Gamma^{mn}_\lambda(U_1 \otimes U_2) , \forall U_1 \in \mathrm{SU}(m) \ , \forall U_2 \in \mathrm{SU}(n) \ .$$ In general this will yield a reducible representation of $\mathrm{SU}(m) \times \mathrm{SU}(n)$. This representation will be built of irreducible representations $\Gamma_\mu^m \boxtimes \Gamma_\nu^{n}$ $$(\Gamma_\mu^m \boxtimes \Gamma_\nu^n) (U_1,U_2) = \Gamma_\mu^m(U_1) \otimes \Gamma_\nu^n(U_2) \ ,$$ We write $\Gamma^{mn}_{\lambda| \mathrm{SU}(m) \times \mathrm{SU}(n)}$ for the restriction of $\Gamma_\lambda^{mn}$ to a $\mathrm{SU}(m) \times \mathrm{SU}(n)$ subgroup. In general: $$\Gamma^{mn}_{ \lambda | \mathrm{SU}(m) \times \mathrm{SU}(n)} = \bigoplus_{\mu , \nu} \Gamma^m_\mu \boxtimes \Gamma^n_\nu \ ,$$ Where there can be repeated copies for a given $\mu , \nu$. In general finding which representations $\Gamma^m_\mu \boxtimes \Gamma^n_\nu$ occur in this restriction is a hard problem. In the following we outline a method which allows us to determine the multiplicity of $\Gamma^m_\mu \boxtimes \Gamma^n_\nu$ in $\Gamma^{mn}_{\lambda| \mathrm{SU}(m) \times \mathrm{SU}(n)}$. $\lambda$ , $\mu$ and $\mu$ will refer to the Young diagrams of $\Gamma^{mn}_\lambda$, $\Gamma^m_\mu$ and $\Gamma^n_\nu$ respectively.
### Inner product of representations of the symmetric group
We consider two representations $\Delta_\mu$ and $\Delta_\nu$ of $S_f$. We construct the Kronecker product of the two matrices $\Delta_\mu(s)$ and $\Delta_\nu(s)$ for all $s \in S_f$. This creates a representation which we call the tensor product $\Delta_\mu \otimes \Delta_\nu$ (sometimes called the inner product). In general this is a reducible representation: $$\Delta_\mu \otimes \Delta_\nu = \bigoplus_\lambda g(\mu, \nu , \lambda) \Delta_\lambda$$ Here we abuse notation slightly to mean that $g(\mu, \nu , \lambda)$ is the multiplicity of $\Delta_\lambda$ in $\Delta_\mu \otimes \Delta_\nu $. These $g(\mu, \nu , \lambda)$ are known as the Clebsch-Gordan coefficients of the symmetric group, and understanding them remains one of the main open problems in classical representation theory. These coefficients are also relevant in quantum information theory, as they are related to the spectra of statistical operators [@Christandl_Nonzero_2007].
### Recipe
What is the multiplicity of $\Gamma^m_\mu \boxtimes \Gamma^n_\nu$ in the restriction of $\Gamma^{mn}_\lambda$ to $ \mathrm{SU}(m) \times \mathrm{SU}(n)$? We adopt the approach from [@itzykson_unitary_1966] to answer this question. Let $f = |\lambda|$ be the number of boxes in the Young diagram $\lambda$. As shown above $\lambda$ also labels a representation of the symmetric group on $f$ objects $S_f$. This representation is $\Delta_\lambda^f$. Take the Young diagram $\mu$ ($\nu$) and add columns of $m$ ($n$) boxes to the left until it has $f$ boxes. The tableau obtained which we call $\mu_f$ ($\nu_f$) labels a representation of $S_f$ denoted $\Delta_{\mu_f}^f$ ($\Delta_{\nu_f}^f$). We remember that adding columns to the left of length $m$ ($n$) keeps $\mu$ ($\nu$) within the equivalence class of Young diagrams labelling the representation $\Gamma_\mu^m$ ($\Gamma_\nu^n$). Hence $\mu_f$ ($\nu_f$) labels the same representation of $\mathrm{SU}(m)$ ($\mathrm{SU}(n)$) as $\mu$ ($\nu$).
Hence the diagrams $\lambda$ , $\mu_f$ and $\nu_f$ refer both to representations of the special unitary group $\Gamma^{mn}_\lambda$ , $\Gamma^m_\mu (= \Gamma^m_{\mu_f} ) $ and $\Gamma^n_\nu (= \Gamma^n_{\nu_f} )$ as well as representations of $S_f$: $\Delta_\lambda^f$, $\Delta_{\mu_f}^f$ and $\Delta_{\nu_f}^f$.
\[recipetheorem\] $\Gamma^m_\mu \boxtimes \Gamma^n_\nu$ occurs as many times in the restriction of $\Gamma^{mn}_\lambda$ to $ \mathrm{SU}(m) \times \mathrm{SU}(n)$ as $\Delta_\lambda^f$ occurs in $\Delta_{\mu_f}^f \otimes \Delta_{\nu_f}^f$, where $f = |\lambda|$ [@itzykson_unitary_1966].
Inductive lemma
---------------
\[inductivelemma\] Consider representations $\Gamma^{mn}_{\bar \lambda}$, $\Gamma^{m}_{\bar \mu}$, $\Gamma^{n}_{\bar \nu}$, $\Gamma^{mn}_{ \lambda}$, $\Gamma^{m}_{ \mu}$, $\Gamma^{n}_{ \nu}$, $\Gamma^{mn}_{\lambda'}$, $\Gamma^{m}_{ \mu'}$ and $\Gamma^{n}_{ \nu'}$ where $\bar \lambda = \lambda + \lambda'$, $\bar \mu = \mu + \mu'$ $\bar \nu = \nu + \nu'$ and $\frac{|\lambda|-|\mu|}{m},\frac{|\lambda|-|\nu|}{n},\frac{|\lambda'|-|\mu'|}{m}$ and $\frac{|\lambda'|-|\nu'|}{n}$ are integers. If $\Gamma^{mn}_{ \lambda| \mathrm{SU}(m) \times \mathrm{SU}(n)}$ contains a term $\Gamma^{m}_{ \mu} \boxtimes \Gamma^{n}_{ \nu}$ and $\Gamma^{mn}_{ \lambda'| \mathrm{SU}(m) \times \mathrm{SU}(n)}$ contains a term $\Gamma^{m}_{ \mu'} \boxtimes \Gamma^{n}_{ \nu'}$ then $\Gamma^{mn}_{ \bar \lambda| \mathrm{SU}(m) \times \mathrm{SU}(n)}$ contains a term $\Gamma^{m}_{ \bar \mu} \boxtimes \Gamma^{n}_{ \bar \nu}$.
$\Gamma^{mn}_{ \lambda| \mathrm{SU}(m) \times \mathrm{SU}(n)}$ containing a term $\Gamma^{m}_{ \mu} \boxtimes \Gamma^{n}_{ \nu}$ implies that $\Delta^f_\lambda$ occurs in $\Delta^f_{\mu_f} \otimes \Delta^f_{\nu_f}$ (by Theorem \[recipetheorem\]). Here $\mu_f$ is the tableau $\mu$ ($\nu$) to which $\frac{f-|\mu|}{m}$($\frac{f-|\nu|}{n}$) columns of length $m$ ($n$) has been added so that the total number of boxes $|\mu_f| = f$ ($|\nu_f|=f$).
$$\begin{aligned}
& \mu_f = \mu + \left(\left(\frac{f-|\mu|}{m}\right)^m\right) \ , \\
& \nu_f = \nu + \left(\left(\frac{f-|\nu|}{n}\right) ^n\right) \ .\end{aligned}$$
Here we recall that $((\frac{f-|\mu|}{m})^m)$ indicates $m$ rows of length $((\frac{f-|\mu|}{m})$. This implies that $g(\lambda, \mu_f, \nu_f)>0$. Similarly $g(\lambda', \mu'_{f'}, \nu'_{f'})>0$.
We now show that $\bar \mu = \mu + \mu'$ and $\bar \nu = \nu + \nu'$ implies that $\bar \mu_{\bar f} = \mu_f + \mu'_{f'}$ and $\bar \nu_{\bar f} = \nu_f + \nu'_{f'}$.
$$\begin{aligned}
\mu_f + \mu'_{f'} & = \mu + \mu' + \left(\left(\frac{f-|\mu|}{m}\right) ^m\right) + \left(\left(\frac{f'-|\mu'|}{m}\right) ^m\right) \\
& = \bar \mu + \left(\left(\frac{(f+f' - |\mu| -|\mu'|)}{m}\right) ^m\right) \\
& = \bar \mu + \left(\left(\frac{\bar f-|\bar \mu|}{m}\right) ^n\right) = \bar \mu_{\bar f} \ .\end{aligned}$$
And similarly for $\bar \nu$.
Let us make use of a property of the Clebsch Gordan coefficients called the semi-group property. If $g(\lambda, \mu_f , \nu_f) > 0 $ and $g(\lambda', \mu'_{f'} ,\nu'_{f'}) > 0$ then $g(\lambda + \lambda',\mu_f + \mu'_{f'} , \nu_f + \nu'_{f'} ) > 0$ [@ikenmeyer_rectangular_2016]. By the semi-group property we have $g(\bar \lambda, \bar \mu_{\bar f}, \bar \nu_{\bar f})>0$. This implies that $\Delta^{\bar f}_{
\bar \lambda}$ occurs in $\Delta^{\bar f}_{\bar \mu_{\bar f}} \otimes \Delta^{\bar f}_{\bar \mu_{\bar f}}$. By Theorem \[recipetheorem\] this implies that $\Gamma^{mn}_{ \bar \lambda| \mathrm{SU}(m) \times \mathrm{SU}(n)}$ contains a term $\Gamma^{m}_{ \bar \mu} \boxtimes \Gamma^{m}_{ \bar \nu}$.
Violation of local tomography in all alternative measurement postulates {#nolocaltom}
=======================================================================
In the following we establish that $\Gamma_{ | \suthree \times \suthree}^{9}$ is not of the form for all representations $\Gamma^{9}$ of $\sunine$ corresponding to non quantum state spaces with pure states $\mathbb C^9$. We show explicitly that every such restriction contains a term of $1^{3} \boxtimes 1^{3}$, where $1^{3}$ is the trivial representation of $\suthree$.
Arbitrary dimension $d$
-----------------------
We now construct a proof by induction to show that the representations $\mathcal D_j^d$ are not compatible with local tomography when restricted to ${\rm SU}(d_\sA) \times {\rm SU}(d_\sB)$. A representation $\mathcal D_j^d$ will violate local tomography if it is not of the form when restricted to ${\rm SU}(d_\sA) \times {\rm SU}(d_\sB)$. It suffices to show that there is a term $1^{d_\sA} \boxtimes 1^{d_\sB}$ in this restriction in order to show that it is not of this form.
\[inductionlemma\] If the representations $\mathcal D_j^d$ and $\mathcal D_2^d$ of $\sud$ contain a term $1^{d_\sA} \boxtimes 1^{d_\sB}$ when restricted to ${\rm SU}(d_\sA) \times {\rm SU}(d_\sB)$ then so does $\mathcal D_{j+2}^d$
Let
$$\begin{aligned}
& \Gamma_\lambda^d = \mathcal D_j^d , \ \Gamma_\mu^{d_\sA} = 1^{d_\sA} , \ \Gamma_\nu^{d_\sB} = 1^{d_\sB} \\
& \Gamma_{\lambda'}^{d} = \mathcal D_2^d , \ \Gamma_{\mu'}^{d_\sA}=1^{d_\sA} , \ \Gamma_{\nu'}^{d_\sB} = 1^{d_\sB} \\
& \Gamma_{\bar \lambda} = \mathcal D_{j+2}^{d} , \ \Gamma_{\bar \mu} = 1^{d_\sA} , \ \Gamma_{\bar \nu}^{d_\sB} = 1^{d_\sB} \ ,\end{aligned}$$
where
$$\begin{aligned}
& \lambda=(2j,j^{d-2}) , \ \mu = 0, \ \nu = 0 \ ,\\
& \lambda' = (4,2^{d-2}) , \ \mu' = 0 , \ \nu' = 0 \ , \\
& \bar \lambda= (2j+4,(j+2)^{d-2}) , \ \bar \mu = 0 , \ \bar \nu = 0 \ .\end{aligned}$$
We observe
$$\begin{aligned}
& \bar \lambda = \lambda + \lambda' , \ \bar \mu = \mu + \mu' , \ \bar \nu = \nu + \nu' \\
& f = |\lambda| = jd , \ f' = |\lambda'| = 2d , \ \bar f = | \bar \lambda| = d(j+2)\end{aligned}$$
Next we check that the quantities below are integer valued:
$$\begin{aligned}
\frac{|\lambda|-|\mu|}{m} = \frac{jd}{d_\sA} = j d_\sB \ ,\\
\frac{|\lambda|-|\nu|}{n} = \frac{jd}{d_\sB} = j d_\sA \ , \\
\frac{|\lambda'|-|\mu'|}{m} = \frac{2d}{d_\sA} = 2d_\sB \ , \\
\frac{|\lambda'|-|\nu'|}{n} = \frac{2d}{d_\sB} =2d_\sA \ .\end{aligned}$$
From Lemma \[inductivelemma\] it follows that if $\mathcal D_j^d$ contains a representation $1^{d_\sA} \boxtimes 1^{d_\sB}$ and $\mathcal D_2^d$ contains a representation $1^{d_\sA} \boxtimes 1^{d_\sB}$ when restricted to ${\rm SU}(d_\sA) \times {\rm SU}(d_\sB)$ then $\mathcal D_{j+2}^d$ contains a representation $1^{d_\sA} \boxtimes 1^{d_\sB}$ when restricted to ${\rm SU}(d_\sA) \times {\rm SU}(d_\sB)$.
Hence it suffices to show that $\mathcal D_2^d$ and $\mathcal D_3^d$ contain $1^{d_\sA} \boxtimes 1^{d_\sB}$ when restricted to ${\rm SU}(d_\sA) \times {\rm SU}(d_\sB)$ to show that $\mathcal D_j^d$ does for any $j >1$.
Existence of $1^3 \boxtimes 1^3$ for all non-quantum representations of $\sunine$
---------------------------------------------------------------------------------
Using Sage software [@sagemath] we can show that $\mathcal D_2^9$ and $\mathcal D_3^9$ have a representation $1^{3} \boxtimes 1^{3}$ when restricted to $\suthree \times \suthree$. By Lemma \[inductionlemma\] all representations $\mathcal D_j^9$, $j>1$ have this property. An arbitrary representation corresponding to an alternative Born rule for $\pcnine$ is of the form: $$\label{nonquantnine}
\Gamma_9 =\bigoplus_{j \in \mathcal J} \mathcal D_j^9$$ Where $\mathcal J$ is a list of positive integers containing at least one integer which is not 1. Since at least one (non-trivial) subrepresentation in $\Gamma^9$ has a $1^{3} \boxtimes 1^{3}$ when restricted to $\suthree \times \suthree$ so does the representation $\Gamma^9$.
Violation of local tomography for all theories
----------------------------------------------
Every representation $\Gamma_9$ of $\sunine$ of the form has a representation $1^{3} \boxtimes 1^{3}$ when restricted to $\suthree \times \suthree$. It follows from this that the restriction of $\Gamma_9$ to $\suthree \times \suthree$ is not of the form required for local tomography. From this it follows that all non-quantum $\pcnine$ systems which are composites of two $\pcthree$ systems violate local tomography.
In order to show that a theory with systems $\pcd$ (for every $d>1$) violates local tomography, it is sufficient to show that one of the systems in the theory violates local tomography. Since all $\pcnine$ non-quantum systems violate local tomography it follows that all non-quantum theories with systems $\pcd$ violate local tomography. We emphasise that here we consider theories for which all values of $d>1$ are possible.
Composition in GPTs
===================
In the earlier appendices we gave a certain primacy to the space of OPFs, and considered the group action on the linear space spanned by the OPFs. However it is equally valid (and more common) to give primacy to the space of states and consider the group action on this space. This standard representation will be helpful for proving the consistency of the toy model. Indeed the consistency constraints required for composition translate naturally to the language of states. This section is largely without proof, as the proofs carry over naturally from the OPF formalism to the standard state formalism. We refer the reader to [@Masanes_lecture_2017] for more detailed proofs.
Representation of states
------------------------
The set of mixed states span a real vector space. A state is a vector: $$\bar \omega = \left(
\begin{array}{c}
F_1(\omega) \\
F_2(\omega)\\
\vdots \\
F_{K_d}(\omega)
\end{array}
\right) \ ,$$ An arbitrary $F \in \mathcal F_d$ can be expressed: $$F = \sum_i c_i F_i$$ To each $F$ we can associate a dual vector (or effect) $\bar {\rm E}_F$ $$\bar {\rm E}_F = (c_1 , ... , c_{K_d}) \ ,$$ By definition $\bar {\rm E}_F \cdot \bar \omega = F(\omega)$. Let $\bar {\bf u}$ be effect associated to the unit OPF. Since $\bar {\bf u} \cdot \bar \omega = 1$ there is a component for all states $\bar \omega$ which is constant. That is to say we can choose $F_1$ to be the unit OPF: $$\bar \omega = \left(
\begin{array}{c}
1 \\
\omega\\
\end{array}
\right) \ ,$$ The group representation $ \Gamma^{d}$ acting on the state space is of the form $ \Gamma^d = 1^d + \Gamma^d$ ($ \Gamma^d$ may be reducible or irreducible). We can write $$\bar {\rm E}_F \cdot \bar \omega = c_1 + {\rm E}_F \cdot \omega$$
In the representation where the trivial component is removed (by an affine transformation) then states are $ \omega$ (whose fiducial outcomes are affinely independent), the group representation is $ \Gamma^d$ and outcome probabilities are affine functions $ {\rm E}_F$ of the state $\omega$. We observe that from the uniqueness of the trivial component in $\Gamma^d$ it follows that $\Gamma^d$ cannot contain any trivial subrepresentation.
Local tomography
----------------
Consider measurement postulates $\mathcal F_{d_\sA d_\sB}$, $\mathcal F_{\sA}$ and $\mathcal F_{\sB}$. States for Alice and Bob can be written as: $$\bar \omega_\sA = \left(
\begin{array}{c}
F_\sA^1(\omega_\sA) \\
F_\sA^2(\omega_\sA)\\
\vdots \\
F_\sA^{K_{d_\sA}}(\omega_\sA)
\end{array}
\right) \ ,$$ and $$\bar \omega_\sB = \left(
\begin{array}{c}
F_\sB^1(\omega_\sB) \\
F_\sB^2(\omega_\sB)\\
\vdots \\
F_\sB^{K_{d_\sB}}(\omega_\sB)
\end{array}
\right) \ .$$
A composite system $\mathcal S_{\sA\sB}$ is locally tomographic if it has fiducial outcomes $\{(F_\sA^i \star F_\sB^j) \}_{i = 1 , j = 1}^{i = K_{d_\sA} , j = K_{d_\sB}}$. A state of the composite system can be written: $$\bar \omega_{\sA\sB} = \left(
\begin{array}{c}
(F_\sA^1 \star F_\sB^1) (\omega_{\sA \sB}) \\
(F_\sA^1 \star F_\sB^2) (\omega_{\sA \sB}) \\
\vdots \\
(F_\sA^{\kda} \star F_\sB^{\kdb}) (\omega_{\sA \sB})
\end{array}
\right) \ ,$$
We observe that $\bar E_{F_\sA^i \star F_\sB^j} = \bar E_{F_\sA^i} \otimes \bar E_{F_\sB^j}$. Indeed any joint local effect $\bar E_{F_\sA \star F_\sB} = \bar E_{F_\sA} \otimes \bar E_{F_\sB}$. Product states are of the form $\bar \omega_\sA \otimes \bar \omega_\sB$.
Non-locally-tomographic theories
--------------------------------
The states of a composite system $\mathcal S_{\sA\sB}$ of a non-locally tomographic (or holistic) theory can be written as [@Hardy_foliable_2009; @Masanes_lecture_2017]: $$\bar \omega_{\sA\sB} = \left(
\begin{array}{c}
\bar \lambda_{\sA\sB}\\
\eta_{\sA\sB}\\
\end{array}
\right) \ ,$$ where $$\bar \lambda_{\sA\sB} = \left(
\begin{array}{c}
(F_\sA^1 \star F_\sB^1) (\omega_{\sA \sB}) \\
(F_\sA^1 \star F_\sB^2) (\omega_{\sA \sB}) \\
\vdots \\
(F_\sA^{\kda} \star F_\sB^{\kdb}) (\omega_{\sA \sB})
\end{array}
\right) \ ,$$ is the locally tomographic part and $$\eta_{\sA\sB} = \left(
\begin{array}{c}
F_{\sA \sB}^1 (\omega_{\sA \sB})\\
\vdots\\
F_{\sA \sB}^{K_{d_\sA d_\sB} - K_{d_\sA} K_{d_\sB}} (\omega_{\sA \sB})
\end{array}
\right) \ ,$$ is the holistic part. The probability for a joint outcome $F_\sA \star F_\sB$ is given by: $$\label{jointlocaleffect}
(F_\sA \star F_\sB) (\omega_{\sA \sB}) = (\bar E_{F_\sA} \otimes \bar E_{F_\sB}) \cdot \bar \lambda_{\sA \sB}$$ Joint local effects are computed using the locally tomographic part of the state space only.
The reduced state space
-----------------------
The reduced states for Alice and Bob can be computed as follows: $$\begin{aligned}
\bar \omega_\sA = (\bar \Gamma^{d_\sA}(\unity_\sA) \otimes {\bf u}_\sB ) \bar \lambda_{\sA\sB} \label{reducedstate} \\
\bar \omega_\sB = ( {\bf u}_\sA \otimes \bar \Gamma^{{d_\sB}}(\unity_\sB) ) \bar \lambda_{\sA\sB}\end{aligned}$$ Reduced states are determined using only the locally tomographic part of the global state.
Toy model {#toymodel}
=========
In this appendix we show that the toy model introduced in section \[Toymodel\] meets consistency constraints [**C1 - C5**]{} (apart from associativity of the $\star$ product).
Constraints C1 and C2
---------------------
It is immediate that consistency constraints [**C1**]{} and [**C2**]{} are met by the toy model.
Constraint C3
-------------
We prove $ (F_\sA \star F_\sB) (\psi_\sA \otimes \phi_\sB) = F_\sA (\psi_\sA) F_\sB (\phi_\sB)\ $:
$$\begin{aligned}
& (F_\sA \star F_\sB) (\psi_\sA \otimes \phi_\sB) \nonumber \\
= & \ {\rm tr}\Big( {
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2}{
| \phi \rangle \! \langle \phi |}_\sB^{\otimes 2}(\hat F_\sA \hat F_\sB + \frac{{\rm tr} (\hat F_\sA \hat F_\sB )}{{\rm tr}(S_\sA S_\sB)}A_\sA A_\sB)\Big) \nonumber \\
= & {\rm tr}\Big( {
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2}{
| \phi \rangle \! \langle \phi |}_\sB^{\otimes 2} \hat F_\sA \hat F_\sB \Big) \nonumber \\
= & \ F_\sA (\psi_\sA) F_\sB (\phi_\sB) \ .\end{aligned}$$
In the penultimate line we have used the fact that the overlap of product states ${
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2}{
| \phi \rangle \! \langle \phi |}_\sB^{\otimes 2}$ and $A_\sA A_\sB$ is $0$.
Constraint C4
-------------
In the following it will occasionally useful to label the two copies of $\mathbb C^{d_\sA}$ with $1$ and $3$ and to label the two copies of $\mathbb C^{d_\sB}$ with $2$ and $4$. We write $\tilde S_\sA$ for the normalised projector onto the symmetric subspace of $(\mathbb C^{d_\sA})^{\otimes 2}$. We make use of the identity $S = \frac{1}{2}(\unity + {\rm SWAP})$ throughout this section.
In this section we show that normalised conditional states for Alice are valid states of a $\mathbb C^{d_\sA}$ system. We first show this for the specific case where the state is conditioned on the unit effect, i.e. is a reduced state. A reduced state of Alice for a bi-partite system in pure state ${
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2}$ is: $$\begin{aligned}
\bar \omega_\sA
=
{\rm tr}_\sB \! \left( S_\sB {
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right)
+ \frac {S_\sA}{{\rm tr} S_\sA}
{\rm tr}_{\sA\sB} \!
\left(A_\sA A_\sB
{
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right) \ .\end{aligned}$$
We show that these reduced states lie in the convex hull of ${
| \psi_\sA \rangle \! \langle \psi_\sA |}^{\otimes 2}$.
\[simpequivlem\] $S_\sB \ket{\psi_{\sA\sB}}^{\otimes 2} = S_\sA S_\sB \ket{\psi_{\sA\sB}}^{\otimes 2}$
$$\ket{\psi_{\sA\sB}}^{\otimes 2} = \alpha_{i_1 i_2} \alpha_{i_3 i_4} \ket{i_1 i_2 i_3 i_4} \ .$$
$$S_\sB \ket{\psi_{\sA\sB}}^{\otimes 2} = \frac{1}{2} \alpha_{i_1 i_2} \alpha_{i_3 i_4} (\ket{i_1 i_2 i_3 i_4} +\ket{i_1 i_4 i_3 i_2}) \ .$$
Let us relabel $i_1 \leftrightarrow i_3$ in the last term: $$\begin{aligned}
S_\sB \ket{\psi_{\sA\sB}}^{\otimes 2} = \frac{1}{2} ( \alpha_{i_1 i_2} \alpha_{i_3 i_4} \ket{i_1 i_2 i_3 i_4} + \alpha_{i_3 i_2} \alpha_{i_1 i_4} \ket{i_3 i_4 i_1 i_2}) \ .
\end{aligned}$$ $$S_\sA S_\sB \ket{\psi_{\sA\sB}}^{\otimes 2} = \frac{1}{4} \alpha_{i_1 i_2} \alpha_{i_3 i_4} (\ket{i_1 i_2 i_3 i_4} +\ket{i_1 i_4 i_3 i_2}+\ket{i_3 i_2 i_1 i_4} +\ket{i_3 i_4 i_1 i_2}) \ .$$ Let us relabel $i_1 \leftrightarrow i_3$ in the second term , $i_2 \leftrightarrow i_4$ in the penultimate term and $i_1 \leftrightarrow i_3$ and $i_2 \leftrightarrow i_4$ in the last term : $$\begin{aligned}
& S_\sA S_\sB \ket{\psi_{\sA\sB}}^{\otimes 2}
= \frac{1}{4} (\alpha_{i_1 i_2} \alpha_{i_3 i_4} \ket{i_1 i_2 i_3 i_4} +\alpha_{i_3 i_2} \alpha_{i_1 i_4} \ket{i_3 i_4 i_1 i_2} \\
+ & \alpha_{i_1 i_4} \alpha_{i_3 i_2} \ket{i_3 i_4 i_1 i_2} + \alpha_{i_1 i_2} \alpha_{i_3 i_4} \ket{i_1 i_2 i_3 i_4})
= \frac{1}{2} ( \alpha_{i_1 i_2} \alpha_{i_3 i_4} \ket{i_1 i_2 i_3 i_4} + \alpha_{i_3 i_2} \alpha_{i_1 i_4} \ket{i_3 i_4 i_1 i_2}) \ .
\end{aligned}$$
From the above lemma we can write a reduced state $\bar \omega$: $$\bar \omega_\sA
=
{\rm tr}_\sB \! \left( S_\sA S_\sB {
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right)
+ \frac {S_\sA}{{\rm tr} S_\sA}
{\rm tr}_{\sA\sB} \!
\left(A_\sA A_\sB
{
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} \right) \ .$$
\[Redstatelem\] The reduced state $\bar \omega_\sA$ can be written as: $$\bar \omega_\sA = S_\sA(\rho_\sA \otimes \rho_\sA) S_\sA + (1 - {\rm tr}( S_\sA(\rho_\sA \otimes \rho_\sA) S_\sA)) \tilde S_\sA \ ,$$ where $\rho_\sA = {\rm tr}_\sB ({
| \psi \rangle \! \langle \psi |}_{\sA \sB})$.
We first show that: $${\rm tr}_\sB ( S_\sA S_\sB {
| \psi \rangle \! \langle \psi |}^{\otimes 2}_{\sA \sB}) = S_\sA (\rho_\sA \otimes \rho_\sA) S_\sA \ .$$ From the proof of Lemma \[simpequivlem\]: $$\begin{aligned}
S_\sA S_\sB \ket{\psi_{\sA\sB}}^{\otimes 2} = \frac{1}{4} \alpha_{i_1 i_2} \alpha_{i_3 i_4} (\ket{i_1 i_2 i_3 i_4} +\ket{i_1 i_4 i_3 i_2}+\ket{i_3 i_2 i_1 i_4} +\ket{i_3 i_4 i_1 i_2}) \ .
\end{aligned}$$ Hence: $$\begin{aligned}
S_\sA S_\sB {
| \psi_{\sA\sB} \rangle \! \langle \psi_{\sA\sB} |}^{\otimes 2} = & \frac{1}{4} \alpha_{i_1 i_2} \alpha_{i_3 i_4} \bar \alpha_{j_1 b_1} \bar \alpha_{j_3 b_2} ({
| i_1 i_2 i_3 i_4 \rangle \! \langle j_1 j_2 j_3 j_4 |} \\
+ & {
| i_1 i_4 i_3 i_2 \rangle \! \langle j_1 j_2 j_3 j_4 |}+{
| i_3 i_2 i_1 i_4 \rangle \! \langle j_1 j_2 j_3 j_4 |} +{
| i_3 i_4 i_1 i_2 \rangle \! \langle j_1 j_2 j_3 j_4 |}) \ .
\end{aligned}$$ which implies: $$\begin{aligned}
{\rm tr}_\sB (S_\sA S_\sB {
| \psi \rangle \! \langle \psi |}_{\sA\sB}^{\otimes 2})
= \frac{1}{2} \alpha_{i_1 b_1} \alpha_{i_3 b_2} \bar \alpha_{j_1 b_1} \bar \alpha_{j_3 b_2} ( {
| i_1 i_3 \rangle \! \langle j_1 j_3 |} + {
| i_3 i_1 \rangle \! \langle j_1 j_3 |} ) \ .
\end{aligned}$$ We now compute $ S_\sA ( \rho_\sA \otimes \rho_\sA) S_\sA$. $${
| \psi \rangle \! \langle \psi |}_{\sA \sB} = \alpha_{i_1 i_2} \bar \alpha_{j_1 j_2} {
| i_1 i_2 \rangle \! \langle j_1 j_2 |} \ .$$ $$\rho_\sA = {\rm tr}_{\sB} ({
| \psi \rangle \! \langle \psi |}_{\sA \sB}) = \alpha_{i_1 b_1} \bar \alpha_{j_1 b_1} {
| i_1 \rangle \! \langle j_1 |} \ .$$ $$\rho_\sA \otimes \rho_\sA = \alpha_{i_1 b_1} \alpha_{i_3 b_2} \bar \alpha_{j_1 b_1} \bar \alpha_{j_3 b_2} {
| i_1 i_3 \rangle \! \langle j_1 j_3 |} \ .$$ $$\begin{aligned}
S_\sA (\rho_\sA \otimes \rho_\sA )
= \frac{1}{2} \alpha_{i_1 b_1} \alpha_{i_3 b_2} \bar \alpha_{j_1 b_1} \bar \alpha_{j_3 b_2} ({
| i_1 i_3 \rangle \! \langle j_1 j_3 |} + {
| i_3 i_1 \rangle \! \langle j_1 j_3 |}) \ .
\end{aligned}$$ $$\begin{aligned}
S_\sA (\rho_\sA \otimes \rho_\sA ) S_\sA
= & \frac{1}{4} \alpha_{i_1 b_1} \alpha_{i_3 b_2} \bar \alpha_{j_1 b_1} \bar \alpha_{j_3 b_2} ({
| i_1 i_3 \rangle \! \langle j_1 j_3 |} + {
| i_3 i_1 \rangle \! \langle j_1 j_3 |}) \\
+ & \frac{1}{4} \alpha_{i_1 b_1} \alpha_{i_3 b_2} \bar \alpha_{j_1 b_1} \bar \alpha_{j_3 b_2} ({
| i_1 i_3 \rangle \! \langle j_3 j_1 |} + {
| i_3 i_1 \rangle \! \langle j_3 j_1 |}) \ .
\end{aligned}$$ In the second term we relabel $j_1 \leftrightarrow j_3$, $i_1 \leftrightarrow i_3$ and $b_1 \leftrightarrow b_2$ to obtain: $$\begin{aligned}
S_\sA (\rho_\sA \otimes \rho_\sA ) S_\sA = \frac{1}{2} (\alpha_{i_1 b_1} \alpha_{i_3 b_2} \bar \alpha_{j_1 b_1} \bar \alpha_{j_3 b_2} ( {
| i_1 i_3 \rangle \! \langle j_1 j_3 |} + {
| i_3 i_1 \rangle \! \langle j_1 j_3 |} ) ) \ .
\end{aligned}$$ Hence, $$\begin{aligned}
\bar \omega_\sA & = S_\sA (\rho_\sA \otimes \rho_\sA) S_\sA + (1-{\rm tr} ({
| \psi \rangle \! \langle \psi |}^{\otimes 2}_{\sA \sB} S_\sA S_\sB)) \tilde S_\sA \nonumber \\
& = S_\sA(\rho_\sA \otimes \rho_\sA) S_\sA + (1 - {\rm tr}( S_\sA(\rho_\sA \otimes \rho_\sA) S_\sA)) \tilde S_\sA \ .
\end{aligned}$$
\[RedstatesLem\] The reduced states $\bar \omega_\sA$ belong to the convex hull of the local pure states ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$.
By Lemma \[Redstatelem\] the reduced state can be written as: $$\bar \omega_\sA = S_\sA(\rho_\sA \otimes \rho_\sA) S_\sA + (1 - {\rm tr}( S_\sA(\rho_\sA \otimes \rho_\sA) S_\sA)) \tilde S_\sA \ ,$$ where $\rho_\sA = {\rm tr}_\sB ({
| \psi \rangle \! \langle \psi |}_{\sA \sB})$. In the following we drop the $\sA$ label. $$\rho = \sum_i \alpha_i {
| i \rangle \! \langle i |} \ ,$$ Here the $\ket i$ are not necessarily orthogonal. The trace of $\rho$ is $\sum_i \alpha_i = 1$, where $\alpha_i >0$. Let us write $\Phi_{ij} = \frac{1}{\sqrt{2}} (\ket{i,j} + \ket{j, i})$ and observe: $$\sum_{i \neq j} {
| \Phi_{ij} \rangle \! \langle \Phi_{ij} |} = 2 \sum_{i<j} {
| \Phi_{ij} \rangle \! \langle \Phi_{ij} |} = \sum_{i \neq j} \left({
| i,j \rangle \! \langle i,j |} + {
| i,j \rangle \! \langle j,i |} \right) \ .$$
Consider the (not necessarily normalised) matrix $$\label{projection}
S (\rho \otimes \rho) S = S(\sum_{ij} \alpha_{i} \alpha_j {
| i,j \rangle \! \langle i,j |})S
=
\sum_{i} \alpha_i^2 |i,i\rangle\! \langle i,i|
+ \sum_{i<j} \alpha_i \alpha_j {
| \Phi_{ij} \rangle \! \langle \Phi_{ij} |}\ ,$$
The trace of this matrix is $1- \sum_{i<j} \alpha_i \alpha_j$; hence: $$\label{target}
\bar \omega = \sum_{i} \alpha_i^2 |i,i\rangle\! \langle i,i|
+ \sum_{i<j} \alpha_i \alpha_j {
| \Phi_{ij} \rangle \! \langle \Phi_{ij} |}
+ \sum_{i<j} \alpha_i \alpha_j \tilde S \ .$$ We now show that this arbitrary mixed state $\bar \omega$ can be expressed as a convex combination of local pure states ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$. Consider the general vector $$|\psi\rangle =
\sum_i e^{i \theta_j} \sqrt{\alpha_j} |j\rangle\ ,$$ where $\alpha_j \geq 0$ and for all $j$. Normalisation implies $\sum_j \alpha_j =1$. Now, let us write the pure product state $$\label{sim prod}
|\psi\rangle\! \langle\psi|^{\otimes 2} =
\sum_{j,k,j',k'}
e^{i \theta_j} e^{i \theta_k} e^{-i \theta_{j'}} e^{-i \theta_{k'}}
\sqrt{\alpha_j \alpha_k \alpha_{j'} \alpha_{k'}}\,
|j,k\rangle\! \langle j',k'|\ .$$ Let us make the following observations. When $j \neq j'$: $$\int_{- \pi}^{\pi} e^{ - i \theta_j} e^{i \theta_{j'}} d \theta_j d \theta_{j'} = 0 \ .$$ When $j = j'$: $$\int_{- \pi}^{\pi} e^{ - i \theta_j} e^{i \theta_{j'}} d \theta_{j} d \theta_{j'} = (2 \pi)^2 \ .$$ Now consider: $$\begin{aligned}
\mathbb E_{\theta_i }
|\psi\rangle\! \langle\psi|^{\otimes 2} =
\frac{1}{(2 \pi)^4}\int_{-\pi}^{\pi} \sum_{j,k,j',k'}
e^{i \theta_j} e^{i \theta_k} e^{-i \theta_{j'}} e^{-i \theta_{k'}}
\sqrt{\alpha_j \alpha_k \alpha_{j'} \alpha_{k'}}\,
|j,k\rangle\! \langle j',k'| d \theta_j d \theta_k d \theta_{j'} d \theta_{k'} \ .
\end{aligned}$$ The non zero contributions will arise from the following terms:
$j=j'=k=k'$: $$\label{phase1}
\int_{- \pi}^{\pi}|e^{i \theta_j}|^4 d \theta_j d \theta_k d \theta_{j'} d \theta_{k'}= (2\pi)^4 \ .$$
$j = j' \neq k = k'$: $$\label{phase2}
\int_{- \pi}^{\pi}|e^{i \theta_j}|^2 |e^{i \theta_k}|^2 d \theta_j d \theta_k d \theta_{j'} d \theta_{k'}= (2\pi)^4 \ .$$
$j = k' \neq k = j'$: $$\label{phase3}
\int_{- \pi}^{\pi}|e^{i \theta_j}|^2 |e^{i \theta_k}|^2 d \theta_j d \theta_k d \theta_{j'} d \theta_{k'}= (2\pi)^4 \ .$$ All other contributions will be zero.
Now, we write the mixed state corresponding to the uniform average over all values of the phases $\theta_i$, $$\bar \omega_1 = \mathbb E_{\theta_i }
|\psi\rangle\! \langle\psi|^{\otimes 2}
=
\sum_{j,k}
\alpha_j \alpha_k
|j,k\rangle\! \langle j,k|
+ \sum_{j\neq k} \alpha_j \alpha_k |j,k\rangle\! \langle k,j| \ ,$$ where the first term arises from contributions and and the second term arises from contribution . Then we can write $$\bar \omega_1
=
\sum_{i} \alpha_i^2 |i,i\rangle\! \langle i,i|
+ 2 \sum_{i<j} \alpha_i \alpha_j {
| \Phi_{ij} \rangle \! \langle \Phi_{ij} |} \ ,$$ Let us take the state: $$\bar \omega_2 = \sum_{i} \alpha_i^2 |i,i\rangle\! \langle i,i| + 2 \sum_{i<j} \alpha_i \alpha_j \tilde S \ .$$ This is a mixture of states of the form ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$ since $\tilde S = \int {
| \psi \rangle \! \langle \psi |}^{\otimes 2}d\psi$ and $\sum_{i} \alpha_i^2 + 2 \sum_{i<j} \alpha_i \alpha_j = 1$. If we take the mixture $\frac{1}{2}(\bar \omega_1 + \bar \omega_2)$ we obtain (\[target\]).
We now consider the more general case where Alice’s state is conditioned on an arbitrary effect $F_\sB$. The conditional state for Alice given one of Bob’s effects $ F_\sB$ is: $$\begin{aligned}
\bar \omega_{\sA | F_\sB} & = {\rm Tr}_\sB(S_\sA \hat F_\sB {
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} ) + {\rm Tr} ({
| \psi \rangle \! \langle \psi |}^{\otimes 2}_{\sA \sB} A_\sA A_\sB)\frac{{\rm Tr}(\hat F_\sB)}{{\rm Tr}(S_\sB)} \tilde S_\sA \ .\end{aligned}$$ Although effects of the form $\hat F_\sB = {
| \phi \rangle \! \langle \phi |}^{\otimes 2}$ are not valid (since the complement effects would not be of the required form), we calculate the conditional state for such effects, as this will allow us to later determine conditional states for general effects $\hat F_\sB = \sum_i \alpha_i {
| \phi_i \rangle \! \langle \phi_i |}^{\otimes 2}$.
\[equlemmacond\] For $\hat F_\sB = {
| \phi \rangle \! \langle \phi |}^{\otimes 2}$: $${\rm Tr}_\sB(S_\sA \hat F_\sB {
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} ) = S_\sA (\rho_{\sA |\phi}^\psi \otimes \rho_{\sA | \phi}^\psi) S_\sA = \rho_{\sA |\phi}^\psi \otimes \rho_{\sA |\phi}^\psi \ ,$$ where $\rho_{\sA |\phi}^\psi = {\rm Tr}((\unity_\sA \otimes {
| \phi \rangle \! \langle \phi |}_\sB) {
| \psi \rangle \! \langle \psi |}_{\sA \sB})$ .
$$\begin{aligned}
{\rm Tr}_\sB(S_\sA \hat F_\sB {
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} ) = S_\sA {\rm Tr}_\sB( \hat F_\sB {
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} ) = S_\sA (\rho_{\sA | \phi}^\psi \otimes \rho_{\sA | \phi}^\psi) \ .
\end{aligned}$$
Let $$\begin{aligned}
\rho_{\sA | \phi}^\psi = {\rm Tr} \big((\unity_\sA \otimes {
| \phi \rangle \! \langle \phi |}_\sB) {
| \psi \rangle \! \langle \psi |}_{\sA \sB} \big) = \sum_{i_1, j_1} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} {
| i_1 \rangle \! \langle j_1 |} \ .
\end{aligned}$$ We assume without loss of generality that $\ket \phi$ is one of the basis vectors $\ket i$. $$\begin{aligned}
\rho_{\sA | \phi}^\psi \otimes \rho_{\sA | \phi}^\psi = \sum_{i_1, i_3, j_1, j_3} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} \alpha_{i_3 , \phi} \bar \alpha_{j_3 , \phi} {
| i_1 i_3 \rangle \! \langle j_1 j_3 |} \ .
\end{aligned}$$ $$\begin{aligned}
S_{\sA} (\rho_{\sA | \phi} \otimes \rho_{\sA | \phi})
= & \frac{1}{2} (\sum_{i_1, i_3, j_1, j_3} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} \alpha_{i_3 , \phi} \bar \alpha_{j_3 , \phi} {
| i_1 i_3 \rangle \! \langle j_1 j_3 |} \nonumber \\
+ & \sum_{i_1, i_3, j_1, j_3} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} \alpha_{i_3 , \phi} \bar \alpha_{j_3 , \phi} {
| i_3 i_1 \rangle \! \langle j_1 j_3 |} ) \ .
\end{aligned}$$ We relabel $i_1 \leftrightarrow i_3$ on the last line to obtain $ S_\sA (\rho_{\sA | \phi} \otimes \rho_{\sA | \phi}) = (\rho_{\sA | \phi} \otimes \rho_{\sA | \phi})$. $$\begin{aligned}
& S_{\sA} (\rho_{\sA | \phi} \otimes \rho_{\sA | \phi}) S_{\sA}
= \frac{1}{4} (\sum_{i_1, i_3, j_1, j_3} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} \alpha_{i_3 , \phi} \bar \alpha_{j_3 , \phi} {
| i_1 i_3 \rangle \! \langle j_1 j_3 |} \nonumber \\
+ & \sum_{i_1, i_3, j_1, j_3} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} \alpha_{i_3 , \phi} \bar \alpha_{j_3 , \phi} {
| i_3 i_1 \rangle \! \langle j_1 j_3 |} + \sum_{i_1, i_3, j_1, j_3} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} \alpha_{i_3 , \phi} \bar \alpha_{j_3 , \phi} {
| i_1 i_3 \rangle \! \langle j_3 j_1 |} \nonumber \\
+ & \sum_{i_1, i_3, j_1, j_3} \alpha_{i_1 , \phi} \bar \alpha_{j_1 , \phi} \alpha_{i_3 , \phi} \bar \alpha_{j_3 , \phi} {
| i_3 i_1 \rangle \! \langle j_3 j_1 |} ) \ .
\end{aligned}$$ We relabel $j_1 \leftrightarrow j_3$ on the last two lines to obtain $ S_\sA (\rho_{\sA | \phi} \otimes \rho_{\sA | \phi}) = S_\sA (\rho_{\sA | \phi} \otimes \rho_{\sA | \phi})S_\sA $.
We need one more lemma before proving that normalised conditional states belong to the convex hull of the local pure states ${
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2}$.
\[stateform\] If $S(\rho \otimes \rho) S = \rho \otimes \rho$ with ${\rm Tr} (\rho) = 1$ then $\rho \otimes \rho = \sum_i p_i {
| \psi_i \rangle \! \langle \psi_i |}^{\otimes 2}$.
By Lemma \[RedstatesLem\] this is a valid reduced state and belongs to ${\rm conv} ({
| \psi \rangle \! \langle \psi |}^\otimes 2)$.
We observe that since all pure states are such that $S ({
| \psi \rangle \! \langle \psi |}^{\otimes 2})S = {
| \psi \rangle \! \langle \psi |}^{\otimes 2}$ and ${\rm Tr}({
| \psi \rangle \! \langle \psi |}^{\otimes 2}) = 1$, we can characterise the state space of the systems of the toy model as being given by ${\rm conv} (\rho \otimes \rho)$ for all normalised density operators such that $S(\rho \otimes \rho) S = \rho \otimes \rho$.
\[CondstatesLem\] The normalised conditional states $\tilde \omega_{\sA|F_\sB}$ belong to the convex hull of the local pure states ${
| \psi \rangle \! \langle \psi |}^{\otimes 2}$.
We first show that the conditional state is a valid local state for effects $\hat F_\sB = {
| \phi \rangle \! \langle \phi |}^{\otimes2}$. As a conditional state, this state can be subnormalised. $$\begin{aligned}
\bar \omega_{\sA | F_\sB} = {\rm Tr}_\sB(S_\sA \hat F_\sB {
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} ) + c \tilde S_\sA \ ,
\end{aligned}$$ where $c = {\rm Tr} ({
| \psi \rangle \! \langle \psi |}^{\otimes 2}_{\sA \sB} A_\sA A_\sB)\frac{{\rm Tr}(\hat F_\sB)}{{\rm Tr}(S_\sB)}$. By Lemma \[equlemmacond\] we have the equivalence: $${\rm Tr}_\sB(S_\sA \hat F_\sB {
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} ) = (\rho_{\sA | \phi}^\psi \otimes \rho_{\sA | \phi}^\psi) \ .$$ The normalised conditional state is: $$\tilde \omega_{\sA | F_\sB} = \frac{\bar \omega_{\sA | F_\sB}}{{\rm Tr}(\bar \omega_{\sA | F_\sB})} \ .$$ Let us set $e= {\rm Tr}(\bar \omega_{\sA | F_\sB})$ and $d = {\rm Tr}(S_\sA \hat F_\sB {
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} ) ={\rm Tr} (\rho_{\sA | \phi}^\psi \otimes \rho_{\sA | \phi}^\psi) $; $e = c+d$. $$\begin{aligned}
\tilde \omega_{\sA | F_\sB} = \frac{1}{e} (\rho_{\sA | \phi} \otimes \rho_{\sA | \phi})
+ \frac{c}{e} \tilde S_\sA \ .
\end{aligned}$$ We use the equality $\rho_{\sA |\phi}^\psi \otimes \rho_{\sA | \phi}^\psi = d (\tilde \rho_{\sA | \phi}^\psi \otimes \tilde \rho_{\sA | \phi}^\psi) $ where $\tilde \rho_{\sA | F}^\psi$ is a standard normalised quantum conditional state. $$\begin{aligned}
\tilde \omega_{\sA | F_\sB} = & \frac{d}{e}(\tilde \rho_{\sA | \phi}^\psi \otimes \tilde \rho_{\sA |\phi}^\psi) + \frac{c}{e} \tilde S_\sA \ .
\end{aligned}$$ By Lemma \[equlemmacond\] $\tilde \rho_{\sA | \phi}^\psi \otimes \tilde \rho_{\sA |\phi}^\psi = S_\sA (\tilde \rho_{\sA | \phi}^\psi \otimes \tilde \rho_{\sA |\phi}^\psi) S_\sA$ hence by Lemma \[stateform\] it is a valid normalised state (i.e. of the form $\sum_i p_i {
| \psi_i \rangle \! \langle \psi_i |}^{\otimes 2}$). Since $0 <\frac{d}{e} < 1$, $0 < \frac{c}{e} < 1 $ and $\frac{d + c}{e} = 1$ the above is a convex combination of $\tilde \rho_{\sA | \phi}^\psi \otimes \tilde \rho_{\sA |\phi}^\psi$ and $ \tilde S_\sA $ which are both valid states. Hence the state $\tilde \omega_{\sA | F_\sB}$ is a valid local state.
Let $F_\sB = \sum \alpha_i {
| \phi_i \rangle \! \langle \phi_i |}^{\otimes 2}$, with $\alpha_i > 0$.
$$\begin{aligned}
\bar \omega_{\sA | F_\sB} & = \sum_{i} \alpha_i ({\rm Tr}_\sB(S_\sA \hat F_\sB {
| \phi_i \rangle \! \langle \phi_i |}^{\otimes 2}{
| \psi \rangle \! \langle \psi |}_{\sA \sB}^{\otimes 2} )
+ {\rm Tr} ({
| \psi \rangle \! \langle \psi |}^{\otimes 2}_{\sA \sB} A_\sA A_\sB)\frac{1}{{\rm Tr}(S_\sB)} \tilde S_\sA) \nonumber \\
& = \sum_{i} \alpha_i \big( (\rho_{\sA |\phi_i}^{\psi} \otimes \rho_{\sA |\phi_i}^{\psi}) + c_i \tilde S_\sA) \ .
\end{aligned}$$
Let $e = {\rm tr} (\omega_{\sA | F_\sB})$ and $d_i = {\rm tr}( \rho_{\sA |\phi_i}^{\psi} \otimes \rho_{\sA |\phi_i}^{\psi})$. We have $e = \sum_i \alpha_i (c_i + d_i)$. From above: $$\begin{aligned}
\tilde \omega_{\sA | F_\sB} = & \sum_i \alpha_i( \frac{d_i}{e}(\tilde \rho_{\sA | \phi_i}^{\psi} \otimes \tilde \rho_{\sA | \phi_i}^{\psi}) + \frac{c_i}{e} \tilde S_\sA) \ .
\end{aligned}$$ Since $0 < \frac{\alpha_ i d_i}{e} < 1$ and $0 < \sum_i \frac{ \alpha_ i c_i}{e}<1$ and $\sum_i \frac{ \alpha_i (c_i + d_i)}{e} = 1$ the above is a convex combination of $(\tilde \rho_{\sA | \phi_i}^{\psi} \otimes \tilde \rho_{\sA | \phi_i}^{\psi})$ with coefficients $\frac{\alpha_ i d_i}{e}$ and the state $\tilde S_\sA$ with coefficient $\sum_i \frac{\alpha_i c_i}{e}$.
Constraint C5
-------------
In this section we show that every OPFs in $\F_{d_\sA d_\sB}^\sg$ applied to a product state $\psi_\sA \otimes \phi_\sB$ has a corresponding OPF $F_\sA'$ in $\F_{d_\sA}^\sl$. Let $F_{\sA \sB}$ be an arbitrary effect in $\F_{d_\sA d_\sB}^\sl$. The corresponding operator is $$\hat F_{\sA \sB} = \sum_i \alpha_i {
| x_i \rangle \! \langle x_i |}_{12} \otimes {
| x_i \rangle \! \langle x_i |}_{34} \ .$$ We evaluate it on product states: $$\begin{aligned}
F_{\sA \sB} (\psi_\sA \otimes \phi_\sB)
& = \sum_i \alpha_i {\rm Tr}\big({
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2} {
| \phi \rangle \! \langle \phi |}_\sB^{\otimes 2} \ ({
| x_i \rangle \! \langle x_i |}_{\sA \sB}^{\otimes 2})\big) \nonumber \\
& = \sum_i \alpha_i {\rm Tr}_{\sA} ({
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2} {\rm Tr}_{\sB} ( {
| \phi \rangle \! \langle \phi |}_\sB^{\otimes 2} ({
| x_i \rangle \! \langle x_i |}_{\sA \sB}^{\otimes 2})) ) \nonumber \\
& = \sum_i \alpha_i {\rm Tr}_{\sA} ({
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2} S_\sA {\rm Tr}_{\sB}( \unity_\sA {
| \phi \rangle \! \langle \phi |}_\sB^{\otimes 2}{
| x_i \rangle \! \langle x_i |}_{\sA \sB}^{\otimes 2} ) ) \nonumber \\
& = {\rm Tr}_{\sA} ({
| \psi \rangle \! \langle \psi |}_\sA^{\otimes 2} ( \sum_i \alpha_i( S_\sA (\rho_{\sA| \phi}^{x_i} \otimes \rho_{\sA| \phi}^{x_i}) S_\sA))) \ .\end{aligned}$$
By Lemma \[stateform\] $( S_\sA (\rho_{\sA| \phi}^x \otimes \rho_{\sA| \phi}^x) S_\sA$ is of the form $\sum_j \beta_j {
| \phi_j \rangle \! \langle \phi_j |}^{\otimes 2}$ with $\beta_j \geq 0$. Hence $ \hat F_\sA' = ( \sum_i \alpha_i( S_\sA (\rho_{\sA| \phi}^{x_i} \otimes \rho_{\sA| \phi}^{x_i}) S_\sA)$ is a valid effect on $\sA$ as long as its complement is also of the form $\sum_i \gamma_i {
| \phi_i \rangle \! \langle \phi_i |}^{\otimes 2}$. Since the complement of $\hat F_{\sA \sB}$ is of the form $\sum_i \alpha_i {
| x_i \rangle \! \langle x_i |}_{12} \otimes {
| x_i \rangle \! \langle x_i |}_{34}$ it follows that the associated effect on $\sA$ (which is the complement of $\hat F_\sA'$) is of the required form. From this it follows that $\hat F_\sA'$ is a valid effect. The set $\mathcal F_{d_\sA d_\sB}^\sg$ also contains effects $F_\sA \star F_\sB$ which are not necessarily of the form given above. However since these are product effects they trivially are consistent with [**C5**]{}.
[^1]: We thank referee 1 for this observation.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to $\mathbb{C}$ in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient manifold has uniformly positive complex sectional curvature. The proof consists of two parts, assuming an “eigenvalue condition” on the $\overline{\partial}$-operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on $\mathbb{C}$; (2) an existence theorem for holomorphic sections with controlled growth by Hörmander’s weighted $L^2$-method.'
address: |
Mathematics Department\
Stanford University\
Stanford, CA 94305
author:
- 'Martin Man-chun Li'
bibliography:
- 'references.bib'
title: 'On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature'
---
Motivation
==========
It has been a central theme in Riemannian geometry to study how the curvatures of a manifold affect its topology. For 2-dimensional surfaces, the Gauss-Bonnet theorem plays a key role. In particular, it implies that any oriented closed surface with positive sectional curvature is diffeomorphic to the 2-sphere. In higher dimensions, Synge theorem says that any closed even-dimensional oriented Riemannian manifold $M^{2n}$ with positive sectional curvature is simply connected. The key idea in the proof is that there exists no stable closed geodesic $\sigma$ in such manifold. To see this, recall the second variation formula for lengths of a 1-parameter family of closed curves $\{\sigma_t\}_{t \in (-\epsilon,\epsilon)}$: $$\left. \frac{d^2 L(\sigma_t)}{dt^2} \right|_{t=0}=\int_{\sigma} [ \| \nabla_{\sigma'} X \|^2 - \langle R(\sigma',X)\sigma',X \rangle ] \; ds$$ where $\sigma=\sigma_0$ is a closed geodesic in $M$, $X=\left. \frac{\partial \sigma_t}{\partial t} \right|_{t=0}$ is a variation field along $\sigma$ and $R(X,Y)Z=\nabla_Y \nabla_X Z - \nabla_X \nabla_Y Z + \nabla_{[X,Y]}Z$ is the Riemann curvature tensor for $M$. Synge observed that if $M$ is even-dimensional and oriented, then there is a non-zero parallel vector field $X$ (i.e. $\nabla_{\sigma'} X=0$), hence positivity of sectional curvature implies that $$\left. \frac{d^2 L(\sigma_t)}{dt^2} \right|_{t=0}<0.$$ Consequently, any closed geodesic in an oriented Riemannian manifold $M^{2n}$ with positive sectional curvature is *unstable*. Therefore, $M$ has to be simply connected. Otherwise, one could minimize the length in any non-trivial free homotopy class in $\pi_1(M)$ to get a stable closed geodesic, which is a contradiction.
The idea of Synge can be extended in various directions. For example, if one considers the second variation formula for area of minimal surfaces, the curvature term that appears in the formula is the “*isotropic curvature*”. Just like positive sectional curvature tends to make geodesics unstable; positive isotropic curvature tends to make minimal surfaces unstable. In [@Micallef-Moore88], M. Micallef and J. Moore proved that any minimal 2-sphere in a manifold with positive isotropic curvature is unstable. It is natural to look at the stability of complete non-compact minimal surface instead. In this paper, we prove (Theorem 2.6) the instability of complete minimal surfaces uniformly conformally equivalent to $\mathbb{C}$ by constructing holomorphic sections (half-parallel sections) with slow growth using Hörmander’s weighted $L^2$-method and then applying a weighted second variation argument.
Definitions and preliminaries
=============================
The purpose of this paper is to study complete minimal surfaces $\Sigma$ in an $n$-dimensional Riemannian manifold $M$ ($n \geq 4$) which minimize area up to second order. In particular, we prove that if $n=4$ and the ambient manifold $M$ is orientable and has uniformly positive isotropic curvature, then there does not exist a complete stable minimal surface which is uniformly conformally equivalent to the complex plane $\mathbb{C}$. Note that we do not require $M$ to be compact. Therefore the result applies to universal covers of compact manifolds with positive isotropic curvature. At the end of the paper, we prove that the same result holds for any $n \geq 4$ but $M$ satisfies the stronger condition that it has uniformly positive complex sectional curvature, where $M$ need not be orientable.
First we recall some definitions. Let $M^n$ be an $n$-dimensional Riemannian manifold. Consider the complexified tangent bundle $T^\mathbb{C}M=TM \otimes_\mathbb{R} \mathbb{C}$ equipped with the Hermitian extension $\langle \cdot, \cdot \rangle$ of the inner product on $TM$, the curvature tensor extends to complex vectors by linearity, and the *complex sectional curvature* of a complex two-dimensional subspace $\pi$ of $T^\mathbb{C}_pM$ at some point $p \in M$ is defined by $K^\mathbb{C}(\pi)=\langle R(v,w)\overline{w},v\rangle$, where $\{v,w\}$ is any unitary basis for $\pi$.
A Riemannian manifold $M^n$ has *uniformly positive complex sectional curvature* if there exists a constant $\kappa>0$ such that $K^\mathbb{C}(\pi) \geq \kappa >0$ for every complex two-dimensional subspace $\pi$ in $T^\mathbb{C}_pM$ at every $p \in M$.
It is clear that having uniformly positive complex sectional curvature implies having uniformly positive sectional curvature. (One simply considers all $\pi$ which comes from the complexification of a real two-dimensional subspace in $T_pM$.) Therefore, by Bonnet-Myers theorem, $M$ is automatically compact if it is complete.
Using Ricci flow techniques, S. Brendle and R. Schoen [@Brendle-Schoen09] proved that any manifold $M$ with uniformly positive complex sectional curvature is diffeomorphic to a spherical space form. In particular, when $M$ is simply-connected, $M$ is diffeomorphic to the $n$-dimensional sphere. In fact, Brendle and Schoen showed that the condition of positive complex sectional curvature is preserved under the Ricci flow, and any manifold equipped with such a metric evolves under the normalized Ricci flow to a spherical space-form. The optimal convergence result so far is obtained by S. Brendle in [@Brendle08], where he proved that any compact manifold $M$ such that $M \times \mathbb{R}$ has positive isotropic curvature would converge to a spherical space form under the normalized Ricci flow.
As a result, having positive complex sectional curvature is a rather restrictive condition. There is a related positivity condition which allows more flexibility called *positive isotropic curvature* (PIC). Instead of extending in a Hermitian way, one can choose to extend the metric on $TM$ to a $\mathbb{C}$-bilinear form $(\cdot,\cdot)$ on $T^\mathbb{C}M$. We say that a vector $v \in T^\mathbb{C}_pM$ is *isotropic* if $(v,v)=0$. A subspace $V \subset T^\mathbb{C}_p$ is *isotropic* if every $v \in V$ is isotropic.
A Riemannian manifold $M^n$, $n \geq 4$, has *uniformly positive isotropic curvature* (uniformly PIC) if there exists a positive constant $\kappa>0$ such that $K^\mathbb{C}(\pi) \geq \kappa >0$ for every isotropic complex two-dimensional subspace $\pi \subset T^\mathbb{C}_pM$ at every $p \in M$.
This condition is clearly weaker than the previous one because it only requires positivity of complex sectional curvature among *isotropic* 2-planes. Moreover, the condition is vacuous when $n\leq 3$ (see [@Micallef-Moore88]), so we restrict ourselves to $n \geq 4$.
In [@Brendle-Schoen09], it was shown that uniformly PIC is also a condition preserved under the Ricci flow. However, it is not true that any manifold equipped with such a metric would converge to a spherical space-form under the normalized Ricci flow. For example, the product manifold $S^{n-1} \times S^1$ is PIC but it has no metric with constant positive sectional curvature. (The universal cover of $S^{n-1} \times S^1$ is $S^{n-1} \times \mathbb{R}$, which is not $S^n$.)
On the other hand, there is a “sphere theorem” for compact manifolds with PIC. Namely, if $M^n$, $n\geq 4$, is a compact manifold with PIC, and $M$ is simply-connected, then $M^n$ is *homeomorphic* to $S^n$. This sphere theorem, which generalizes the classical sphere theorem since any $1/4$-pinched manifold is PIC, was proved by M. Micallef and J. Moore in [@Micallef-Moore88], where the notion of PIC was first defined. In other words, all simply-connected PIC manifolds are topologically “trivial”. This, of course, raises the natural question: what are the topological obstructions on the fundamental group $\pi_1M$ for a compact manifold $M$ to admit a metric with PIC?
Along this direction, A. Fraser [@Fraser03] proved that for $n \geq 5$, $\pi_1M$ cannot contain a subgroup isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$, the fundamental group of a torus. Her proof uses the existence theory of stable minimal surfaces by R. Schoen and S.T. Yau [@Schoen-Yau79], and the Riemann-Roch theorem to construct *almost holomorphic* variations which, in turn, contradicts stability. A few years later, S. Brendle and R. Schoen [@Brendle-Schoen] proved that the same result holds for $n=4$. On the other hand, M. Micallef and M. Wang [@Micallef-Wang93] proved that if $(M_1^n,g_1)$ and $(M_2^n,g_2)$ are manifolds with positive isotropic curvature, then $M_1 \sharp M_2$ also admits a metric with positive isotropic curvature. In particular, $\sharp_{i=1}^k(S^1 \times S^{n-1})$ admits a metric with positive isotropic curvature for any positive integer $k$. Therefore, the fundamental group of a manifold with positive isotropic curvature can be quite large.
To state the main result in this paper, we need the following definition.
A Riemannian surface $(\Sigma,h)$ is *uniformly conformally equivalent* to the complex plane $\mathbb{C}$ if there is a (conformal) diffeomorphism $\phi:\mathbb{C} \to \Sigma$, a positive smooth function $\lambda$ on $\mathbb{C}$ and a constant $C>0$ such that $$\phi^*h=\lambda^2 |dz|^2 \qquad \text{with} \qquad \frac{1}{C} \leq \lambda^2.$$
Our theorem states that surfaces of this type cannot arise as stable minimal surfaces in any orientable 4-manifold with uniformly positive isotropic curvature.
Let $M$ be a 4-dimensional complete orientable Riemannian manifold with uniformly positive isotropic curvature. Let $\mathbb{C}$ be the complex plane equipped with the standard flat metric. Then there does not exist a stable immersed minimal surface $\Sigma$ in $M$ which is uniformly conformally equivalent to $\mathbb{C}$.
If we assume that $M$ has uniformly positive complex sectional curvature, then the result holds in any dimension, and without the orientability assumption on $M$.
Let $M$ be an $n$-dimensional complete (not necessarily orientable) Riemannian manifold with uniformly positive complex sectional curvature. Let $\mathbb{C}$ be the complex plane equipped with the standard flat metric. Then there does not exist a stable immersed minimal surface $\Sigma$ in $M$ which is uniformly conformally equivalent to $\mathbb{C}$.
An outline of the paper is as follows. In section 3, we prove a vanishing theorem for the $\overline{\partial}$-operator on holomorphic bundles satisfying an “eigenvalue condition”. In section 4, we describe Hörmander’s weighted $L^2$-method and use it to construct holomorphic sections with controlled growth, assuming the “eigenvalue condition”. In section 5, we apply the results to prove Theorem 2.6 and Theorem 2.7.
*Acknowledgement.* The author would like to thank his advisor, Professor Richard Schoen, for all his continuous support and encouragement throughout the progress of this work. He would also like to thank Professor Simon Brendle for many helpful discussions. He is thankful to the referee for many useful comments.
A vanishing theorem
===================
Throughout the paper, $\mathbb{C}$ will denote the standard complex plane with the flat metric $$ds^2=dx^2+dy^2=|dz|^2.$$ Suppose $E$ is a holomorphic vector bundle over $\mathbb{C}$ with a compatible Hermitian metric. Let $\overline{\partial}$ denote the $\overline{\partial}$-operator associated to $E$. Assume that $(E,\overline{\partial})$ satisfies the following “eigenvalue condition”: these exists some positive constant $\kappa_0>0$, and a sufficiently small constant $\epsilon_0>0$ (depending on $\kappa_0$) such that for all $0< \epsilon<\epsilon_0$, we have $$\kappa_0 \int_\mathbb{C} |s|^2 e^{-\epsilon |z|} \; dxdy \leq \int_\mathbb{C} |\overline{\partial} s|^2 e^{-\epsilon |z|} \; dxdy$$ for all compactly supported smooth sections $s$ of $E$, we write $s \in C^\infty_c(E)$.
For any positive continuous function $\varphi$ on $\mathbb{C}$, we can define an $L^2$-norm on $C^\infty_c(E)$ by $$\|s\|_{L^2(E,\varphi)}=\left( \int_\mathbb{C} |s|^2 \varphi \; dxdy \right)^{\frac{1}{2}} .$$ Let $L^2(E,\varphi)$ be the Hilbert space completion of $C^\infty_c(E)$ with respect to the weighted $L^2$-norm $\| \cdot \|_{L^2(E,\varphi)}$. In other words, $s \in L^2(E,\varphi)$ if and only if $s$ is a measurable section of $E$ with $\|s\|_{L^2(E,\varphi)}<\infty$.
In this section, we prove a vanishing theorem for the $\overline{\partial}$-operator on these weighted $L^2$ spaces of sections of holomorphic bundles over $\mathbb{C}$ satisfying the “eigenvalue condition” above. Roughly speaking, a holomorphic section cannot grow too slowly unless it is identically zero.
Suppose $E$ is a holomorphic vector bundle over $\mathbb{C}$ satisfying the “eigenvalue condition” (3.1) with constants $\kappa_0$ and $\epsilon_0$. Then, there exists no non-trivial holomorphic section $s \in L^2(E,e^{-4\epsilon |z|})$ for any $0 < \epsilon<\epsilon_0/4$.
The proof is a direct cutoff argument. Suppose $s$ is a holomorphic section of $E$ which belongs to $L^2(E,e^{-4\epsilon |z|})$ for some $0< \epsilon<\epsilon_0/4$. We will show that $s \equiv 0$.
For every real number $R>0$, choose a cutoff function $\phi_R \in C^\infty_c(\mathbb{C})$ such that
- $\phi_R(z)=1$ for $|z| \leq R$;
- $\phi_R(z)=0$ for $|z| \geq 2R$;
- $|\nabla \phi_R| \leq \frac{2}{R}$.
Let $\hat{s}=\phi_R s$. Note that $\hat{s} \in C^\infty_c(E)$. Therefore, by (3.1), properties of $\phi_R$ and holomorphicity of $s$, $$\begin{aligned}
\kappa_0 \int_{|z| \leq R} |s|^2 e^{-4\epsilon|z|} \; dxdy & \leq \kappa_0 \int_\mathbb{C} |\hat{s}|^2 e^{-4\epsilon|z|} \; dxdy \\
&\leq \int_\mathbb{C} |\overline{\partial}\hat{s}|^2 e^{-4\epsilon |z|} \; dxdy \\
&=\int_\mathbb{C} |\overline{\partial}\phi_R|^2 |s|^2 e^{-4\epsilon|z|} \; dxdy \\
& \leq \frac{1}{R^2} \int_{R \leq |z| \leq 2R} |s|^2 e^{-4\epsilon |z|} \; dxdy \\
&\leq \frac{1}{R^2} \|s\|^2_ {L^2(E,e^{-4\epsilon |z|})}.\end{aligned}$$ By our assumption, $s \in L^2(E,e^{-4\epsilon |z|})$. As $R \to \infty$, the right hand side goes to zero while the left hand side goes to $\kappa_0 \|s\|_{L^2(E,e^{-4\epsilon |z|})}$. Since $\kappa_0>0$, we conclude that $\|s\|_{L^2(E,e^{-4\epsilon |z|})}=0$, hence $s \equiv 0$. This completes the proof.
Hörmander’s weighted $L^2$ method
=================================
In this section, we will use Hörmander’s weighted $L^2$ method to construct non-trivial weighted $L^2$ holomorphic sections on $E$. Some basic facts on unbounded operators between Hilbert spaces can be found in [@Chen-Shaw].
Assume $E$ is a holomorphic vector bundle on $\mathbb{C}$ satisfying the “eigenvalue condition” (3.1) with constants $\epsilon_0$ and $\kappa_0$. For this section, we also assume that $E$ is the complexification of some real vector bundle $\xi$, hence $\overline{E}=E$.
Recall that there is a natural first order differential operator $\overline{\partial}$ defined on the space of smooth compactly supported sections of $E$: $$\overline{\partial}: C^\infty_c(E) \to C^\infty_c(E \otimes T^{0,1}\mathbb{C}),$$ where $$\overline{\partial}s=(\nabla_{\frac{\partial}{\partial \overline{z}}} s )\otimes d\overline{z}.$$ Let $L^2(E,e^{-2\epsilon |z|})$ be the Hilbert space completion of $C^\infty_c(E)$ with respect to the weighted $L^2$-norm $$\|s\|_{L^2(E,e^{-2\epsilon |z|})} = \left( \int_\mathbb{C} |s|^2 e^{-2 \epsilon |z|} \; dxdy \right)^{\frac{1}{2}}.$$ Similarly, let $L^2(E \otimes T^{0,1} \mathbb{C}, e^{-2\epsilon |z|})$ denote the Hilbert space completion of $C^\infty_c(E \otimes T^{0,1}\mathbb{C})$ with respect to the weighted $L^2$-norm $$\| \sigma \|_{L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})} = \left( \int_\mathbb{C} |\sigma|^2 e^{-2 \epsilon |z|} \; dxdy \right)^{\frac{1}{2}} .$$ Now, let $$\overline{\partial}: L^2(E,e^{-2\epsilon |z|}) \to L^2(E \otimes T^{0,1}\mathbb{C}, e^{-2 \epsilon |z|})$$ be the maximal closure of $\overline{\partial}$ defined as follows: an element $s \in L^2(E,e^{-2\epsilon |z|})$ is in the domain of $\overline{\partial}$ if $\overline{\partial}s$, defined in the distributional sense, belongs to $L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})$. Then, $\overline{\partial}$ defines a linear, closed, densely defined unbounded operator. Note that $\overline{\partial}$ is closed because differentiation is a continuous operation in distribution theory. It is densely defined since Dom($\overline{\partial}$) contains all compactly supported smooth sections $C^\infty_c(E)$, which is clearly dense in $L^2(E,e^{-2\epsilon |z|})$.
By standard Hilbert space theory, the Hilbert space adjoint of $\overline{\partial}$, denoted by $\overline{\partial}^*$, is a linear, closed, densely defined unbounded operator and $$\overline{\partial}^* : L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|}) \to L^2(E,e^{-2 \epsilon |z|}).$$ An element $\sigma$ belongs to Dom$(\overline{\partial}^*)$ if there is an $s \in L^2(E,e^{-2 \epsilon |z|})$ such that for every $t \in $Dom$(\overline{\partial})$, we have $$(\sigma,\overline{\partial}t)_{L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})}=(s,t)_{L^2(E,e^{-2\epsilon |z|})}.$$ We then define $\overline{\partial}^* \sigma=s$.
The key theorem in this section is the surjectivity of $\overline{\partial}$.
Suppose $E$ is a holomorphic vector bundle over $\mathbb{C}$ satisfying the “eigenvalue condition” (3.1) with constants $\kappa_0$ and $\epsilon_0$. Moreover, assume that $E$ is the complexification of some real vector bundle $\xi$ over $\mathbb{C}$. Then, $$\overline{\partial} : L^2(E,e^{-2\epsilon |z|}) \to L^2(E \otimes T^{0,1}\mathbb{C}, e^{-2 \epsilon |z|})$$ is surjective for all $0 < \epsilon < \min (\frac{\epsilon_0}{2},\frac{\sqrt{\kappa_0}}{2})$.
By standard Hilbert space theory (see section 4.1 in [@Chen-Shaw]), it suffices to show that
- the adjoint operator $\overline{\partial}^*$ is injective, and
- the range of $\overline{\partial}$ is closed.
First of all, we need to compute $\overline{\partial}^*$ explicitly.
*Claim 1:* For any $\sigma=s \otimes d\overline{z} \in C^\infty_c(E \otimes T^{0,1}\mathbb{C})$, $$\overline{\partial}^* \sigma = - \nabla_{\frac{\partial}{\partial z}} s + \epsilon \frac{\overline{z}}{|z|} s.$$
*Proof of Claim 1:* This is just integration by parts. Let $t \in $Dom$(\overline{\partial})$, and $\langle \cdot, \cdot \rangle_E$, $\langle \cdot, \cdot \rangle_{E \otimes T^{0,1}\mathbb{C}}$ be the pointwise Hermitian metric on $E$ and $E \otimes T^{0,1}\mathbb{C}$ respectively. $$\begin{aligned}
(\sigma,\overline{\partial} t)_{L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})}&=\int_\mathbb{C} \langle \sigma, \overline{\partial} t \rangle_{E \otimes T^{0,1}\mathbb{C}} \;e^{-2 \epsilon |z|}\; dxdy \\
&= \int_\mathbb{C} \langle e^{-2 \epsilon |z|} \sigma, \overline{\partial} t \rangle_{E \otimes T^{0,1}\mathbb{C}}\; dxdy
\\
&= \int_\mathbb{C} \langle e^{-2 \epsilon |z|} s, \nabla_{\frac{\partial}{\partial \overline{z}}} t \rangle_E \; dxdy
\\
&= \int_\mathbb{C} \frac{\partial}{\partial z} \langle e^{-2\epsilon |z|} s,t \rangle_E \; dxdy -
\int_\mathbb{C} \langle \nabla_{\frac{\partial}{\partial z}}(e^{-2 \epsilon |z|}
s),t \rangle_E \; dxdy \\
&= -\int_\mathbb{C} \langle \nabla_{\frac{\partial}{\partial z}} s -\epsilon \frac{\overline{z}}{|z|}s,t
\rangle_E \; e^{-2\epsilon |z|} \; dxdy. \\
&=(-\nabla_{\frac{\partial}{\partial z}} s + \epsilon \frac{\overline{z}}{|z|} s,t)_{L^2(E,e^{-2\epsilon |z|})}.\end{aligned}$$ The first term in the second to last line vanishes because we have integrated by parts and used the fact that $s$ is compactly supported. This proves Claim 1.
Next, we need to establish a basic estimate for the adjoint operator $\overline{\partial}^*$.
*Claim 2:* For every $0 < \epsilon < \min(\frac{\epsilon_0}{2},\frac{\sqrt{\kappa_0}}{2})$, there exists a constant $\kappa_1>0$ such that $$\kappa_1 \int_\mathbb{C} |\sigma|^2 e^{-2 \epsilon |z|} \; dxdy \leq \int_\mathbb{C} | \overline{\partial}^* \sigma|^2 e^{-2 \epsilon |z|} \; dxdy$$ for all $\sigma \in $Dom($\overline{\partial}^*)$.
*Proof of Claim 2:* Since $E$ is the complexification of some real vector bundle, (2.1) implies that $$\kappa_0 \int_\mathbb{C} |s|^2 e^{-2\epsilon |z|} \; dxdy \leq \int_\mathbb{C} |\nabla_{\frac{\partial}{\partial z}} s |^2 e^{-2\epsilon |z|} \; dxdy$$ for any $s \in C^\infty_c(E)$. First, we establish claim 2 for $\sigma=s \otimes d\overline{z} \in C^\infty_c(E \otimes T^{0,1}\mathbb{C})$. By (3.1), the triangle inequality, and that $\epsilon < \frac{\sqrt{\kappa_0}}{2}$, we have $$\begin{aligned}
\| \overline{\partial}^* \sigma\|_{L^2(E,e^{-2\epsilon |z|})}
&=\| -\nabla_{\frac{\partial}{\partial z}} s + \epsilon \frac{\overline{z}}{|z|} s \|_{L^2(E,e^{-2\epsilon |z|})} \\
&\geq \| \nabla_{\frac{\partial}{\partial z}} s\|_{L^2(E,e^{-2\epsilon |z|})} - \epsilon \| s \|_{L^2(E,e^{-2\epsilon |z|})} \\
&\geq \sqrt{\kappa_0} \|s \|_{L^2(E,e^{-2\epsilon |z|})} - \epsilon \| s \|_{L^2(E,e^{-2\epsilon |z|})} \\
&\geq \frac{\sqrt{\kappa_0}}{2} \|s \|_{L^2(E,e^{-2\epsilon |z|})} \\
&= \frac{\sqrt{\kappa_0}}{2} \|\sigma \|_{L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})}. \end{aligned}$$ Squaring both sides give the inequality we want, with $\kappa_1=\frac{\kappa_0}{4}$. To prove the inequality for arbitrary $\sigma \in $Dom($\overline{\partial}^*)$, it suffices to show that $C^\infty_c(E\otimes T^{0,1}\mathbb{C})$ is dense in Dom($\overline{\partial}^*$) in the graph norm $$\sigma \mapsto \|\sigma\|_{L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})} + \| \overline{\partial}^* \sigma \|_{L^2(E,e^{-2\epsilon |z|})}.$$ A proof of this elementary fact can be found in Appendix A. Therefore, we have completed the proof of claim 2.
Now, claim 2 clearly implies both (i) and (ii) (lemma 4.1.1 in [@Chen-Shaw]). This finishes the proof of Theorem 4.1.
An important corollary of Theorem 4.1 is the following existence theorem of holomorphic sections of $E$ with controlled growth.
Suppose $E$ is a holomorphic vector bundle over $\mathbb{C}$ satisfying the “eigenvalue condition” (3.1) with constants $\kappa_0$ and $\epsilon_0$. Assume that $E$ is the complexification of some real vector bundle $\xi$ over $\mathbb{C}$.
Then, for any $0 < \epsilon <\min (\frac{\epsilon_0}{2},\frac{\sqrt{\kappa_0}}{2})$, there exists a non-trivial holomorphic section $s \in L^2(E,e^{-4\epsilon |z|})$, that is, $$\overline{\partial}s=0$$ and $$\int_\mathbb{C} |s|^2 e^{-4 \epsilon |z|} \; dxdy < \infty.$$
First, notice that any holomorphic vector bundle on a non-compact Riemann surface is holomorphically trivial ([@Forster]). Therefore, we can choose a nowhere vanishing holomorphic section $u$ of $E$. However, such a $u$ maybe not be in $L^2(E,e^{-4\epsilon |z|})$. We will correct it by a cutoff argument and solving an inhomogeneous equation of the form $\overline{\partial}s=\sigma$ to construct a holomorphic section in $L^2(E,e^{-4\epsilon |z|})$.
Take a smooth compactly supported cutoff function $\psi \in C^\infty_c(\mathbb{C})$ such that
- $\psi(z)=1$ for $|z| \leq 1$, and
- $\psi(z)=0$ for $|z| \geq 2$.
Define $$v=\frac{1}{z} \left(\frac{\partial \psi}{\partial \overline{z}} \right) u.$$ Observe that $v \in C^\infty_c(E)$ even though $1/z$ is singular at $z=0$. By Theorem 4.1, there exists $w \in L^2(E,e^{-2\epsilon |z|})$ such that $$\overline{\partial}w=v \otimes d\overline{z} .$$ Since $v$ is smooth and compactly supported, by elliptic regularity, $w$ is a smooth section of $E$ (but not necessarily compactly supported).
Next, we let $$s=\psi u-zw.$$
*Claim:* $s$ is a non-trivial holomorphic section in $L^2(E,e^{-4\epsilon |z|})$.
*Proof of Claim:* First of all, $$s(0)=\psi(0)u(0)=u(0) \neq 0$$ since $u$ is nowhere vanishing. $s$ is holomorphic since $$\overline{\partial}s=\overline{\partial} (\psi u)-z\overline{\partial}w
=\frac{\partial \psi}{\partial \overline{z}} u \otimes d\overline{z} - z \overline{\partial}w=0.$$ Finally, to see that $s$ is in $L^2(E,e^{-4\epsilon |z|})$, we see that $\psi u$ is smooth and compactly supported, hence, in $L^2(E,e^{-4\epsilon |z|})$. Moreover, $$\int_\mathbb{C} |z|^2 |w|^2 e^{-4\epsilon |z|} \; dxdy = \int_\mathbb{C} |z|^2e^{-2\epsilon|z|}
|w|^2 e^{-2\epsilon |z|} \; dxdy$$ Since $|z|^2e^{-2\epsilon |z|} \leq 1$ on $|z| \geq R$ for some $R>0$ sufficiently large, it follows that $zw \in L^2(E,e^{-4\epsilon |z|})$, and so does $s$. This proves our claim and hence establishes the corollary.
Proof of Theorem 2.6 and 2.7
============================
In this section, we prove Theorems 2.6 and 2.7 using the results in section 3 and 4.
*Proof of Theorem 2.6:* We argue by contradiction. Suppose Theorem 2.6 is false. Then there exists a stable minimal immersion $u: \Sigma \to M$ into an oriented Riemannian 4-manifold $M$ with uniformly positive isotropic curvature bounded from below by $\kappa>0$, with $\Sigma$ uniformly conformally equivalent to $\mathbb{C}$. Recall that $u$ is *minimal* if it is a critical point of the area functional with respect to compactly supported variations, and $u$ is *stable* if and only if the second variation of area for any compactly supported variation is nonnegative. Note that we assume $u$ to be an immersion, therefore we do not allow the existence of branch points.
We consider the normal bundle of the surface $u(\Sigma)$. We denote by $F$ the bundle on $\Sigma$ given by the pullback under $u$ of the normal bundle of $u(\Sigma)$. Note that $F$ is a smooth real vector bundle of rank 2. Since $M$ and $\Sigma$ are orientable, we conclude that $F$ is orientable. Let $F^\mathbb{C}=F \otimes_\mathbb{R} \mathbb{C}$ be the complexification of $F$. Since $F$ is orientable, the complexified bundle $F^\mathbb{C}$ splits as a direct sum of two holomorphic line bundles $F^{1,0}$ and $F^{0,1}$. Here, $F^{1,0}$ consists of all vectors of the form $\mu(v-iw) \in F^\mathbb{C}$, where $\mu \in \mathbb{C}$ and $\{v,w\}$ is a positively oriented orthonormal basis of $F$. An important observation here is that every section $s \in C^\infty(F^{1,0})$ is automatically isotropic, i.e. $(s,s)=0$. Moreover, the splitting $F=F^{1,0}\oplus F^{0,1}$ is parallel, i.e. invariant under $\nabla$.
Since $u:\Sigma \to M$ is stable, the complexified stability inequality (see [@Fraser03]) says that $$\int_\Sigma \left\langle R\left(s,\frac{\partial u}{\partial z}\right)\frac{\partial u}{\partial \overline{z}}, s \right\rangle
\; dxdy \leq \int_\Sigma (|\nabla^\perp_{\frac{\partial}{\partial \overline{z}}} s|^2-
|\nabla^\top_{\frac{\partial}{\partial z}} s|^2) \; dxdy$$ for all compactly supported sections $s \in C^\infty_c(F)$, where $z=x+iy$ is a local isothermal coordinate on $\Sigma$. In particular, it holds for all $s \in C^\infty_c(F^{0,1})$. Note that $s \perp \frac{\partial u}{\partial z}$, $s$ is isotropic and $\frac{\partial u}{\partial z}$ is also isotropic (since $z$ is an isothermal coordinate and $u$ is an isometric immersion). Therefore, $\{s,\frac{\partial u}{\partial z}\}$ span a two dimensional isotropic subspace. Using the lower bound on the isotropic curvature and throwing away the second term on the right, we get the following inequality $$\kappa \int_\Sigma |s|^2 \; da \leq \int_\Sigma |\overline{\partial}s|^2 \; da$$ for every $s \in C^\infty_c(F^{1,0})$, where $da$ is the area element of $\Sigma$.
Since $\Sigma$ is uniformly conformally equivalent to $\mathbb{C}$, by definition, there exists a diffeomorphism $\phi:\mathbb{C} \to \Sigma$ and a constant $C>0$ such that $$\phi^*h=\lambda^2 |dz|^2 \qquad \text{with} \qquad \frac{1}{C} \leq \lambda^2,$$ where $h$ is the induced metric on $\Sigma$. Define $E=\phi^*(F^{1,0})$ be the pullback of the holomorphic bundle $F^{1,0}$ by $\phi$. Since $\phi$ is a conformal diffeomorphism, $E$ is again a holomorphic bundle over $\mathbb{C}$. By (5.1) and (5.2), $$\frac{\kappa}{C} \int_\mathbb{C} |s|^2 \; dxdy \leq \int_\mathbb{C} |\overline{\partial}s|^2 \; dxdy$$ for every $s \in C^\infty_c(E)$. We then show that $E$ satisfies the “eigenvalue condition” (3.1).
*Claim:* there exists a constant $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$, $$\frac{\kappa}{4C} \int_\mathbb{C} |s|^2 e^{-\epsilon |z|} \; dxdy \leq \int_\mathbb{C} |\overline{\partial} s|^2 e^{-\epsilon |z|} \; dxdy$$ for all compactly supported smooth sections $s \in C^\infty_c(E)$.
*Proof of Claim:* Let $s \in C^\infty_c(E)$. Take $t=e^{-\epsilon |z|/2}s$, notice that $$\begin{aligned}
|\overline{\partial}t|^2&=|\overline{\partial}(e^{-\epsilon |z|/2})s+e^{-\epsilon |z|/2}\overline{\partial}s|^2 \\
&\leq 2|\overline{\partial}(e^{-\epsilon |z|/2})s|^2+2|e^{-\epsilon |z|/2}\overline{\partial}s|^2 \\
&= 2 e^{-\epsilon |z|} ( \frac{\epsilon^2}{16} |s|^2+|\overline{\partial}s|^2)\end{aligned}$$ Applying (5.3) to $t \in C^\infty_c(E)$ and using the above estimate, $$\begin{aligned}
\frac{\kappa}{C} \int_\mathbb{C} |s|^2 e^{-\epsilon |z|} \; dxdy &= \frac{\kappa}{C} \int_\mathbb{C} |t|^2 \; dxdy \\
&\leq \int_\mathbb{C} |\overline{\partial}t|^2 \; dxdy \\
&\leq \frac{\epsilon^2}{8} \int_\mathbb{C} |s|^2 e^{-\epsilon |z|} \; dxdy+ 2 \int_\mathbb{C} |\overline{\partial}s|^2 e^{- \epsilon |z|} \; dxdy. \end{aligned}$$ Hence, if we take $\epsilon_0=\frac{2 \sqrt{\kappa}}{\sqrt{C}}$, for every $0< \epsilon <\epsilon_0$, we get $$\frac{\kappa}{4C} \int_\mathbb{C} |s|^2 e^{-\epsilon |z|} \; dxdy \leq \int_\mathbb{C} |\overline{\partial}s|^2 e^{- \epsilon |z|} \; dxdy.$$ This proves our claim.
To summarize, we have constructed a holomorphic vector bundle $E$ over $\mathbb{C}$ which satisfies the “eigenvalue condition” (3.1) with $\kappa_0=\frac{\kappa}{4C}$ and $\epsilon_0=\frac{2 \sqrt{\kappa}}{\sqrt{C}}$. So we can apply our results in section 3. Moreover, even though $E$ is not a complexification of a real vector bundle, we see that the result in section 4 still holds for $E$ because $E$ satisfies (4.3) (This follows from the fact that the inequality (5.1) holds with $\overline{\partial}$ replaced by $\partial$, we have to use the fact that $F=F^{1,0}\oplus F^{0,1}$ is a parallel splitting). Hence, if we fix $\epsilon>0$ sufficiently small ($\epsilon< \min(\frac{\epsilon_0}{4},\frac{\sqrt{\kappa_0}}{2})$), then Corollary 4.2 gives a non-trivial holomorphic section $s \in L^2(E,e^{-4\epsilon |z|})$, which contradicts Theorem 3.1. This contradiction completes the proof of Theorem 2.6.
*Proof of Theorem 2.7:* The proof is very similar to the above. Again we argue by contradiction. Suppose Theorem 2.7 is false. Then there exists a stable minimal immersion $u: \Sigma \to M$ into a Riemannian n-manifold $M$ with uniformly positive complex sectional curvature bounded from below by $\kappa>0$, with $\Sigma$ uniformly conformally equivalent to $\mathbb{C}$.
Let $F$ be the complexified normal bundle of $u(\Sigma)$ as before and let $E=\phi^*F$. By our assumption on $M$, (5.1) holds for every $s \in C^\infty_c(F)$. Exactly the same argument as above gives our desired contradiction.
Appendix A: A density lemma
===========================
We give a proof of the following density lemma used in the proof of Theorem 4.1. The proof is very similar to that of Lemma 4.1.3 in [@Hormander].
The subspace $C^\infty_c(E\otimes T^{0,1}\mathbb{C})$ is dense in Dom($\overline{\partial}^*$) in the graph norm $$\sigma \mapsto \|\sigma\|_{L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})} + \| \overline{\partial}^* \sigma \|_{L^2(E,e^{-2\epsilon |z|})}.$$
Let $\sigma=s\otimes d\overline{z}\in $Dom$(\overline{\partial}^*)$. First of all, we will show that the set of $\tau \in $Dom$(\overline{\partial}^*)$ with compact support is dense in Dom$(\overline{\partial}^*)$. For each $R>0$, let $\varphi=\varphi_R \in C^\infty_c(\mathbb{C})$ be a smooth cutoff function such that
- $\varphi(z)=1$ for $|z| \leq R$;
- $\varphi(z)=0$ for $|z| \geq 2R$;
- $|\nabla \varphi| \leq \frac{2}{R}$.
*Claim 1:* When $R \to \infty$, $\varphi_R \; \sigma$ converges to $\sigma$ in the graph norm.
*Proof of Claim 1:* First of all, we observe that $\varphi_R \; \sigma \in$Dom$(\overline{\partial}^*)$. In fact, for any $t \in $Dom$(\overline{\partial})$, $$\begin{aligned}
(\varphi_R \sigma, \overline{\partial}t)
&= (\sigma, \overline{\partial}(\varphi_R t))-(\sigma,(\overline{\partial}\varphi_R)t) \\
&= (\overline{\partial}^* \sigma, \varphi_R t) - ((\frac{\partial \varphi_R}{\partial z})s,t) \\
&= (\varphi_R \overline{\partial}^* \sigma-(\frac{\partial \varphi_R}{\partial z})s,t)\end{aligned}$$ It follows that $(\varphi_R \sigma, \overline{\partial}t)$ is continuous in $t$ for the norm $\|t\|_{L^2(E,e^{-2\epsilon |z|})}$, so $\varphi_R \; \sigma \in$Dom$(\overline{\partial}^*)$ and $$\overline{\partial}^*(\varphi_R \sigma)= \varphi_R \overline{\partial}^* \sigma-(\frac{\partial \varphi_R}{\partial z})s.$$ Therefore, we have the estimate $$|\overline{\partial}^*(\varphi_R \sigma)- \varphi_R \overline{\partial}^* \sigma| \leq \frac{2}{R}|\sigma|.$$ Since $\sigma \in L^2(E \otimes T^{0,1}\mathbb{C},e^{-2\epsilon |z|})$, and $\varphi_R \overline{\partial}^* \sigma \to \overline{\partial}^* \sigma$ in $L^2(E,e^{-2\epsilon |z|})$ as $R \to \infty$, therefore, we have $\varphi_R \sigma$ converges to $\sigma$ in the graph norm as $R \to \infty$. This finishes the proof of Claim 1.
Next, we need to approximate (in the graph norm) any $\tau \in $Dom$(\overline{\partial}^*)$ with compact support by elements in $C^\infty_c(E \otimes T^{1,0}\mathbb{C})$. Take any $\chi \in C^\infty_c(\mathbb{C})$ with $\int_\mathbb{C} \chi \; dxdy=1$, and set $\chi_\delta(z)=\delta^{-2} \chi(z/\delta)$. Take any $\tau \in $Dom$(\overline{\partial}^*)$ with compact support, the convolution $\tau * \chi_\delta$ is a smooth section of $E \otimes T^{1,0}\mathbb{C}$ with compact support. Since we can fix a compact set so that all $\tau * \chi_\delta$ are supported inside the compact set, we see that $\tau * \chi_\delta$ converges to $\tau$ in $L^2(E \otimes T^{1,0}\mathbb{C},e^{-2\epsilon |z|})$ as $\delta \to 0$. It is easy to check that $$\overline{\partial}^* (\tau * \chi_\delta)=(\overline{\partial}^* \tau)*\chi_\delta + \epsilon \frac{\overline{z}}{|z|}(\tau * \chi_\delta) -(\epsilon \frac{\overline{z}}{|z|} \tau) * \chi_\delta.$$ Therefore, $$\overline{\partial}^* (\tau * \chi_\delta)-(\overline{\partial}^* \tau)*\chi_\delta= \epsilon \frac{\overline{z}}{|z|}(\tau * \chi_\delta) -(\epsilon \frac{\overline{z}}{|z|} \tau) * \chi_\delta,$$ and the right hand side converges to $0$ in $L^2(E \otimes T^{1,0}\mathbb{C},e^{-2\epsilon |z|})$ as $\delta \to 0$. Hence, we conclude that $\tau * \chi_\delta$ converges to $\tau$ in the graph norm as $\delta \to 0$. Hence the proof of Lemma 6.1 is completed.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper substantially generalizes our paper “Foliation Cones” (KirbyFest, 1999) while simplifying many proofs and correcting some errors and gaps. In essence, we classify finite depth, foliated 3-manifolds $(M,{\mathcal{F}})$ with a given “substructure” $S$. The components $W$ of $M{\smallsetminus}S$ are stably foliated and the possible such foliations are classified, up to isotopy, by the rays through the integer lattice points in the interiors of finitely many closed, convex, non-overlapping, finite-sided, polyhedral cones in a suitable cohomology of $W$. The other rays classify dense-leaved refoliations of $W$ without holonomy, also up to isotopy, as will be shown in a paper “Foliation Cones III” now in preparation. While the cones are generally infinite dimensional, they have only finitely many faces. Results of this type are given both for the case that $(M,{\mathcal{F}})$ is smooth and that in which it is of class ${C^{0+}}$ (i.e., ${\mathcal{F}}$ is integral to a ${C^{0}}$ plane field).'
address:
- |
Department of Mathematics\
St. Louis University\
St. Louis, MO 63103
- |
Department of Mathematics\
Washington University, St. Louis, MO 63130
author:
- John Cantwell
- Lawrence Conlon
title: Foliation Cones II
---
Introduction
============
We assume that $M$ is a compact 3-manifold and that ${\mathcal{F}}$ is a transversely oriented, codimension 1, finite depth foliation of $M$. We will assume that ${\mathcal{F}}$ has depth $k\ge1$, hence does not fiber $M$ over ${S^{1}}$ or $I$. The manifold and hence the leaves may be nonorientable. If the foliated manifold $(M,{\mathcal{F}})$ is smooth of class ${C^{2}}$, then it is homeomorphic to a ${C^{\infty}}$-foliated manifold [@cc:prop1; @cc:smth2]. Indeed, implicit in these references is the fact that each leaf of the ${C^{\infty}}$-foliated manifold can be assumed to have holonomy ${C^{\infty}}$-tangent to the identity $ \iota$ (such a foliation will be said to be of class ${C^{\infty,\iota}}$). Accordingly, all ${C^{2}}$ foliations hereafter will be assumed to be of class ${C^{\infty,\iota}}$. We will also consider the case of minimal regularity, the foliation only being tangent to a ${C^{0}}$ plane field, saying that such a foliation is of class ${C^{0+}}$. (In [@condel1] such foliations were said to be of class $C^{1,0+}$ to emphasize that the leaves are individually of class ${C^{1}}$. In fact, for codimension 1 foliations, the leaves are individually of class ${C^{\infty}}$ [@cc:smth2].)
Our foliated manifolds will be allowed to have tangential boundary ${{\partial}_{\tau}}M$, with components compact leaves, and/or transverse boundary ${{\partial}_{\pitchfork}}M$ with components annuli, tori, and/or Klein bottles. (These are sutured manifolds in the sense of D. Gabai [@ga0].) The tangential and transverse boundary are separated by convex corners along their common boundary. The foliation induced by ${\mathcal{F}}$ on ${{\partial}_{\pitchfork}}M$ will be assumed to have no Reeb components. We will assume that no compact leaf is an annulus, Möbius strip, torus or Klein bottle. Because of Reeb stability, we rule out the totally trivial cases of a compact leaf being a disk, sphere, or projective plane. For proper, ${C^{2}}$ foliations, our hypotheses imply that no leaf has a simple end [@cc:hm Definition 2.1]. By [@cc:hm Lemma 2.19] this remains true in the ${C^{0+}}$ case. In either case, our hypotheses imply that the foliation is taut [@cc:hm Lemma 2.18] and that there is a Riemannian metric on $M$ such that all leaves of ${\mathcal{F}}$ are hyperbolic [@cand:hyp; @cc:prop2].
A leaf of ${\mathcal{F}}$ is locally stable if it has a saturated neighborhood foliated as a product.
The union ${\mathcal{O}}$ of the locally stable leaves is open and dense. The complement $S=M{\smallsetminus}{\mathcal{O}}$ is called the *substructure* of ${\mathcal{F}}$. While ${\mathcal{O}}$ may have infinitely many components $W$, at most finitely many fail to be “foliated products”. We say that $W$ is a foliated product if its transverse completion ${\widehat}W$ has the structure of an interval bundle with ${\mathcal{F}}|{\widehat}W$ transverse to the fibers. In any event, ${\mathcal{F}}|W$ fibers $W$ over a 1-manifold. The transverse completion of $W$ is obtained by adding finitely many border leaves making up ${{\partial}_{\tau}}{\widehat}W$. (It is possible that pairs of distinct components of ${{\partial}_{\tau}}{\widehat}W$ are identified to the same leaf of the substructure $S{\subset}M$.) Remark that, for finite depth foliations, the leaves at maximal depth are locally stable. These facts are all standard, being found, for instance, in [@condel1].
The subspace of $H^{1}({\widehat}{W})$ (real coefficients) consisting of classes that can be represented by compactly supported cocycles will be denoted by ${H^{1}_{\kappa}}({\widehat}{W})$.
This is not the same as the cohomology with compact supports, denoted by $H^{1}_{c}({\widehat}{W})$. The natural homomorphism from this latter space to the former is surjective but not generally injective.
The subspaces ${H^{1}}({\widehat}{W};{\mathbb{Z}})$ and ${H^{1}_{\kappa}}({\widehat}{W};{\mathbb{Z}})={H^{1}_{\kappa}}({\widehat}{W})\cap{H^{1}}({\widehat}{W};{\mathbb{Z}})$ of ${H^{1}}({\widehat}{W})$ and ${H^{1}_{\kappa}}({\widehat}{W})$, respectively, are called the integer lattices in these vector spaces.
The rays in $H^{1}({\widehat}{W})$ and ${H^{1}_{\kappa}}({\widehat}{W})$ issuing from the origin and meeting nonzero points of the integer lattice are called rational rays. Rays not meeting nonzero points of the integer lattice are termed irrational.
If ${\widehat}W$ is not compact, the spaces ${H^{1}}({\widehat}{W})$ and ${H^{1}_{\kappa}}({\widehat}{W})$ are infinite dimensional and will be given appropriate topologies.
Our principal goal is the proof of the following result.
\[cone\] Let $(M,{\mathcal{F}})$ be of class ${C^{\infty,\iota}}$ and depth $k\ge1$. If $W$ is a component of ${\mathcal{O}}$, there is a finite family ${\mathcal{K}}_{W}$ of nonoverlapping, closed, convex, finite-sided, polyhedral cones in ${H^{1}_{\kappa}}({\widehat}{W})$, with nonempty interiors, having the following property the set of rational rays in the interiors of these cones is in natural one-one correspondence with the set of isotopy classes of taut foliations ${\mathcal{F}}'$ of ${\widehat}W$, having holonomy only along ${{\partial}_{\tau}}{\widehat}W$, each of which completes ${\mathcal{F}}|(M{\smallsetminus}W)$ to a ${C^{\infty,\iota}}$ foliation of depth $k$. This isotopy is smooth in $W$ but may only be ${C^{0}}$ on ${\widehat}{W}$. When $(M,{\mathcal{F}})$ is of class ${C^{0+}}$, the entirely analogous result holds, where the cones are contained in the ordinary cohomology $H^{1}({\widehat}{W})$.
The elements of ${\mathcal{K}}_{W}$ are called *foliation cones*. We will denote the ray corresponding to ${\mathcal{F}}'$ by $\left<{\mathcal{F}}'\right>$ and call it a foliated ray. If it is a rational ray, it will also be called a proper foliated ray, the adjective “proper” referring to the fact that the leaves of ${\mathcal{F}}'$ are proper. In the case that ${\mathcal{F}}$ is of class ${C^{\infty,\iota}}$, we will say that the foliation ${\mathcal{F}}'$ “appropriately extends” ${\mathcal{F}}|(M{\smallsetminus}W)$ if the new foliation obtained by replacing ${\mathcal{F}}|{\widehat}W$ with ${\mathcal{F}}'$ is also of class ${C^{\infty,\iota}}$.
In principle, this theorem allows us to classify homologically all finite depth foliations with a given substructure $S$. In the smooth case, it is known that the foliation in each component $W$ of ${\mathcal{O}}$ must be trivial outside of some compact subset of ${\widehat}W$ (compactness of junctures [@condel1 Theorem 8.1.26]), which is exactly why the cones live in ${H^{1}_{\kappa}}({\widehat}{W})$ in that case.
It will be necessary to allow the possibility that the entire vector space ${H^{1}_{\kappa}}({\widehat}{W})$ is a foliation cone, this happening if and only if ${\widehat}W$ is a foliated product. This is the one case in which the vertex $0$ of the cone lies in its interior. The class $0$ will correspond to the product foliation and $\{0\}$ will be a (degenerate) foliated ray.
The case in which the foliation is depth one with only the components of ${{\partial}_{\tau}}M$ as compact leaves is worth special mention. The problem of classifying them was proposed by W. Thurston in the closing paragraph of his Memoir paper [@th:norm]. This paper found polyhedral cones classifying fibrations transverse to ${\partial}M$ (depth 0 foliations) subtended by the “fibered faces” of the Thurston ball. There was the clear expectation there that the classification in the depth one case would again use his norm on ${H^{1}}(M)$ or, perhaps, a similar norm. The solution to this classification problem, given in [@cc:cone] and generalized here, does not use a norm. Indeed, examples such as the complement of the knot $8_{15}$ and the link $(2,2,2)$ show that, in general, the cones cannot be defined by a norm. There are indications that our cone structures for this case relate to the polytopes constructed by A. Juhász in his work on sutured Floer homology [@Ju:polytope].
In a paper [@cc:cone3], now in preparation, we prove that the irrational rays in the interiors of the cones correspond one-to-one to the ${C^{0}}$ isotopy classes of foliations of ${\widehat}W$ which are dense-leaved without holonomy in $W$. More precisely, every such foliation is ${C^{0}}$ isotopic to one defined by a “foliated form” (Definition \[ff\]), and those defined by foliated forms are unique up to an isotopy that is ${C^{\infty}}$ in $W$. Thus, we will obtain a generalization of the Laudenbach-Blank theorem [@LB]. That theorem stated that, on a compact 3-manifold, nonsingular, cohomologous, closed 1-forms are smoothly isotopic, and we will prove this for the foliated forms on $W$.
From the point of view of 3-manifold topology, classification of the possible substructures would seem to be of considerable interest. Gabai’s sutured manifold decompositions [@ga1] appear basically to do that. It would be interesting if the methods of this paper would lead to a more elementary algorithm. These methods might also clarify the obstructions to achieving smoothness in the resulting foliations.
The proof of Theorem \[cone\] uses the Handel-Miller classification of endperiodic homeomorphisms [@cc:hm] and the Schwartzmann-Sullivan theory of asymptotic cycles [@sch_cycles; @sull:cycles]. The results proven exclusively by asymptotic cycles are valid for compact manifolds of arbitrary dimension $n\ge3$ with finite depth foliations. These results promise to be useful for further research. The Handel-Miller theory is strictly for 2-dimensional leaves and there is no evident replacement for it in higher dimensions. Although Handel and Miller never published the details, a complete account is now available [@cc:hm].
In the case of depth 1 foliations arising from disk decompositions [@ga0], if the disks can be chosen in $M$ from the start, our classification program is quite effective. Indeed, the disks of the decomposition typically split up in a natural way into the rectangles of a Markov partition associated to the Handel-Miller monodromy and the foliation cones are easily determined from this information.
This paper can be read almost independently of its predecessor [@cc:cone]. The one exception is that the proof of finiteness of the set of cones is so easily adapted to our present context that we will refer the reader to [@cc:cone Section 6] for details. The reader may also find it helpful to study the computations of foliation cones in [@cc:cone Section 7] and [@cc:norm Section 5] for a number of depth 1 examples.
Endperiodic monodromy
=====================
Let ${\mathcal{F}}$ be depth $k$ of class ${C^{2}}$ or class ${C^{0+}}$. For the time being, we allow $\dim M=n\ge3$. The definitions of the locally stable set ${\mathcal{O}}$ and the substructure $S$ are valid in all dimensions. By [@cc:prop1; @cc:smth2], we lose no generality in the ${C^{2}}$ case by assuming that ${\mathcal{F}}$ is of class ${C^{\infty,\iota}}$. Let ${\mathcal{L}}$ be a smooth 1-dimensional foliation of $M$ that is transverse to ${\mathcal{F}}$ and is tangent to ${{\partial}_{\pitchfork}}M$, is oriented by the transverse orientation of ${\mathcal{F}}$, and induces the product foliation by compact intervals on all components of ${{\partial}_{\pitchfork}}M$ that have boundary and a foliation by circles on those components without boundary. (This, of course, restricts the topology of ${\mathcal{F}}$. In 3-manifolds it is guaranteed by our assumption that the induced foliation of ${{\partial}_{\pitchfork}}M$ has no Reeb components.) Let $W$ be a component of ${\mathcal{O}}$.
There is a sublamination ${\mathcal{X}}{\subset}{\mathcal{L}}|{\widehat}W$ consisting of those leaves that do not meet ${{\partial}_{\tau}}{\widehat}W$. This sublamination is a compact sublamination of ${\mathcal{L}}|W$. Indeed, ${\mathcal{X}}$ is closed since every leaf of ${\mathcal{L}}|{\widehat}W$ that gets sufficiently close to ${{\partial}_{\tau}}{\widehat}W$ meets ${{\partial}_{\tau}}{\widehat}W$. The octopus decomposition of ${\widehat}W$ [@condel1 Definition 5.2.13] implies that ${\mathcal{X}}$ is contained in a compact core $K$ of ${\widehat}W$, hence is compact. We call ${\mathcal{X}}$ the *core lamination* of ${\mathcal{L}}|W$. We will assume that $W$ is not a foliated product, hence the core lamination is nonempty. Indeed, if some leaf $\ell$ of ${\mathcal{L}}$ issues from one component of ${{\partial}_{\tau}}{\widehat}W$ but never reaches another component, the asymptote of $\ell$ in ${\widehat}W$ will be a nonempty subset of ${\mathcal{X}}$. Thus, if ${\mathcal{X}}={\emptyset}$, every leaf of ${\mathcal{L}}$ issues from one component of ${{\partial}_{\tau}}{\widehat}W$ and terminates at another component, implying that ${\widehat}W\cong F\times I$, where $F$ is a leaf of ${\mathcal{F}}$. In this case, we will see that Theorem \[cone\] is rather trivial, hence we will assume that ${\mathcal{X}}\ne{\emptyset}$.
The foliation ${\mathcal{L}}|W$ can be parametrized as a flow $\Phi_{t}$ preserving ${\mathcal{F}}|W$ such that $\Phi_{1}$ sends each leaf to itself. Indeed, ${\mathcal{F}}|W$ defines a fibration $$\pi:W\to{S^{1}}$$ and one can lift the standard locally defined parameter $ \theta$ of ${S^{1}}$ to ${\mathcal{L}}$. Equivalently, this is a smooth, transverse, holonomy invariant measure for ${\mathcal{F}}$. Remark that the parameter becomes unbounded near ${{\partial}_{\tau}}{\widehat}W$ and that ${\mathcal{L}}\pitchfork {{\partial}_{\tau}}{\widehat}W$, hence our flow extends to a flow on ${\widehat}W$ that fixes ${{\partial}_{\tau}}{\widehat}W$ pointwise. If $L$ is a leaf of ${\mathcal{F}}|W$, the flow induces a first return map, $\Phi_{1}=f:L\to L$, called the monodromy diffeomorphism of $L$. If ${\mathcal{F}}$ is of class ${C^{2}}$, the monodromy is endperiodic ([@cc:hm Definition 2.10 and Proposition 2.16] which extend to the higher dimensional cases). In the case that ${\mathcal{F}}$ is of class ${C^{0+}}$, however, the restriction ${\mathcal{F}}|K$ to the nucleus of an octopus decomposition has the property that the monodromy of any noncompact leaf is endperiodic and this will be sufficient for our purposes in that case.
Let ${\mathcal{L}}_{f}$ denote ${\mathcal{L}}|W$ and (by abuse) ${\mathcal{L}}|{\widehat}W$, recording thereby the monodromy $f$ that the flow induces on $L$. We will also denote ${\mathcal{X}}{\subset}{\mathcal{L}}$ by ${\mathcal{X}}_{f}$. If $g:L\to L$ is an endperiodic homeomorphism (*not* assumed to be smooth) isotopic to $f$ through endperiodic homeomorphisms that agree with $f$ on ${\partial}L$, we can again find a 1-dimensional foliation ${\mathcal{L}}_{g}$ of ${\widehat}W$, having the fibers of $\pi$ as cross-sections, agreeing with ${\mathcal{L}}_{f}$ on ${{\partial}_{\pitchfork}}{\widehat}W$ and inducing monodromy $g:L\to L$. If $g$ is not smooth, ${\mathcal{L}}_{g}$ cannot be smooth. We again get a core lamination ${\mathcal{X}}_{g}{\subset}{\mathcal{L}}_{g}$, the saturation of the compact set $X_{g}{\subset}L$ consisting of those points whose $g$-orbits are bounded away from all ends of $L$. If $g$ is smooth (or, more generally, regular as defined below), we can apply the Schwartzmann-Sullivan theory to ${\mathcal{X}}_{g}$. Otherwise, only a very attenuated version of that theory applies.
The Schwartzmann-Sullivan asymptotic cycles
===========================================
We continue to work in $n$-manifolds, $n\ge3$, save mention to the contrary. For compact, 1-dimensional laminations such as ${\mathcal{X}}_{g}{\subset}{\mathcal{L}}_{g}$, where $g$ is smooth, Sullivan’s theory of foliation cycles [@sull:cycles] specializes to the Schwartzmann theory of asymptotic cycles [@sch_cycles]. In fact, it is only required that ${\mathcal{X}}_{g}$ be integral to a ${C^{0}}$ vector field. If $g$ is an endperiodic homeomorphism admitting ${\mathcal{L}}_{g}$ of class ${C^{0+}}$, we will say that $g$ is *regular* and assume this save mention to the contrary. We will take as our main reference the exposition of Sullivan’s theory in [@condel1 Chapter 10], but taking into account the full theory of de Rham currents in [@deRham Chapitre III] on noncompact manifolds such as ${\widehat}W$. As developed in [@condel1], this theory was concerned primarily with compact foliated manifolds having all leaves without boundary, but as noted there, everything goes through for compact laminations of possibly noncompact manifolds, provided that, again, all leaves have empty boundary.
Forms and currents
------------------
Slightly modifying the notation of [@deRham] so as to keep track of the degrees of forms and currents, we set ${\mathcal{D}}_{p}={\mathcal{D}}_{p}({\widehat}{W})$, the locally convex topological vector space of *compactly supported* $p$-forms of class ${C^{\infty}}$. That is, the underlying vector space is $A^{p}_{c}({\widehat}{W})$ and the topology is generated by the increasing union of the topologies ${\mathfrak{T}}_{k}$ defined by the $C^{k}$ norm $\|\cdot\|_{k}$ on compact sets, $0\le k<\infty$. This norm arises from a choice of ${C^{\infty}}$ atlas on ${\widehat}W$, and basic open neighborhoods of 0 in ${\mathfrak{T}}_{k}$ are of the form $V(C,{\varepsilon})=\{{\varphi}\mid\operatorname{supp}{\varphi}{\subset}C\text{ and }\|{\varphi}\|_{k}<{\varepsilon}\}$, where $C{\subset}{\widehat}{W}$ is compact and ${\varepsilon}>0$. Following de Rham [@deRham p. 44], we say that a subset $B{\subset}{\mathcal{D}}_{p}$ is bounded if all of its elements have support in a common compact set and $B$ is bounded in the $C^{k}$ norm, $0\le k<\infty$.
We set ${\mathcal{D}}'_{p}$ equal to the strong dual of ${\mathcal{D}}_{p}$, the space of continuous linear functionals on ${\mathcal{D}}_{p}$. This is the space of $p$-*currents* on ${\widehat}W$. The topology on ${\mathcal{D}}'_{p}$ can now be defined exactly as in [@condel1 Definition 10.1.17]. This makes ${\mathcal{D}}'_{p}$ into a locally convex, topological vector space. It has a notion of bounded subset [@condel1 Definition 10.1.20]. Both ${\mathcal{D}}_{p}$ and ${\mathcal{D}}'_{p}$ are strong duals of one another [@deRham p. 89, Théorème 13].
De Rham also introduces a locally convex topological vector space ${\mathcal{E}}_{p}$ with underlying vector space $A^{p}({\widehat}{W})$. The topology uses a notion of boundedness which is local boundedness in the $C^{k}$ norms, enabling one to define when a sequence of forms converges to $0$. The support of a $p$-current $T$ is the smallest closed subset $C{\subset}{\widehat}W$ such that $T(\alpha)=0$ whenever $\operatorname{supp}(\alpha)\cap C={\emptyset}$. The subset ${\mathcal{E}}_{p}'{\subset}{\mathcal{D}}'_{p}$ of currents with compact support is exactly the strong dual of ${\mathcal{E}}_{p}$ and vice-versa [@deRham p. 89, Théorème 13]. Using the notion of bounded set in ${\mathcal{E}}_{p}$, one mimics [@condel1 Definition 10.1.17] to define the locally convex topology on ${\mathcal{E}}'_{p}$. All of these spaces are Montel, meaning that every bounded subset is precompact. For the case $p=0$, this is proven in [@schwartz p. 70, Théorèm VII and p. 74, Théorème XII], the general case being similar.
The exterior derivative $d:{\mathcal{D}}_{p}\to {\mathcal{D}}_{p+1}$ is continuous, hence has continuous adjoint ${\partial}:{\mathcal{D}}_{p}'\to{\mathcal{D}}_{p-1}'$. Similarly, one has that $d:{\mathcal{E}}_{p}\to{\mathcal{E}}_{p+1}$ is continuous with continuous adjoint ${\partial}:{\mathcal{E}}'_{p}\to{\mathcal{E}}'_{p-1}$. Since $d^{2}=0$, we see that ${\partial}^{2}=0$. The kernel ${\mathcal{Z}}_{p}{\subset}{\mathcal{E}}'_{p}$ of ${\partial}$ is the space of $p$-cycles and the image ${\mathcal{B}}_{p}={\partial}({\mathcal{E}}'_{p+1}){\subset}{\mathcal{E}}'_{p}$ is the space of $p$-boundaries. These are closed subspaces of ${\mathcal{E}}'_{p}$. The space $H_{p}({\widehat}{W})={\mathcal{Z}}_{p}/{\mathcal{B}}_{p}$ is the de Rham homology of ${\widehat}{W}$, canonically isomorphic to the singular homology. The homology of the complex $({\mathcal{D}}'_{*},{\partial})$ gives the dual space to $H^{p}_{c}({\widehat}{W})$, for each $p\ge0$, which we may think of as homology computed with locally finite, but possibly infinite, chains and denote it by $H_{p}^{\infty}({\widehat}{W})={\mathcal{Z}}^{\infty}_{p}/{\mathcal{B}}^{\infty}_{p}$, where, of course, ${\mathcal{Z}}^{\infty}_{p}$ is the kernel of ${\partial}$ in ${\mathcal{D}}_{p}'$ and ${\mathcal{B}}_{p}^{\infty}$ its image.
Since the inclusion $W{\hookrightarrow}{\widehat}W$ is a homotopy equivalence, we can identify the homology and cohomology of $W$ with that of ${\widehat}W$.
Topologies on homologies and cohomologies. {#tops}
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In homology, we topologize $H_{p}({\widehat}{W})={\mathcal{Z}}_{p}/{\mathcal{B}}_{p}$ and $H_{p}^{\infty}({\widehat}{W})={\mathcal{Z}}^{\infty}_{p}/{\mathcal{B}}^{\infty}_{p}$ with the quotient topology.
The manifold ${\widehat}W$, if noncompact, is an increasing union $$K_{0}{\subset}K_{1}{\subset}\cdots{\subset}K_{i}{\subset}\cdots={\widehat}W,$$ where each $K_{i}$ is a compact, sutured manifold which is the nucleus of an octopus decomposition of ${\widehat}W$. This can be used to define natural topologies on $H^{p}({\widehat}{W})$ and $H^{p}_{\kappa}({\widehat}{W})$.
We topologize $H^{p}({\widehat}{W})={\underleftarrow}{\lim}\,H^{p}(K_{i})$ [@hatch Proposition 3F.5] with the inverse limit topology. This is the topology induced from the Tychonov topology by the inclusion $${\underleftarrow}{\lim}\,H^{p}(K_{i}){\subset}H^{p}(K_{0}){\times}H^{p}(K_{1}){\times}\cdots{\times}H^{p}(K_{i}){\times}\cdots$$ In fact, this is also the quotient topology induced by the complex $({\mathcal{E}}_{*},d)$ (exercise). In any event, it is clear that the elements of $H^{p}({\widehat}{W})$, viewed as linear functionals on $H_{p}({\widehat}{W})$, are continuous and so, this being the vector space dual, it is also the strong dual of $H_{p}({\widehat}{W})$. Similarly, the elements of $H_{p}({\widehat}{W})$, viewed as linear functionals on $H^{p}({\widehat}{W})$, are continuous (exercise), so homology and cohomology are strong duals of each other.
Set $K_{i}^{{\circ}}$ equal to the relative interior of $K_{i}$ in ${\widehat}W$, noting that the inclusions $H^{p}_{\kappa}(K_{i}^{{\circ}}){\hookrightarrow}H^{p}_{\kappa}(K_{i+1}^{{\circ}})$ induce a natural identification $H^{p}_{\kappa}({\widehat}{W})={\underrightarrow}{\lim}\,H^{p}_{\kappa}(K_{i}^{{\circ}})$. We give this space the direct limit (weak) topology. Equivalently, the topology on $H^{p}_{\kappa}({\widehat}{W}){\subset}H^{p}({\widehat}{W})$ is the induced topology (exercise).
The asymptotic currents for ${\mathcal{X}}_{g}$.
------------------------------------------------
The Dirac currents for ${\mathcal{X}}_{g}$ are the positively oriented, nontrivial tangent vectors to ${\mathcal{X}}_{g}$. These currents clearly have compact support. The closure in ${\mathcal{D}}_{1}'$ of the union of all positive linear combinations of Dirac currents is a closed, convex cone ${\mathcal{C}}_{g}{\subset}{\mathcal{D}}'_{1}$, called the cone of *asymptotic currents*. This cone lies on one side of a hyperplane $H= \omega^{-1}(0)$, where $ \omega:{\mathcal{D}}'_{1}\to{\mathbb{R}}$ is a compactly supported 1-form integrable on a neighborhood $U$ of ${\mathcal{X}}_{g}$ and defining ${\mathcal{F}}|U$. (The proof in [@condel1 Lemma 10.2.3] goes through practically unchanged, just noting that the compactness of ${\mathcal{X}}_{g}$ replaces that of the ambient manifold.) Since ${\mathcal{X}}_{g}$ is compact, the asymptotic currents are compactly supported and we can view ${\mathcal{C}}_{g}{\subset}{\mathcal{E}}'_{1}$. When working in ${\mathcal{D}}'_{1}$, the continuous linear functionals are the compactly supported 1-forms, but when working in ${\mathcal{E}}'_{1}$, they are all 1-forms.
The base ${\widehat}{{\mathcal{C}}}_{g}={\mathcal{C}}_{g}\cap \omega^{-1}(1)$ of the cone ${\mathcal{C}}_{g}$ is compact (both in ${\mathcal{E}}'_{1}$ and in ${\mathcal{D}}'_{1}$) by [@condel1 Lemma 10.2.3]. Once again the proof goes through by the compactness of ${\mathcal{X}}_{g}$ and the fact that our spaces are Montel. Those continuous linear functionals $ \eta:{\mathcal{D}}_{1}'\to{\mathbb{R}}$ which are strictly positive on ${\widehat}{{\mathcal{C}}}_{g}$ are exactly the smooth, compactly supported 1-forms on ${\widehat}{W}$ which are transverse to ${\mathcal{X}}_{g}$ (meaning that they take a positive value on each Dirac current). Similarly, the continuous linear functionals $\eta:{\mathcal{E}}_{1}'\to{\mathbb{R}}$, strictly positive on ${\widehat}{{\mathcal{C}}}_{g}$, are the smooth 1-forms on ${\widehat}W$ which are transverse to ${\mathcal{X}}_{g}$. Sullivan applies the Hahn-Banach theorem, using compactness of the base, to produce interesting 1-forms that are transverse to ${\mathcal{X}}_{g}$ (see [@condel1 Subsection 10.2]).
Cones defined by the asymptotic cycles
--------------------------------------
The cone ${\mathcal{C}}_{g}\cap {\mathcal{Z}}_{1}$ of *asymptotic cycles* is also a closed, convex cone with compact base. The natural continuous linear surjection ${\mathcal{Z}}_{1}\to H_{1}({\widehat}{W})$ carries the cone of asymptotic cycles onto a convex cone ${\mathfrak{C}}'_{g}{\subset}H_{1}({\widehat}{W})$ with compact base. Compactness of the base implies that this cone is closed. There are dual closed, convex cones in ${H^{1}_{\kappa}}({\widehat}{W})$ and ${H^{1}}({\widehat}{W})$: $$\begin{aligned}
{\mathfrak{C}}_{g}^{\kappa} &= \{[ \eta]\in{H^{1}_{\kappa}}({\widehat}{W})\mid [ \eta]([z])\ge0, \forall[z]\in {\mathfrak{C}}'_{g})\},\\
{\mathfrak{C}}_{g} &= \{[ \eta]\in{H^{1}}({\widehat}{W})\mid [ \eta]([z])\ge0, \forall[z]\in {\mathfrak{C}}'_{g})\}.\end{aligned}$$ Generally, these do not have compact base. Indeed, there are interesting cases in which ${\mathfrak{C}}'_{g}$ reduces to a single ray, hence the dual cone will be a full half-space.
Examples of asymptotic cycles are nonnegative, transverse, holonomy invariant measures $\mu$ on ${\mathcal{X}}_{g}$ that are finite on (transverse) compact sets. By Sullivan [@condel1 Theorem 10.2.12], these are the only ones.
\[measures\] The asymptotic cycles for ${\mathcal{X}}_{g}$ are exactly the nonnegative, transverse, holonomy invariant measures on ${\mathcal{X}}_{g}$ that are finite on compact sets.
By a well known theorem of J. F. Plante [@plante:meas] and the fact that the leaves of ${\mathcal{X}}_{g}$, being 1-dimensional, have linear growth, we obtain the following.
\[nontriv:cycles\]
There are nontrivial asymptotic cycles for ${\mathcal{X}}_{g}$.
\[transverseform\] There is a closed $1$-form on $W$, transverse to ${\mathcal{X}}_{g}$, hence no nontrivial asymptotic cycle bounds in $({\mathcal{E}}'_{*},{\partial})$. If ${\mathcal{F}}$ is of class ${C^{2}}$, this form can be chosen to be compactly supported and no nontrivial asymptotic cycle bounds either in $({\mathcal{D}}'_{*},{\partial})$ or $({\mathcal{E}}'_{*},{\partial})$.
Whether the foliation is of class ${C^{2}}$ or ${C^{0+}}$, ${\mathcal{F}}|W$, as a fiber bundle over ${S^{1}}$ with smooth leaves, has a smooth structure. Let $ \omega$ be a closed, nonsingular 1-form defining ${\mathcal{F}}|W$. This is clearly transverse to ${\mathcal{X}}_{g}$ and $\omega$ takes positive values on all nontrivial asymptotic cycles which, therefore, cannot bound in $({\mathcal{E}}'_{*},{\partial})$. Relative to a suitable octopus decomposition of ${\widehat}W$, ${\mathcal{X}}_{g}$ is contained in the interior of the compact nucleus $K_{0}$ of ${\widehat}W$. If ${\mathcal{F}}$ is of class ${C^{2}}$, then ${\mathcal{F}}$ is trivial in ${\widehat}{W}{\smallsetminus}K_{i}$, for $i\ge0$ sufficiently large (the ${C^{2}}$ hypothesis is critical here). Thus $ \omega= d\gamma$ outside of $K_{i}$. Extending and damping $ \gamma$ off smoothly to be 0 in a neighborhood of ${\mathcal{X}}_{g}$, we see that the compactly supported form $ \omega-d \gamma$ is as desired. Again $\omega$ takes positive values on all nontrivial asymptotic cycles, hence these cannot bound either in $({\mathcal{D}}'_{*},{\partial})$ or $({\mathcal{E}}'_{*},{\partial})$.
We will need the following key result, the proof of which is an adaptation of the proof of [@condel1 Lemma 10.2.8], which assumed the foliated manifold was compact. The other hypotheses of that lemma are satisfied because of Lemma \[nontriv:cycles\] and Lemma \[transverseform\].
\[intr:cone\] Every nontrivial asymptotic cycle defines a nontrivial class in ${\mathfrak{C}}'_{g}$. Furthermore, $\operatorname{int}{\mathfrak{C}}_{g}\ne{\emptyset}$ and consists of exactly those classes $[ \eta]\in{H^{1}}({\widehat}{W})$ that are represented by closed $1$-forms $\eta$ transverse to ${\mathcal{X}}_{g}$. Furthermore, if ${\mathcal{F}}$ is of class ${C^{2}}$, the completely analogous assertion holds for ${{\mathfrak{C}}^{\kappa}}_{g}$.
The nontriviality is given by Lemma \[transverseform\]. Let ${\mathfrak{C}}_{g}^{{\circ}}{\subset}{\mathfrak{C}}_{g}$ be the set of classes that take strictly positive values on the nonzero elements of ${\mathfrak{C}}'_{g}$. This is nonempty by Lemma \[transverseform\]. If $$[\eta]=([\eta_{0}],[\eta_{1}],\dots,[\eta_{i}],\dots)\in{\underleftarrow}{\lim}\,{H^{1}}(K_{i})={H^{1}}({\widehat}{W})$$ is in ${\mathfrak{C}}^{{\circ}}_{g}$, then $\eta_{0}$ takes positive values on all nontrivial asymptotic cycles. These cycles are all supported in the compact manifold $K_{0}$, hence the analogous result for ${\mathcal{F}}|K_{0}$ is proven in [@condel1 Lemma 10.2.8]. (Compactness is used to guarantee that ${H^{1}}(K_{0})$ is finite dimensional.) Thus $[\eta_{0}]\in U{\subset}{H^{1}}(K_{0})$, where $U$ is the interior of the dual cone there. Then $${\mathcal{U}}=U{\times}{H^{1}}(K_{1}){\times}{H^{1}}(K_{2}){\times}\cdots{\times}{H^{1}}(K_{i}){\times}\cdots$$ is an open set in the Tychonov topology and ${\mathcal{U}}\cap{H^{1}}({\widehat}{W})={\mathfrak{C}}^{{\circ}}_{g}$, hence this set is open. We prove that it is the entire interior of ${\mathfrak{C}}_{g}$ by showing that every $[\alpha]\in{\mathfrak{C}}_{g}$ that vanishes on some nontrivial asymptotic cycle $T$ is in the frontier. Indeed, if $[\eta]\in{\mathfrak{C}}^{{\circ}}_{g}$, form the line of classes $t[\eta]+(1-t)[\alpha]$, $-1\le t\le 1$. For $0<t\le1$, these classes are in ${\mathfrak{C}}^{{\circ}}_{g}$, but for $-1\le t<0$, they take negative values on $T$, hence are not in ${\mathfrak{C}}_{g}$.
It remains to show that if $[\eta]\in{\mathfrak{C}}^{{\circ}}_{g}$, then $\eta$ is cohomologous to a closed form $\eta'$ that takes positive values on the entire base ${\widehat}{{\mathcal{C}}}_{g}$ of the cone of asymptotic currents. The kernel of $[\eta]:H_{1}({\widehat}{W})\to{\mathbb{R}}$ is a closed hyperplane in that vector space and its pre-image $V{\subset}{\mathcal{Z}}_{1}$ is the kernel of $\eta:{\mathcal{Z}}_{1}\to{\mathbb{R}}$, again a closed vector subspace containing ${\mathcal{B}}_{1}$ and meeting ${\mathcal{C}}_{g}\cap{\mathcal{Z}}_{1}$ only at the vertex of that cone, hence meeting ${\mathcal{C}}_{g}$ itself only at the vertex. Using the standard Hahn-Banach argument (cf. [@condel1 Section 10.2]), we find a continuous linear functional $\eta':{\mathcal{E}}'\to{\mathbb{R}}$ which is strictly positive on ${\widehat}{{\mathcal{C}}}_{g}$ and vanishes identically on $V$. Since ${\mathcal{B}}_{1}{\subset}V$, $\eta'$ is a closed form transverse to ${\mathcal{X}}_{g}$. Since $[\eta']$ vanishes on the kernel of $[\eta]$ and is not trivial, $[\eta]$ and $[\eta']$ are positive multiples of each other, completing the proof for the case where ${\mathcal{F}}$ may only be of class ${C^{0+}}$.
If ${\mathcal{F}}$ is of class ${C^{2}}$, the proof is quite similar, but there are sufficient differences that we will give details. The fact that ${\mathfrak{C}}^{\kappa{\circ}}_{g}\ne{\emptyset}$ is guaranteed by Lemma \[transverseform\]. The fact that this is the interior of ${{\mathfrak{C}}^{\kappa}}_{g}$ follows from the corresponding fact for ${\mathfrak{C}}_{g}$ via the relative topology. If $[\eta]\in\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{g}$, we take $\eta$ compactly supported, hence it is a continuous linear functional $\eta:{\mathcal{D}}'_{1}\to{\mathbb{R}}$. Restricting to the space ${\mathcal{Z}}_{1}^{\infty}$ of closed currents, we obtain a closed subspace $V=\eta^{-1}\{0\}$ which contains the space ${\mathcal{B}}_{1}^{\infty}$ of boundaries. Viewing ${\mathcal{C}}_{g}$ as a cone in ${\mathcal{D}}'_{1}$, we again apply the Hahn-Banach theorem to produce a compactly supported form $\eta'$ strictly positive on ${\widehat}{{\mathcal{C}}}_{g}$ and vanishing on $V$. Thus $\eta'$ is compactly supported and transverse to ${\mathcal{X}}_{g}$. Since ${\mathcal{B}}_{1}^{\infty}{\subset}V$, $\eta'$ is closed. Via the continuous injection ${\mathcal{E}}'_{1}{\hookrightarrow}{\mathcal{D}}'_{1}$, $V$ pulls back to a space having as image in $H_{1}({\widehat}{W})$ the kernel of $[\eta]:H_{1}({\widehat}{W})\to{\mathbb{R}}$. Since $[\eta']$ also vanishes on this space and is nontrivial, $[\eta]$ and $[\eta']$ are positive multiples of each other.
Varying the regular monodromy $g$ within its isotopy class will give different cones ${\mathfrak{C}}_{g}$ and ${{\mathfrak{C}}^{\kappa}}_{g}$. One of our main goals is to show that, when $\dim M=3$, there is a maximal one and that, in the smooth case, it corresponds to the Handel-Miller monodromy $h$ in this isotopy class. Roughly speaking, this means that the Handel-Miller monodromy has the “tightest” dynamics in its isotopy class, determining the “narrowest” homology cone ${\mathfrak{C}}'_{h}$.
The notations ${\mathfrak{C}}_{g}$, ${{\mathfrak{C}}^{\kappa}}_{g}$ and ${\mathfrak{C}}'_{g}$, suggest dependence only on $g$, rather than on the choice of ${\mathcal{X}}_{g}$. In Subsection \[indchoice\], we will see that this is correct when $\dim M=3$. At the level of currents, the notation ${\mathcal{C}}_{g}$ is a bit of an abuse.
Homology directions
-------------------
It will be important to characterize a particularly simple spanning set of ${\mathfrak{C}}_{g}'$, the so called “homology directions” of Fried [@fried p. 260]. Assuming that ${\mathcal{L}}_{g}$ has been parametrized as a nonsingular ${C^{0+}}$ flow ${\Phi}_{t}$ that preserves ${\mathcal{F}}|W$, select a point $x\in{\mathcal{X}}_{g}$ and let $\Gamma $ denote the ${\Phi}$-orbit of $x$. If this is a closed orbit, it defines an asymptotic cycle which we will denote by ${\overline}{\Gamma }$. If it is not a closed orbit, let $\Gamma _{\tau
}=\{{\Phi}_{t} (x)\mid 0\le t\le \tau \}$. Let $\tau
_{k}\uparrow\infty$ and set $\Gamma _{k}=\Gamma _{\tau _{k}}$. After passing to a subsequence, we obtain an asymptotic current $${\overline}{\Gamma}
= \lim_{k{\rightarrow}\infty}\frac{1}{\tau _{k}}\int_{\Gamma _{k}}^{}.$$ In fact this is a *cycle*. One calls $\Gamma_{k}$ a “long, almost closed orbit” of ${\mathcal{X}}_{h}$. Its endpoints lie in the compact set ${\mathcal{X}}_{g}$ and it can be closed by adding a uniformly bounded curve in $M$. These are averaged out in the limit and the corresponding singular cycles, also called “long, almost closed orbits”, are denoted by $ \Gamma'_{k}$. The cycles $(1/\tau_{k}) \Gamma'_{k}$ also limit on ${\overline}{\Gamma}$, proving that it is a cycle.
\[long\_aco\] The asymptotic current ${\overline}{\Gamma }$, obtained as above, is an asymptotic cycle.
Another proof can be given by appealing to Stokes’s theorem.
All asymptotic cycles ${\overline}{\Gamma} $, obtained as above, and their homology classes are called homology directions of ${\mathcal{X}}_{g}$.
By abuse, we will denote both the cycle and its homology class by ${\overline}\Gamma$.
An elementary application of ergodic theory proves the following (see [@sull:cycles Proposition II.25] and [@condel1 Proposition 10.3.11]).
\[span\] Any asymptotic cycle $\mu$ can be arbitrarily well approximated by finite, nonnegative linear combinations $\sum_{i=1}^{r}a_{i}{\overline}{\Gamma }_{i}$ of homology directions. If $\mu \ne0$, the coefficients $a_{i}$ are strictly positive and their sum is bounded below by a constant $b_{\mu }>0$ depending only on $\mu
$.
The independence of the cones from various choices {#indchoice}
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Let us first note that long, almost closed orbits and homology directions, as classes in the singular homology $H_{1}({\widehat}{W})$, are clearly well defined even when ${\mathcal{L}}_{g}$ and ${\mathcal{X}}_{g}$ are only ${C^{0}}$ (which may well be the case when $g$ is only a homeomorphism). While most of the theory of asymptotic classes fails when these objects are not at least integral to a ${C^{0}}$ line field, we can still define the cone ${\mathfrak{C}}'_{g}$ as the closure in $H_{1}({\widehat}{W})$ of the set of nonnegative linear combinations of homology directions. This is a closed, convex cone which, when $g$ is regular, coincides with the cone already defined (Lemma \[span\]). We are going to show that, if $\dim M=3$, this cone depends only on $g$, not on the choice of ${\mathcal{L}}_{g}$. In this subsection we require no regularity.
Let $L$ be a leaf of ${\mathcal{F}}|W$, and let ${\mathcal{L}}$ and ${\mathcal{L}}_{\sharp}$ be $1$-dimensional foliations of ${\widehat}W$ of class ${C^{0}}$, transverse to ${\mathcal{F}}|{\widehat}W$ and having each leaf of ${\mathcal{F}}|W$ as a section. Let ${\mathcal{X}}$ and ${\mathcal{X}}_{\sharp}$ be the respective core laminations and let ${\mathfrak{C}}'_{{\mathcal{X}}}$ and ${\mathfrak{C}}'_{{\mathcal{X}}_{\sharp}}$ denote the corresponding cones in $H_{1}({\widehat}{W})$, ${\mathfrak{C}}_{{\mathcal{X}}}$ and ${\mathfrak{C}}_{{\mathcal{X}}_{\sharp}}$ those in ${H^{1}}({\widehat}{W})$, and ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{X}}}$ and ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{X}}_{\sharp}}$ those in ${H^{1}_{\kappa}}({\widehat}{W})$.
In [@cc:cone], the following elementary theorem was deduced as a corollary of a much deeper result (Lemma 4.10 in that reference) which we attempted to deduce from results of M. E. Hamstrom [@ham:disk-holes; @ham:torus; @ham:homeo]. A correct proof of that lemma needs a deep result of T. Yagasaki [@yag], but we omit this because we do not need it.
\[samecones\] Let $L$ be a leaf of ${\mathcal{F}}|W$, and let ${\mathcal{L}}$ and ${\mathcal{L}}_{\sharp}$ be $1$-dimensional foliations of ${\widehat}W$, transverse to ${\mathcal{F}}|{\widehat}W$ and having each leaf of ${\mathcal{F}}|W$ as a section, having respective core laminations ${\mathcal{X}}$ and ${\mathcal{X}}_{\sharp}$, and inducing the same endperiodic monodromy $g:L\to L$. Then ${\mathfrak{C}}'_{{\mathcal{X}}}={\mathfrak{C}}'_{{\mathcal{X}}_{\sharp}}$, ${\mathfrak{C}}_{{\mathcal{X}}}={\mathfrak{C}}_{{\mathcal{X}}_{\sharp}}$ and ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{X}}}={{\mathfrak{C}}^{\kappa}}_{{\mathcal{X}}_{\sharp}}$.
We will show that the homology directions determined by the long, almost closed orbits of ${\mathcal{X}}$ are exactly the same as the ones for ${\mathcal{X}}_{\sharp}$ and the theorem will follow. Our arguments are carried out in the ${C^{0}}$ category for arbitrary surfaces $L$ *which have nonabelian fundamental group*. Our leaf $L$, being noncompact with no simple end satisfies this requirement.
Let $I$ be the compact interval $[0,1]$ and consider the product $L\times I$. (One obtains such a product, for instance, by cutting $W$ apart along $L$ and taking as the interval fibers the resulting segments of the leaves of ${\mathcal{L}}$.) For each $x\in L$, denote by $I_{x}$ the interval fiber with endpoints $\{x\}\times\{0,1\}$. Consider a second fibration of $L\times I$ by intervals $J_{x}$, requiring that the endpoints of $J_{x}$ coincide with those of $I_{x}$, for all $x\in L$. (By the hypothesis on ${\mathcal{L}}_{\sharp}$, this second fibration arises in our case by cutting apart along $L$ and using the segments of leaves of ${\mathcal{L}}_{\sharp}$ as fibers.) For each $x\in L$, let $\alpha_{x}$ denote the loop in $L\times I$ obtained by following $I_{x}$ from $(x,0)$ to $(x,1)$ and then following $J_{x}$ from $(x,1)$ to $(x,0)$. Finally, if $p:L\times I\to L$ is the canonical projection, let $\beta_{x}=p{\circ}\alpha_{x}$, a loop in $L$.
\[commutes\] Let $x_{0}\in L$ and set $\delta=\beta_{x_{0}}$. If $\gamma(s)$, $0\le s\le 1$, is any other closed curve in $L$ based at $x_{0}$, then $\gamma\cdot\delta =
\delta\cdot\gamma$ in $\pi_{1}(L,x_{0})$.
Define $F(s,t) = \beta_{\gamma(s)}(t)$. Then $F(s,0) = \gamma(s) =
F(s,1)$. Also $F(0,t) = F(1,t) = \beta_{x_{0}}(t) = \delta(t)$. Because of this last, we can view $F$ as a map from the cylinder $S^1\times [0,1]$ into $L$. The curve obtained by following $F(0,t)$, $0\le t\le 1$, followed by $F(s,1)$, $0\le s\le 1$, and then $F(0,1-t)$, $0\le t\le 1$, is the composite loop $\delta\cdot\gamma\cdot\delta^{-1}$. We show how to deform this curve continuously to $\gamma$, keeping the basepoint $x_{0}$ fixed throughout the deformation.
Let $\sigma_{t}$ be the curve obtained by following $F(0,\tau)$, $0\le\tau\le t$, followed by $F(s,t)$, $0\le s\le 1$, followed by $F(0,t-\tau)$, $0\le \tau\le t$. Since $\sigma_{0} = \gamma$ and $\sigma_{1} = \delta\cdot\gamma\cdot\delta^{-1}$, we have the desired deformation.
\[0homotopic\] If there exists an $x_{0}\in L$ so that $J_{x_{0}}$ cannot be deformed into $I_{x_{0}}$ keeping the endpoints fixed, then $L$ is either the torus or has the homotopy type of the circle.
The hypothesis implies that $\alpha_{x_{0}}$ is essential in $L\times I$, and so $\beta_{x_{0}}$ is essential in $L$ and thus is a nontrivial element of $\pi_{1}(L,x_{0})$. By Lemma \[commutes\], every element of $\pi_{1}(L,x_{0})$ commutes with $\beta_{x_{0}}$. The only closed orientable surface that $L$ could be is the torus. If $L$ is not closed, $L$ is homotopically equivalent to a bouquet $B$ of circles. The only bouquet of circles that contains a nontrivial element of $\pi_1(B,*)$ that commutes with every other element of $\pi_1(B,*)$ is one circle.
Let ${\overline}{\Gamma }\in H_{1}
({\widehat}{W};{\mathbb{R}}) $ be a homology direction for ${\mathcal{X}}$.
First assume that ${\overline}{\Gamma }$ is not represented by a closed orbit in ${\mathcal{X}}$ and write $${\overline}{\Gamma } = \lim_{k\to\infty}\frac{1}{\tau _{k}}[\Gamma' _{k}],$$ a limit in $H_{1} (M)$ of the homology classes of long, almost closed orbits. The numbers $\tau _{k}$ are the “lengths” of $\Gamma _{k}$ (measured by the transverse, invariant measure for ${\mathcal{F}}|W$) and increase to $\infty$ with $k$. Thus, except for a uniformly bounded arc in $L$, $\Gamma '_{k}$ is a sequence of segments, $\sigma _{1},\dots,\sigma _{n_{k}}$ of an orbit in ${\mathcal{X}}$, each starting and ending in $L$. There is a corresponding sequence $\sigma' _{1},\dots,\sigma' _{n_{k}}$ of segments of an orbit in ${\mathcal{X}}_{\sharp}$ such that $\sigma _{i}\text{ and }\sigma '_{i}$ have the same endpoints and the same lengths, $1\leq i\leq n_{k}$. By Corollary \[0homotopic\], these respective segments are homotopic by a homotopy that keeps their endpoints fixed. Thus, we see that ${\overline}{\Gamma }$ is also a homology direction for ${\mathcal{X}}_{\sharp}$. In the case that ${\overline}{\Gamma }$ is represented by a closed orbit, the argument adapts and is simpler. Finally, the roles of ${\mathcal{X}}\text{ and }{\mathcal{X}}_{\sharp}$ can be interchanged, proving that the two laminations have the same homology directions.
\[C0\] Let ${\mathcal{L}}\text{ and }{\mathcal{L}}_{\sharp}$ be two $1$-dimensional foliations transverse to ${\mathcal{F}}|{\widehat}{W}$. Suppose that the respective core laminations ${\mathcal{X}}\text{ and }{\mathcal{X}}_{\sharp}$ are ${C^{0}}$-isotopic by an isotopy $\varphi _{t}:{\mathcal{X}}{\hookrightarrow}M$, $\varphi _{0}=\operatorname{id}_{{\mathcal{X}}}$ and $\varphi _{1} ({\mathcal{X}})={\mathcal{X}}_{\sharp}$, such that $\varphi _{t} (x)$ lies in the same leaf of ${\mathcal{F}}|W$ as $x$, for $0\le t\le 1$, $\forall\,x\in {\mathcal{X}}$. Then ${\mathfrak{C}}'_{{\mathcal{X}}}={\mathfrak{C}}'_{{\mathcal{X}}_{\sharp}}$ and ${\mathfrak{C}}_{{\mathcal{X}}}={\mathfrak{C}}_{{\mathcal{X}}_{\sharp}}$.
Parametrize the two foliations as flows using the same transverse invariant measure $ \theta$ for ${\mathcal{F}}$. Since ${\mathcal{F}}$ is leafwise invariant under the isotopy, the flow parameter is preserved and the long, almost closed orbits of ${\mathcal{X}}$ are isotoped to the long, almost closed orbits of ${\mathcal{X}}_{\sharp}$. Homotopic singular cycles are homologous and the assertions follow.
The property that points of $W$ remain in the same leaf of ${\mathcal{F}}|W$ throughout the isotopy will be indicated by saying that ${\mathcal{F}}|W$ is leafwise invariant by $\varphi _{t}$.
\[conj\] Let $g:L{\rightarrow}L$ be the endperiodic first return homeomorphism induced on a leaf $L$ of ${\mathcal{F}}|W$ by a transverse $1$-dimensional foliation ${\mathcal{L}}_{g}$ of class ${C^{0}}$. If $\varphi_{t}:L{\rightarrow}L$ is an isotopy of ${\varphi}_{0}=\operatorname{id}$ to a homeomorphism ${\varphi}_{1}$, then ${\mathfrak{C}}'_{g}={\mathfrak{C}}'_{\varphi_{1} g\varphi_{1}^{-1}}$.
Let $N$ be a closed normal neighborhood of $L$ in $W$ which is a foliated product with leaves the leaves of ${\mathcal{F}}$ meeting $N$ and normal fibers the arcs of ${\mathcal{L}}\cap N$. Write $N=L{\times}[-{\varepsilon},{\varepsilon}]$ and consider each arc $\ell_{x}$ of a leaf of ${\mathcal{L}}$ issuing in the positive direction from $(x,{\varepsilon})\in L{\times}\{{\varepsilon}\}$ and first returning to $N$ at $(g(x),-{\varepsilon})$. In $N$, replace each arc $\tau_{y}=\{y\}{\times}[-{\varepsilon},{\varepsilon}]$ of ${\mathcal{L}}\cap N$ with an arc $\sigma_{y}:[-1,1]\to N$ defined by $$\begin{aligned}
\sigma_{y}(t) &= ({\varphi}_{t+1}(y),{\varepsilon}t),\,-1\le t\le0,\\
\sigma_{y}(t) &= ({\varphi}_{t}^{-1}({\varphi}_{1}(y)),{\varepsilon}t),\,0\le t\le1.\end{aligned}$$ Notice that this still connects $(y,-{\varepsilon})$ to $(y,{\varepsilon})$. We construct an ambient leaf-preserving isotopy $\psi$, supported in $N$ and carrying each $\tau_{y}$ to $\sigma_{y}$, by $$\begin{aligned}
\psi_{s}(y,{\varepsilon}t) &= ({\varphi}_{s(t+1)}(y),{\varepsilon}t),\,-1\le t\le0,\,0\le s\le1,\\
\psi_{s}(y,{\varepsilon}t) &= ({\varphi}_{st}^{-1}({\varphi}_{s}(y)),{\varepsilon}t),\,0\le t\le1,\,0\le s\le1.\end{aligned}$$ We obtain ${\mathcal{L}}'$ from ${\mathcal{L}}$ by replacing $\tau_{y}$ with $\sigma_{y}$, $\forall y\in L$, observing that the monodromy induced by ${\mathcal{L}}'$ on $L=L{\times}\{0\}$ is ${\varphi}_{1}{\circ}g{\circ}{\varphi}_{1}^{-1}$. The assertion follows by Proposition \[C0\].
Foliated forms {#fforms}
--------------
In this subsection we assume that $g:L\to L$ is a smooth endperiodic monodromy homeomorphism isotopic to $f$ and we choose ${\mathcal{L}}_{g}$ to be smooth. In fact (and this is an important remark), all we need is that ${\mathcal{L}}_{g}{\smallsetminus}{\mathcal{X}}_{g}$ is smooth and that ${\mathcal{L}}_{g}$ itself is of class ${C^{0+}}$. Our discussion will be valid for all dimensions.
\[ff\] A 1-form $\eta\in A^{1}(W)$ is a foliated form if it is closed and nowhere vanishing and becomes unbounded at ${{\partial}_{\tau}}{\widehat}W$ in such a way that the corresponding foliation ${\mathcal{F}}_{\eta}$ that it defines on $W$ extends to a ${C^{\infty,\iota}}$ foliation ${\widehat}{{\mathcal{F}}}_{\eta}$ of ${\widehat}W$ by adjunction of ${{\partial}_{\tau}}{\widehat}W$.
We will prove the following.
\[folforms\] The open cone $\operatorname{int}{\mathfrak{C}}_{g}$ consists of classes in $H^{1}({\widehat}{W})$ that can be represented by foliated forms transverse to ${\mathcal{L}}_{g}$. When ${\mathcal{F}}$ is of class ${C^{\infty,\iota}}$, the interior of ${{\mathfrak{C}}^{\kappa}}_{g}$ is characterized analogously, except that the forms become exact in $W{\smallsetminus}K_{i}$ for large enough values of $i$, defining a product foliation there identical with ${\mathcal{F}}|(W{\smallsetminus}K_{i})$.
Remark that foliated forms only live in $W$, not in ${\widehat}W$, but ${H^{1}}({\widehat}W)={H^{1}}(W)$ and any form representing a class in $W$ can be taken to be equal to that representing the class in ${\widehat}W$ outside of any small neighborhood of ${{\partial}_{\tau}}{\widehat}W$. If the foliated form is exact in the arms of some octopus decomposition, it is clear that it represents a class in ${H^{1}_{\kappa}}({\widehat}{W})$.
Thus, we call ${\mathfrak{C}}_{g}$ and ${{\mathfrak{C}}^{\kappa}}_{g}$ foliation cones associated to $g$. The rays out of the origin meeting the interior of ${{\mathfrak{C}}^{\kappa}}_{g}$ correspond to foliations of ${\widehat}W$ that have holonomy only along ${{\partial}_{\tau}}{\widehat}W$ (${C^{\infty}}$ tangent to the identity), are transverse to ${\mathcal{L}}_{g}$ and extend ${\mathcal{F}}|(M{\smallsetminus}W)$ over $M$ to a foliation of class ${C^{\infty,\iota}}$, provided ${\mathcal{F}}$ was ${C^{\infty,\iota}}$ smooth initially. The rays in ${\mathfrak{C}}_{g}$ correspond to foliations (of class ${C^{\infty,\iota}}$ in ${\widehat}W$) that may only extend ${\mathcal{F}}|(M{\smallsetminus}W)$ to a ${C^{0+}}$ foliation. In both cases, the rational rays correspond to foliations defined by forms $\eta$ with period group infinite cyclic. These foliations are exactly the ones in which we are interested, being the ones that replace ${\mathcal{F}}|W$ to give a new finite depth foliation of $M$. The rest of the rays in $\operatorname{int}{\mathfrak{C}}_{g}$ consist of classes having period group dense in ${\mathbb{R}}$ and so define foliations that are dense leaved in $W$. Recall that compactness of junctures in ${C^{2}}$ foliations is what forces ${\mathcal{F}}|{\widehat}W$ to be trivial in the arms when ${\mathcal{F}}$ is smooth and finite depth.
Fix a class $[\eta]\in\operatorname{int}{\mathfrak{C}}_{g}$, the 1-form $\eta\in[\eta]$ being defined on ${\widehat}W$ and transverse to ${\mathcal{X}}_{g}$ (Theorem \[intr:cone\]). If ${\mathcal{F}}$ is of class ${C^{\infty,\iota}}$, select this form to be compactly supported in $W$, $[\eta]\in{{\mathfrak{C}}^{\kappa}}_{g}$. Select a neighborhood $U$ of ${\mathcal{X}}_{g}$ such that $\eta{\pitchfork}{\mathcal{L}}_{g}|U$. We need to show that $\eta$ is cohomologous to a foliated form. Note that, if $\eta$ is compactly supported in $K_{i}$, the foliated form will automatically be exact in $W{\smallsetminus}K_{i}$.
Given $x\in W{\smallsetminus}{\mathcal{X}}_{g}$, let $s(t)$ be the smooth trajectory along ${\mathcal{L}}_{g}$ in that set, smoothly reparametrized so that $x=s(0)$ and $s(\pm1)\in F_{\pm}$. Here, $F_{+}$ is the union of outwardly oriented leaves of ${{\partial}_{\tau}}{\widehat}{W}$ and $F_{-}$ the union of inwardly oriented ones. For some choices of $x$ both signs may be possible and for others only one. For definiteness, consider the case $s(-1)\in F_{-}$. Define a tubular neighborhood $V_{x}=D{\times}[-1,3/4)$ of $s$ so that $s(t)=(0,t)$ and $\{z\}{\times}[-1,3/4)$ is an arc in ${\mathcal{L}}_{g}$, $\forall z\in D$. Here, $D$ is the open unit $(n-1)$-ball with polar coordinates $(r, \theta_{1},\dots,\theta_{n-2})$, $0\le r<1$. This gives cylindrical coordinates $(t,r, \theta_{1},\dots,\theta_{n-2})$ on $V_{x}$. On $V_{x}$, define a smooth, real valued function $$\ell_{x}(t,r, \theta_{1},\dots,\theta_{n-1})=\ell_{x}(t,r)=\ell(t)\lambda(r),$$ where $\ell(t)=t-1$, $-1\le t\le1/2$, and damps off to 0 smoothly and with positive derivative as $t\to3/4$, and $ \lambda(r)\equiv1$, $0\le r\le1/2$, and damps off to 0 smoothly through positive values as $r\to1$. Thus, $ \ell _{x}(t,r)$ vanishes outside of $V_{x}$ and $d \ell _{x}$ is transverse to ${\mathcal{L}}_{g}$ in $V_{x}$. Let $V'_{x}{\subset}V_{x}$ be the neighborhood of $x$ defined by $-1\le t<1/2$ and $0\le r<1/2$. Perform an analogous construction for trajectories out of $x$ with $s(1)\in F_{+}$.
Suitable choices of these open cylinders (using the local compactness) give a locally finite open cover $\{U,V'_{x_{1}},V'_{x_{2}},\dots\}$ of ${\widehat}W$. For suitable choices of positive constants $c_{i}$, set $ \ell=\sum_{i=1}^{\infty} c_{i}\ell_{x_{i}}$, a smooth function, supported in $W{\smallsetminus}{\mathcal{X}}_{g}$, with $d\ell{\pitchfork}{\mathcal{L}}_{g}$ outside of a compact neighborhood of ${\mathcal{X}}_{g}$ in $U$. Since $\eta$ is bounded in any compact region of ${\widehat}W$ and is transverse to ${\mathcal{L}}_{g}$ in $U$, we can choose the coefficients $c_{i}>0$ large enough that $\eta'=\eta+ d \ell$ is a closed form in ${\widehat}W$, cohomologous to $\eta$ and transverse to ${\mathcal{L}}_{g}$. This form might be badly behaved at ${{\partial}_{\tau}}{\widehat}W$, hence we must modify it by adding on a suitable exact form supported in a neighborhood of the boundary leaves.
Let $V=F_{-}{\times}[0,1)$ be a normal neighborhood of $F_{-}$ in ${\widehat}W$, the fibers being arcs in leaves of ${\mathcal{L}}_{g}$. Let $\lambda$ be a smooth function on the deleted normal neighborhood $V{\smallsetminus}F_{-}$, depending only on the normal parameter $t$, and having $\lambda'(t)\ge0$, with $\lambda'(t)=e^{1/t^{2}}$ near $F_{-}$. We claim that ${\widetilde}{\eta}=\eta'+d\lambda$ is the desired foliated form in the cohomology class $[\eta]$. Indeed, it is everywhere transverse to ${\mathcal{L}}_{g}$, hence nonsingular on $W$, it is cohomologous to $\eta'$ and it becomes unbounded at ${{\partial}_{\tau}}{\widehat}W$. We must show that $\ker{\widetilde}\eta$ extends ${C^{\infty,\iota}}$-smoothly to a plane field on ${\widehat}W$ by adding on the tangent planes to ${{\partial}_{\tau}}{\widehat}W$. For this, set ${\overline}{\eta}={\widetilde}{\eta}/\lambda'=\eta'/\lambda'+dt$, a form defined on a small enough deleted neighborhood of $F_{-}$. This form is no longer closed but satisfies $\ker{\overline}{\eta}=\ker{\widetilde}\eta$ in that neighborhood. Since $\eta'$ is bounded on ${\widehat}W$, it is clear that ${\overline}{\eta}$ approaches $dt $ in the ${C^{\infty}}$ topology as $t\to0$ and that the resulting foliation of ${\widehat}W$ is of class ${C^{\infty,\iota}}$. After a similar construction in a normal neighborhood of $F_{+}$, we obtain a foliated form, again denoted by ${\widetilde}\eta$, transverse to ${\mathcal{L}}_{g}$.
While all of this works equally well whether or not $\eta$ is compactly supported in $W$, in the case that ${\mathcal{F}}$ is of class ${C^{\infty,\iota}}$, we choose $\eta$ compactly supported and want to produce the foliated form ${\widetilde}\eta$ so that, outside of some compact region in ${\widehat}W$, it agrees with the (exact) foliated form $\omega$ that defined ${\mathcal{F}}$ there. For $i$ large, $\omega|({\widehat}{W}{\smallsetminus}K_{i})=d\gamma$ and $\operatorname{supp}\eta{\subset}K_{i}^{{\circ}}$. Choose $r>0$ so that $\{U,V'_{x_{1}},V'_{x_{2}},\dots,V'_{x_{r}}\}$ covers $K_{i+1}$. Smoothly damp $\gamma$ off to $0$ in a neighborhood $N$ of $K_{i}$ in $K_{i+1}$ so that it becomes $0$ near $K_{i}$ and is unchanged in ${\widehat}{W}{\smallsetminus}N$, extending it by $0$ over $K_{i}$. Call this new function ${\widetilde}\gamma$. Similarly, damp $\lambda$ off to $0$ in $V{\smallsetminus}K_{i+1}$ so that $d\lambda$ remains transverse to ${\mathcal{L}}_{g}$ in $V{\smallsetminus}K_{i+1}$ and becomes $0$ in $V{\smallsetminus}K_{i+2}$, calling this new function ${\widetilde}\lambda$. Now, ${\widetilde}\eta=\eta+\sum_{j=1}^{r}c_{j}d\lambda_{x_{j}}+d{\widetilde}\lambda+d{\widetilde}\gamma$ is as desired for suitably large choices of the constants $c_{j}.$ The proof of Theorem \[folforms\] is complete.
In the case that ${\mathcal{F}}$ is of class ${C^{\infty,\iota}}$, refoliating $W$ by the new foliated form ${\widetilde}\eta$ as above only changes ${\mathcal{F}}$ in a compact part of ${\widehat}W$. Thus, the *germinal* holonomy along all leaves of ${\mathcal{F}}|(M{\smallsetminus}{\widehat}{W})$ has been unchanged, while the infinitesimal holonomy along the leaves of ${{\partial}_{\tau}}{\widehat}W$ is ${C^{\infty}}$-tangent to the identity. Thus, the new foliation of $M$ is again smooth of class ${C^{\infty,\iota}}$. Refoliating finitely many components of ${\mathcal{O}}$ in this way continues to preserve this smoothness class. If one wants to simultaneously refoliate *all* components of ${\mathcal{O}}$, preserving smoothness gets a bit delicate. Instead of tackling this here, we remark that the new foliation, of class ${C^{0+}}$ and finite depth, satisfies the hypotheses of [@cc:smth2], hence is homeomorphic to a foliation of class ${C^{\infty,\iota}}$.
The Handel-Miller cones
=======================
In this section, we restrict our attention to the case that ${\mathcal{F}}$ is smooth and $\dim M=3$. Consequently, the monodromy $f:L\to L$ is endperiodic [@cc:hm Proposition 2.16].
For a detailed treatment of the Handel-Miller theory of endperiodic maps of surfaces, see [@cc:hm]. Notation will be based on that paper save mention to the contrary.
Let $h:L\to L$ be the Handel-Miller representative of the isotopy class of the monodromy $f$ of $L$. We will show that the cone ${{\mathfrak{C}}^{\kappa}}_{h}$ is the maximal foliation cone corresponding to that isotopy class. One difficulty is that “the” Handel-Miller representative is a misnomer. There are infinitely many, but we will show that ${{\mathfrak{C}}^{\kappa}}_{h}$ is independent of the choice. For this, we need to show that the homology cone ${\mathfrak{C}}'_{h}$, hereafter called the *Handel-Miller cone*, is independent of the choice. We need to investigate the asymptotic cycles for ${\mathcal{X}}_{h}$ more carefully.
Generally, the choices of $h$ are not smooth, perhaps not even regular, hence the interpretation of ${{\mathfrak{C}}^{\kappa}}_{h}$ as a foliation cone seems problematic. However, by [@cc:hm Theorem 12.1], there is a choice of $h$ which has associated ${\mathcal{L}}_{h}$ satisfying the hypotheses of Subsection \[fforms\].
The invariant set
-----------------
For the time being, no smoothness hypotheses are imposed on $h$ nor on ${\mathcal{L}}_{h}$. The lamination ${\mathcal{X}}_{h}$ is the ${\mathcal{L}}_{h}$-saturation of the invariant set $X_{h}=L{\smallsetminus}({\mathcal{U}}_{+}\cup{\mathcal{U}}_{-})$, the set of points that do not escape to ends of $L$ under forward or backward iteration of $h$. We recall that $h$ leaves invariant a pair of pseudo-geodesic laminations $ \Lambda_{\pm}$ and that $ X^{*}_{h}=\Lambda_{+}\cap \Lambda_{-}{\subseteq}X_{h}$ is called the *meager* invariant set. Generally, $X_{h}^{*}\ne X_{h}$. Referring to [@cc:hm] (where $X_{h}$ was denoted by ${\mathcal{I}}$ and $X_{h}^{*} $ by ${\mathcal{K}}$), we describe the set $X_{h}{\smallsetminus}X^{*}_{h}$.
The complement $L{\smallsetminus}(\Lambda_{+}\cup \Lambda_{-})$ has infinitely many components, “most” of which lie in the $\pm$-escaping set ${\mathcal{U}}_{+}\cup{\mathcal{U}}_{-}$, hence are disjoint from $X_{h}$. Those components that do not lie in the $\pm$-escaping set are the nuclei $N_{i} $ of finitely many principal regions $P_{i}$, $1\le i\le n$. The nucleus $N_{i}$ is a compact, connected surface meeting $Z^{*}_{h}$ in finitely many vertices [@cc:hm Lemma 6.40]. The principal regions and their nuclei are permuted amongst themeselves by $h$. For a complete treatment of principal regions and their nuclei, see [@cc:hm Subsection 6.5].
Thus, if we set $N=N_{1}\cup\cdots\cup N_{n}$, we can write $X_{h}=X_{h}^{*}\cup N$. If $N_{i}$ is simply connected, $h|N_{i}$ can be defined so that $h^{p}|N_{i}$ is the identity, for a minimal integer $p\ge1$. The ${\mathcal{L}}_{h}$-saturation of such $N_{i}$ contributes only one ray to the Handel-Miller cone ${\mathfrak{C}}'_{h}$, spanned by the closed orbit of any vertex of ${\partial}N_{i}$. Thus, this ray already was contributed by the ${\mathcal{L}}_{h}$-saturation of $X_{h}^{*}$. If not simply connected, $N_{i}$ is the union of closed annuli $A_{i,1},\dots,A_{i,r_{i}}$ and a “core” $N'_{i}$. The union of the annuli, taken over all admissible index pairs $(i,j)$, is $h$-invariant and each $A_{i,j}$ is cobounded by a polygonal circle $s_{i,j}$, with edges lying alternately in leaves of $ \Lambda_{+}$ and $ \Lambda_{-}$, and another simple closed reducing curve $ \rho_{i,j}$. These reducing curves constitute the boundary of $N'_{i}$. If $p>0$ is an integer such that $h^{p}(A_{i,j})=A_{i,j}$ and $h^{p}$ fixes each vertex of $s_{i,j}$, then $h^{p}|A_{i,j}$ is isotopic to the identity. In the ${\mathcal{L}}_{h}$-saturation of the annuli $A_{i,j}$, it is evident that the homology directions are homologous to positive multiples of the closed orbits of the vertices of the polygonal circles $s_{i,j}$. These vertices lie in $X^{*}_{h}$. Thus, the ${\mathcal{L}}_{h}$-saturations of the annuli contribute no classes to ${\mathfrak{C}}'_{h}$ other than those contributed by the ${\mathcal{L}}_{h}$-saturation ${\mathcal{X}}^{*}_{h}{\subset}{\mathcal{X}}_{h}$ of $X^{*}_{h}$. All new classes in the Handel-Miller cone will be contributed only by homology directions of the ${\mathcal{X}}_{h}$-saturation of the cores $N'_{i}$.
There is a power $h^{p}$, for a minimal integer $p\ge1$, that maps each core $N'_{i}$ into itself and is the identity on the set $N\cap X^{*}_{h}$ of vertices. By the Nielsen-Thurston theory [@bca; @FLP; @HandT], there is a family of simple closed curves (reducing circles) in $N'_{i}$, splitting that surface into subsurfaces in each of which some power $h^{kp}$ is either isotopic to a pseudo-Anosov homeomorphism or a periodic homeomorphism. As part of our definition of “Handel-Miller homeomorphism” we will require that $h^{kp}$ be exactly pseudo-Anosov or periodic on each of these pieces. (We will say that $h|N$ is of Nielsen-Thurston type.) Actually, the reducing circles need to be slightly thickened to invariant annuli. These annuli contribute no cycles to the cone not already contributed by the periodic and/or pseudo-Anosov pieces. Each periodic piece contributes a ray of homology classes to ${\mathfrak{C}}'_{h}$. Each pseudo-Anosov piece contributes a closed, convex subcone. Although the choice of $h|N$ is not unique, any two choices $h$ and $h'$ are related by $(h')^{kp}={\varphi}{\circ}h^{kp}{\circ}{\varphi}^{-1}$, where ${\varphi}:L\to L$ is a homeomorphism isotopic to the identity and supported on $N$ (cf. [@th:surfaces page 421]). By Corollary \[conj\], the Handel-Miller cone ${\mathfrak{C}}'_{h}={\mathfrak{C}}'_{h^{kp}}$ is independent of the allowable choices of $h|N$.
Uniqueness of the Handel-Miller cone
------------------------------------
The laminations $ \Lambda_{\pm}$ associated to a Handel-Miller homeomorphism $h$ are augmented by the $h$-junctures [@cc:hm] to give mutually transverse, $h$-invariant laminations $ \Gamma_{\pm}$. Let $Y_{h}= \Gamma_{+}\cap \Gamma_{-}$. Each arc of $ \Gamma_{+}\cup \Gamma_{-}{\smallsetminus}Y_{h}$ completes to a compact arc meeting $Y_{h}$ only in its endpoints. If we replace each such arc with the unique homotopic geodesic joining the endpoints, we obtain new laminations $ \Gamma^{\flat}_{\pm}$. Note that $ \Gamma_{+}^{\flat}\cap \Gamma_{-}^{\flat}=Y_{h}$. We can now extend $h|Y_{h}$ to a Handel-Miller endperiodic automorphism $h^{\flat}:L\to L$ by essentially the same procedure as in the proof of [@cc:hm Theorem 8.1], the laminations $ \Gamma_{\pm}^{\flat}$ being the extended laminations for $h^{\flat}$.
In defining $h^{\flat}|N^{\flat}$ by a Nielsen-Thurston homeomorphism, there is a certain latitude. One first finds a homeomorphism $\psi:L\to L$, isotopic to the identity and carrying ${\partial}N$ to ${\partial}N^{\flat}$ [@Epstein:isotopy]. Then $\psi:N\to N^{\flat}$ is isotopic to the inclusion. We set $h^{\flat}|N^{\flat}=\psi{\circ}h|N{\circ}\psi^{-1}$. By the discussion in the previous subsection and the fact that $h$ and $h^{\flat}$ agree on $X_{h}^{*}=X_{h^{\flat}}^{*}$, we see that ${\mathfrak{C}}'_{h}={\mathfrak{C}}'_{h^{\flat}}$.
If $g$ is another Handel-Miller representative of the isotopy class of $h$, we produce $g^{\flat}$ as above and corresponding extended laminations $ \Gamma_{\pm}^{g^{\flat}}$. Similarly, the extended laminations for $h^{\flat}$ will be denoted by $ \Gamma_{\pm}^{h^{\flat}}$. Since $h^{\flat}$ and $g^{\flat}$ are isotopic, they have lifts to the universal cover of $L$ which induce the same homeomorphisms on the ideal boundary at infinity. Thus, in standard fashion (cf. [@cc:hm Theorem 8.1]), there is a natural homeomorphism $ \theta: L\to L$, isotopic to the identity, that carries $ \Gamma_{\pm}^{g^{\flat}}$ to $ \Gamma_{\pm}^{h^{\flat}}$. Then $ \theta^{-1}{\circ}h^{\flat}{\circ}\theta$ is a legitimate choice of $g^{\flat}$. Thus, appealing to Corollary \[conj\], we get $${\mathfrak{C}}'_{h}={\mathfrak{C}}'_{h^{\flat}}={\mathfrak{C}}'_{ \theta^{-1}{\circ}h^{\flat}{\circ}\theta}={\mathfrak{C}}'_{g^{\flat}}={\mathfrak{C}}'_{g}.$$ We have completed the proof of the following.
\[hmindep\] The Handel-Miller cone ${\mathfrak{C}}'_{h}{\subset}H_{1}({\widehat}{W})$ is independent of the choice of the Handel-Miller representative of the isotopy class of the endperiodic monodromy of $L$.
Because of this theorem, we will denote the cone ${\mathfrak{C}}'_{h}$ by ${\mathfrak{C}}'_{{\mathcal{F}}}$ and the dual ${H^{1}_{\kappa}}({\widehat}{W})$-cohomology cone by ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$. We will call this latter the *Handel-Miller foliation cone*.
Handel-Miller foliation cones are polyhedral
--------------------------------------------
We fix a Handel-Miller monodromy map $h:L\to L$. By [@cc:hm Theorem 12.1], this will now be chosen to be a diffeomorphism except at the finitely many $p$-pronged singularities in the interior of pseudo-Anosov components (if any) of $N$. Then ${\mathcal{L}}_{h}$ can be chosen to be smooth except, perhaps, at finitely many closed orbits corresponding to the $p$-pronged singularities. Standard models of the neighborhoods of these orbits (cf. [@cc:LB Appendix B]) make it easy to arrange that ${\mathcal{L}}_{h}$ is of class ${C^{0+}}$ at those closed orbits.
As in [@cc:hm Section 9], the dynamical system $h:X^{*}_{h}\to X^{*}_{h}$ is conjugate to a 2-ended subshift of finite type. As is standard, on the pseudo-Anosov pieces in $N'$, the action of $h$ is semi-conjugate to such a 2-ended subshift, where we use new letters. On the periodic pieces, only an orbit of minimal period is needed to contribute to ${\mathfrak{C}}'_{{\mathcal{F}}}$. This orbit is encoded, up to a shift, by a periodic, bi-infinite sequence composed of new letters, and the action of $h$ on that orbit is conjugate to the shift. Thus the closed sublamination ${\mathcal{X}}^{\bullet}_{h}$ of ${\mathcal{X}}_{h}$, given by the saturations of $X^{*}_{h}$, of each pseudo-Anosov piece in $N'$, and of a single orbit from each periodic piece in $N'$, carries all the asymptotic cycles needed to span ${\mathfrak{C}}'_{{\mathcal{F}}}$ and corresponds to an $h$-invariant subset $X_{h}^{\bullet}{\subset}X_{h}$ on which $h$ is semi-conjugate to a 2-ended subshift of finite type. Write this subshift as $ \sigma_{A}:{\mathcal{S}}_{A}\to{\mathcal{S}}_{A}$, where ${\mathcal{S}}_{A}$ is a collection of bi-infinite sequences $\iota=(i_{k})_{k\in{\mathbb{Z}}}$ of finitely many letters, $A$ is a matrix of 0’s and 1’s encoding which letter can follow which, and $\sigma_{A}$ shifts each letter one position to the right. The finitely many distinct letters $i$ each label a rectangle $R_{i}{\subset}L$ of a Markov partition. (As in [@cc:hm Section 9], we allow degenerate rectangles. Those corresponding to a periodic piece in $N'$ degenerate to a singleton.) These rectangles cover $X^{\bullet}_{h}$ and do not overlap. An element $\iota=(i_{k})_{k\in{\mathbb{Z}}}\in{\mathcal{S}}_{A}$ represents a unique point $x_{ \iota}\in R_{i_{0}}\cap X^{\bullet}_{h}$ such that $h^{k}(x_{ \iota})\in R_{i_{k}}$, $\forall k\in{\mathbb{Z}}$. In terms of the lamination ${\mathcal{X}}^{\bullet}_{h}$, this means that the leaf issuing from $x_{ \iota}$ meets $L$ successively in $R_{i_{0}},R_{i_{1}},\dots,R_{i_{k}},\dots$ in forward time, with a corresponding statement for backward time. Points in $X^{*}_{h}$ have a unique representative sequence $\iota\in{\mathcal{S}}_{A}$ [@cc:hm Section 9], but some points in the pseudo-Anosov pieces of $N'$ may have finitely many such representatives. The problem is that distinct Markov rectangles in $N'$ may meet along parts of their boundaries. Thus the map $\iota\mapsto x_{ \iota}$ is finite to one, defining a semi-conjugacy of $ \sigma_{A}$ to $h|X^{\bullet}_{h}$.
The periodic elements of ${\mathcal{S}}_{A}$ are those carried to themselves by some power $ \sigma_{A}^{q}$, $q\ge1$. These correspond to closed leaves in ${\mathcal{X}}^{\bullet}_{h}$. The substring $( i_{0}, i_{1},\dots, i_{q-1})$ of a periodic sequence $ \iota$, $ \sigma_{A}^{q}( \iota)={ \iota}$, where $q\ge1$ is minimal, will be called the period of $ \iota$. The substring $(i_{0},i_{1},\dots,i_{q-1},i_{0})$ will be called a periodic string. If no proper substring of a period is a periodic string, we say that the period is *minimal*. Since there are only finitely many distinct entries occurring in the sequences $\iota\in{\mathcal{S}}_{A}$, it is evident that there are only finitely many minimal periods. Those closed leaves $\gamma $ of ${\mathcal{X}}^{\bullet}_{h}$ that correspond to minimal periods in the symbolic system will be called minimal loops in ${\mathcal{X}}^{\bullet}_{h}$ and denoted by $ \gamma_{1}, \gamma_{2},\dots, \gamma_{r}$. Our goal is to prove the following.
\[hull\] The Handel-Miller cone ${\mathfrak{C}}'_{{\mathcal{F}}}$ is the convex hull of the union of rays $${\mathbb{R}}^{+}[ \gamma_{1}]\cup{\mathbb{R}}^{+}[ \gamma_{2}]\cup\cdots\cup{\mathbb{R}}^{+}[ \gamma_{r}].$$ Thus, the dual cone ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$ is polyhedral, defined by the linear inequalities $[ \gamma_{i}]\ge0$, $1\le i\le r$, hence has only finitely many faces.
Let $\Phi_{t}$ denote the flow on ${\widehat}W$ that stabilizes ${{\partial}_{\tau}}{\widehat}W$ pointwise, has flow lines in $W$ coinciding with the leaves of ${\mathcal{L}}_{h}$ and is parametrized so as to preserve ${\mathcal{F}}|W$ and so that $\Phi_{1}|L=h$.
Let $\iota =(i_{k})_{k=-\infty}^{\infty}\in {\mathcal{S}}_{A}$ and suppose that $i_{q}=i_{0}$ for some $q>0$. Let $x\in R_{\iota }=\bigcap_{j=-\infty}^{\infty}R_{i_{j}}$. Then there is a corresponding singular cycle $\Gamma _{q}$ formed from the orbit segment $\gamma _{q}=\{{\Phi}_{t}(x)\}_{0\le t\le q}$ and an arc $\tau {\subset}R_{i_{0}}$ from ${\Phi}_{q}(x)=h^{q}(x) \text{ to }
x$. Also, since $i_{q}=i_{0}$, there is a periodic element $\iota
'\in
\Sigma _{A} $ with period $i_{0},\dots,i_{q-1}$ and a corresponding closed leaf $\Gamma _{\iota '}=\Gamma '$ of ${\mathcal{X}}^{\bullet}_{h}$.
\[periodic\] The singular cycle $\Gamma _{q}$ and closed leaf $\,\Gamma '$, obtained as above, are homologous in ${\widehat}W$. In particular, the homology class of $\Gamma _{q}$ depends only on the periodic element $\iota '$.
The loop $\Gamma '$ is the orbit segment $\{{\Phi}_{t}(y)\}_{0\le
t\le q}$, for a periodic point $$y\in R_{i_{0}}\cap
h^{-1}(R_{i_{1}})\cap\dots\cap h^{-q}(R_{i_{q}})=R'.$$ Remark that $x\in R'$ also. Let $\tau '$ be an arc in the rectangle $R'$ from $x$ to $y$ and set $\tau ''=h^{q}(\tau ')$, an arc in $h^{q}(R')$ from $h^{q}(x)$ to $y$. Since $i_{q}=i_{0}$, $h^{q}(R'){\subset}R_{i_{0}}$ and the cycle $\tau +\tau '-\tau ''$ in the rectangle $R_{i_{0}}$ is homologous to 0. That is, we can replace the cycle $\Gamma _{q}=
\gamma _{q}+ \tau $ by the homologous cycle $\gamma _{q}-\tau '+\tau
''$. Finally, a homology between this cycle and $\Gamma '$ is given by the map $$H:[0,1]\times [0,q]{\rightarrow}{\widehat}W,$$ defined by parametrizing $\tau '$ on $[0,1]$ and setting $$H(s,t)={\Phi}_{t}(\tau '(s)).$$
\[minloops\] Every closed leaf $\Gamma $ of ${\mathcal{X}}^{\bullet}_{h}$ is homologous in ${\widehat}W$ to a linear combination of the minimal loops in ${\mathcal{X}}^{\bullet}_{h}$ with non–negative integer coefficients.
The closed leaf $\Gamma$ corresponds to a period $(i_{0},\dots,i_{q-1})$. If this period is minimal, we are done. Otherwise, after a cyclic permutation, we can assume that the period is of the form $(i_{0},i_{1},\dots,i_{p}=i_{0},i_{p+1},\dots,i_{q-1})$. We then see that $\Gamma$ is homologous to the sum of two loops, one being the arc $\gamma$ of $\Gamma$ corresponding to the periodic string $(i_{0},\dots,i_{p}=i_{0})$ followed by an arc $\tau$ from the endpoint of $\gamma$ to its initial point, and one being $-\tau+\gamma'$, where $\gamma'$ is the subarc of $\Gamma$ corresponding to the periodic string $(i_{0},i_{p+1},\dots,i_{q-1},i_{q}=i_{0})$. By Lemma \[periodic\], both $\gamma+\tau$ and $-\tau+\gamma'$ are homologous to closed orbits corresponding to periods strictly shorter than $(i_{0},\dots,i_{q-1})$. Thus, finite iteration of this procedure proves the corollary.
Let $\iota=(i_{k})_{k=-\infty}^{\infty},\iota'=(i'_{k})_{k=-\infty}^{\infty}\in{\mathcal{S}}_{A}$ and suppose that $$(i_{0},i_{1},\dots,i_{q})=(i'_{0},i'_{1},\dots,i'_{q}),$$ not necessarily a period. Let $x=x_{\iota}$ and $x'=x_{\iota'}$. Both of these points are in $$R' = R_{i_{0}}\cap
h^{-1}(R_{i_{1}})\cap\dots\cap h^{-q}(R_{i_{q}}).$$ Choose a path $\tau$ in $R_{i_{0}}$ from $x'$ to $x$ and a path $\tau'{\subset}R_{i_{q}}$ from $h^{q}(x')$ to $h^{q}(x)$. Consider the orbit segments $\Gamma=\{\Phi_{t}(x)\}_{t=0}^{q}$ and $\Gamma'=\{\Phi_{t}(x')\}_{t=0}^{q}$. Let $K$ be an upper bound of the diameters of $R_{i}$, $1\le i\le n$. Then the paths $\tau$ and $\tau'$ can always be chosen to have length less than $K$.
The following is proven analogously to Lemma \[periodic\].
\[homologouschains\] The singular chains $\Gamma'$ and $\tau+\Gamma-\tau'$ are homologous. In particular, for each closed $1$-form $\eta$ on ${\widehat}W$, $$\int_{\Gamma'}\eta=\int_{\tau+\Gamma-\tau'}\eta.$$ Here, the paths $\tau$ and $\tau'$ have length less than $K$.
\[combsofminloops\] Every homology direction can be arbitrarily well approximated by nonnegative linear combinations of the minimal loops.
Let $\Gamma=\{\Phi(t)(x)\}_{t=-\infty}^{\infty}$ be an orbit and suppose that $x$ corresponds to the symbol $\iota=\{i_{r}\}_{r=-\infty}^{\infty}$. By a suitable shift, we can assume that $i_{0}$ occurs infinitely often in forward time in this symbol. Consequently, for each index $i$ in $\iota$, there is a positive integer $k_{i}$ such the the $(i,i_{0})$-entry in $A^{k_{i}}$ is strictly positive. Let $k$ be the largest of the $k_{i}$. Thus, given a substring $(i_{0},i_{1},\dots,i_{q})$ of $\iota$, there is a periodic element $\iota'\in\Sigma_{A}$ with period $(i_{0},i_{1},\dots,i_{q},i_{q+1},\dots,i_{q+s})$, where $s\le k$. Let $\Gamma'_{q}$ denote the corresponding periodic orbit. If we parametrize the flow $\Phi_{t}$ by the invariant measure for ${\mathcal{F}}$ of period 1, then the length of the segment $\Gamma_{q}$ of $\Gamma$ corresponding to the string $(i_{0},i_{1},\dots,i_{q})$ is $q$. Choosing a suitable sequence $q\uparrow\infty$, we obtain the general homology direction $$\mu=\lim_{q\to\infty}\frac{1}{q}\int_{\Gamma_{q}}.$$ Passing to a subsequence, we also obtain a cycle $$\mu'=\lim_{q\to\infty}\frac{1}{q}\int_{\Gamma'_{q}}.$$ Since $s$ is bounded independently of $q$, Lemma \[homologouschains\] implies that $\mu$ and $\mu'$ agree on all closed 1-forms, and so $\mu'$ is a cycle homologous to $\mu$ (both in $({\mathcal{D}}'_{*},{\partial})$ and $({\mathcal{E}}'_{*},{\partial})$). Corollary \[minloops\] then implies the assertion.
By Lemma \[span\], Theorem \[hull\] follows
Maximality of the Handel-Miller cones
-------------------------------------
Our next goal is to show that, if $g$ is an endperiodic map in the isotopy class of $h$, then ${\mathfrak{C}}_{g}^{\kappa}{\subseteq}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$. In fact, we will show that if $g$ is a monodromy map for *any* fibration ${\mathcal{G}}$ of $W$ that appropriately extends ${\mathcal{F}}|(M{\smallsetminus}W)$, then either $(\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{g})\cap{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}={\emptyset}$, or ${{\mathfrak{C}}^{\kappa}}_{g}{\subseteq}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$.
We begin with an analysis of the group $G=H^{1}_{\kappa}({\widehat}{W})\cap H^{1}({\widehat}{W};{\mathbb{Z}})$. This will play the role of the integer lattice in ${H^{1}_{\kappa}}({\widehat}{W})$. The “rational rays” in this vector space will be those rays issuing from the origin that meet $G$ in nonzero points. It will be crucial that the union of the rational rays be dense in ${H^{1}_{\kappa}}({\widehat}{W})$.
Recall from Subsection \[tops\] that the topology on ${H^{1}_{\kappa}}({\widehat}{W})={\underrightarrow}{\lim}{H^{1}_{\kappa}}(K_{i}^{{\circ}})$ can be taken to be the weak topology.
By ${\partial}K_{i}$ we will mean the relative boundary of $K_{i}$ in ${\widehat}W$. It consists of finitely many disjoint rectangles and annuli. No component can be a Möbius strip since it will be fibered by *oriented* intervals that are arcs of ${\mathcal{L}}_{h}$.
\[vanonbd’\] The subspace ${H^{1}_{\kappa}}(K_{i}^{{\circ}}){\subset}{H^{1}}(K_{i})$ consists of those classes which restrict to $0$ in ${H^{1}}({\partial}K_{i})$.
If $[\omega]\in{H^{1}}(K_{i})$ restricts to $0$ on ${\partial}K_{i}$, then $\omega|{\partial}K_{i}$ is exact, hence is also exact in a normal neighborhood $N$ of ${\partial}K_{i}$. That is, $\omega|N=d\lambda$ for a smooth function $\lambda$ which can be damped off to $0$ and extended by $0$ to a smooth function $\lambda$ on all of $K_{i}$. Then $\omega-d\lambda$ has compact support in $K_{i}^{{\circ}}$. The converse is immediate.
Set $G_{i}={H^{1}_{\kappa}}(K_{i}^{{\circ}})\cap{H^{1}}(K_{i};{\mathbb{Z}})$, a finitely generated, free abelian group.
\[fullattice\] The subgroup $G_{i}{\subset}{H^{1}_{\kappa}}(K_{i}^{{\circ}})$ is a full lattice subgroup of i.e., spans this finite dimensional vector space.
The core loops of the annular components of ${\partial}K_{i}$ represent elements (not necessarily linearly independent) of the finitely generated, free abelian group $\Gamma_{i}=H_{1}(K_{i};{\mathbb{Z}})/\operatorname{torsion}$. We call them “peripheral” elements. Let $\Gamma'_{i}{\subseteq}\Gamma_{i}$ be the smallest direct summand in this lattice that contains the peripheral elements. In particular, by Lemma \[vanonbd’\], the classes in ${H^{1}}(K_{i})$ which vanish on $\Gamma'_{i}$ are exactly the elements of ${H^{1}_{\kappa}}(K_{i}^{{\circ}})$. If $[\sigma_{1}],[\sigma_{2}],\dots,[\sigma_{r}]$ are a basis of $\Gamma'_{i}$, then extend to an ordered basis $([\sigma_{1}],\dots,[\sigma_{r}],[\sigma_{r+1}],\dots,[\sigma_{q}])$ of $\Gamma_{i}$, hence of $H_{1}(K_{i})$, Let $([\omega_{1}],\dots,[\omega_{r})],[\omega_{r+1}],\dots,[\omega_{q}])$ be the dual basis of ${H^{1}}(K_{i})$. Then, by Lemma \[vanonbd’\], the classes $[\omega_{j}]$ lie in $G_{i}$, $r+1\le j\le q$, and form a basis of ${H^{1}_{\kappa}}(K_{i}^{{\circ}})$.
It is standard that the rays in the finite dimensional vector space ${H^{1}_{\kappa}}(K_{i}^{{\circ}})$ which meet the full lattice $G_{i}$ at nonzero points (the rational rays) unite to form a dense subset of that vector space. Furthermore, note that $G_{0}{\subseteq}G_{1}{\subseteq}\cdots{\subseteq}G_{i}{\subseteq}\cdots$ has increasing union $G{\subset}{H^{1}_{\kappa}}({\widehat}{W})$. An appeal to the definition of the weak topology makes the following clear.
\[ratdense\] The union of the rational rays is dense in ${H^{1}_{\kappa}}({\widehat}{W})$.
\[samecone\] The proper foliated ray $\left<{\mathcal{G}}\right>$ lies in $\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$ if and only if ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{G}}}={{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$.
Suppose that ${\left<}{\mathcal{G}}{\right>}{\subset}\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$. By Theorem \[folforms\], we take the foliation ${\mathcal{G}}$ to be transverse to ${\mathcal{L}}_{h}$. By the “transfer theorem” [@cc:hm Theorem 11.1], ${\mathcal{L}}_{h}$ induces Handel-Miller monodromy $g$ on each leaf of ${\mathcal{G}}|W$. A comment is needed since, in [@cc:hm], it was not required that Handel-Miller monodromy be exactly Nielsen-Thurston on $N'$. But, as Fried [@fried] proves the transfer theorem for the Nielsen-Thurston pseudo-Anosov pieces, the discussion in Subsection 11.4 of [@cc:hm] is easily augmented to accomodate that requirement. Thus, the cones ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{G}}}$ and ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$ are determined by the same core lamination ${\mathcal{X}}_{g}={\mathcal{X}}_{h}$ and so are identical. For the converse, ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{G}}}={{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$, for proper foliated rays ${\left<}{\mathcal{G}}{\right>}$ and ${\left<}{\mathcal{F}}{\right>}$, clearly implies that ${\left<}{\mathcal{G}}{\right>}{\subset}\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$.
\[notinbd\] No proper foliated ray is contained in ${\partial}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$.
If there is a proper foliated ray ${\left<}{\mathcal{G}}{\right>}{\subset}{\partial}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$, then $\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{G}}}\cap\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}\ne{\emptyset}$. By Corollary \[ratdense\], there is a proper foliated ray ${\left<}{\mathcal{H}}{\right>}{\subset}\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{G}}}\cap\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$. By Proposition \[samecone\], we see that ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{G}}}={{\mathfrak{C}}^{\kappa}}_{{\mathcal{H}}}={{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$. That is, ${\left<}{\mathcal{G}}{\right>}{\subset}\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$, contrary to our hypothesis.
The boundary ${\partial}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$ is made up of $r$ top faces $F_{1},\dots,F_{r}$, where $F_{i}$ is a convex, polyhedral cone with nonempty (relative) interior in the hyperplane $[ \gamma_{i}]=0$.
\[denseinFi\] Each $F_{i}$ contains a dense family of rays that meet points of the integer lattice ${H^{1}_{\kappa}}({\widehat}W;{\mathbb{Z}})$.
Since $[ \gamma_{i}]$ is an integral homology class, it takes integer values on the integer lattice. The corresponding claim is standard in the finite dimensional spaces ${H^{1}_{\kappa}}(K_{j}^{{\circ}})$ and remains true in the direct limit by the definition of the weak topology.
\[max\] If $g$ is a monodromy map endperiodic for a fibration ${\mathcal{G}}$ of $W$ that appropriately extends ${\mathcal{F}}|(M{\smallsetminus}W)$, then either $(\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{g})\cap{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}={\emptyset}$, or ${{\mathfrak{C}}^{\kappa}}_{g}{\subseteq}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$. In particular, ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}={{\mathfrak{C}}^{\kappa}}_{h}$ is the maximal foliation cone for monodromies in the isotopy class of $h$.
If $(\operatorname{int}{{\mathfrak{C}}^{\kappa}}_{g})\cap{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}\ne{\emptyset}$ and ${{\mathfrak{C}}^{\kappa}}_{g}\not{\subseteq}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$, then Lemma \[denseinFi\] implies that there is a proper foliated ray in ${\partial}{{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}}$, contradicting Corollary \[notinbd\].
Correspondingly, the dual homology cone ${\mathfrak{C}}'_{{\mathcal{F}}}={\mathfrak{C}}'_{h}$ is the minimal ${\mathfrak{C}}'_{g}$ for all monodromies $g$ isotopic to $h$. In this sense, we can say that the Handel-Miller monodromy has the “tightest” dynamics in its isotopy class.
Foliated products
-----------------
We have been assuming that $W$ is not a foliated product. If ${\widehat}{W}=F{\times}I$, the cohomological classification of the foliations is quite trivial. Given any closed, compactly supported 1-form $\eta$ on ${\widehat}W$, one modifies it to a foliated form, transverse to the interval fibers, as in Section \[fforms\]. In this case, one can apply the construction to the 1-form $0$, producing an exact foliated form $dg$. The class $0\in{H^{1}_{\kappa}}({\widehat}{W})$ represents the product foliation. As remarked in the introduction, we regard ${H^{1}_{\kappa}}({\widehat}{W})$ itself as a foliation cone and $\{0\}$ as a degenerate proper foliated ray.
Uniqueness of the foliations up to isotopy
------------------------------------------
Let ${\mathcal{F}}$ and ${\mathcal{H}}$ be foliations of ${\widehat}W$ which fiber $W$ and appropriately extend ${\mathcal{F}}|(M{\smallsetminus}W)$ and suppose that ${\left<}{\mathcal{F}}{\right>}={\left<}{\mathcal{H}}{\right>}$. These two foliations are trivial outside $K_{i}$, for large enough $i$. Both ${\mathcal{F}}$ and ${\mathcal{H}}$ restrict to depth one foliations of the sutured manifold $K_{i}$ that are defined by cohomologous foliated forms there. By [@cc:isotopy], these foliations are isotopic in $K_{i}$ and it is easy to make this isotopy global in ${\widehat}W$. (In the case of foliated products, it should be remarked that the fact that the foliations classified by $0$ are all isotopic to the product foliation is, in fact, quite deep. It is true in each $K_{j+1}{\smallsetminus}K^{{\circ}}_{j}$, $i\le j<\infty$ by the Laudenbach-Blank theorem [@LB] or, equivalently, by a theorem of J. Cerf [@cf:Gamma4]. It is then propagated to all of ${\widehat}W$ as $i\to\infty$.) The isotopy is smooth in $W$, but not at ${{\partial}_{\tau}}W$.
Finiteness of the set of Handel-Miller cones
--------------------------------------------
The nucleus $K_{0}$ of the octopus decomposition of ${\widehat}W$ cuts off finitely many arms which are of the form $B{\times}I$, where $B{\subset}{{\partial}_{\tau}}{\widehat}W$ is connected and noncompact. All of our foliations ${\mathcal{G}}$ that appropriately extend ${\mathcal{F}}|(M{\smallsetminus}W)$, when restricted to an arm, are transverse to the $I$ fibers. Thus, for all the appropriate foliations ${\mathcal{G}}$ with Handel-Miller monodromy $g$, the core lamination ${\mathcal{X}}_{g}$ lies in $K_{0}$. The cone ${\mathfrak{C}}_{{\mathcal{G}}}^{\kappa}$ is defined by inequalities $[\gamma_{i}]\ge0$, $1\le i\le r$, where each $\gamma_{i}$ is a periodic orbit in ${\mathcal{X}}_{g}$. These same inequalities define the cone ${\mathfrak{C}}_{{\mathcal{G}}|K_{0}}{\subset}H^{1}(K_{0})$ determined by the depth one foliation ${\mathcal{G}}|K_{0}$ of the sutured manifold $K_{0}$. This sets up a one-to-one correspondence between the set ${\mathcal{K}}_{W}$ of foliation cones in ${H^{1}_{\kappa}}({\widehat}{W})$ and a subset of the set of foliation cones in $H^{1}(K_{0})$. By [@cc:cone Theorem 6.4], it follows that there are only finitely many of these cones.
The proof of Theorem \[cone\] for foliations of class ${C^{\infty,\iota}}$ and the rational rays is complete.
Foliations of class ${C^{0+}}$
==============================
The isomorphism ${H^{1}}({\widehat}W)={\underleftarrow}{\lim}\,{H^{1}}(K_{i})$ is induced by the natural homomorphisms $\psi_{i}:{H^{1}}({\widehat}{W})\to{H^{1}}(K_{i})$. Recall from Subsection \[tops\] that the topology on ${H^{1}}({\widehat}{W})$ is the standard inverse limit topology, relativized from the Tychonov topology via the inclusion $${\underleftarrow}{\lim}{H^{1}}(K_{i}){\subset}{H^{1}}(K_{0}){\times}{H^{1}}(K_{1}){\times}\cdots{\times}{H^{1}}(K_{i}){\times}\cdots.$$ Since $\psi_{i}$ is induced by projection of the product onto its $i$th factor, it is continuous.
Let $V_{i}{\subset}{H^{1}}(K_{i})$ denote the image of $\psi_{i}$ and remark that ${H^{1}}({\widehat}{W})={\underleftarrow}{\lim}V_{i}$. Also note that, since ${H^{1}}({\widehat}{W})={H^{1}}({\widehat}{W};{\mathbb{Z}})\otimes{\mathbb{R}}$, the integer lattice ${H^{1}}({\widehat}{W};{\mathbb{Z}})$ is carried onto a full lattice subgroup of $V_{i}$, $i\ge0$. Indeed, it is carried onto $V_{i}\cap{H^{1}}(K_{i};{\mathbb{Z}})$ and this must be a full lattice subgroup since $\psi_{i}$ surjects onto $V_{i}$. Recall that the rays issuing from the origin in ${H^{1}}({\widehat}{W})$ that meet nonzero integer lattice points are called rational rays.
\[uniondense\] The union of rational rays in ${H^{1}}({\widehat}{W})$ is everywhere dense.
The open sets in ${H^{1}}({\widehat}{W})={\underleftarrow}{\lim}V_{i}$ are unions of sets of the form $$U={\underleftarrow}{\lim}V_{i}\cap (U_{0}{\times}U_{1}{\times}\cdots U_{n}{\times}V_{n+1}{\times}V_{n+2}{\times}\cdots),$$ where $U_{i}{\subseteq}V_{i}$ is open, $0\le i\le n$. If this set is nonempty, we must prove that it meets a rational ray. Let $\theta_{i}:V_{n}\to V_{i}$ be the natural surjection, $0\le i<n$. Then the set $$Y=U_{n}\cap\theta_{n-1}^{-1}(U_{n-1})\cap\cdots\cap\theta_{0}^{-1}(U_{0})$$ is open and the assumption that $U\ne{\emptyset}$ implies that $Y\ne{\emptyset}$. Since $V_{n}$ is finite dimensional, the rational rays do have dense union there, hence we select such a ray $\rho$ that meets $Y$. There is a rational ray $\rho'$ in ${H^{1}}({\widehat}{W})$ which is mapped onto $\rho$ by $\psi_{n}$. Viewing $\rho'$ in ${\underleftarrow}{\lim}V_{i}$, one sees that it meets $U$.
If ${\mathcal{F}}$ is only of class ${C^{0+}}$, Handel-Miller does not apply directly to the foliations of ${\widehat}W$ that are fibrations extending ${\mathcal{F}}|(M{\smallsetminus}W)$. Indeed, such a foliation may be nontrivial outside every $K_{i}$, hence the monodromy will not be endperiodic. It will be necessary to work in the progressively expanding $K_{i}$’s, in each of which the endperiodic theory works fine, in order to produce the foliation cones in ${H^{1}}({\widehat}W)={\underleftarrow}{\lim}\,{H^{1}}(K_{i})$. The reader will note that the lack of smoothness is only for the foliations in $M$, the foliation in each ${\widehat}{W}$ being itself of class ${C^{\infty,\iota}}$.
Let $ \zeta\in H_{1}({\widehat}{W})$. This class is represented by a singular cycle which necessarily lives in $K_{k}$, for large enough $k\ge0$. Then $\zeta:{H^{1}}(K_{k})\to{\mathbb{R}}$ is a continuous linear functional. We can define $\zeta:{H^{1}}({\widehat}{W})\to{\mathbb{R}}$ by the composition $${H^{1}}({\widehat}{W}){\xrightarrow}{\psi_{k}}{H^{1}}(K_{k}){\xrightarrow}{\zeta}{\mathbb{R}},$$ remarking that this is independent of the choice of large enough $k$ and defines a continuous linear functional.
We consider ${\mathcal{F}}|{\widehat}W$. While this foliation may never trivialize outside any $K_{i}$, ${\mathcal{F}}|K_{i}$ is a depth one foliation, hence is smooth and our previous discussion can be applied to this foliation. Note that, since we are working in the compact manifold $K_{i}$ rather than $K_{i}^{{\circ}}$, all differential forms are compactly supported. We can assume that the arms are the connected components of ${\widehat}{W}{\smallsetminus}K_{0}^{{\circ}}$, hence that ${\mathcal{F}}$ is transverse to the interval fibers in these arms. We can choose the monodromy $h$ so that, in the arms, it is given by the flow along these interval fibers and is Handel-Miller for ${\mathcal{F}}|K_{0}$, hence also for ${\mathcal{F}}|K_{i}$, $i\ge0$. (One should note that a leaf of ${\mathcal{F}}$ may intersect $K_{i}$ in a finite family of leaves of ${\mathcal{F}}|K_{i}$, but this really doesn’t matter.) Thus, the core lamination ${\mathcal{X}}_{h}$ of ${\mathcal{L}}_{h}|K_{i}$ lives in $K_{0}$ and is independent of $i$. Exactly as before, this lamination gives a set $[ \gamma_{1}],[ \gamma_{2}],\dots,[ \gamma_{r}]\in H_{1}({\widehat}{W})$, hence the linear inequalities $[ \gamma_{i}]\ge0$ define a polyhedral cone in ${H^{1}}({\widehat}{W})$, the foliation cone we have designated by ${\mathfrak{C}}_{h}$ and now designate by ${\mathfrak{C}}_{{\mathcal{F}}}$.
The foliation cone ${\mathfrak{C}}_{{\mathcal{F}}}$ is polyhedral with finitely many faces.
We have seen that $\operatorname{int}{\mathfrak{C}}_{{\mathcal{F}}}$ consists of classes of foliated forms. Any such form in the integer lattice restricts to an element of the integer lattice in each ${H^{1}}(K_{i})$, hence has as period group a subgroup of ${\mathbb{Z}}$. Thus the rational rays in $\operatorname{int}{\mathfrak{C}}_{{\mathcal{F}}}$ correspond to fibrations of $W$ that appropriately extend ${\mathcal{F}}|(M{\smallsetminus}W)$.
\[samecone’\] A proper foliated ray ${\left<}{\mathcal{G}}{\right>}$ lies in $\operatorname{int}{\mathfrak{C}}_{{\mathcal{F}}}$ if and only if ${\mathfrak{C}}_{{\mathcal{G}}}={\mathfrak{C}}_{{\mathcal{F}}}$.
Suppose that ${\left<}{\mathcal{G}}{\right>}{\subset}\operatorname{int}{\mathfrak{C}}_{{\mathcal{F}}}$. By Theorem \[folforms\], we choose ${\mathcal{G}}$ transverse to ${\mathcal{L}}_{h}$. The proof of Proposition \[samecone\] carries through for ${\left<}{\mathcal{G}}|K_{i}{\right>}$ and ${{\mathfrak{C}}^{\kappa}}_{{\mathcal{F}}|K_{i}}$, $0\le i<\infty$, showing that ${\mathcal{L}}_{h}$ induces Handel-Miller monodromy for ${\mathcal{G}}|K_{i}$ also, $0\le i<\infty$. In particular, the same family $\gamma_{1},\dots,\gamma_{r}$ defines the Handel-Miller cone for both ${\mathcal{F}}|K_{i}$ and ${\mathcal{G}}|K_{i}$, $0\le i<\infty$, hence ${\mathfrak{C}}_{{\mathcal{F}}}={\mathfrak{C}}_{{\mathcal{G}}}$. Again, the converse is trivial.
\[notinbd’\] No proper foliated ray is contained in ${\partial}{\mathfrak{C}}_{{\mathcal{F}}}$.
Indeed, this follows from Proposition \[samecone’\] exactly as Corollary \[notinbd\] follows from Proposition \[samecone\], using the denseness of the rational rays (Lemma \[uniondense\]).
The foliation cone ${\mathfrak{C}}_{{\mathcal{F}}}$ is maximal.
As before, this will follow From Corollary \[notinbd’\] if the union of rays through the integer lattice that lie in the face $F_{i}$ are dense in $F_{i}$, $1\le i\le r$. This is proven analogously to Corollary \[uniondense\]. Simply add the constraint $[ \gamma_{i}]=0$.
\[ratuniq\] The rational rays in $\operatorname{int}{\mathfrak{C}}_{{\mathcal{F}}}$ each determine the corresponding foliation uniquely up to a ${C^{0}}$ isotopy in ${\widehat}W$ which is smooth in $W$.
As before, this follows from the main result of [@cc:isotopy]. That theorem must now be applied not only in $K_{0}$, but in each component of $K_{i}{\smallsetminus}K^{{\circ}}_{i-1}$, $i\ge1$. For this, let ${\mathcal{F}}$ and ${\mathcal{F}}'$ correspond to the same proper foliated ray and choose ${\mathcal{F}}$ transverse to ${\mathcal{L}}_{h}$. Note that each component of ${\partial}K_{i}$, the interface between $K_{i}$ and ${\widehat}{W}{\smallsetminus}K^{{\circ}}_{i}$, is either an ${\mathcal{L}}_{h}$-saturated rectangle or an ${\mathcal{L}}_{h}$-saturated annulus, hence transverse to ${\mathcal{F}}$. An application of the version of the Roussarie-Thurston theorem stated in [@cc:isotopy Theorem 4.6] gives an ambient isotopy of ${\partial}K_{i}$ in $K_{i}$, fixing the boundary of each component pointwise, to a position transverse to ${\mathcal{F}}'$. The image of this isotopy may be assumed, via a small perturbation, to be disjoint from ${\partial}K_{i-1}$. Reversing the isotopy deforms ${\mathcal{F}}'$ to a foliation transverse to ${{\partial}_{\pitchfork}}K_{i-1}$, $i\ge1$. Now the main result of [@cc:isotopy] can be applied in $K_{0}$ and in each component of $K_{i}{\smallsetminus}K^{{\circ}}_{i-1}$, $i\ge1$.
The finiteness of the cones follows as before from the finiteness of the cones for $K_{0}$. The case in which $W$ is a foliated product again gives that ${H^{1}}({\widehat}{W})$ is itself the unique foliation cone. The proof of Theorem \[cone\] for the rational rays is complete.
Recall that the substructure of ${\mathcal{F}}$ is the compact lamination $S=M{\smallsetminus}{\mathcal{O}}$. Refoliating in the product components of ${\mathcal{O}}$ is well understood, as noted above. In the finitely many non-product components, we have classified cohomologically the possible depth $k$ refoliations. Thus the classification of finite depth foliations sharing the same substructure has been reduced to an essentially finite procedure in both the ${C^{\infty,\iota}}$ and ${C^{0+}}$ categories.
Simple disk decompositions and foliation cones
==============================================
An original goal of this work was to quantify the depth one foliations ${\mathcal{F}}$ of sutured manifolds $M$ constructed by Gabai’s process of disk decomposition [@ga0]. In many cases, the foliation cones can be read off of the disks in a *simple* disk decomposition.
If the disks $D_{1},\dots,D_{n}$ of the decomposition all live in $M$ at the beginning, the disk decomposition of $M$ is said to be *simple*
In other words, the decompositions can be done simultaneously rather than sequentially. Obviously, for this to be true, $M$ must be a sutured handlebody and $\{[D_{1}],\dots,[D_{n}]\}{\subset}H_{2}(M,{\partial}M)$ is a basis. If $\{[\alpha_{1}],\dots,[\alpha_{n}]\}{\subset}H^{1}(M)$ is the Poincaré dual basis, $\left<[\alpha_{1}],\dots,[\alpha_{n}]\right>$ will denote the convex polyhedral cone spanned by these vectors.
Let ${\mathcal{F}}$ be a depth one foliation constructed via the simple disk decomposition as above. Then every ray in $\operatorname{int}\left<[\alpha_{1}],\dots,[\alpha_{n}]\right>$ is a foliated ray and ${\left<}{\mathcal{F}}{\right>}$ is one of these.
By Gabai’s construction, each disk $D_{i}$ is transverse to ${\mathcal{F}}$ except at a single positive (multi-pronged) saddle tangency. By the fact that the decomposition is simple, one sees that there are positive closed transversals $\sigma_{i}$ to ${\mathcal{F}}$ meeting $D_{i}$ precisely at the saddle tangency with $\int_{\sigma_{i}}\alpha_{j}=\delta_{ij}$, $1\le i,j\le n$. Here, $\alpha_{j}$ is the usual Poincaré dual form to $D_{j}$, supported in a small normal neighborhood of $D_{j}$, $1\le j\le n$. Since the intersection number of $ \sigma_{i}$ with a depth one leaf of ${\mathcal{F}}$ is strictly positive, it follows that the proper foliated ray ${\left<}{\mathcal{F}}{\right>}$ lies in $\operatorname{int}{\left<}[\alpha_{1}],\dots,[\alpha_{n}]{\right>}$. We wish to show that every ray in the interior of this cone through the integer lattice is a proper foliated ray.
Let $\omega$ be a foliated form defining ${\mathcal{F}}$. Then, since the saddle tangencies are positive, one constructs a smooth, nonvanishing vector field $v$ on $M$ such that $$(t\omega+(1-t)\alpha_{i})(v)>0$$ on $M$, $0<t\le1$, $1\le i\le n$. This can be done so that the 1-dimensional foliation ${\mathcal{L}}$ tangent to $v$ is transverse both to ${\mathcal{F}}$ and $D_{i}$ and, even though the disks meet the annular components of ${{\partial}_{\pitchfork}}M$, one can arrange that ${\mathcal{L}}$ fibers these components over $S^{1}$. If ${\mathcal{X}}$ is the core lamination of ${\mathcal{L}}$, it follows that $t[\omega]+(1-t)[\alpha_{i}]$ takes strictly positive values on all nontrivial asymptotic cycles of ${\mathcal{X}}$, $0<t\le1$, $1\le i\le n$. If $f$ is the monodromy induced by ${\mathcal{L}}$ on a leaf of ${\mathcal{F}}$, we see that $t[\omega]+(1-t)[\alpha_{i}]\in\operatorname{int}{\mathfrak{C}}_{f}$, $0<t\le1$, $1\le i\le n$. But these classes fill up the interior of $\left<[\alpha_{1}],\dots,[\alpha_{n}]\right>$, proving that this is a subcone of ${\mathfrak{C}}_{f}$.
In many examples, this proposition makes it possible to completely determine the foliation cones (cf. [@cc:cone Section 7] and [@cc:norm Section 5]). This involves relating the disks in the simple decomposition to the Markov process induced by the Handel-Miller monodromy.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Resolution studies of test problems set baselines and help define minimum resolution requirements, however, resolution studies must also be performed on scientific simulations to determine the effect of resolution on the specific scientific results. We perform a resolution study on the formation of a protostar by modelling the collapse of gas through 14 orders of magnitude in density. This is done using compressible radiative non-ideal magnetohydrodynamics. Our suite consists of an ideal magnetohydrodynamics (MHD) model and two non-ideal MHD models, and we test three resolutions for each model. The resulting structure of the ideal MHD model is approximately independent of resolution, although higher magnetic field strengths are realised in higher resolution models. The non-ideal MHD models are more dependent on resolution, specifically the magnetic field strength and structure. Stronger magnetic fields are realised in higher resolution models, and the evolution of detailed structures such as magnetic walls are only resolved in our highest resolution simulation. In several of the non-ideal MHD models, there is an off-set between the location of the maximum magnetic field strength and the maximum density, which is often obscured or lost at lower resolutions. Thus, understanding the effects of resolution on numerical star formation is imperative for understanding the formation of a star.'
author:
-
bibliography:
- 'SPHERICXIV\_Wurster\_Bate.bib'
title: |
Resolving numerical star formation:\
A cautionary tale
---
Introduction
============
When modelling a phenomenon, such as the formation of a protostar, there are many aspects that must be considered. First, one must consider the relevant and important physical processes that actually describe the phenomenon; these are often selected based upon observational and empirical evidence. Next, one must chose the numerical algorithms and the resolution; the numerical algorithms include how to model both the physical processes required for realism and the artificial algorithms required for numerical stability. Even within the smoothed particle hydrodynamics (SPH) framework, there are multiple numerical formalisms for each algorithm, and the final results will differ based upon this choice. For example, we previously showed the effect that different artificial resistivity algorithms had on the formation of a protostellar disc [@Wurster+2017]. Thus, the choice of numerical algorithms is equally as important and the choice of which physical processes to include.
Choosing a resolution requires a trade-off between accuracy and speed. There are always minimum criteria that must be resolved, such as the Jeans mass [@BateBurkert1997] for molecular clouds, or the Toomre-mass and scale-height [@Nelson2006] for protostellar discs. However, even if these minimum criteria are initially met, information may still be lost in simulations where certain regions require even higher minimum resolution criteria, or in regions where these criteria become violated due to the adaptive nature of compressible SPH. The preferable resolution is the convergence limit – i.e. the minimum resolution simulation to which all higher resolution simulations yield the same results. However, this convergence limit can be difficult or impossible to determine due to the computational cost of running numerous simulations at increasing resolution. To date, there are many processes in astrophysics where numerical convergence has not been reached due to computational limitations, including fragmentation of protostellar discs [( [@MeruBate2011criteria; @MeruBate2011converge; @MeruBate2012; @Meyer+2018])]{}, turbulence [( [@Price2012turb; @TriccoPriceFederrath2016; @BoothClarke2019])]{}, and also star formation [( [@BateTriccoPrice2014; @WursterPriceBate2016; @WursterBatePrice2018ff])]{}.
Investigating resolution in simplified tests, such as MHD shock tubes [( [@BrioWu1988; @RyuJones1995])]{} or the Orszag-Tang vortex [@OrszagTang1979] yield useful information and baselines, but it is not until resolution is investigated in practical models that its effect is fully understood. This is due to the complexity of the practical models where many different physical processes operate simultaneously, unlike in the simplified tests. Moreover, the resolution effects, and hence the convergence limit, will likely be different for different phenomena.
As a practical investigation of the effects of resolution, we will investigate the formation of a protostar. To self-consistently model this, we will model the gravitational collapse of gas through 14 orders of magnitude in density; this will be done in the presence of weakly ionised gas [@MestelSpitzer1956] and strong magnetic fields [@HeilesCrutcher2005], which are observed in star forming regions. We include radiation hydrodynamics [@BateKeto2015], magnetohydrodynamics (MHD) [@Price2012], and the non-ideal MHD processes [( [@WardleNg1999; @BraidingWardle2012accretion; @BraidingWardle2012sf])]{}, specifically Ohmic resistivity, ambipolar diffusion and the Hall effect. Ohmic resistivity and ambipolar diffusion are dissipative terms that weaken the magnetic field, and the Hall effect is a dispersive term that modifies the geometry of the magnetic field. In non-ideal MHD, the gas is composed of charged gas that is coupled to the magnetic field and neutral gas that is independent of the magnetic field; the neutral gas can slip through the magnetic field, but is influenced by it via collisions with the charged gas. Non-ideal MHD represents physical resistivity, but we also include artificial resistivity [@Phantom2018] in our models for numerical stability.
We will begin in by describing our methods, followed by our initial conditions in . We will investigate the effect of resolution in , and we will briefly discuss and summarise our results in .
Numerical Methods {#sec:num}
=================
Radiation non-ideal magnetohydrodynamics
----------------------------------------
The equations of self-gravitating, compressible, radiation non-ideal magnetohydrodynamics are $$\begin{aligned}
\frac{{\rm d}\rho}{{\rm d}t} & = & -\rho \nabla\cdot \bm{v}, \label{eq:cty} \\
\frac{{\rm d} \bm{v}}{\rm{d} t} & = & -\frac{1}{\rho}\nabla \cdot \left[\left(p+\frac{B^2}{2}\right)\mathbb{I} - \bm{B}\bm{B}\right] \notag \\
&-& \nabla\Phi + \frac{\kappa \mbox{\boldmath$F$}}{c} + \left.\frac{{\rm d} \bm{v}}{\rm{d} t} \right|_\text{art}, \label{eq:mom} \\
\frac{\rm d}{{\rm d}t} \left(\frac{\bm{B} }{\rho} \right) & = & \left( \frac{\bm{B}}{\rho} \cdot \nabla \right) \bm{v} \notag \\
&+& \left.\frac{\rm d}{{\rm d}t} \left(\frac{\bm{B} }{\rho} \right) \right|_\text{non-ideal} + \left.\frac{\rm d}{{\rm d}t} \left(\frac{\bm{B} }{\rho} \right) \right|_\text{art} \label{eq:ind}, \\
\rho \frac{\rm d}{{\rm d}t}\left( \frac{E}{\rho}\right) & = & -\nabla\cdot \bm{F} - \mbox{$\nabla \bm{v}${\bf :P}} + 4\pi \kappa \rho B_\text{P} - c \kappa \rho E, \label{eq:radiation} \\
\frac{{\rm d}u}{{\rm d}t} & = & -\frac{p}{\rho} \nabla\cdot\bm{v} - 4\pi \kappa B_\text{P} + c \kappa E\notag \\
& + & \left.\frac{{\rm d} u}{\rm{d} t} \right|_\text{non-ideal} + \left.\frac{{\rm d} u}{\rm{d} t} \right|_\text{art}, \label{eq:matter} \\
\nabla^{2}\Phi & = & 4\pi G\rho, \label{eq:grav}\end{aligned}$$ where ${\rm d}/{{\rm d}t} \equiv \partial/\partial t + \bm{v}\cdot \nabla$ is the Lagrangian derivative, $\rho$ is the density, ${\bm v}$ is the velocity, $p$ is the gas pressure, ${\bm B}$ is the magnetic field, $\Phi$ is the gravitational potential, $B_\text{P}$ is the frequency-integrated Plank function, $E$ is the radiation energy density, $\mbox{\boldmath $F$}$ is the radiative flux, [**P**]{} is the radiation pressure tensor, $c$ is the speed of light, $G$ is the gravitational constant, and $\mathbb{I}$ is the identity matrix. The terms with subscript ‘non-ideal’ are the contribution from the non-ideal MHD processes, and the terms with subscript ‘art’ are artificial terms required for numerical stability. We assume units for the magnetic field such that the Alfv[é]{}n speed is $v_{\rm A} = \vert B\vert/\sqrt{\rho}$ [@PriceMonaghan2004b].
To evolve these equations, we use the three-dimensional smoothed particle hydrodynamics code <span style="font-variant:small-caps;">sphNG</span> that originated from W. Benz [@Benz1990]; over the past few decades, it has been substantially modified to include individual particle time-steps [@BateBonnellPrice1995], variable smoothing lengths [( [@PriceMonaghan2004b; @PriceMonaghan2007])]{}, radiation [@BateKeto2015], magnetohydrodynamics [@Price2012] and the non-ideal MHD processes [( [@WursterPriceAyliffe2014; @WursterPriceBate2016; @Wurster2016])]{}. We use the cubic spline smoothing kernel, thus each particle has approximately 58 neighbours.
Artificial algorithms
---------------------
Artificial algorithms are required for numerical stability, and include artificial viscosity, resistivity and conductivity that are applied to the momentum , induction and internal energy equations, respectively. These terms are used to smooth discontinuities such that simulations remain stable. However, applying too much artificial viscosity/resistivity/conductivity will smooth the domain too much such that the results are no longer physical; applying too little may not prevent instabilities from occurring, thus preventing reliable results. For this paper, we will focus on the magnetic field and on artificial resistivity.
The discretised form of artificial resistivity we use ([@PriceMonaghan2004], [@PriceMonaghan2005], [@Phantom2018]) is $$\begin{aligned}
\left.\frac{\text{d} }{\text{d} t}\left(\frac{B^i_a}{\rho_a}\right)\right|_\text{art} = \frac{1}{\Omega_{a}\rho_{a}^{2}} \sum_{b} m_{b} v_{\text{sig},ab}B^i_{ab} \hat{r}^j_{ab} \nabla^j_aW_{ab}(h_{a}), \label{eq:artificialB}\end{aligned}$$ where we sum over all particles $b$ within the kernel radius, $W_{ab}$ is the smoothing kernel, $m_b$ is the particle mass, $\Omega_a$ is a dimensionless correction term to account for a spatially variable smoothing length $h_a$, $B^i_{ab} \equiv B^i_a - B^i_b$, and the signal velocity is $v_{\text{sig},ab} = |\bm{v}_{ab} \times \hat{\bm{r}}_{ab} |$. This artificial resistivity is second-order accurate away from shocks i.e. $\propto h^2$, thus, will decrease as $h$ decreases (i.e. as resolution increases). For an investigation in to artificial resistivity prescriptions, see [@Wurster+2017].
The remaining terms in the induction equation are also dependent on resolution (i.e. on $h$), but to a lesser degree; the ideal component of the induction equation (i.e. the first term of Equation \[eq:ind\]) is given by $$\begin{aligned}
\left.\frac{\text{d} }{\text{d} t}\left(\frac{B^i_a}{\rho_a}\right)\right|_\text{ideal} = -\frac{1}{\Omega_a \rho_a^2} \sum_b m_b v^i_{ab} B^j_a \nabla^j_a W_{ab}\left(h_a\right).\end{aligned}$$ Higher resolution models can calculate higher values of the magnetic field (if physically motivated) since the smoothing will occur over a smaller region and due to the decreased contribution from the artificial resistivity. Thus, increased resolution will contribute to a more precise value.
Initial conditions {#sec:ics}
==================
Our initial conditions are similar to our previous studies [( [@BateTriccoPrice2014; @WursterBatePrice2018sd; @WursterBatePrice2018hd; @WursterBatePrice2018ff])]{}. A sphere of radius $R=4\times~10^{16}$ cm and mass $M=1$ is placed in a low-density box of edge length $l = 4R$ and a density contrast of 30:1. The cloud has an initial rotational velocity of $\Omega = 1.77\times 10^{-13}$ rad s$^{-1}$ and an initial sound speed of $c_\text{s,0} = 2.19\times 10^4$ cm s$^{-1}$. The entire domain is threaded with a uniform magnetic field strength of $B_0 = 163\mu$G which is parallel to the rotation axis. The low-density box is used to provide boundary conditions at the edge of the cloud so that the entire cloud can be self-consistently modelled; periodic boundary conditions are used for the magnetohydrodynamic processes at the edge of the box, which is sufficiently far from the sphere such that the boundary conditions will have no effect on the results.
The Hall effect, one of the non-ideal MHD processes, is dependent on the direction of the magnetic field [@WardleNg1999], thus we present three models: an ideal MHD model (where the non-ideal terms are set to zero; named [*iB*$_{\text{-}z}^{}$]{}), a non-ideal model where the magnetic field vector is aligned with the rotation vector (named [*nB*$_{\text{+}z}^{}$]{}) and a non-ideal model where the vectors are anti-aligned (named [*nB*$^{}_{\text{-}z}$]{}); collectively, the non-ideal models are [*nB*$_{\pm z}^{}$]{}. For each model, we present resolutions of [$N=3\times10^5$]{}, [$10^6$]{} and [$3\times10^6$]{} particles in the sphere. Model names without superscripts refer to all three models of the given initial conditions; names with a superscript refer to a particular model with the resolution stated in the superscript.
Results {#sec:results}
=======
From previous star formation studies [( [@Tsukamoto+2015hall; @Tsukamoto+2015oa; @Tsukamoto+2017; @WursterBatePrice2018sd; @WursterBatePrice2018hd; @WursterBatePrice2018ff])]{}, the expected outcome of each model is known. Model [*iB*$_{\text{-}z}^{}$]{} is performed in strong, ideal magnetic fields and will succumb to the magnetic braking catastrophe [@AllenLiShu2003], thus the collapse will be axi-symmetric and no protostellar disc will form. When including non-ideal MHD, the Hall effect will extract angular momentum from near the centre if the magnetic field is aligned with the rotation axis as in [*nB*$_{\text{+}z}^{}$]{}, yielding an approximately axi-symmetric collapse and forming a small disc at late times; the Hall effect will contribute to the angular momentum near the centre if the magnetic field is anti-aligned with the rotation axis as in [*nB*$^{}_{\text{-}z}$]{}, forming a large disc early during the collapse. Although these are the expected outcomes, it has yet to be investigated how they will be affected by resolution.
Our analysis will be split into three components. First, we will discuss the general evolution, which will follow the gravitational collapse of the gas from the initial density of through the first hydrostatic core phase (also first core phase; [@Larson1969]) to the formation of the protostar at (). We will then investigate the gas and magnetic field structure near the end of the first core phase (; ) and just prior to the birth of the protostar ().
Global evolution {#sec:res:global}
----------------
As a molecular cloud gravitationally collapses, the central density increases, which is used to determine the phase of evolution. Figure \[fig:rvt\] shows the maximum density as a function of time for each model.
![Evolution of the maximum density during the gravitational collapse of a molecular cloud to form a protostar. The lower and upper horizontal lines represent the beginning and end of the first hydrostatic core phase, respectively. We define the birth of the protostar to occur at . At each resolution, there is a similar off-set in time between models reaching , and this off-set is decreasing with increasing resolution. This suggests that the [$3\times10^6$]{} models yield a prediction of the end of the first core phase that is only slightly early than the ‘true’ time.[]{data-label="fig:rvt"}](Figures/fig_RhoVsTime.eps){width="0.8\columnwidth"}
As expected from previous studies, all models at a given resolution initially collapse at similar rates, reaching the first core phase at at similar times. The evolution then diverges, with [*iB*$_{\text{-}z}^{}$]{} collapsing faster to the end of the first core phase at than [*nB*$_{\text{+}z}^{}$]{} which collapses faster than [*nB*$^{}_{\text{-}z}$]{}. At each resolution, [*iB*$_{\text{-}z}^{}$]{} reaches 30 yr before [*nB*$_{\text{+}z}^{}$]{} and 400 yr before [*nB*$^{}_{\text{-}z}$]{}. Thus, the relative difference in collapse times between models is independent of resolution.
For each model, the first core phase ends 290 yr in the [$3\times10^5$]{} version prior to the [$10^6$]{} version, which ends 60 yr prior to the [$3\times10^6$]{} version. Although these models have not converged on an end time, the trend suggests that the ‘true’ end-time is only slightly later than that calculated in the [$3\times10^6$]{} models, thus end time of the [$3\times10^6$]{} models yield a reasonable prediction.
Given the varying collapse times, it is convenient to use maximum density as a proxy for time. For the duration of this paper, we will compare the simulations at similar maximum densities, rather than at similar times.
As gas collapses in ideal MHD, the magnetic field is dragged into the centre with the gas, thus the strongest field strength is coincident with the maximum density [@BateTriccoPrice2014]. Once non-ideal MHD processes are included, the collapse of the charged gas is slowed by the magnetic fields, while the neutral gas continues to collapse to the centre of the system. The charged gas is stalled in a torus around the central density peak, and the magnetic fields pile up here, creating a so-called ‘magnetic wall’ [( [@LiMckee1996; @TassisMouschovias2005b])]{}.
Figure \[fig:bvr\] shows the maximum ([$B_\text{max}$]{}) and central ([$B_\text{cen}$]{}; coincident with the maximum density ) magnetic field strengths for each model; where only one line exists for a given model, then $B_\text{max} = B_\text{cen}$.
![Evolution of the maximum and central magnetic field strengths as a function of maximum density, which is a proxy for time. Each model has two lines per resolution, where the upper line represents the maximum magnetic field strength $B_\text{max}$, and the lower line represents the central field strength $B_\text{cen}$ which is coincident with . Where only one line exists, $B_\text{max} = B_\text{cen}$. The vertical lines represents the beginning and end of the first hydrostatic core phase. $B_\text{max} = B_\text{cen}$ in [*iB*$_{\text{-}z}^{}$]{} (top panel) and for in the non-ideal models (bottom two panels). The field strength is approximately independent of resolution for [*iB*$_{\text{-}z}^{}$]{}, but strongly resolution-dependent for [*nB*$_{\text{+}z}^{3e6}$]{} and [*nB*$^{}_{\text{-}z}$]{} after .[]{data-label="fig:bvr"}](Figures/fig_BVsRho.eps){width="0.8\columnwidth"}
In ideal MHD, [*iB*$_{\text{-}z}^{}$]{}, $B_\text{max} = B_\text{cen}$ as expected. However, $B_\text{max}$ is approximately independent of resolution, and by , $B_\text{max}$ differs by only a factor of 3 between [*iB*$_{\text{-}z}^{3e5}$]{} and [*iB*$_{\text{-}z}^{3e6}$]{}, and 1.4 between [*iB*$_{\text{-}z}^{1e6}$]{} and [*iB*$_{\text{-}z}^{3e6}$]{}, suggesting that the value of [$B_\text{max}$]{} is converging. Since [$B_\text{max}$]{} coincides with , then much of this decrease can be attributed to the artificial resistivity algorithms. Despite a factor of 3 difference in magnetic field strengths, this difference is small in astrophysical terms. Thus, we conclude, for ideal MHD, the maximum magnetic field strength up to the formation of a protostar is relatively insensitive to resolution.
When using non-ideal MHD, we expect a magnetic wall to form during the first core phase, thus [$B_\text{max}$]{} and [$B_\text{cen}$]{} should diverge; this occurs at in [*nB*$_{\pm z}^{}$]{} (bottom two panels of Figure \[fig:bvr\]). From this figure, it appears that a wall is formed and sustained, forms and dissipates, and never forms in the high, mid and low resolution [*nB*$_{\text{+}z}^{}$]{} models, respectively; thus, by the formation of the protostar at , $B_\text{max} \ne B_\text{cen}$ in [*nB*$_{\text{+}z}^{3e6}$]{} only. [$B_\text{cen}$]{} is approximately the same in the high and mid resolution model, which is approximately 2 times higher than the low resolution model. Although $B_\text{cen}$ is similar at the birth of the protostar, different conclusions are reached regarding [$B_\text{max}$]{} and the magnetic wall based upon the initial resolution.
At each resolution in [*nB*$^{}_{\text{-}z}$]{} (bottom panel of Figure \[fig:bvr\]), there is a clear distinction between $B_\text{max}$ and $B_\text{cen}$ for . In this model, $B_\text{max}$ ($B_\text{cen}$) differs by a factor of 5000 (50) amongst the three resolutions. As the resolution decreases, the ratio $B_\text{max}/B_\text{cen}$ also decreases. Thus, this model clearly has a strong dependence on resolution; the current results suggest that we cannot predict to what values [$B_\text{max}$]{} and [$B_\text{cen}$]{} will converge.
These non-ideal MHD plots suggest that resolution is very important in reaching the correct conclusions, and that the convergence limit has not yet been reached. This resolution-dependence is somewhat counter-intuitive since the non-ideal models include additional resolution-insensitive resistivity processes that should make the results less susceptible to resolution effects. However, we need to be cautious with the above analysis since it involves only one or two values at each time or density. In each of the next two sections, we will focus on the gas structure at a single time, which will complement the above time-sequence analysis.
First hydrostatic core {#sec:res:FHC}
----------------------
The first hydrostatic core phase is $10^{-12} \lesssim \rho_\text{max}/($$) \lesssim 10^{-8}$ [@Larson1969]. From Figure \[fig:bvr\], the models appear similar the beginning of this phase, thus we will analyse the density and magnetic field at the end of this phase once [$B_\text{max}$]{} and [$B_\text{cen}$]{} become distinct. Figure \[fig:FHC\] shows the gas density and magnetic field strength through the mid-plane near the end of the first core phase at .
![Gas density (top panel) and magnetic field strength (bottom panel) at the end of the first hydrostatic core phase at . The frames sizes are selected to show the structure of the magnetic wall (middle row, bottom panel) and of the bar (bottom row, both panels). The density profiles have minimal resolution dependence. The non-ideal magnetic field geometries are strongly dependent on resolution, in particular the strength and structure of the magnetic wall (model [*nB*$_{\text{+}z}^{}$]{}) and the location and structure of the maximum magnetic field strength (model [*nB*$^{}_{\text{-}z}$]{}).[]{data-label="fig:FHC"}](Figures/fig8sml_Rhoxy.eps "fig:"){width="\columnwidth"} ![Gas density (top panel) and magnetic field strength (bottom panel) at the end of the first hydrostatic core phase at . The frames sizes are selected to show the structure of the magnetic wall (middle row, bottom panel) and of the bar (bottom row, both panels). The density profiles have minimal resolution dependence. The non-ideal magnetic field geometries are strongly dependent on resolution, in particular the strength and structure of the magnetic wall (model [*nB*$_{\text{+}z}^{}$]{}) and the location and structure of the maximum magnetic field strength (model [*nB*$^{}_{\text{-}z}$]{}).[]{data-label="fig:FHC"}](Figures/fig8sml_Bxy.eps "fig:"){width="\columnwidth"}
For the inner $r\lesssim 10$ au, the density profiles of [*iB*$_{\text{-}z}^{}$]{} and [*nB*$_{\text{+}z}^{}$]{} are approximately axi-symmetric and approximately independent of resolution. The additional physical processes in [*nB*$^{}_{\text{-}z}$]{} have permitted a gravitationally unstable disc to form at ; the disc has a radius of $r\approx25$ au, thus only the central bar is shown in Figure \[fig:FHC\]. The bar structure is similar at each resolution, although the current rotation angle is different simply due to the different lengths of time it has taken to reach the current density (recall Section \[sec:res:global\]); note that the rotational velocities are similar at each resolution. Once the rotation angle is taken into account, then near the end of the first core phase, the density structure of the inner $r\lesssim 10$ au is approximately independent of resolution.
For all three models for $r \gtrsim10$ au, the density profiles become more broad for increasing resolution; specifically, the higher resolution models can capture more detail further from the core. In [*iB*$_{\text{-}z}^{}$]{} and [*nB*$_{\text{+}z}^{}$]{}, this likely has only moderate consequence to any analyses since studies such as this typically focus on the first core itself (i.e. $r\lesssim 10$ au); however, studies such as [*nB*$^{}_{\text{-}z}$]{} typically focus on the disc thus require good resolution out to several tens of au, thus resolution at larger radii must also be carefully considered.
It is clear from Figure \[fig:FHC\] that both the magnetic strength and structure are affected by the both the physical processes and resolution. The magnetic field structure of [*iB*$_{\text{-}z}^{}$]{} is least dependent on resolution. However, the non-ideal MHD models [*nB*$_{\pm z}^{}$]{} are strongly dependent on resolution, despite their additional resistive processes being physically motivated. In each of the [*nB*$_{\text{+}z}^{}$]{} models (middle row), there is a torus of gas with a strong magnetic field that comprises of the magnetic wall, whose maximum extent is $r \sim4$ au. In [*nB*$_{\text{+}z}^{3e5}$]{}, the wall has a slightly higher magnetic field strength than the surrounding gas, but is still lower than [$B_\text{cen}$]{} = [$B_\text{max}$]{} (recall Figure \[fig:bvr\]). Increasing the resolution to [$10^6$]{}, the magnetic wall becomes well-defined. This is a small wall of width d$r\sim1.5$ au, but its maximum magnetic field strength is only slightly higher than then central strength. From this snapshot, the processes that are causing the magnetic wall are just defined at this resolution. However, given the weakness of the wall, their full effect cannot be fully understood.
The magnetic wall is very well-defined in our highest resolution model, [*nB*$_{\text{+}z}^{3e6}$]{}. As in [*nB*$_{\text{+}z}^{1e6}$]{}, the primary wall has width d$r\sim1.5$ au and the outermost extent is $r\sim4$ au. In the high resolution wall, the magnetic field strength is 5 times stronger than [$B_\text{cen}$]{}. More importantly, spiral structures are visible interior to $r \lesssim 4$ au, and these spirals have field strength 10 times higher than [$B_\text{cen}$]{}. These spirals are caused by the rotating, highly magnetised gas on the interior of the torus losing angular momentum, detaching from the torus and migrating towards the centre. This spiral structure is absent in the density plot (top panel) since the majority of the gas is neutral and acting mostly independently of the magnetic field. Thus, these spirals are created by the small fraction of charged gas that drags the magnetic field into the centre, which cannot be distinguished in the density plots.
Thus, for [*nB*$_{\text{+}z}^{}$]{}, there is no discernible magnetic wall for [*nB*$_{\text{+}z}^{3e5}$]{}, a defined wall for [*nB*$_{\text{+}z}^{1e6}$]{}, and a very well-defined wall for [*nB*$_{\text{+}z}^{3e6}$]{} where the gas can be seen detaching from the wall and spiralling towards the centre. Although [*nB*$_{\text{+}z}^{1e6}$]{} yields the wall, these results suggest that the higher resolution of [$3\times10^6$]{} is required to resolve the behaviour of the charged gas in and leaving the torus. Since we did not perform a higher resolution model, we cannot confidently say that even [$3\times10^6$]{} is high enough to properly resolve the formation and evolution of the magnetic wall.
The gravitationally unstable disc that forms during the first core phase when in [*nB*$^{}_{\text{-}z}$]{} complicates the evolution of the magnetic field; the formation of this disc coincides with the divergence of [$B_\text{max}$]{} and [$B_\text{cen}$]{}. Contrary to what was discussed in Section \[sec:res:global\] and predicted by Figure \[fig:bvr\], there is no magnetic wall. In this model, the magnetic field tries to anchor the gas, thus a bar of charged gas slightly lags behind the primary, mostly neutral bar; this is unseen in the density plot of Figure \[fig:FHC\] since the density of charged gas is much less than that of neutral gas. Thus, the motion of the charged and neutral gas caused by the bars prevent the formation of a magnetic wall; rather, the most magnetised gas initially piles up near the edge of the bar. Thus, the divergence of [$B_\text{max}$]{} and [$B_\text{cen}$]{} in this model is due to the rotation of the bar rather than a magnetic wall.
In [*nB*$^{3e5}_{\text{-}z}$]{}, the bar has nearly a uniform magnetic field strength, but in [*nB*$^{1e6}_{\text{-}z}$]{} and [*nB*$^{3e6}_{\text{-}z}$]{}, a strongly magnetised clump of gas has collapsed to the west side of the bar. Although symmetric results would have been expected given the initial conditions, even slight asymmetries caused by numerical processes or rounding could cause the discrepancy that has lead to the asymmetric bar. Independent of the cause, the asymmetry has formed in similar locations in both models, with the structure being better well-defined and having a stronger magnetic field strength in [*nB*$^{3e6}_{\text{-}z}$]{}.
In very general terms, the three [*nB*$^{}_{\text{-}z}$]{} models are qualitatively similar, yielding a 25 au disc during the first core phase. The bar structure is similar amongst them, however, the asymmetry only appears at the higher two resolutions, suggesting a minimum resolution of [$10^6$]{}.
Stellar birth {#sec:res:birth}
-------------
Between the end of the first core phase at and the birth of the protostar at , only a few years pass (recall Figure \[fig:rvt\]). However, there is noticeable evolution during this time, specifically in the centre of the cloud and in the non-ideal models. Figure \[fig:sb\] shows the gas density and magnetic field strength for each model at .
![Gas density (top panel) and magnetic field strength (bottom panel) at the birth of the protostar at . In [*nB*$_{\text{+}z}^{}$]{} (middle row, bottom panel), the magnetic wall persists, although its magnetic field strength is weaker than the central field strength. The bar has begun to collapse in [*nB*$^{}_{\text{-}z}$]{} (bottom row, both panels), beginning to form a spherical core with a strong magnetic field strength.[]{data-label="fig:sb"}](Figures/fig4sml_Rhoxy.eps "fig:"){width="\columnwidth"} ![Gas density (top panel) and magnetic field strength (bottom panel) at the birth of the protostar at . In [*nB*$_{\text{+}z}^{}$]{} (middle row, bottom panel), the magnetic wall persists, although its magnetic field strength is weaker than the central field strength. The bar has begun to collapse in [*nB*$^{}_{\text{-}z}$]{} (bottom row, both panels), beginning to form a spherical core with a strong magnetic field strength.[]{data-label="fig:sb"}](Figures/fig4sml_Bxy.eps "fig:"){width="\columnwidth"}
Similar to the end of the first core phase, the density profiles of [*iB*$_{\text{-}z}^{}$]{} and [*nB*$_{\text{+}z}^{}$]{} are approximately independent of resolution. During this time, the bar in [*nB*$^{}_{\text{-}z}$]{} collapses and this collapse is resolution-dependent: it stays approximately symmetric for [*nB*$^{3e5}_{\text{-}z}$]{}, collapses asymmetrically to near the centre in [*nB*$^{1e6}_{\text{-}z}$]{}, and collapses to the west end of the bar in [*nB*$^{3e6}_{\text{-}z}$]{}. Thus, for the first time in this study, we see noticeable resolution effects on the structure of the density.
The magnetic field structure of [*iB*$_{\text{-}z}^{}$]{} is similar at all resolutions, but the field strength decreases with resolution, especially in the inner 0.01 au. This decrease is due to a combination of the lower resolution smoothing out the field strength, and the higher artificial resistivity dissipating more of the magnetic field.
Model [*nB*$_{\text{+}z}^{}$]{} has retained its magnetic wall (and spiral structure interior to it for [*nB*$_{\text{+}z}^{3e6}$]{}). The magnetic wall remains well defined in the two higher resolution cases, and in all cases, its field strength is stronger than that in the surrounding gas. Thus, independent of resolution, we can conclude that the magnetic wall is a persistent feature. In [*nB*$_{\text{+}z}^{3e5}$]{} and [*nB*$_{\text{+}z}^{1e6}$]{}, the [$B_\text{max}$]{} = [$B_\text{cen}$]{}, again with a lower strength in the former model. In [*nB*$_{\text{+}z}^{3e6}$]{}, the maximum field strength is 3 times stronger than the central strength, but the maximum strength is only slightly mis-coincident with the maximum density. Thus, the expected location of [$B_\text{max}$]{} at this time cannot be determined; although it is coincident with the maximum density in two of the models, it is separate from the maximum density in the highest resolution model. Thus even higher resolution is required to reach a robust conclusion on the expected location of [$B_\text{max}$]{}.
As with the density evolution, the magnetic field structure of [*nB*$^{}_{\text{-}z}$]{} is highly dependent on resolution. None of these models had a well-defined magnetic wall, nor has one formed after the end of the first core phase. In all cases, the maximum magnetic field strength resides 0.01-0.1 au from the maximum density, and the off-set decreases with resolution. Thus, although the gas is collapsing along the bar, the magnetic field is not necessarily tracing this collapse. Although the off-set decreases with increasing resolution, there is likely a minimum, non-zero off-set which will be reached at the convergence limit since there is an initial off-set during the first core phase and non-zero off-sets are possible given the ideal MHD model, [*iB*$_{\text{-}z}^{}$]{}. The values of [$B_\text{max}$]{} and [$B_\text{cen}$]{} increase with increasing resolution, as does the factor between them, indicating that artificial resistivity is less important with increasing resolution. Again, we cautiously conclude that [$B_\text{max}$]{} $\ne$ [$B_\text{cen}$]{} is a real effect, but we cannot comment on the values that these terms should take, nor the expected location of [$B_\text{max}$]{}.
Discussion and Summary {#sec:conc}
======================
Computational expense
---------------------
We have repeatedly stated that higher resolution simulations are required to reach the convergence limit and robust conclusions. However, higher resolution simulations necessarily take more resources, which will eventually become a prohibiting factor. Figure \[fig:cpu\] shows the runtime for each model at each resolution.
![The cumulative CPU time used for each model. The vertical lines are at the beginning and end of the first hydrostatic core phase. Independent of the non-ideal processes, all models at a given resolution take a similar length of computational time to reach , after which [*nB*$^{}_{\text{-}z}$]{} and [*nB*$_{\text{+}z}^{}$]{} require considerable more computational resources to evolve through the first core phase.[]{data-label="fig:cpu"}](Figures/fig_cputime_evol.eps){width="\columnwidth"}
For [*iB*$_{\text{-}z}^{}$]{}, the cumulative CPU time increases smoothly, and [*iB*$_{\text{-}z}^{3e6}$]{} takes only 7000 CPU-hours to reach the formation of the protostar. Thus, if we were to increase the resolution by a factor of 10 to $N=3\times10^7$, we can extrapolate the runtime to be $\sim5\times10^5$ CPU-hours. However, the non-ideal models have greater physical motivation, and Figure \[fig:cpu\] shows that evolving the first core phase is computationally demanding in [*nB*$_{\pm z}^{}$]{}. Even at [$N=3\times10^6$]{} it takes [*nB*$_{\text{+}z}^{3e6}$]{} and [*nB*$^{3e6}_{\text{-}z}$]{} $4.9\times 10^4$ and $2.4\times 10^5$ CPU-hours to reach , respectively. Increasing resolution by a factor of 10 would require nearly $10^7$ CPU-hours for [*nB*$^{3e7}_{\text{-}z}$]{}. Thus, convergence studies on star formation will be computationally expensive, thus comparisons such as those presented here are required to determine the acceptable trade-off between accuracy and computational expense.
Although already expensive, there is great interest in modelling the early evolution of the protostar itself [( [@BateTriccoPrice2014; @WursterBatePrice2018hd; @WursterBatePrice2018ff])]{}, which requires reaching densities of at least . However, the computational time to model the evolution between $10^{-2} \lesssim \rho_\text{max}/($$) \lesssim 10^{-1}$ increases considerable for all models [@WursterBatePrice2018ff], making the search for the convergence limit in this scientifically interesting regime even more prohibitive.
Counter-intuitively, there is less resolution convergence in the non-ideal models than in the ideal model. The non-ideal processes introduce a restrictive time-stepping criteria $\propto~h^2$ [( [@Maclow+1995; @ChoiKimWiita2009; @WursterPriceAyliffe2014])]{} which is always satisfied by construction and accounts for the steep increase in runtime during the first core phase; for reference, the Courant time-step is longer and is $\propto~h$. These processes also introduce a new characteristic length scale [( [@WursterPriceAyliffe2014; @MarchandCommerconChabrier2018])]{}, which we have determined is resolved in the magnetic wall and the bar. Given the lack of convergence, it is possible that these new length scales are not restrictive enough to properly resolve the non-ideal MHD processes, and that new characteristic scales need to be determined. However, this will likely require finding the convergence limit, which may be prohibitively expensive.
Summary
-------
We ran three star formation models each at three different resolutions. We performed one ideal MHD simulation ([*iB*$_{\text{-}z}^{}$]{}), and two non-ideal MHD simulations ([*nB*$_{\text{+}z}^{}$]{} and [*nB*$^{}_{\text{-}z}$]{}) using two different initial alignments of the magnetic field to test the extreme effects of the non-ideal process of the Hall effect. In all models, the higher resolution simulations collapsed later than lower resolution simulations, but at a given resolution, the relative collapse time between models was consistent.
The general structure of the ideal MHD models was similar at each resolution, but with decreasing maximum magnetic field strength for decreasing resolution. Both the lower resolution and the greater contribution from artificial resistivity contributed to this decrease.
The magnetic field structure of the non-ideal MHD models was resolution-dependent. At the end of the first hydrostatic core phase, [*nB*$_{\text{+}z}^{}$]{} formed a magnetic wall which persisted to the formation of the protostar (although only very weakly in [*nB*$_{\text{+}z}^{3e5}$]{}). The magnetic field strength of the wall and the ability to model the gas being accreted from the inner edge of the wall was strongly dependent on resolution. As the gas collapsed to form the protostar, the maximum magnetic field strength remained off-set from the maximum density only in the highest resolution simulation and the maximum strength was interior to the magnetic wall.
A magnetic wall did not form prior to the birth of the protostar in [*nB*$^{}_{\text{-}z}$]{}. The maximum magnetic field strength was never coincident with the maximum density after the formation of the bar, and the maximum and central magnetic field strengths and the ratio between them was dependent on resolution. The central bar in the disc collapsed after the first core phase, and the nature of the collapse was also sensitive to resolution.
Despite additional physical processes in the non-ideal models, these models were more dependent on resolution than the ideal MHD model. Thus, these new processes necessarily introduced new time and length scales that must be resolved; although these are resolved in all our non-ideal models, they may not be restrictive enough to fully resolve these processes.
The accuracy obtained from higher resolution is always a trade-off with computational expense, and some very high resolution simulations are infeasible to run. In any resolution study, the convergence limit would ideally be reached, but this limit has often proved elusive in astrophysical studies. Therefore, we must always be cautious of the results, and perform simulations as feasible or as required to examine the impact of resolution on all conclusions.
Acknowledgment {#acknowledgment .unnumbered}
==============
JW and MRB acknowledge support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007- 2013 grant agreement no. 339248). Calculations and analyses for this paper were performed on the University of Exeter Supercomputer, Isca, which is part of the University of Exeter High-Performance Computing (HPC) facility, and on the DiRAC Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1. DiRAC is part of the National e-Infrastructure. The research data supporting this publication are openly available from the University of Exeter’s institutional repository at https://doi.org/10.24378/exe.607. Several figures were made using <span style="font-variant:small-caps;">splash</span> [@Price2007].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have studied EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ by the measurements of x-ray diffraction, electrical resistivity, thermopower, magnetic susceptibility, magnetoresistance and specific heat. Partial substitution of As with P results in the shrinkage of lattice, which generates chemical pressure to the system. It is found that EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ undergoes a superconducting transition at 26 K, followed by ferromagnetic ordering of Eu$^{2+}$ moments at 20 K. This finding is the first observation of superconductivity stabilized by internal chemical pressure, and supplies a rare example showing coexistence of superconductivity and ferromagnetism in the ferro-arsenide family.'
author:
- 'Zhi Ren,$^{1}$ Qian Tao,$^{1}$ Shuai Jiang,$^{1}$ Chunmu Feng,$^{2}$ Cao Wang,$^{1}$ Jianhui Dai,$^{1}$ Guanghan Cao$^{1}$ and Zhu’an Xu'
title: 'Superconductivity induced by phosphorus doping and its coexistence with ferromagnetism in EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$'
---
Recently, high-temperature superconductivity has been discovered in a family of materials containing FeAs layers. The superconductivity is induced by doping charge carriers into a parent compound, characterized by an antiferromagnetic (AFM) spin-density-wave (SDW) transition associated with the FeAs layers.[@WNL; @Dai] Electron doping has been realized by the partial substitutions of F-for-O,[@Kamihara08] vacancy-for-O,[@OV] Th-for-Ln,[@Wang] Co/Ni-for-Fe,[@Co; @Co-122; @Ni; @Ni-122]. Examples of hole doping include the partial substitutions of Sr-for-Ln[@Wen] and K-for-Ba/Sr/Eu[@Rotter; @122-Sr; @EuK]. All the chemical doping suppresses the long-range SDW order, eventually resulting in the emergence of superconductivity. Up to now, no superconductivity has been reported through doping at the As site. Apart from carrier doping, application of hydrostatic pressure is also capable of stabilizing superconductivity.[@P-Ca122; @P-SrBa122; @P-Eu122]
Among the parent compounds in ferro-arsenide family, EuFe$_2$As$_2$ exhibits peculiar behavior because the moments of Eu$^{2+}$ ions order antiferromagnetically at relatively high temperature of 20 K.[@Eu122; @Ren; @Jeevan] Magnetoresistance measurements on EuFe$_2$As$_2$ crystals[@Jiang] suggest a strong coupling between the magnetism of Eu$^{2+}$ ions and conduction electrons, which may affect or even destroy superconductivity. For example, though Ni doping in BaFe$_{2}$As$_{2}$ leads to superconductivity up to 21 K,[@Ni-122] ferromagnetism rather than superconductivity was found in EuFe$_2$As$_2$ by a systematic Ni doping[@Ren-Ni]. Another relevant example is that the superconducting transition temperature of (Eu,K)Fe$_{2}$As$_{2}$[@EuK] is 32 K, substantially lower than those of (Ba,K)Fe$_{2}$As$_{2}$ ($T_{c}$=38 K)[@Rotter] and (Sr,K)Fe$_{2}$As$_{2}$ ($T_{c}$=37 K)[@122-Sr]. Resistivity measurement under hydrostatic pressures[@P-Eu122] on EuFe$_{2}$As$_{2}$ crystals showed a resistivity drop at 29.5 K. However, no zero resistivity could be achieved, which was ascribed to the AFM ordering of the Eu$^{2+}$ moments[@P-Eu122].
While hetero-valent substitution generally induces charge carriers, iso-valent substitution may supply chemical pressure. The latter substitution is of particular interest in EuFe$_2$As$_2$ when As is partially replaced by P, as suggested theoretically in order to search for the magnetic quantum criticality without changing the number of Fe 3$d$-electrons[@JDai]. In this Letter, we demonstrate bulk superconductivity at 26 K in EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$. For the first time, superconductivity has been realized through the doping at the As site in the iron arsenide system. Strikingly, we observe coexistence of ferromagnetic ordering of Eu$^{2+}$ moments with superconductivity below 20 K.
Polycrystalline samples of EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ were synthesized by solid state reaction with EuAs, Fe$_{2}$As and Fe$_{2}$P. EuAs was presynthesized by reacting Eu grains and As powders at 873 K for 10 h, then 1073 K for 10 h and finally 1223 K for another 10 h. Fe$_{2}$As was prepared by reacting Fe powers and As powders at 873 K for 10 h then 1173 K for 0.5 h. Fe$_{2}$P was presynthesized by heating Fe powders and P powders very slowly to 873 K and holding for 10 h. Powders of EuAs, Fe$_{2}$As and Fe$_{2}$P were weighed according to the stoichiometric ratio, ground and pressed into pellets in an argon-filled glove-box. The pellets were sealed in evacuated quartz tubes and annealed at 1273 K for 20 h then cooled slowly to room temperature. The phase purity of the samples was investigated by powder X-ray diffraction, using a D/Max-rA diffractometer with Cu-K$_{\alpha}$ radiation and a graphite monochromator. The XRD data were collected with a step-scan mode in the 2$\theta$ range from 10$^{\circ}$ to 120$^{\circ}$. The structural refinement was performed using the programme RIETAN 2000.[@Izumi]
The electrical resistivity was measured on bar-shaped samples using a standard four-probe method. The applied current density was $\sim$ 0.5 A/cm$^{2}$. The measurements of magnetoresistance, specific heat, ac magnetic susceptibility and thermopower were performed on a Quantum Design Physical Property Measurement System (PPMS-9). DC magnetic properties were measured on a Quantum Design Magnetic Property Measurement System (MPMS-5).
Figure 1 show the XRD pattern of EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ at room temperature, together with the profile of the Rietveld refinement using the space group of I4/*mmm*. No additional diffraction peak is observed. The refined lattice parameters are *a*=3.889(1)[Å]{} and $c$=11.831(3)[Å]{}. Compared with those of the undoped EuFe$_{2}$As$_{2}$[@Ren], the *a*-axis is decreased by 0.35%, the *c*-axis is shortened by 1.8% and the cell volume shrinks by 3.2% for EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$. These results suggest that the iso-valent substitution of As with P indeed generates chemical pressure to the system. In addition, the As(P) position *z* decreases, indicating that the As(P) atoms move toward the Fe planes. As a consequence, the Fe-As(P)-Fe angle increases from 110.15$^{\circ}$ to 111.48$^{\circ}$.
{width="8cm"}
Figure 2(a) shows the temperature dependence of resistivity ($\rho$) for EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ under zero field. The anomaly associated with the SDW transition in undoped EuFe$_{2}$As$_{2}$[@Ren] is completely suppressed. The resistivity is linear with temperature down to $\sim$ 90 K and shows upward deviation from the linearity at lower temperatures. The resistivity ratio of *R*(300K)/*R*(30K) is 5.2, indicating high quality of the present sample. Below 29 K, the resistivity drops steeply, suggesting a superconducting transition. The midpoint of the transition is 26 K. On closer examination shown in the inset of Fig. 1(a), however, a small resistivity peak is observed around 16 K, which coincides with the ferromagnetic ordering of Eu$^{2+}$ moments (to be shown below). This observation is reminiscent of the reentrant superconducting behavior as observed, for example, in *R*Ni$_{2}$B$_{2}$C$_{2}$ (*R*=Tm, Er, Ho)[@Eisaki].
Figure 2(b) shows the temperature dependence of thermopower ($S$). The thermopower in the whole temperature range is negative, indicating that the dominant charge carriers are electron-like. $\mid$$S$$\mid$ decreases sharply below 29 K, corresponding to the superconducting transition. However, the $\mid$$S$$\mid$ value does not drop to zero until $\sim$ 13 K, in relation with the ferromagnetic ordering of the Eu$^{2+}$ moments.
{width="8cm"}
In figure 2(c), we show the temperature dependence of field-cooling dc magnetic susceptibility for EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ under $\mu_{0}H_{dc}$=0.1 T. The obvious deviation of linearity in $\chi^{-1}$ above 230 K is probably due to the presence of trace amount of ferromagnetic Fe$_{2}$P impurity[@Cava]. The $\chi$ data between 50 K and 200 K can be well described by the modified Curie-Weiss law, $$\chi=\chi_0+\frac{C}{T-\theta},$$ where $\chi_0$ denotes the temperature-independent term, $C$ the Curie-Weiss constant and $\theta$ the Weiss temperature. The fitting yields *C*= 8.1(1) emu$\cdot$K/mol and $\theta$=22(1) K. The corresponding effective moment $P_{eff}$=8.0(1) $\mu_{B}$ per formula unit, close to the theoretical value of 7.94 $\mu_{B}$ for a free Eu$^{2+}$ ion. $\chi$ increases steeply with decreasing temperature below 20 K and becomes gradually saturated. Field dependence of magnetization gives a saturated magnetic moment of 6.9(1)$\mu_{B}$/f.u., as expected for fully paralleled Eu$^{2+}$ moments with *S*=7/2. Therefore, it is concluded that the moments of Eu$^{2+}$ order ferromagnetically in EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$, in analogy with EuFe$_{2}$P$_{2}$[@EuFe2P2]. Due to the proximity of superconducting transition and ferromagnetic ordering, the superconducting diamagnetic signal is hard to observe unless the applied magnetic field is very low. As shown in the inset of Fig. 2(c), ac magnetic susceptibility measured under $\mu_{0}H_{ac}$=1 Oe shows a kink around 26 K. After subtraction of the paramagnetic contribution from Eu$^{2+}$ moments, a clear diamagnetism is seen, confirming superconductivity in EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$.
{width="8cm"}
The temperature dependence of resistivity under magnetic fields is shown in figure 3. With increasing magnetic fields, the resistive transition shifts towards lower temperature and becomes broadened, further affirming the superconducting transition. The $T_{c}$(*H*), defined as a temperature where the resistivity falls to 50% of the normal state value, is plotted as a function of magnetic field in the inset of Fig. 3. The $\mu_{0}H_{c2}$-*T* diagram shows a slight upward curvature, which is probably due to the multi-band effect[@twoband]. The initial slope $\mu_{0}$$\partial$$H_{c2}$/$\partial$$T$ near $T_{c}$ is -1.18 T/K, giving an upper critical field of $\mu_{0}H_{c2}$(0) $\sim$ 30 T by linear extrapolation. It is also noted that the reentrant superconducting behavior is not obvious under magnetic field, in contrast with that in *R*Ni$_{2}$B$_{2}$C$_{2}$ superconductors[@Eisaki].
{width="8cm"}
Figure 4 shows the specific heat ($C$) for the EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ sample. Two anomalies below 30 K are identified. One is a $\lambda$-shape peak with the onset at 20 K, indicating a second-order transition. With increasing magnetic fields, the anomaly shifts to higher temperatures and becomes broadened, in accordance with ferromagnetic nature of the transition. The other anomaly is much weaker but detectable at $\sim$26 K (shown in the upper-left panel of Fig. 4), which coincides with the superconducting transition.
To analyze the $C(T)$ data further, it is assumed that the total specific heat consists of the electronic, phonon and magnetic components. At high temperatures, the dominant contribution comes from the phonon component, which can be well described by the Debye model with only one adjustable parameter (i. e., Debye temperature $\Theta_{D}$). The data fitting above 100 K gives $\Theta_{D}$ of 345 K. Since $\Theta_{D}$$\sim$1/$\sqrt{M}$ (*M* the molecular weight), the $\Theta_{D}$ for EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ is calculated to be 348 K, in good agreement with the fitted value, using the $\Theta_{D}$ data of 380 K for EuFe$_{2}$P$_{2}$[@EuFe2P2]. The upper bound of electronic specific heat coefficient is estimated to be 10 mJ/mol$\cdot$K$^{2}$, which is comparable with those of other iron-arsenide superconductors. The electronic specific heat contribution is less than 2% of the total specific heat even at low temperatures, and thus is not taken into consideration in the following analysis.
By subtracting the lattice contribution, the specific anomaly due to the superconducting transition is more prominent. The jump in specific heat *$\Delta$C*$\approx$300 mJ/mol$\cdot$K at $T_{c}$, which is of the same order as that in Ba$_{0.55}$K$_{0.45}$Fe$_{2}$As$_{2}$ crystals[@NN]. Meanwhile, the magnetic entropy associated with the ferromagnetic transition is 16.5 J/mol$\cdot$K, which amounts to 95% of *R*ln(2*S*+1) with *S*=7/2 for Eu$^{2+}$ ions. The thermodynamic properties indicate that the superconducting transition and the ferromagnetic ordering are both of bulk nature.
A broad specific-heat hump below 90 K is evident after deduction of the lattice contribution, indicating existence of additional magnetic contribution. This anomaly is accompanied with the upward deviation from the linear temperature dependence in resistivity as shown in Fig. 1(a). Meanwhile, negative magnetoresistance was observed in the same temperature range, and reaches -6% at 30 K under $\mu_{0}H$=8 T (data not shown here). All these features are probably attributed to the interaction between the moments of Eu$^{2+}$ ions and conduction electrons. Such interaction may be responsible for the observed ferromagnetic ordering of Eu$^{2+}$ moments below 20 K.
The isovalent substitution of As with P does not change the number of Fe 3$d$ electrons, but generates chemical pressure, as manifested by the shrinkage of lattice. According to a coherent-incoherent scenario[@JDai], the low energy physics of the FeAs-containing system is described by the interplay of the coherent excitations (associated with the itinerant carriers) and incoherent ones (modeled in terms of Fe localized magnetic moments). The internal chemical pressure generated via P doping results in the enhancement of coherent spectral weight, which weakens the SDW ordering and probably induces a magnetic quantum critical point (QCP). In the structural point of view, the P doping results in closer distance between As(P) and Fe planes. As a consequence, the low-energy band width becomes larger (correspondingly the coherent spectral weight is enhanced), according to the related band calculations[@band]. Furthermore, the magnetic fluctuations near the QCP may induce superconductivity, as has been well documented in literatures.[@QCP] This explains the simultaneous suppression of SDW transition and emergence of superconductivity in EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ assuming that the QCP locates near P content of $\sim$30%.
The superconducting properties of EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$ bear similarity with those of EuFe$_{2}$As$_{2}$ crystals under hydrostatic pressure. As a matter of fact, the onset temperature of resistive drop is nearly the same in both cases, suggesting that the effect of internal chemical pressure is in analogy with application of external physical pressure. Thus it is expectable to find superconductivity in other iron arsenide systems via the P-doping strategy.
In summary, we have found superconductivity at 26 K in EuFe$_{2}$(As$_{0.7}$P$_{0.3}$)$_{2}$. Moreover, ferromagnetic ordering of Eu$^{2+}$ moments coexists with the superconductivity below 20 K. The observation of sizable anomalies in thermodynamic properties, concomitant with the transitions, indicates that both of them are bulk phenomena. The interplay of superconductivity and ferromagnetism may bring about exotic properties and provide clues to the superconductivity mechanism, which renders EuFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ worthy of further exploration.
This work is supported by the NSF of China, National Basic Research Program of China (No. 2007CB925001) and the PCSIRT of the Ministry of Education of China (IRT0754).
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The Okamoto-Nolen-Schiffer (ONS) anomaly is the long-standing discrepancy between the calculated and measured binding-energy differences of mirror nuclei [@Oka64; @NolSch69]. The anomaly is likely to arise from charge symmetry breaking (CSB) in the strong interaction [@MNS90], itself believed to originate from the up-down quark mass difference and electromagnetic effects in the Standard Model. Thus, the study of CSB is a useful tool to elucidate the structure of strongly-interacting nuclear systems.
The ONS anomaly can be calculated on several levels. Perhaps the simplest is the observation by B. A. Brown [@BABro98] that the magnitude of the anomaly is approximately equal to the Coulomb exchange energy. If one adds an extra proton to a nucleus in a simple Hartree-Fock picture, there will be both a direct (Hartree) and exchange (Fock) Coulomb interaction with the other protons. If one—arbitrarily—neglects the Fock term, one obtains a better agreement with experiment.
At a different level, Blunden and Iqbal compute the ONS anomaly by calculating the contribution from $\rho$-$\omega$ mixing to the CSB component of the nucleon-nucleon (NN) interaction [@BluIqb87]. Their CSB interaction can explain part of the anomaly. However, at present this is controversial, both in the choice of meson couplings [@SKB93] and in the momentum dependence of $\rho$-$\omega$ mixing [@GHT92]. There have been also a number of calculations of the anomaly based on the contribution from the up-down quark mass difference ($\Delta m$). Indeed, Nakamura and coworkers [@Nak96] calculate a CSB NN interaction using a constituent quark model where the short-range color hyperfine interaction depends explicitly on the quark masses. Moreover, the mass difference between the neutron and proton may be density dependent [@HK89]. Finally, there are other more recent model calculations, such as the one reported in Ref. [@TST99].
Although the observation by Brown is not a dynamical explanation, it is an interesting characterization of the size of the anomaly. Could there be something wrong with the exchange term? As the nucleon is a composite object, could it be that the exchange energy of composite objects yields results significantly different from the exchange of point nucleons? One expects identical results if the composite scale of the nucleon is much smaller than the inter-particle spacing. However these scales are similar in nuclei. Moreover, although various calculations based on the up-down quark mass difference exist, we are not aware of any calculation of electromagnetic (EM) effects between quarks to the ONS anomaly.
The neutron-proton mass difference in free space is made up from comparable contributions of $\Delta m$ and EM effects. Note that EM and $\Delta m$ terms contribute with opposite signs to the neutron-proton mass difference. However, the ONS anomaly is sensitive to the density dependence of these contributions so the relative sign is unknown. In this letter we study EM effects involving the Coulomb exchange interactions of composite nucleons.
To clarify the importance of EM and $\Delta m$ terms we consider a high-density limit of the ONS anomaly. We will show, with some mild assumptions, that in the high-density limit: (1) there is an ONS anomaly and (2) that it is dominated by EM effects with $\Delta m$ being unimportant. Further, (3) the magnitude of the anomaly is simply related to the Coulomb exchange energy and (4) its sign is the same as that observed at lower densities. Finally, we will perform model calculations to see how relevant this high-density limit is to normal-density nuclei.
Consider very high-density symmetric nuclear matter. We assume that an electron gas makes the system electrically neutral. Thus the direct Coulomb interaction vanishes. Yet Coulomb exchange effects are still present. Now add either one proton or one neutron to the system and calculate the change in energy. First, model the system as a Fermi gas of elementary nucleons. An added proton will have a Coulomb exchange energy of $$V_p = -e^2 {k_F\over \pi} \;.
\label{Vproton}$$ Here $k_F$ is the Fermi momentum of the proton and $e$ is its electric charge. In contrast, an added neutron has zero Coulomb exchange energy: $V_n\!=\!0$. Thus, the energy difference between an added proton and a neutron is just: $$E_p - E_n = V_p+ M_p - M_n = -e^2{k_F\over \pi}-\Delta M \;,
\label{DeltaEH}$$ where $\Delta M=M_n-M_p=1.29$ MeV is the neutron-proton mass difference. Equation (\[DeltaEH\]) is the simple expectation of a model with unexcited point nucleons.
Next we consider a quark-gluon plasma. We assume because of asymptotic freedom, that at very high density the system is nearly a free Fermi gas of quarks. This is because the strong coupling $\alpha_S(k_F^2)$ becomes small at the large momentum scale characterized by $k_F$ . When a proton is added it will dissociate into two up and one down quark. Therefore, the Coulomb exchange energy of these three quarks is $$V_p^{(q)}=-\left(\sum_{i=1}^3 e_i^2\right) {k_F\over \pi} \;,
\label{Vprotonq}$$ where $e_i$ denotes the quark electric charge and $k_F$ is the quark Fermi momentum. Note that there are three times as many (valence) quarks as nucleons. However the quarks have an extra color degeneracy of three. As a result, the quark Fermi momentum in Eq. (\[Vprotonq\]) is the same as the proton’s Fermi momentum in Eq. (\[Vproton\]). The sum of the squares of the valence charges in a proton is $(4/9\!+\!4/9\!+\!1/9)e^2\!=\!e^2$. Because of this “numerical accident" the quark Coulomb exchange energy is equal to the Coulomb exchange energy of an elementary proton.
An interesting difference arises when we add a neutron. In a quark-gluon plasma the Coulomb exchange energy is no longer zero because a neutron is made up of charged constituents. Moreover, the exchange energy is always negative independent of the sign of the charges; the contributions from positive and negative charges add rather than cancel. Indeed, the sum of the squares of the valence quark charges in a neutron is $(4/9\!+\!1/9\!+\!1/9)e^2\!=\!2e^2/3$. Thus, the neutron Coulomb energy is fully two thirds of that of a proton: $V_n^{(q)}\!=\!-{2\over 3}e^2 {k_F/\pi}$. The energy difference between an added proton and a neutron becomes: $$E_p^{(q)}-E_n^{(q)} =
- e^2{k_F\over \pi} + {2\over 3} e^2{k_F\over \pi} =
- {1\over 3} e^2{k_F\over \pi}\;.
\label{DeltaEq}$$
We choose to define an ONS anomaly $\Delta E_{\rm ONS}$ as the actual energy difference, which we assume is given by Eq. (\[DeltaEq\]), minus the hadronic-model expectation of Eq. (\[DeltaEH\]) $$\Delta E_{\rm ONS}\!\equiv\!
(E_p^{(q)}\!-\!E_n^{(q)})\!-\!(E_p-E_n)\!=\!
{2\over 3} e^2 {k_F\over \pi}\!+\!\Delta M \;.
\label{DeltaEONS}$$ This anomaly arises, not because of an error in the proton’s energy but, because there is a nonzero Coulomb contribution for a (dissociated) neutron. In principle we should add to the above equation the contribution from the up-down quark mass difference. However, in the high-density limit, all contributions from $\Delta m$ are suppressed by the large Fermi momentum. Indeed, the difference in the Fermi energy of free down and up quarks is: $\sqrt{k_F^2+m_d^2}\!-\!\sqrt{k_F^2+m_u^2}\!\approx\!(m_d^2-m_u^2)/2k_F$. Thus, in the limit of very high density the total anomaly—including contributions from $\Delta m$—becomes dominated by Eq. (\[DeltaEONS\]). Moreover, the original mass difference between the neutron and proton ($\Delta M$) “disappears”at high density because the contributions from $\Delta m$ are suppressed and the Coulomb self-energies of the neutron and the proton are no longer relevant, as the quarks have rearranged themselves into a uniform free Fermi gas.
In summary, we expect that at high density there will be an ONS anomaly with a magnitude that is two-thirds that of the proton Coulomb exchange energy. Furthermore, EM effects dominate over the contribution from $\Delta m$ and the sign of the anomaly is the same as that observed at normal density.
Our earlier discussion suggests that the Coulomb energy of pure neutron matter is nonzero. Below we focus on neutron matter because of the simple expectation that for point neutrons the Coulomb energy is zero. This may provide a signature of substructure.
Since the above statements are only strictly true in the limit of very high density, we investigate their implications at normal density by performing a model calculation of neutron matter composed of valence quarks. While a model is necessary, our philosophy is to use a “minimal” one by demanding the following general features that any realistic model must posses. We require the many-quark wave function to (1) be explicitly anti-symmetric even for the exchange of quarks from different nucleons and (2) have cluster separability: the quark wave function of a nucleon removed to infinity must reduce to that of a free nucleon, without any unphysical long-range interactions. Finally, we demand that (3) quarks be confined and (4) for the wave function to reduce to free nucleons at low density and to a quark Fermi gas at high density. Perhaps, any model satisfying these general features can be used.
Conventional quark potential models with two-body confining interactions do not satisfy cluster separability as they generate unphysical long-range van der Waals interactions between nucleons. String-flip models on the other hand, do satisfy the four properties described above [@Lenz86; @HMN85]. Unfortunately, we are not aware of any other models which both satisfy these properties and allow a simple calculation. Thus, we employ the three-quark string-flip model discussed in Ref [@HorPie92]. The model has nonrelativistic constituent quarks of mass $m_c$ of fixed red, green, and blue colors. A system of $A$ nucleons is modeled with $N=3A$ quarks interacting via the following many-body potential: $V\!=\!V_{RG}\!+\!V_{GB}\!+\!V_{BR}$, where each term represents the optimal pairing of quarks. For example, the “red-green” component of the potential is defined as $$V_{RG}={\rm Min} \left\{\sum_{j=1}^A v({\bf r}_j^{(R)}-
{\bf r}_{P_j}^{(G)})\right\} \;.
\label{VRG}$$ Here ${\bf r}_j^{(R)}$ is the coordinate of the $j_{th}$ red quark and ${\bf r}_{P_j}^{(G)}$ is its green partner in the neutron. The minimum is over all $A!$ permutations $P_j$ of the set of $A$ green quarks. A harmonic string potential $v(r)=kr^2/2$ is used to confine the quarks and the Hamiltonian for the model becomes $$H=\sum_{i=1}^{N}\frac{{\bf P}_{i}^{2}}{2m_c}+V
=-\sum_{i=1}^N \frac{\nabla_i^2}{2m_c}+V \;.
\label{Hamiltonian}$$ Each red quark is connected by harmonic strings to one and only one green and to one and only one blue quark. This insures that quarks will be confined into “color-neutral” clusters. For three quarks the model reduces to the well-known harmonic oscillator quark model. For neutron matter there is a very large number of permutations or ways to connect the strings. We employ an implementation of the linear sum assignment algorithm by Burkard and Derigs that efficiently finds the optimal permutation in a time proportional to $N^3$ [@BurDer80]. This allows Monte Carlo simulations with hundreds of quarks.
The model has two dimension-full parameters: $k$ and $m_c$. Yet we are only interested in the harmonic-oscillator length $b=(km_c)^{-1/4}$, as this sets the length scale for quark confinement. The root mean square radius of a nucleon is $\langle r^2\rangle^{1/2}=3^{-1/4}b$. Hence, to reproduce the experimental charge radius of the proton $\langle r^2\rangle^{1/2}=0.86$ fm we choose $b=1.13$ fm. At the end we can rescale our results for other values of $b$.
We are interested in simulating neutron matter. Therefore we assign to red and green quarks an electromagnetic charge of $-e/3$ and to blue quarks a charge of $2e/3$. For simplicity we do not include any other intrinsic degree of freedom, such as spin or isospin. The electromagnetic self-energy of an isolated neutron is ($\alpha\!=\!e^2\!=\!1/137$) $$V_n^0=-\sqrt{\frac{2}{9\pi}}
\frac{\alpha}{\langle r^2\rangle^{1/2}}=-0.446~{\rm MeV} \;.
\label{vnself}$$ A simple variational wave function for the many-quark system has been constructed in Ref. [@HorPie92]. It is given by $$\Psi = \exp\left(-\lambda {V\over kb^2}\right) \Phi \;,
\label{Psivar}$$ with $\Phi$ a product of Slater determinants for the red, green, and blue quarks. In Ref. [@HorPie92] $\lambda$ is a variational parameter characterizing the length scale for quark confinement. At low density a value of $\lambda\!=\!1/\sqrt{3}$ allows Eq. (\[Psivar\]) to reproduce the gaussian wave function of a free nucleon. For simplicity we keep lambda fixed at $\lambda\!=\!1/\sqrt{3}$ for all densities. This insures that any change in the Coulomb energy of a neutron does not arise from an artificial change in this length scale.
We calculate the total Coulomb energy $$V_{\rm Coul}^{\rm tot}=\sum_{i<j}^N
\frac{e_ie_j}{|{\bf r}_i-{\bf r}_j|} \;,
\label{VCoulomb}$$ of a system of $N\!=\!3A$ quarks in a box of volume $V$ with antiperiodic boundary conditions. To minimize finite size effects we use periodic distances to compute the quark separation. The neutron density of the system is $\rho_n\!=\!{A/V}$. We use standard Metropolis Monte Carlo techniques to calculate the expectation value of the total Coulomb energy for the wave function given in Eq. (\[Psivar\]).
Figure 1 shows the change in the Coulomb energy per neutron $$\Delta V\equiv
{1\over A}\langle V_{\rm Coul}^{\rm tot}\rangle - V_n^0 \;,
\label{DeltaV}$$ as a function of density for systems with $N$=96 and 264 quarks. We have subtracted the neutron self-energy $V_n^0$ of Eq. (\[vnself\]) because this is included in the experimental neutron-proton mass difference. We find $\Delta V$ to be nonzero.
to 2.6in[to 8in]{}
At normal density $\rho_n\!=\!0.08$ fm$^{-3}$ and $N=96$: $\Delta V = -78 \pm 1~{\rm keV}$. The scale of this result suggests that changes in the Coulomb energies of quarks can make a significant contribution to the ONS anomaly. More refined models may give results which are of the same order of magnitude, given the ratio of the nucleon size to interparticle spacing. Furthermore, we expect an additional contribution from the up-down quark mass difference $\Delta m$. Our result is somewhat smaller than the total observed anomaly of the order 200 keV in mass 15 and 300 keV in mass 39 [@MHSha94]. Note that, for simplicity, we have calculated the average Coulomb energy per neutron rather than the self-energy of a single valence neutron. These quantities are expected to be similar. Indeed, in a free Fermi gas the average Coulomb energy per proton is three fourths of that of Eq. (\[Vproton\]).
Figure 2 shows $\Delta V$ as a function of the nucleon root mean square radius or oscillator length at the fixed density of $\rho_n=0.08$ fm$^{-3}$. Making the quark core of a nucleon smaller reduces $\Delta V$, but not by much. Further, as the oscillator length is made very small the scale of the neutron self-energy $V_n^0$ grows and this can increase $\Delta V$. Of course, if the nucleon core is small one must use a large meson cloud to account for the full proton charge radius. This meson cloud, which we have not included, could further increase $\Delta V$.
to 2.6in[to 8in]{}
One should extend our results by using more elaborated quark models. It is important to study models with more intrinsic spin and flavor degrees of freedom along with more complete treatments of color. However, in these more complete models we still expect an exchange or dynamical correlation between quarks associated with the nucleon’s hard core. This correlation could lead to a nonzero Coulomb energy for neutrons. Note that we have used harmonic oscillator confining strings. Thus, our wave function has gaussian tails. Linear confinement may increase the tails and this should enhance the Coulomb exchange energy.
In conclusion, we have considered a high-density limit of the Okamoto-Nolen-Schiffer anomaly to clarify the role of electromagnetic interactions (EM) and of the up-down quark mass difference $\Delta
m$. We have added a single neutron or proton to a quark gluon plasma. In this high-density limit we find that: (1) there is an ONS anomaly, (2) it is dominated by EM interactions rather than by $\Delta
m$, and (3) its magnitude is two-thirds of the proton Coulomb exchange energy. We find an attractive Coulomb exchange energy for an added neutron because of the neutron’s charged constituents. This suggests that the ONS anomaly could be closely related to the nucleon substructure. We use a minimal string-flip quark model to calculate the Coulomb energy of pure neutron matter. The model wave function is fully anti-symmetric and satisfies cluster separability and quark confinement. At normal density, we find a nonzero Coulomb energy for neutron matter that could make a significant contribution to the ONS anomaly.
This work was supported in part by DOE grants DE-FG02-87ER40365, DE-FC05-85ER250000, and DE-FG05-92ER40750.
K. Okamoto, Phys. Lett. [**11**]{} (1964) 150. J. A. Nolen and J. P. Schiffer, Ann. Rev. Nucl. Sci. [**19**]{} (1969) 471. G. A. Miller, B. M. K. Nefkens and I. Slaus, Phys. Rep. [**194**]{} (1990) 1. B. Alex Brown, Phys. Rev. [**C58**]{} (1998) 220. P. G. Blunden and M. J. Iqbal, Phys. Lett. [**B198**]{} (1987) 14. Thomas Schafer, Volker Koch and Gerald E. Brown, Nuc. Phys. [**A562**]{} (1993) 644. T. Goldman, J. A. Henderson and A. W. Thomas, Few-Body Systems, [**12**]{} (1992) 123. S. Nakamura, K. Muto, M. Oka, S. Takeuchi and T. Oda, Phys. Rev. Lett. [**76**]{} (1996) 881. E. M. Henley and G. Krein, Phys. Rev. Lett. [**62**]{} (1989) 2586. K. Tsushima, K. Saito and A. W. Thomas, Phys. Lett. [**B465**]{} (1999) 36. F. Lenz et al., Ann. of Phys. (N.Y.) [**170**]{} (1986) 65. C. J. Horowitz, E. Moniz and J. Negele, Phys. Rev. [**D31**]{} (1985) 1689. C. J. Horowitz and J. Piekarewicz, Nuc. Phys. [**A536**]{} (1992) 669. R.E. Burkard and U. Derigs, [*Lecture Notes in Economics and Math Systems*]{}, [**184**]{} (Springer-Verlag) 1980, Chapter 1. M. H. Shahnas, Phys. Rev. [**C50**]{} (1994) 2346.
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author:
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title: 'GNA: new framework for statistical data analysis'
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Introduction
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Interpretation of observables
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Software
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GNA
===
Design principles
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GNA core
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GNA usage
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Transformations, bundles and expressions
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Prospects
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Multithreading
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GPU
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Conclusion
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Acknowledgements {#acknowledgements .unnumbered}
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[^1]:
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=-1.3cm -2cm
[**AMPLIFICATION EFFECTS ON THE TRANSMISSION AND REFLEXION PHASES IN 1D PERIODIC SYSTEMS**]{}\
Nouredine Zekri [^1], Karima Bennabi and Souad Maarouf
[*U.S.T.O., Département de Physique, L.E.P.M.,\
B.P.1505 El M’Naouar, Oran, Algeria*]{}.[*\
and* ]{}
[*Abdus Salam International Centre for Theoretical Physics, Trieste (Italy)*]{}.
**Abstract**
We investigate the localization observed recently for locally non-hermitian Hamiltonians by studying the effect of the amplification on the scaling behavior of the transmission and reflection phases in 1D periodic chains of $\delta$-potentials. The amplification here is represented by an imaginary term added to the on-site potential. It is found that both phases of the transmission and reflection amplitudes are strongly affected by the amplification term. In particular, the phases in the region of amplification become independent of the length scale while they oscillate strongly near the maximum transmission (or reflection). The interference effects on the phase in passive systems are used to interpret those observed in the presence of amplification. The phases of the transmission and reflection are found to oscillate in passive systems whith increasing periods in the allowed band for the transmission phase while for the reflection phase, its initial value is always less than $\pi /2$ in this band.
[Keywords:]{} Non hermiticity, Localization, Amplification.
PACS Nos. 05.40.+j, 42.25.Bs, 71.55.Jv, 72.15.Rn
Introduction
=============
Recently, there was an increase of interest in non-hermitian hamiltonians and quantum phase transitions (typically localized to extended wavefunctions) in systems characterized by them. There are in general two classes of problems in this context: one in which the non-hermiticity is in the nonlocal part [@nlnh1; @nlnh2] and the other in which it is in the local part \[3-8\]. In the first category, one considers an imaginary vector potential added to the momentum operator in the Schrödinger hamiltonian. In the second category (non-hermiticity in the local term), an imaginary term is introduced in the one-body potential. It is well-known from textbooks on quantum mechanics that depending on the sign of the imaginary term, this means the presence of a sink (absorber) or a source (amplifier) in the system. It may be noted that this second category does also have a counterpart in classical systems characterized by a Helmholtz (scalar) wave equation as well, where the practical application is in the studies of the effects of classical wave (light) localization due to backscattering in the presence of an amplifying (lasing) medium that has a complex dielectric constant with spatial disorder in its real part [@pradhan; @zhao]. There is a common thread binding both the problems though, namely that the spectrum for both becomes complex (the hamiltonian being non-hermitean or real non-symmetric), but can admit real eigenvalues as well. The common property is that the real eigenvalues represent localized states and the eigenvalues off the real lines extended states. That it is so in the first category has been shown in the recent works starting with Hatano and Nelson and followed by others [@nlnh1; @nlnh2]. In the rest of the paper we would be concerned with non-hermitian hamiltonians of the second category only. For this category with sources at each scatterer and in the absence of impurities, it seems counter-intuitive that there are localized solutions; but it has been shown in a simple way \[5,8,9\] that the real eigenvalues are always localized. However, up to now the physical origins of this effect have not been provided. Since the localization is a consequence of the backscattering and the destructive interferences, we expect this effect to be related to the scaling behavior of the phases of the transmission and reflection amplitudes. This is the aim of this letter where we examine numerically the effect of the amplification on the phase of the transmission and reflection amplitudes. We use for this end the Kronig-Penney model which is a continuous multiband model. We first consider a periodic passive system in order to understand the behavior of the phase for localized and extended states. This allow us to explain the phase behavior in such amplifying systems.
Model description
=================
We consider a non interacting electron of energy $E$ moving through a linear chain of $\delta $-potentials strengths strength $\beta_{n}$, $n$ is the site position. In each site an imaginary term $\eta$ is included leading to a Non Hermitian Hamiltonian. The Schrödinger equation then reads
$$\left\{ -\frac{d^{2}}{dx^{2}}+\sum_{n}(\beta _{n}+\eta)\delta (x-n)\right\}
\Psi (x)=E\Psi (x)$$
Here $\Psi (x)$ is the single particle wavefunction at $x$, and $E$ is expressed in units of $\hbar ^{2}/2m$ with $m$ being the free electron effective mass. For simplicity, the lattice spacing is taken to be unity in all this work. Since we are interested only in periodic systems, the potential strength $\beta_{n}$ is a constant $\beta_{0}$. The complexe potential appearing in the local part of the Hamiltonian in (1) leads either to complex eigenvalues and real wavenumbers or real eigenvalues and complex vavenumbers. We consider the system Ohmically connected to ideal leads so that the second case is used since the total energy is conserved. In this case the imaginary part acts either as a sink (absorber) if $\eta<0$ or as a source (amplifier) if $\eta>0$ [@zekri1]. From the computational point of view it is more useful to consider the discrete version of the Schrödinger equation which is called the generalized Poincaré map and can be derived without any approximation from (1). It reads [@Bellis]
$$\Psi _{n+1} = \left[ 2\cos k + \frac{\sin k}{k}( \beta_0 + i \eta
) \right] \Psi _{n} - \Psi _{n-1}$$
where $\Psi _{n}$ is the value of the wavefunction at site $n$ and $k=\sqrt{E}$. This representation relates the values of the wavefunction at three successive discrete locations along the x-axis without restriction on the potential shape at those points and is very suitable for numerical computations. The solution of equation (2) is done iteratively by taking for our initial conditions the following values at sites $1$ and $2$ : $\Psi
_{1}=$ $\exp (-ik)$ and $\Psi _{2}=$ $\exp (-2ik)$. We consider here an electron having a wave number $k_{F}$ (at Fermi energy) incident at site $%
N+3 $ from the right (by taking the chain length $L=N$, i.e. $N+1$ scatterers). The transmission and reflection amplitudes ($t$ and $r$) can then be expressed as
$$t=\frac{-2i\exp (-ik(N+3))\sin k}{\Psi _{N+3}\exp (-ik)-\Psi _{N+2}},$$
and
$$r=\frac{\exp (-2ik(N+3))\left( \Psi _{N+2}-\exp (ik)\Psi _{N+3}\right) }{%
\Psi _{N+3}\exp (-ik)-\Psi _{N+2}},$$
where the terms $\exp (-ik(N+3))$ and $\exp (-2ik(N+3))$ apprearing respectively in the transmission and reflection amplitudes originate from the fact that the electron is incident at site $N+3$ with an incident phase $-k(N+3)$. Therefore, these fictious phases are to be disgarded. Note here that the wave number $k$ appearing in the last expressions is that of the free electron moving in the leads and is different from that inside the system (which is complex). From Eqs. (3 and 4) the phases of the transmission and reflection amplitudes depend only on the values of the wavefunction at the end sites, $%
\Psi _{N+2}$, $\Psi _{N+3}$ which are evaluated from the iterative equation (2). The phases of the transmission and reflection amplitudes ($\Phi_t$ and $\Phi_r$) are then the arguments of $t$ and $r$ respectively. These phases vary obviously between $0$ and $2 \pi$.
Results and discussion
======================
As discussed below, the observed asymptotic localization in amplifying periodic systems [@zekri1] should come from the phase interferences and the backscattering. Indeed, the maximum transmission length ($L_{max}$) in this case can be seen as the characteristic length separating the region where the amplification dominates from that where the interferences and backscattering dominate (below $L_{max}$). Let us first consider the effect on the transmission and reflection phases.
In order to understand the phase behavior in the case of constructive and destructive interferences, we start examining its scaling in a passive periodic system. We fix in this case $\beta_0 =8$ which, from Eqs. (1) and (2) leads us to the energy spectrum shown in Fig.1. In this spectrum, we choose the energies $E=1$, $E=3$ and $E=5$ to scan the phase scaling either in the gap and the allowed band (Figs.2). The transmission phase in Fig.2a oscillates around $\pi$ with decreasing periods for energies away from the allowed band while they increase inside this band. Therefore a higher frequency oscillating phase means a localization. In Fig.2b, the initial reflection phase seems to be always between $\pi /2$ and $3 \pi
/2$ for energies in the gap which corresponds to localized states for such finite systems.
Let us now examine the phase scaling for amplifying systems $\eta >0$ (see Figs.3). For simplicity we consider that the on-site potential is purely imaginary (i.e., $\beta =0$). We see in particular in these figures that both the reflection and amplification phases remain constant in the region where the transmission coefficient grows. It is important to notice that the reflection phase is greater than $\pi /2$ which indicates that there are destructive interferences in the region of growing transmission but they seem to not affect it. In the region of maximum transmission (and reflection) both phases oscillate and the transport properties of the system seems to become sensitive to them.
Conclusion
==========
We used in this letter the effect of the amplification on the scaling behavior of both transmission and reflection phases in order to interpret the recently observed effect on the coefficients. The main results show a constant phase in the growth region while it starts oscillating near the maximum transmission and reflection. However, the amplification effect has been studied here only in the allowed band of the corresponding passive periodic system (since $\beta_0 = 0$ when the amplification $\eta$ is applied, all the spectrum of the passive system is Bloch like). Therefore, it is interesting to examine this effect in the gap of the corresponding passive system. In this case the transmission coefficient is exponentially decaying (the system being finite) and the Lyapunov exponent should be affected differently by the amplification.
[**Acknowledgements**]{} NZ would like to thank the ICTP for its hospitality and the Arab Funds for its support during the progress of this work.
[99]{}
N. Hatano and D.R. Nelson, Phys. Rev. Lett. [**77**]{}, 570 (1996); see preprint cond-mat/9705290 for further details.
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****
[**Figure Captions** ]{}
[**Fig.1**]{} Transmission coefficient (in a log scale) versus energy for $\beta_0 = 8$ and $\eta =0$ (passive system).
Variation of the reflection and transmission phase with the length scale for $\eta=0$, $\beta_0 = 8$ and different energies $1, 3$ and $5$. a) $\Phi_t$, b) $Phi_r$
variations of the reflexion and transmission phases and the transmission coefficient with the length scale $L$ for $\beta =0$, $\eta =0.05$ and $0.1$ and the energy $E=1$. a) phase of the transmission, b) phase of the reflection, c) transmission coefficient.
[^1]: Corresponding author, e-mail: [email protected]
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{
"pile_set_name": "ArXiv"
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---
author:
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Mark O’Neill, Scott Ruoti, Kent Seamons, Daniel Zappala\
\
\
\
bibliography:
- 'bib.bib'
title: '**TLS Proxies: Friend or Foe?**'
---
Acknowledgments
===============
This work is supported by a 2014 Google Faculty Research Award.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Recovering the support of sparse vectors in underdetermined linear regression models, *aka*, compressive sensing is important in many signal processing applications. High SNR consistency (HSC), i.e., the ability of a support recovery technique to correctly identify the support with increasing signal to noise ratio (SNR) is an increasingly popular criterion to qualify the high SNR optimality of support recovery techniques. The HSC results available in literature for support recovery techniques applicable to underdetermined linear regression models like least absolute shrinkage and selection operator (LASSO), orthogonal matching pursuit (OMP) etc. assume *a priori* knowledge of noise variance or signal sparsity. However, both these parameters are unavailable in most practical applications. Further, it is extremely difficult to estimate noise variance or signal sparsity in underdetermined regression models. This limits the utility of existing HSC results. In this article, we propose two techniques, *viz.*, residual ratio minimization (RRM) and residual ratio thresholding with adaptation (RRTA) to operate OMP algorithm without the *a priroi* knowledge of noise variance and signal sparsity and establish their HSC analytically and numerically. To the best of our knowledge, these are the first and only noise statistics oblivious algorithms to report HSC in underdetermined regression models.'
bibliography:
- 'compressive.bib'
title: High SNR Consistent Compressive Sensing Without Signal and Noise Statistics
---
Introduction
============
Consider a linear regression model $${\bf y}={\bf X}\boldsymbol{\beta}+{\bf w},$$ where ${\bf y} \in \mathbb{R}^n$ is the observation vector, ${\bf X}\in \mathbb{R}^{n \times p}$ is the $n \times p$ design matrix, $\boldsymbol{\beta}\in \mathbb{R}^{p}$ is the unknown regression vector and ${\bf w} \in \mathbb{R}^{n}$ is the noise vector. We consider a high dimensional or underdetermined scenario where the number of observations ($n$) is much less than the number of variables/predictors ($p$). We also assume that the entries in the noise vector ${\bf w}$ are independent and identically distributed Gaussian random variables with mean zero and variance $\sigma^2$. Such regression models are widely studied in signal processing literature under the compressive sensing or compressed sensing paradigm [@eldar2012compressed]. Subset selection in linear regression models refers to the identification of support $\mathcal{S}=supp(\boldsymbol{\beta})=\{k:\boldsymbol{\beta}_k\neq 0\}$, where $\boldsymbol{\beta}_k$ refers to the $k^{th}$ entry of $\boldsymbol{\beta}$. Identifying supports in underdetermined or high dimensional linear models is an ill posed problem even in the absence of noise ${\bf w}$ unless the design matrix ${\bf X}$ satisfies regularity conditions [@eldar2012compressed] like restricted isometry property (RIP), mutual incoherence property (MIC), exact recovery condition (ERC) etc. and $\boldsymbol{\beta}$ is sparse. A vector $\boldsymbol{\beta}$ is called sparse if the cardinality of support $\mathcal{S}$ given by $k_0=card(\mathcal{S})\ll p$. In words, only few entries of a sparse vector $\boldsymbol{\beta}$ will be non-zero. Identification of sparse supports in underdetermined linear regression models have many applications including and not limited to detection in multiple input multiple output (MIMO)[@choi2017detection] and generalised MIMO systems[@yu2012compressed; @kallummil2016combining], multi user detection[@shim2012multiuser], subspace clustering[@you2016scalable] etc. This article discusses this important problem of recovering sparse supports in high dimensional linear regression models. After presenting the notations used in this article, we provide a brief summary of sparse support recovery techniques discussed in literature and the exact problem discussed in this article.
Notations used
--------------
$col({\bf A})$ the column space of matrix ${\bf A}$. ${\bf A}^T$ is the transpose and ${\bf A}^{\dagger}=({\bf A}^T{\bf A})^{-1}{\bf A}^T$ is the Moore-Penrose pseudo inverse of ${\bf A}$. ${\bf P}_{\bf A}={\bf A}{\bf A}^{\dagger}$ is the projection matrix onto $col({\bf A})$. ${\bf A}_{\mathcal{J}}$ denotes the sub-matrix of ${\bf A}$ formed using the columns indexed by $\mathcal{J}$. When ${\bf A}$ is clear from the context, we use the shorthand ${\bf P}_{\mathcal{J}}$ for ${\bf P}_{{\bf A}_{\mathcal{J}}}$. Both ${\bf a}_{\mathcal{J}}$ and ${\bf a}(\mathcal{J})$ denote the entries of vector ${\bf a}$ indexed by $\mathcal{J}$. $\mathcal{N} ({\bf u},{\bf C})$ is a Gaussian random vector (R.V) with mean ${\bf u}$ and covariance ${\bf C}$. $\mathbb{B}(a,b)$ represents a Beta R.V with parameters $a$ and $b$ and $B(a,b)$ represents the Beta function. $F_{a,b}(x)=\dfrac{1}{B(a,b)}\int_{t=0}^x t^a(1-t)^b$ is the CDF of a $\mathbb{B}(a,b)$ R.V. ${\bf a}\sim{\bf b}$ implies that R.Vs ${\bf a}$ and ${\bf b}$ are identically distributed. $\|{\bf a}\|_m=(\sum\limits_{j}|{\bf a}_j|^m)^{\frac{1}{m}}$ for $1\leq m\leq \infty$ is the $l_m$ norm and $\|{\bf a}\|_0=card(supp({\bf a}))$ is the $l_0$ quasi norm of ${\bf a}$. For any two index sets $\mathcal{J}_1$ and $\mathcal{J}_2$, the set difference $\mathcal{J}_1/\mathcal{J}_2=\{j \in \mathcal{J}_1: j\notin \mathcal{J}_2\}$. $X\overset{p}{\rightarrow } Y$ denotes the convergence of random variable $X$ to $Y$ in probability. $\mathbb{P}()$ and $\mathbb{E}()$ represent probability and expectation. Signal to noise ratio (SNR) for the regression model (1) is given by $SNR=\dfrac{\mathbb{E}(\|{\bf X}\boldsymbol{\beta}\|_2^2)}{\mathbb{E}(\|{\bf w}\|_2^2)}=\dfrac{\|{\bf X}\boldsymbol{\beta}\|_2^2}{n\sigma^2}$.
High SNR consistency in linear regression {#intro}
-----------------------------------------
The quality of a support selection technique delivering a support estimate $\hat{\mathcal{S}}$ is typically quantified in terms of the probability of support recovery error $PE=\mathbb{P}(\hat{\mathcal{S}}\neq \mathcal{S})$ or the probability of correct support recovery $PCS=1-PE$. The high SNR behaviour (i.e. behaviour as $\sigma^2 \rightarrow 0$ or $SNR\rightarrow \infty$) of support recovery techniques in general and the concept of high SNR consistency (HSC) defined below in particular has attracted considerable attention in statistical signal processing community recently[@ding2011inconsistency; @schmidt2012consistency; @stoica2012proper; @stoica2013model; @SNLShighSNR; @tsp; @spl; @elsevier].
[**Definition 1:-**]{} A support recovery technique is defined to be high SNR consistent (HSC) iff $\underset{\sigma^2 \rightarrow 0}{\lim}PE=0$ or equivalently $\underset{SNR \rightarrow \infty}{\lim}PE=0$.
In applications where the problem size $(n,p)$ is small and constrained, the support recovery performance can be improved only by increasing the SNR. This makes HSC and high SNR behaviour in general very important in certain practical applications.
Most of the existing literature on HSC deal with overdetermined ($n>p$) or low dimensional linear regression models. In this context, high SNR consistent model order selection techniques like exponentially embedded family (EEF)[@ding2011inconsistency][@eefenumeration], normalised minimum description length (NMDL)[@schmidt2012consistency], forms of Bayesian information criteria (BIC)[@stoica2013model], penalised adaptive likelihood (PAL)[@stoica2013model], sequentially normalised least squares (SNLS)[@SNLShighSNR] etc. when combined with a $t$-statistics based variable ordering scheme were shown to be HSC [@tsp]. Likewise, the necessary and sufficient conditions (NSC) for the HSC of threshold based support recovery schemes were derived in [@spl]. However, both these HSC support recovery procedures are applicable only to overdetermined ($n>p$) regression models and are not applicable to the underdetermined ($n<p$) regression problem discussed in this article. Necessary and sufficient conditions for the high SNR consistency of compressive sensing algorithms like OMP[@tropp2004greed; @cai2011orthogonal; @tropp2007signal] and variants of LASSO [@tropp2006just] are derived in [@elsevier]. However, for HSC and good finite SNR estimation performance, both OMP and LASSO require either the *a priori* knowledge of noise variance $\sigma^2$ or sparsity level $k_0$. Both these quantities are unknown *a priori* in most practical applications. However, unlike the case of overdetermined regression models where unbiased estimates of $\sigma^2$ with explicit finite sample guarantees are available, no estimate of $\sigma^2$ with such finite sample guarantees are available in underdetermined regression models to the best of our knowledge. Similarly, we are also not aware of any technique to efficiently estimate the sparsity level $k_0$. Hence, the application of HSC results in [@elsevier] to practical underdetermined support recovery problems are limited.
Contribution of this article
----------------------------
[ Residual ratio thresholding (RRT)[@icml; @robust; @mos] is a concept recently introduced to perform sparse variable selection in linear regression models without the *a priori* knowledge of nuisance parameters like noise variance, sparsity level etc. This concept was initially developed to operate support recovery algorithms like OMP, orthogonal least squares (OLS) etc, in underdetermined linear regression models with explicit finite SNR and finite sample guarantees [@icml]. Later, this concept was extended to outlier detection problems in robust regression [@robust] and model order selection in overdetermined linear regresssion [@mos]. A significant drawback of RRT in the context of support recovery in underdetermined regression models (as we establish in this article) is that it is inconsistent at high SNR. In other words, inspite of having a decent finite SNR performance, RRT is suboptimal in the high SNR regime. ]{} In this article, we propose two variants of RRT, *viz.*, residual ratio minimization (RRM) and residual ratio thresholding with adaptation (RRTA) to operate algorithms like OMP, OLS etc. without the *a priori* knowledge of $k_0$ or $\sigma^2$. Unlike RRT, these two schemes are shown to be high SNR consistent both analytically and numerically. In addition to HSC which is an asymptotic result, we also derive finite sample and finite SNR support recovery guarantees for RRM based on RIP. These support recovery results indicate that the SNR required for successfull support recovery using RRM increases with the dynamic range of $\boldsymbol{\beta}$ given by $DR(\boldsymbol{\beta})=\dfrac{\boldsymbol{\beta}_{max}=\underset{j \in \mathcal{S}}{\max}|\boldsymbol{\beta}_j|}{\boldsymbol{\beta}_{min}=\underset{j \in \mathcal{S}}{\min}|\boldsymbol{\beta}_j|}$, whereas, numerical simulations indicate that the SNR required by RRTA (like RRT and OMP with *a priori* knowledge of $\sigma^2$ or $k_0$) depends only on the minimum non zero value $\boldsymbol{\beta}_{min}$. Consequently, the finite SNR utility of RRM is limited to wireless communication applications like [@kallummil2016combining] where $DR(\boldsymbol{\beta})$ is close to one. In contrast to RRM, RRTA is useful in both finite and high SNR applications irrespective of the dynamic range of $\boldsymbol{\beta}$.
Organization of this article
----------------------------
Section [slowromancap2@]{} presents the existing results on OMP. Section [slowromancap3@]{} introduces RRT and develope RRM and RRTA techniques along with their analytical guarantees. Section [slowromancap4@]{} presents numerical simulations.
High SNR consistency of OMP with *a priori* knowledge of $\sigma^2$ or $k_0$
============================================================================
-------------------------------------------------------------------------------------------------------------------------
[**Input:**]{} Observation ${\bf y}$, design matrix ${\bf X}$ and stopping condition.
[**Step 1:-**]{} Initialize the residual ${\bf r}^{(0)}={\bf y}$.
$\hat{\boldsymbol{\beta}}={\bf 0}_p$, Support estimate ${\mathcal{S}_0}=\emptyset$, Iteration counter $k=1$;
[**Step 2:-**]{} Update support estimate: ${\mathcal{S}_k}={\mathcal{S}_{k-1}}\cup t^k$,
where $t^k=\underset{t \in [p]}{\arg\max}|{\bf X}_t^T{\bf r}^{k-1}|.$
[**Step 4:-**]{} Estimate $\boldsymbol{\beta}$ using current support:
$ \ \ \ \ \ \ \ \ \ \hat{\boldsymbol{\beta}}(\mathcal{S}_k)={\bf X}_{\mathcal{S}_k}^{\dagger}{\bf y}$.
[**Step 5:-**]{} Update residual: ${\bf r}^{k}={\bf y}-{\bf X}\hat{\boldsymbol{\beta}}=({\bf I}_n-{\bf P}_{k}){\bf y}$.
${\bf P}_k={\bf X}_{\mathcal{S}_k}{\bf X}_{\mathcal{S}_k}^{\dagger}$.
[**Step 6:-**]{} Increment $k$. $k \leftarrow k+1$.
[**Step 7:-**]{} Repeat Steps 2-6, until the stopping condition is satisfied.
[**Output:-**]{} Support estimate $\hat{\mathcal{S}}=\mathcal{S}_k$ and signal estimate $\hat{\boldsymbol{\beta}}$.
-------------------------------------------------------------------------------------------------------------------------
: OMP algorithm.[]{data-label="tab:omp"}
OMP [@tropp2004greed] in TABLE \[tab:omp\] is a widely used greedy and iterative sparse support recovery algorithm. OMP algorithm starts with a null set as support estimate and observation ${\bf y}$ as the initial residual. At each iteration, OMP identifies the column that is the most correlated with the current residual ($t^k=\underset{j}{\arg\max}|{\bf X}_j^T{
\bf r}^{k-1}|$) and expand the support estimate by including this selected column index $(\mathcal{S}_k=\mathcal{S}_{k-1} \cup t^k)$. Later, the residual is updated by projecting the observation vector ${\bf y}$ orthogonal to the column space produced by the current support estimate (i.e., $col({\bf X}_{\mathcal{S}_k})$). Since ${\bf r}^k$ is orthogonal to the column space of ${\bf X}_{\mathcal{S}_k}$, ${\bf X}_t^T{\bf r}^k=0$ for all $t\in \mathcal{S}_k$. Consequently, an index selected in an initial stage will not be selected again later. Consequently, the support estimate sequence monotonically increases with iteration $k$, i.e., $\mathcal{S}_k\subset \mathcal{S}_{k+1}$ and $card(\mathcal{S}_k)=k$.
OLS iterations are also similar to that of OMP except that OLS select the column that results in the maximum decrease in residual energy $\|{\bf r}^k\|_2^2$, i.e., $t^k=\underset{j}{\arg\min}\|({\bf I}_n-{\bf P}_{\mathcal{S}_{k-1}\cup j}){\bf y}\|_2^2$. OLS support estimate sequence also satisifes $\mathcal{S}_k\subset \mathcal{S}_{k+1}$ and $card(\mathcal{S}_k)=k$. The techniques developed in this article will be discussed using OMP algorithm. However, please note that these techniques are equally applicable to OLS also.
The iterations in OMP are continued until a user defined stopping condition is met. The performance of OMP depends crucially on this stopping condition. When the sparsity level $k_0$ is known *a priori*, many articles suggest stopping OMP exactly after $k_0$ iterations. When $k_0$ is unknown *a priori*, one can stop OMP when the residual power $\|{\bf r}^k\|_2$ is sufficiently small. Two such residual based stopping conditions are popular in literature[@cai2011orthogonal]. One rule proposes to stop OMP iterations once the residual power drops below $\|{\bf r}^k\|_2\leq \|{\bf w}\|_2$, whereas, another rule proposes to stop OMP when the residual correlation drops below $\|{\bf X}^T{\bf r}^k\|_{\infty}\leq \|{\bf X}^T{\bf w}\|_{\infty}$. When ${\bf w} \sim \mathcal{N}({\bf 0}_n,\sigma^2{\bf I}_n)$ and the columns ${\bf X}_j$ have unit $l_2$ norm, it was shown in [@cai2011orthogonal] that $$\begin{array}{ll}
\mathbb{P}\left(\|{\bf w}\|_2\geq \sigma\sqrt{n+2\sqrt{n\log(n)}}\right)\leq 1/n
\ \text{and}\\
\mathbb{P}\left(\|{\bf X}^T{\bf w}\|_{\infty}\geq \sigma\sqrt{2\log(p)}\right)\leq 1/p.
\end{array}$$ Consequently, one can stop OMP iterations in Gaussian noise once $\|{\bf r}^k\|_2\leq \sigma\sqrt{n+2\sqrt{n\log(n)}}$ or $\|{\bf X}^T{\bf w}\|_{\infty}\leq \sigma\sqrt{2\log(p)}$.
A number of deterministic recovery guarantees are proposed for OMP. Among these guarantees, the conditions based on restricted isometry constants (RIC) are the most popular for OMP. RIC of order $j$ denoted by $\delta_j$ is defined as the smallest value of $\delta$ such that $$(1-\delta)\|{\bf b}\|_2^2\leq \|{\bf X}{\bf b}\|_2^2\leq (1+\delta)\|{\bf b}\|_2^2$$ hold true for all ${\bf b} \in \mathbb{R}^p$ with $\|{\bf b}\|_0=card(supp({\bf b}))\leq j$. A smaller value of $\delta_j$ implies that ${\bf X}$ act as a near orthogonal matrix for all $j$ sparse vectors ${\bf b}$. Such a situation is ideal for the recovery of a $j$-sparse vector ${\bf b}$ using any sparse recovery technique. The latest RIC based finite SNR support recovery guarantee and HSC results for OMP are given in Lemma \[lemma:latest\_omp\].
\[lemma:latest\_omp\] Suppose that the matrix ${\bf X}$ satisfies $\delta_{k_0+1}<{1}/{\sqrt{k_0+1}}$. Then,\
1). OMP with $k_0$ iterations or stopping condition $\|{\bf r}^k\|_2\leq \|{\bf w}\|_2$ can recover any $k_0$ sparse vector $\boldsymbol{\beta}$ once $\|{\bf w}\|_2\leq \epsilon_{omp}=\boldsymbol{\beta}_{min}\sqrt{1-\delta_{k_0+1}}\left[\dfrac{1-\sqrt{k_0+1}\delta_{k_0+1}}{1+\sqrt{1-\delta_{k_0+1}^2}-\sqrt{k_0+1}\delta_{k_0+1}}\right]$ [@latest_omp].\
2). Define $\epsilon_{\sigma}=\sigma\sqrt{n+2\sqrt{n\log(n)}}$. Then, OMP with $k_0$ iterations or stopping condition $\|{\bf r}^k\|_2\leq \epsilon_{\sigma}$ can recover any $k_0$ sparse vector $\boldsymbol{\beta}$ with a probability greater than $1-1/n$ once $\epsilon_{\sigma}\leq \epsilon_{omp}$.\
3). OMP running precisely $k_0$ iterations is high SNR consistent, i.e., $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{k_0}=\mathcal{S})=1$ [@elsevier].\
4). OMP with stopping rule $\|{\bf r}^k\|_2 \leq \sigma g(\sigma) $ is HSC iff $\underset{\sigma^2\rightarrow 0}{\lim}g(\sigma)=\infty$ and $\underset{\sigma^2\rightarrow 0}{\lim}\sigma g(\sigma)=0$ [@elsevier].\
5). OMP with stopping rule $\|{\bf X}^T{\bf r}^k\|_{\infty} \leq \sigma g(\sigma) $ is HSC iff $\underset{\sigma^2\rightarrow 0}{\lim}g(\sigma)=\infty$ and $\underset{\sigma^2\rightarrow 0}{\lim}\sigma g(\sigma)=0$ [@elsevier].
Lemma \[lemma:latest\_omp\] implies that OMP with the *a priori* knowledge of $k_0$ or $\sigma^2$ can recover support $\mathcal{S}$ once the matrix satisfies the regularity condition $\delta_{k_0+1}<{1}/{\sqrt{k_0+1}}$ and the SNR is sufficiently high. Lemma \[lemma:latest\_omp\] also implies that OMP with *a priori* knowledge of $k_0$ is always HSC. Further, stopping conditions $\|{\bf r}^{(k)}\|_2<\sigma\sqrt{n+2\sqrt{n\log(n)}}$ [@cai2011orthogonal; @omp_rip_noise] or $\|{\bf X}^T{\bf r}^{(k)}\|_{\infty}<\sigma \sqrt{2\log(p)}$[@cai2011orthogonal] which fail to satisfy 4) and 5) of Lemma \[lemma:latest\_omp\] are inconsistent at high SNR.
Residual ratio techniques
=========================
As one can see from Lemma \[lemma:latest\_omp\], good finite SNR support recovery guarantees and HSC using OMP require either the *a priori* knowledge of $k_0$ or $\sigma^2$. However, as mentioned earlier, both $k_0$ and $\sigma^2$ are not available in most practical applications. Recently, we demonstrated in [@icml] that one can achieve high quality support recovery using OMP without the *a priori* knowledge of $\sigma^2$ or $k_0$ by using the properties of residual ratio statistic defined by $RR(k)=\dfrac{\|{\bf r}^k\|_2}{\|{\bf r}^{k-1}\|_2}$, where ${\bf r}^k=({\bf I}_n-{\bf P}_{k}){\bf y}$ is the residual corresponding to OMP support at the $k^{th}$ iteration, i.e., $\mathcal{S}_k$. The technique developed in [@icml] was based on the behaviour of $RR(k)$ for $k=1,2,\dotsc,k_{max}$, where $k_{max}\geq k_0$ is a fixed quantity independent of data. $k_{max}$ is a measure of the maximum sparsity level expected in a support recovery experiment. Since the maximum sparsity level upto which support recovery can be guaranteed for any sparse recovery algorithm (not just OMP) is $\floor{\dfrac{n+1}{2}}$, [@icml] suggests fixing $k_{max}=\floor{\dfrac{n+1}{2}}$. Note that this is a fixed value that is independent of the data and the algorithm (OMP or OLS) under consideration. The residual ratio statistic has many interesting properties as derived in [@icml]. Since the support sequence is monotonic i.e., $\mathcal{S}_k\subset \mathcal{S}_{k+1}$, the residual ${\bf r}^k$ is obtained by projecting ${\bf y}$ onto a subspace of decreasing dimension. Hence, $\|{\bf r}^{k+1}\|_2 \leq \|{\bf r}^k\|_2$ which inturn implies that $0\leq RR(k)\leq 1$. Please note that while residual norms are monotonically decreasing, residual ratios $RR(k)$ are not monotonic in $k$. A number of properties regarding the residual ratio statistic are based on the concept of minimal superset.
[**Definition 2:-**]{} The minimal superset in the OMP support sequence $\{\mathcal{S}_{k}\}_{k=1}^{k_{max}}$ is given by $\mathcal{S}_{k_{min}}$, where $k_{min}=\min\left(\{k:\mathcal{S}\subseteq \mathcal{S}_k\}\right)$. When the set $\{k:\mathcal{S}\subseteq \mathcal{S}_k\}=\emptyset$, we set $k_{min}=\infty$ and $\mathcal{S}_{k_{min}}=\phi$.
In words, minimal superset is the smallest superset of support $\mathcal{S}$ present in a particular realization of the support estimate sequence $\{\mathcal{S}_{k}\}_{k=1}^{k_{max}}$. Note that both $k_{min}$ and $\mathcal{S}_{k_{min}}$ are unobservable random variables. Since $card(\mathcal{S}_k)=k$ and $card(\mathcal{S})=k_0$, $\mathcal{S}_{k}$ for $k<k_0$ cannot satisfy $\mathcal{S}\subseteq \mathcal{S}_k$ and hence $k_{min}\geq k_0$. Further, the monotonicity of $\mathcal{S}_k$ implies that $\mathcal{S} \subset \mathcal{S}_k$ for all $k\geq k_{min}$.\
[**Case 1:-**]{} When $k_{min}=k_0$, then $\mathcal{S}_{k_0}=\mathcal{S}$ and $\mathcal{S}_{k}\supset \mathcal{S}$ for $k\geq k_0$, i.e., $\mathcal{S}$ is present in the solution path. Further, when $k_{min}=k_0$, it is true that $\mathcal{S}_k\subseteq \mathcal{S} $ for $k\leq k_0$.\
[**Case 2:-**]{} When $k_0<k_{min}\leq k_{max}$, then $\mathcal{S}_{k}\neq \mathcal{S}$ for all $k$ and $\mathcal{S}_{k}\supset \mathcal{S}$ for $k\geq k_{min}$, i.e., $\mathcal{S}$ is not present in the solution path. However, a superset of $\mathcal{S}$ is present.\
[**Case 3:-**]{} When $k_{min}=\infty$, then $\mathcal{S}_{k}\not \supseteq \mathcal{S}$ for all $k$, i.e., neither $\mathcal{S}$ nor a superset of $\mathcal{S}$ is present in $\{\mathcal{S}_{k}\}_{k=1}^{k_{max}}$.\
To summarize, exact support recovery using any OMP/OLS based scheme is possible only if $k_{min}=k_0$. Whenever $k_{min}>k_0$, it is possible to estimate true support $\mathcal{S}$ without having any false negatives. However, one then has to suffer from false positives. When $k_{min}=\infty$, any support in $\{\mathcal{S}_{k}\}_{k=1}^{k_{max}}$ has to suffer from false negatives and all supports $\mathcal{S}_{k}$ for $k>k_0-1$ has to suffer from false positives also. Note that the matrix and SNR conditions required for exact support recovery in statements 1) and 2) of Lemma \[lemma:latest\_omp\] implies that $k_{min}=k_0$ and $\mathcal{S}_{k_0}=\mathcal{S}$ at high SNR. The main distributional properties of residual ratio statistic are stated in the following lemma [@icml].
\[lemma:RR\_properties\] 1). Define $\Gamma_{RRT}^{\alpha}(k)=\sqrt{F^{-1}_{\frac{n-k}{2},\frac{1}{2}}\left(\dfrac{\alpha}{k_{max}(p-k+1)}\right)}$, where $F^{-1}_{a,b}(.)$ is the inverse function of the CDF $F_{a,b}(.)$ of a Beta R.V $\mathbb{B}(a,b)$. $0<\alpha<1$ is a fixed quantity independent of the data. Then under no assumption on the matrix ${\bf X}$ and for all $\sigma^2>0$, $RR(k)$ satisfies the following. $$\mathbb{P}(RR(k)>\Gamma_{RRT}^{\alpha}(k),\forall k>k_{min})\geq 1-\alpha.$$ 2). Suppose that the design matrix ${\bf X}$ satisfies a regularity condition which ensures that $k_{min}=k_0$ once $\|{\bf w}\|_2\leq \epsilon$ for some $\epsilon>0$ (for example, $\delta_{k_0+1}<1/\sqrt{k_0+1}$ and $\|{\bf w}\|_2\leq \epsilon_{omp}$ in Lemma \[lemma:latest\_omp\]). Then, $$\mathbb{P}(k_{min}=k_0)=\mathbb{P}(\mathcal{S}_{k_{min}}=\mathcal{S})\rightarrow 1\ \text{as}\ \sigma^2\rightarrow 0 \ \text{and}$$ $$RR(k_0)\overset{P}{\rightarrow} 0\ \text{as}\ \sigma^2\rightarrow 0.$$
Lemma \[lemma:RR\_properties\] implies that under appropriate matrix conditions and sufficiently high SNR, $RR(k)$ for $k>k_0$ will be higher than the positive quantity $\Gamma_{RRT}^{\alpha}(k)$ with a high probability, whereas, $RR(k_0)$ will be smaller than $\Gamma_{RRT}^{\alpha}(k_0)$. Consequently, the last index for which $RR(k)<\Gamma_{RRT}^{\alpha}(k)$ will be equal to the sparsity level $k_0$. This motivates the RRT support estimate given by $$\mathcal{S}_{RRT}=\mathcal{S}_{k_{RRT}} \ \text{where} \ k_{RRT}=\max\{k: RR(k)<\Gamma_{RRT}^{\alpha}(k)\}.$$ The performance guarantees for RRT are stated in Lemma \[lemma:icml\].
\[lemma:icml\] Let $k_{max}\geq k_0$ and suppose that the matrix ${\bf X}$ satisfies $\delta_{k_0+1}<\frac{1}{\sqrt{k_0+1}}$. Then [@icml],\
1). RRT can recover the true support $\mathcal{S}$ with probability greater than $1-1/n-\alpha$ provided that $\epsilon_{\sigma}<\min(\epsilon_{omp},\epsilon_{rrt})$, where $\epsilon_{omp}$ is given in Lemma \[lemma:latest\_omp\] and $$\epsilon_{rrt}=\dfrac{\Gamma_{RRT}^{\alpha}(k_0)\sqrt{1-\delta_{k_{0}}}\boldsymbol{\beta}_{min}}
{1+\Gamma_{RRT}^{\alpha}(k_0)}.$$ 2). $\underset{\sigma^2 \rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{k_{RRT}}\neq \mathcal{S})\leq \alpha.$\
3). $\mathbb{P}(\mathcal{S}_{k_{RRT}}\supseteq \mathcal{S})\leq \alpha$ in the moderate to high SNR regime (empirical result).
Lemma \[lemma:icml\] implies that RRT can identify the true support $\mathcal{S}$ under the same set of matrix conditions required by OMP with *a priori* knowledge of $k_0$ or $\sigma^2$, albeit at a slightly higher SNR. Further, the probability of support recovery error is upper bounded by $\alpha$ at high SNR. Even in the low to moderately high SNR regime, empirical results indicate that the probability of false discovery (i.e., $card(\mathcal{S}_{k_{RRT}}/\mathcal{S})>0$) is upper bounded by $\alpha$. Hence, in RRT, the hyper parameter $\alpha$ has an operational interpretation of being the high SNR support recovery error and finite SNR false discovery error. However, no lower bound on the probability of support recovery error at high SNR is reported in [@icml]. In the following lemma, we establish a novel high SNR lower bound on the probability of support recovery error for RRT.
\[lemma:inconsistency\] Suppose that the design matrix ${\bf X}$ satisfies the RIC condition $\delta_{k_0+1}<1/\sqrt{k_0+1}$, MIC or the ERC. Then $$\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{RRT}\supset \mathcal{S})\geq \dfrac{\alpha}{k_{max}(p-k_0)}.$$
Please see Appendix A.
In words, Lemma \[lemma:inconsistency\] implies that RRT is inconsistent at high SNR. [ In particular, Lemma \[lemma:inconsistency\] states that RRT suffers from false discoveries at high SNR. Indeed, one can reduce the lower bound on support recovery error by reducing the value of $\alpha$. Since $\Gamma_{RRT}^{\alpha}(k_0)$ is an increasing function of $\alpha$[@icml], reducing the value of $\alpha$ will result in decrease in $\epsilon_{rrt}$. Hence, a decrease in $\alpha$ to reduce the high SNR will result in an increase in the SNR required for accurate support recovery according to Lemma \[lemma:icml\]. In other words, it is impossible to improve the high SNR performance in RRT without compromising on the finite SNR performance. This is because of the fact that $\alpha$ is a user defined parameter that has to be set independent of the data. A good solution would be to use a value of $\alpha$ like $\alpha=0.1$ ( recommended in [@icml] based on finite SNR estimation performance) for low to moderate SNR and a low value of $\alpha$ like $\alpha=0.01$ or $\alpha=0.001$ in the high SNR regime. Since it is impossible to estimate SNR or $\sigma^2$ in underdetermined linear models, the statistician is unaware of operating SNR and cannot make such adaptations on $\alpha$. Hence, achieving very low values of PE at high SNR or HSC using RRT is extremely difficult. ]{} This motivates the novel RRM and RRTA algorithms discussed next which can achieve HSC using the residual ratio statistic itself.
Residual ratio minimization
---------------------------
The analysis of $RR(k)$ in [@icml] (see Lemma \[lemma:RR\_properties\]) discussed only the behaviour of $RR(k)$ for $k\geq k_{min}$. However, no analysis of $RR(k)$ for $k<k_{min}$ is mentioned in [@icml]. In the following lemma, we charecterize the behaviour of $RR(k)$ for $k<k_{min}$.
\[lemma:kless\] Suppose that the matrix ${\bf X}$ satisfies the RIC condition $\delta_{k_0+1}<1/\sqrt{k_0+1}$. Then, for $k<k_0$, $$\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}\left(RR(k)>\dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})}\right)=1.$$
Please see Appendix B.
Combining Lemma \[lemma:RR\_properties\] and Lemma \[lemma:kless\], one can see that with increasing SNR, $RR(k)$ for $k<k_0$ is bounded away from zero, whereas, $RR(k)$ for $k>k_0$ behave like a R.V that is bounded from below by a constant with a very high probability. In contrast to the behaviour of $RR(k)$ for $k\neq k_0$, $RR(k_0)$ converges to zero with increasing SNR. Consequently, under appropriate regularity conditions on the design matrix ${\bf X}$, $\underset{k}{\arg\min}RR(k)$ will converge to $k_0$ with increasing SNR. Also from Lemmas \[lemma:latest\_omp\] and \[lemma:RR\_properties\], we know that $k_{min}=k_0$ and $\mathcal{S}_{k_0}=\mathcal{S}$ with a very high probability at high SNR. Consequently, the support estimate given by $$\mathcal{S}_{RRM}=\mathcal{S}_{k_{RRM}}, \ \text{where} \ k_{RRM}= \underset{k=1,\dotsc,k_{max}}{\arg\min}RR(k)$$ will be equal to the true support $\mathcal{S}$ with a probability increasing with increasing SNR. $\mathcal{S}_{RRM}$ is the residual ratio minimization based support estimate proposed in this article. The following theorem states that RRM is a high SNR consistent estimator of support $\mathcal{S}$.
\[thm:rmm\_hsc\] Suppose that the matrix ${\bf X}$ satisfies RIC condition $\delta_{k_0+1}<\dfrac{1}{\sqrt{k_0+1}}$. Then RRM is high SNR consistent, i.e., $\underset{\sigma^2 \rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{RRM}=\mathcal{S})=1$.
Please see Appendix D.
While HSC is an important qualifier for any support recovery technique, it’s finite SNR performance is also very important. The following theorem quantifies the finite SNR performance of RRM.
\[thm:rrm\_finite\] Suppose that the design matrix ${\bf X}$ satisfies $\delta_{k_{0}+1}<\dfrac{1}{\sqrt{k_{0}+1}}$ and $0<\alpha<1$ is a constant. Then RRM can recover the true support $\mathcal{S}$ with a probability greater than $1-1/n-\alpha$ once $\epsilon_{\sigma}=\sigma\sqrt{n+2\sqrt{n\log(n)}}<\min\left(\epsilon_{omp},\tilde{\epsilon_{rrt}},\epsilon_{rrm}\right)$, where $\epsilon_{omp}$ is given in Lemma \[lemma:latest\_omp\], $$\label{epsb}
{\epsilon_{rrm}}= \dfrac{\sqrt{1-\delta_{k_{0}}}\boldsymbol{\beta}_{min}}{1+\frac{\sqrt{1+\delta_{k_{0}}}}{\sqrt{1-\delta_{k_{0}}}} \left(2+ \frac{\boldsymbol{\beta}_{max}}{\boldsymbol{\beta}_{min}} \right)} \ \text{and}$$ $$\label{epsb}
\tilde{\epsilon}_{rrt}=\dfrac{\underset{1\leq k\leq k_{max}}{\min}\Gamma_{RRT}^{\alpha}(k)\sqrt{1-\delta_{{k_{0}}}}\boldsymbol{\beta}_{min}}
{1+\underset{1\leq k\leq k_{max}}{\min}\Gamma_{RRT}^{\alpha}(k)}.$$
Please see Appendix C.
Please note that the presence of $\alpha$ in Theorem \[thm:rrm\_finite\] is an artefact of our analytical framework. Very importantly, unlike RRT, there are no user specified hyperparameters in RRM. The following remark compares the finite SNR support recovery guarantee for RRM in Theorem \[thm:rrm\_finite\] with that of RRT.
For the same $(1-\alpha-1/n)$ bound on the probability of error, RRM requires higher SNR level than RRT. This is true since $\tilde{\epsilon}_{rrt}=\dfrac{\underset{1\leq k\leq k_{max}}{\min}\Gamma_{RRT}^{\alpha}(k)\sqrt{1-\delta_{{k_{0}}}}\boldsymbol{\beta}_{min}}
{1+\underset{1\leq k\leq k_{max}}{\min}\Gamma_{RRT}^{\alpha}(k)}$ in Theorem \[thm:rrm\_finite\] is lower than the $\epsilon_{rrt}=\dfrac{\Gamma_{RRT}^{\alpha}(k_0)\sqrt{1-\delta_{{k_0}}}\boldsymbol{\beta}_{min}}
{1+\Gamma_{RRT}^{\alpha}(k_0)}$ of Lemma \[lemma:icml\] for RRT. Further, unlike RRT where the support recovery guarantee depends only on $\boldsymbol{\beta}_{min}$, the support recovery guarantee for RRM involves the term $DR(\boldsymbol{\beta})=\boldsymbol{\beta}_{max}/\boldsymbol{\beta}_{min}$. In particular, the SNR required for exact support recovery using RRM increases with increasing $DR(\boldsymbol{\beta})$. This limits the finite SNR utility of RRM for signals with high $DR(\boldsymbol{\beta})$.
The detiorating performance of RRM with increasing $DR(\boldsymbol{\beta})$ can be explained as follows. Note that both RRM and RRT try to identify the sudden decrease in $\|{\bf r}^k\|_2$ compared to $\|{\bf r}^{k-1}\|_2$ once $\mathcal{S}\subseteq \mathcal{S}_k$ for the first time, i.e., when $k=k_{min}$. This sudden decrease in $\|{\bf r}^k\|_2$ at $k=k_{min}$ is due to the removal of signal component in ${\bf r}^k$ at the $k_{min}^{th}$ iteration. However, a similar dip in $\|{\bf r}^k\|_2$ compared to $\|{\bf r}^{k-1}\|_2$ can also happens at a $k<k_{min}^{th}$ iteration if $\boldsymbol{\beta}_{t^k}$ ($t^k$ is the index selected by OMP in it’s $k^{th}$ iteration) contains most of the energy in the regression vector $\boldsymbol{\beta}$. This intermediate dip in $RR(k)$ can be more pronounced than the dip happening at the $k_{min}^{th}$ iteration when the SNR is moderate and $DR(\boldsymbol{\beta})$ is high. Consequently, the RRM estimate has a tendency to underestimate $k_{min}$ (i.e., $k_{RRM}<k_{min}$) when $DR(\boldsymbol{\beta})$ is high. Note that by Lemmas \[lemma:latest\_omp\] and \[lemma:RR\_properties\], $k_{min}=k_0$ and $\mathcal{S}_{k_{min}}=\mathcal{S}$ with a very high probability once $\epsilon_{\sigma}<\epsilon_{omp}$. Hence, this tendency of RRM to underestimate $k_{min}$ results in a support recovery error when $DR(\boldsymbol{\beta})$ is high. This behaviour of RRM is reflected in the higher SNR required for recovering the support of $\boldsymbol{\beta}$ with higher $DR(\boldsymbol{\beta})$. Please note that RRM will be consistent at high SNR irrespective of the value of $DR(\boldsymbol{\beta})$. Since RRT is looking for the “last" significant dip instead of the “most significant" dip in $RR(k)$, RRT is not affected by the variations in $DR(\boldsymbol{\beta})$.
Residual ratio thresholding with adaptation (RRTA)
--------------------------------------------------
As aforementioned, inspite of it’s HSC and noise statistics oblivious nature, RRM has poor finite SNR support recovery performance for signals with high dynamic range and good high SNR performance for all types of signals. In contrast to RRM, RRT has good finite SNR performance and inferior high SNR performance. This motivates the RRTA algorithm which tries to combine the strengths of both RRT and RRM to produce a high SNR consistent support recovery scheme that also has good finite SNR performance. Recall from Lemma \[lemma:RR\_properties\] that when the matrix ${\bf X}$ satisfies the RIC condition $\delta_{k_0+1}<\dfrac{1}{\sqrt{k_0+1}}$, then the minimum value of $RR(k)$ decreases to zero with increasing SNR. Also recall that the reason for the high SNR inconsistency of RRT lies in our inability to adapt the RRT hyperparameter $\alpha$ with respect to the operating SNR. In particular, for the high SNR consistency of RRT, we need to enable the adaptation $\alpha\rightarrow 0$ as $\sigma^2\rightarrow 0$ without knowing $\sigma^2$. Even though $\sigma^2$ is a parameter very difficult to estimate, given that the minimum value of $RR(k)$ decreases to zero with increasing SNR or decreasing $\sigma^2$, it is still possible to adapt $\alpha\rightarrow 0$ with increasing SNR by making $\alpha$ a monotonically increasing function of $\underset{k}{\min}RR(k)$. This is the proposed RRTA technique which “adapts" the $\alpha$ parameter in the RRT algorithm using $\underset{k}{\min}RR(k)$. The support estimate in RRTA can be formally expressed as $$\mathcal{S}_{RRTA}=\mathcal{S}_{k_{RRTA}},\ \text{where} \ k_{RRTA}=\max\{k:RR(k)<\Gamma_{RRT}^{\alpha*}(k)\}$$ $$\text{and} \ \ \alpha^*=\min\left(PFD_{finite},\underset{k}{\min}RR(k)^q\right)$$ for some $q>0$ and $PFD_{finite}>0$. RRTA algorithm has two user defined parameters,i.e., $PFD_{finite}$ and $q$ which control the finite SNR and high SNR behaviours respectively.
We first discuss the choice of hyperparameter $PFD_{finite}$. Note that $\underset{k}{\min}RR(k)^q$ will take small values only at high SNR. Hence, with a choice of $\alpha^*=\min\left(PFD_{finite},\underset{k}{\min}RR(k)^q\right)$, RRTA will operate like RRT with $\alpha=PFD_{finite}$ in the low to moderate high SNR regime, whereas, RRTA will operate like RRT with $\alpha=\underset{k}{\min}RR(k)^q$ in the high SNR regime. As discussed in Lemma \[lemma:icml\], $RRT$ with a date independent parameter $\alpha$ can control the probability of false discovery (PFD) in the finite SNR regime within $\alpha$. Hence, the user specified value of $PFD_{finite}$ specifies the maximum finite SNR probability of false discovery allowed in RRTA. This is a design choice. Following the empirical results and recommendations in [@icml] we set this parameter to $PFD_{finite}=0.1$.
As aforementioned, the choice of second hyperparameter $q$ determines the high SNR behaviour of RRTA. The following theorem specifies the requirements on the hyperparameter $q$ such that RRTA is a high SNR consistent estimator of the true support $\mathcal{S}$.
\[thm:RRTA\] Suppose that the matrix satisfies RIP condition of order $k_0+1$ and $\delta_{k_0+1}<1/\sqrt{k_0+1}$. Also suppose that $\alpha^{*}=\min\left(PFD_{finite}, \underset{k}{\arg\min}RR(k)^q\right)$ for some $q>0$. Then RRTA is high SNR consistent, i.e., $\underset{\sigma^2 \rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{RRTA}=\mathcal{S})=1$ for any fixed $PFD_{finite}>0$ once $n>k_0+q$.
Please see Appendix D.
Note that $\alpha^*=\min\left(PFD_{finite}, \underset{k}{\arg\min}RR(k)^q\right)$ is a monotonic function of $\underset{k}{\arg\min}RR(k)$ at high SNR for all values of $q>0$. However, the rate at which $\underset{k}{\arg\min}RR(k)^q$ decreases to zero increases with increasing values of $q$. The constriants $q>0$ and $q<n-k_0$ ensures that $\underset{k}{\arg\min}RR(k)^q$ should decrease to zero at a rate that is not too high. A very small value of $q$ implies that $\underset{k}{\arg\min}RR(k)^q$ will be greater than $PFD_{finite}=0.1$ for the entire operating SNR range thereby denying RRTA the required SNR adaptation, whereas, a very large value of $q$ implies that $\underset{k}{\arg\min}RR(k)^q<0.1$ even for low SNR. Operating RRT with a very low value of $\alpha$ at low SNR results in inferior performance. Hence, the choice of $q$ is important in RRTA. Through extensive numerical simulations, we observed that a value of $q=2$ delivers the best overall performance in the low and high SNR regimes. Such subjective choices are also involved in the noise variance aware HSC results developed in [@elsevier].
With the choice of $q=2$, the constraint $n>k_0+2$ has to be satisifed for the HSC of RRTA. Note that $k_0$ is unknown *a priori* and hence it is impossible to check the condition $n>k_0+2$. However, successfull sparse recovery using any sparse recovery technique requires $k_0<\floor{\frac{n+1}{2}}$ or equivalently $n>2k_0-1$. Note that $2k_0-1>k_0+2$ for all $k_0>3$. In addition to this, the regularity condition $\delta_{k_0+1}<\dfrac{1}{\sqrt{k_0+1}}$ required for sparse recovery using OMP will be satisfied in many widely used matrices ${\bf X}$ if $n=O\left(k_0^2\log(p)\right)$. Hence, the condition $n>k_0+2$ will be satisfied automatically in all problems where OMP is expected to carry out successfull sparse recovery.
Note that the poor performance of RRM with increasing $DR(\boldsymbol{\beta})$ is pronounced in the low to moderate SNR regime. Note that with $q=2$, RRTA works exactly like RRT with $\alpha=PFD_{finite}=0.1$ in the low to moderate SNR regime. Since, RRT is not affected by the value of $DR(\boldsymbol{\beta})$, RRTA also will not be affected by high $DR(\boldsymbol{\beta})$.
Note that we are setting $\alpha^*=\min\left(PFD_{finite},\underset{k}{\min}RR(k)^q\right)$ instead of $\alpha^*=\underset{k}{\min}RR(k)^q$. This is to ensure that RRTA work as RRT with parameter $\alpha^*=PFD_{finite}$ when the SNR is in the small to moderate regime. In other words, this particular form of $\alpha^*$ will help keep the values of $\alpha$ motivated by HSC arguments applicable only at high SNR and values of $\alpha$ motivated by finite SNR performance applicable to finite SNR situations.
Given the stochastic nature of the hyperparameter $\alpha^*$ in RRTA, it is difficult to derive finite SNR guarantees for RRTA. However, numerical simulations given in Section [slowromancap4@]{} indicate that RRTA delivers a performance very close to that of RRT with parameter $\alpha=0.1$ and OMP with *a priori* knowledge of $k_0$ or $\sigma^2$ in the low to moderately high SNR regime.
RRM and RRTA algorithms: A discussion
-------------------------------------
In this section, we compare and contrast RRM and RRTA algorithms proposed in this article with the results and algorithms discussed in existing literature. These comparisons are organized as seperate remarks. We first compare the RRTA and RRM algorithms with exisiting algorithms.
Note that RRT has a user specified parameter $\alpha$ which was set to $\alpha=0.1$ in [@icml] based on finite sample estimation and support recovery performance. This choice of $\alpha=0.1$ is also carried over to RRTA which uses $\alpha^*=\min\left(PFD_{finite},\underset{k}{\min}RR(k)^q \right)$ and $PFD_{finite}=0.1$. In addition to this, RRTA also involves a hyperparameter $q$ which is also set based on analytical results and empirical performance. In contrast, RRM does not involve any user specified parameter. Hence, while RRT and RRTA are signal and noise statistics oblivious algorithms, i.e., algorithms that does not require *a priori* knowledge of $k_0$ or $\sigma^2$ for efficient finite or high SNR operation, RRM is both signal and noise statistics oblivious and hyper parameter free. Recently, there has been a significant interest in the development of such hyper parameter free algorithms. Significant developments in this area of research include algorithms related to sparse inverse covariance estimator, *aka* SPICE [@spicenote; @spice_connection; @spice; @spice_like; @Stoica20141], sparse Bayesian learning (SBL)[@wipf2004sparse] etc. However, to the best of our knowledge, no explict finite sample and finite SNR support recovery guarantees are developed for SPICE, it’s variants or SBL. The HSC of these algorithms are also not discussed in literature. In contrast, RRM is a hyper parameter free algorithm which is high SNR consistent. Further, RRM has explicit finite sample and finite SNR support recovery guarantees. Also, for signals with low dynamic range, the performance of RRM is shown analytically to be comparable with OMP having *a priori* knowledge of $k_0$ or $\sigma^2$.
As aforementioned, all previous literature regarding HSC in low dimensional and high dimensional regression models were applicable only to situations where the noise variance $\sigma^2$ is known *a priori* or easily estimable. In contrast, RRM and RRTA can achieve high SNR consistency in underdetermined regression models even without requiring an estimate of $\sigma^2$. To the best of our knowledge, RRM and RRTA are the first and only noise statistics oblivious algorithms that are shown to be high SNR consistent in underdetermined regression models.
The HSC results developed in this article rely heavily on the bound $\mathbb{P}\left(RR(k)>\Gamma_{RRT}^{\alpha}(k),\forall k>k_{min}\right)\geq 1-\alpha$ in Lemma \[lemma:RR\_properties\]. This bound is valid iff the support sequence $\mathcal{S}_k$ is monotonic, i.e., $\mathcal{S}_k\subset\mathcal{S}_{k+1}$. Unfortunately, the support sequences produced by sparse recovery algorithms like LASSO, SP, CoSaMP etc. are not monotonic. Hence, the RRT technique in [@icml] and the RRM/RRTA techniques proposed in this article are not applicable to non monotonic algorithms like LASSO, SP, CoSaMP etc. Developing versions of RRT, RRM and RRTA that are applicable to non monotonic algorithms like LASSO, CoSaMP, SP etc. is an important direction for future research.
Next we discuss the matrix regularity conditions involved in deriving RRTA and RRM algorothms.
\[rem:mic\] Please note that the HSC of RRM and RRTA are derived assuming only the existence of a matrix regularity condition which ensures that $\mathcal{S}_{k_0}=\mathcal{S}$ once $\|{\bf w}\|_2\leq \epsilon$ for some $\epsilon>0$. RIC based regularity conditions are used in this article because they are the most widely used in analysing OMP. However, two other regularity conditions, viz., mutual incoherence condition (MIC) and exact recovery condition (ERC) are also used for analysing OMP. The mutual incoherence condition $\mu_{\bf X}=\underset{j\neq k}{\max}|{\bf X}_j^T{\bf X}_k| <\dfrac{1}{2k_0-1}$ along with $\|{\bf w}\|_2\leq \dfrac{\boldsymbol{\beta}_{min}(1-(2k_0-1)\mu_{\bf X})}{2} $ implies that $\mathcal{S}_{k_0}=\mathcal{S}$. Similarly, the exact recovery condition (ERC) $\underset{j \notin \mathcal{S}}{\max}\|{\bf X}_{\mathcal{S}}^{\dagger}{\bf X}_j\|_1<1$ along with $\|{\bf w}\|_2\leq \dfrac{\boldsymbol{\beta}_{min}\lambda_{min}(1-\|{\bf X}_{\mathcal{S}}^{\dagger}{\bf X}_j\|_1)}{2}$ also ensures that $\mathcal{S}_{k_0}=\mathcal{S}$ [@cai2011orthogonal]. Consequently, both RRTA and RRM are high SNR consistent once MIC or ERC are satisfied.
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\[rem:random\] When the matrix is generated by randomly sampling from a $\mathcal{N}(0,1)$ distribution, then for every $\delta\in (0,0.36)$, $n\geq ck_0\log\left(\dfrac{p}{\delta}\right)$ where $c\leq 20$ is a constant ensures that $\mathcal{S}_{k_0}=\mathcal{S}$ with a probability greater than $1-2\delta$ (when ${\bf w}={\bf 0}_n$) [@tropp2007signal]. Hence, even in the absence of noise ${\bf w}$, there is a fixed probability $\approx 2\delta$ that $\mathcal{S}_{k_0}\neq \mathcal{S}$. This result implies that even OMP running exactly $k_0$ iterations cannot recover the true support with arbitrary high probability as $\sigma^2\rightarrow 0$ in such situations. Consequently, no OMP based scheme can be HSC when the matrix is randomly generated. However, numerical simulations indicate that the PE of RRM/RRTA and OMP with *a priori* knowledge of $k_0$ match at high SNR, i.e., RRM achieves the best possible performance that can be delivered by OMP.
Numerical Simulations
=====================
In this section, we numerically verify the HSC results derived for RRM and RRTA. We also evaluate and compare the finite SNR performance of RRM and RRTA with respect to other popular OMP based support recovery schemes. We consider two models for the design matrix ${\bf X}$. Model 1 has ${\bf X}=[{\bf I}_n,{\bf H}_n]\in \mathbb{R}^{n \times 2n}$, i.e., ${\bf X}$ is formed by the concatenation of a $n\times n$ identity matrix and a $n \times n$ Hadamard matrix. This matrix has $\mu_{\bf X}=\dfrac{1}{\sqrt{n}}$[@elad_book]. Consequently, this matrix satisfies the MIC $\mu_{\bf X}<\dfrac{1}{2k_0-1}$ for all vectors $\boldsymbol{\beta}\in \mathbb{R}^{2n}$ with sparsity $k_0\leq \floor{\dfrac{1+\sqrt{n}}{2}}$. Model 2 is a $n \times p$ random matrix formed by sampling the entries independently from a $\mathcal{N}(0,1/n)$ distribution. For a given sparsity $k_0$, this matrix satisfies $\mathcal{S}_{k_0}=\mathcal{S}$ at high SNR with a high probability whenever $k_0 =O(\frac{n}{\log(p)})$. Please note that there is a nonzero probability that a random matrix fails to satisfy the regularity conditions required for support recovery. Hence, no OMP scheme is expected to be high SNR consistent in matrices of model 2. In both cases the dimensions are set as $n=32$ and $p=2n=64$.
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Next we discuss the regression vector $\boldsymbol{\beta}$ used in our experiments. The support $\mathcal{S}$ is obtained by randomly sampling $k_0$ entries from the set $\{1,2,\dotsc,p\}$. Sparsity level $k_0$ is fixed at $k_0=3$. Since the performance of RRM is influenced by $DR(\boldsymbol{\beta})$, we consider two types of regression vectors $\boldsymbol{\beta}$ in our experiments with significantly different values of $DR(\boldsymbol{\beta})$. For the first type of $\boldsymbol{\beta}$, the non zero entries are randomly assigned $\pm 1$. When $\boldsymbol{\beta}_j=\pm 1$ for $j\in \mathcal{S}$, $DR(\boldsymbol{\beta})$ is at it’s lowest value, i.e., one. For the second type of $\boldsymbol{\beta}$, the non zero entries are sampled from the set $\{1,1/3,\dotsc,1/3^{k_0-1}\}$ without resampling. Here the dynamic range $DR(\boldsymbol{\beta})=3^{k_0-1}=9$ is very high. All results presented in this section are obtained after performing $10^4$ Monte carlo iterations.
Algorithms under consideration
------------------------------
Figures \[fig:pm1\]-\[fig:effectq\] present the $PE$ versus $SNR$ curves for the four possible combinations of design matrix/regression vectors discussed earlier. OMP($k_0$) in Fig \[fig:pm1\]-\[fig:effectq\] represent the OMP scheme that performs exactly $k_0$ iterations. OMP with residual power stopping condition (RPSC) stops iterations once $\|{\bf r}^{(k)}\|_2\leq \sigma\Gamma_1$. RPSC in Fig.\[fig:pm1\]-\[fig:decay\] represent this scheme with $\Gamma_1=\sqrt{n+2\sqrt{n\log(n)}}$[@cai2011orthogonal]. OMP with residual correlation stopping condition (RCSC) stops iterations $\|{\bf X}^T{\bf r}^{(k)}\|_{\infty}\leq \sigma\Gamma_2$. RCSC in Fig.\[fig:pm1\]-\[fig:decay\] represent this scheme with $\Gamma_2=\sqrt{2\log(p)}$. RPSC-HSC and RCSC-HSC represent RPSC and RCSC with $\Gamma_1=\dfrac{1}{\sigma^{\eta}}\sqrt{n+2\sqrt{n\log(n)}}$ and $\Gamma_2=\dfrac{1}{\sigma^{\eta}}\sqrt{2\log(p)}$ respectively[@cai2011orthogonal]. By Lemma \[lemma:latest\_omp\], both RPSC-HSC and RCSC-HSC are high SNR consistent once $0<\eta<1$[@elsevier]. In our simulations, we set $\eta= 0.1$ as suggested in [@elsevier]. RRT($\alpha$) in Fig.\[fig:pm1\]-\[fig:decay\] represent RRT with $\alpha=0.1$ and $\alpha=0.01$. RRTA in Fig.\[fig:pm1\]-\[fig:decay\] represent RRTA with $\alpha^*=\min(0.1,\underset{k}{\min}RR(k)^2)$, i.e., the parameters $PFD_{finite}$ and $q$ are set to $0.1$ and $2$ respectively. RRTA($q= $) in Fig.\[fig:effectq\] represents RRTA with $\alpha^*=\min(0.1,\underset{k}{\min}RR(k)^q)$.
Comparison of RRM/RRTA with existing OMP based support recovery techniques
--------------------------------------------------------------------------
Fig.\[fig:pm1\] presents the PE performance of algorithms when $DR(\boldsymbol{\beta})$ is low. When the condition $\mu_{\bf X}<\dfrac{1}{2k_0-1}$ is met, one can see from Fig.\[fig:pm1\] that the PE of RRM, RRTA, RCSC-HSC, RPSC-HSC and OMP($k_0$) decreases to zero with increasing SNR. This validates the claims made in Lemma \[lemma:latest\_omp\], Theorem \[thm:rmm\_hsc\] and Theorem \[thm:RRTA\]. Please note that unlike OMP($k_0$) which has *a priori* knowledge of $k_0$ and RCSC-HSC and RPSC-HSC with *a priori* knowledge of $\sigma^2$, RRM and RRTA achieve HSC without having *a priori* knowledge of either signal or noise statistics. OMP($k_0$) achieves the best PE performance, whereas, the performance of other schmes are comparable to each other in the low to moderate SNR. This validates the claim made in Theorem \[thm:rrm\_finite\] that RRM performs similar to the noise statistics aware OMP schemes when $DR(\boldsymbol{\beta})$ is low. However, at high SNR, the rate at which the PE of RRTA converges to zero is lower than the rate at which the PE of RRM, RPSC-HSC etc. decrease to zero. PE of RPSC, RCSC and RRT are close to OMP($k_0$) at low SNR. However, the PE versus SNR curves of these algorithms exhibit a tendency to floor with increasing SNR resulting in high SNR inconsistency. When the design matrix ${\bf X}$ is randomly generated, no OMP based scheme achieves HSC. However, the PE level at which RRTA, RRM etc. floor is same as the PE level at which OMP($k_0$), RPSC-HSC and RCSC-HSC floor. In other words, when HSC is not achievable, RRTA and RRM will deliver a high SNR PE performance similar to the signal and noise statistics aware OMP schemes.
Next we consider the performance of algorithms when $DR(\boldsymbol{\beta})$ is high. Comparing Fig.\[fig:pm1\] and Fig.\[fig:decay\], one can see that the $PE$ versus SNR curves for all algorithms shift towards the high SNR region with increasing $DR(\boldsymbol{\beta})$. Note that for a fixed SNR, $\boldsymbol{\beta}_{min}/\sigma$ decreases with increasing $DR(\boldsymbol{\beta})$. Since all OMP based schemes require $\boldsymbol{\beta}_{min}/\sigma$ to be sufficiently high, the relatively poor performance with increasing $DR(\boldsymbol{\beta})$ is expected. However, as one can see from Fig.\[fig:decay\], the deterioration in performance with increasing $DR(\boldsymbol{\beta})$ is very severe in RRM compared to other OMP based algorithms. This verifies the finite sample results derived in Theorem \[thm:rrm\_finite\] for RRM which states that RRM has poor finite SNR performance when $DR(\boldsymbol{\beta})$ is high. Note that the performance of RRTA with increased $DR(\boldsymbol{\beta})$ is similar to that of OMP($k_0$), RPSC, RCSC etc. in the finite SNR regime. This implies that unlike RRM, the performance of RRTA depends only on $\boldsymbol{\beta}_{min}$ and not $DR(\boldsymbol{\beta})$.
Effect of parameter $q$ on RRTA performance
-------------------------------------------
Theorem \[thm:RRTA\] states that RRTA with all values of $q$ satisfying $0<q<n-k_0$ are high SNR consistent. However, the finite SNR performance of RRTA with different values of $q$ will be different. In Fig.\[fig:effectq\], we evaluate the performance of RRTA for different values of $q$. As one can see from Fig.\[fig:effectq\], the rate at which PE decreases to zero with increasing SNR becomes faster with the increase in $q$. The rate at which the PE of RRTA with $q=10$ decreases to zero is similar to the steep decrease seen in the PE versus SNR curves of signal and noise statistics aware schemes like OMP($k_0$), RPSC-HSC, RCSC-HSC etc. In contrast, the rate at which the PE of RRTA with $q=1$ decreases to zero is not steep. RRTA with $q=2$ exhibit a much steeper PE versus SNR curve. However, as one can see from the R.H.S of Fig.\[fig:effectq\], the finite SNR performance of RRTA with $q=1$ and $q=2$ is better than that of RRTA with $q=5$, $q=10$ etc. In other words, RRTA with a larger value of $q$ can potentially yield a better PE than RRTA with a smaller value of $q$ in the high SNR regime. However, this come at the cost of an increased PE in the low to medium SNR regime. Since the objective of any good HSC scheme should be to achieve HSC while guaranteeing good finite SNR performance, one can argue that $q=2$ is a good design choice for the hyperparameter in RRTA.
Conclusion
==========
This article proposes two novel techniques called RRM and RRTA to operate OMP without the *a priori* knowledge of signal sparsity or noise variance. RRM is hyperparameter free in the sense that it does not require any user specified tuning parameters, whereas, RRTA involve hyperparameters. We analytically establish the HSC of both RRM and RRTA. Further, we also derive finite SNR guarantees for RRM. [ Numerical simulations also verify the HSC of RRM and RRTA. RRM and RRTA are the first signal and noise statistics oblivious techniques to report HSC in underdetermined regression models. ]{}
Appendix A: Proof of Lemma \[lemma:inconsistency\] {#appendix-a-proof-of-lemma-lemmainconsistency .unnumbered}
===================================================
The event $\mathcal{S}_{RRT}\supset \mathcal{S}$ in RRT estimate $k_{RRT}=\max\{k:RR(k)<\Gamma_{RRT}^{\alpha}(k)\}$ is true once $\exists k>k_{min}$ such that $RR(k)<\Gamma_{RRT}^{\alpha}(k)$. Hence, $$\label{eq:inconsistency1}
\mathbb{P}(\mathcal{S}_{RRT}\supset \mathcal{S}) \geq \mathbb{P}\left(\underset{k>k_{min}}{\bigcup}\{RR(k)<\Gamma_{RRT}^{\alpha}(k)\}\right).$$ One can further bound (\[eq:inconsistency1\]) as follows. $$\label{eq:inconsistency2}
\begin{array}{ll}
\mathbb{P}(\mathcal{S}_{RRT}\supset \mathcal{S}) \overset{(a)}{\geq} \mathbb{P}\big(\{RR(k_{min}+1)<\Gamma_{RRT}^{\alpha}(k_{min}+1)\} \big) \\ \overset{(b)}{\geq} \mathbb{P}\big(\{RR(k_{min}+1)<\Gamma_{RRT}^{\alpha}(k_{min}+1)\}\cap \{k_{min}=k_0\}\big) \\
= \mathbb{P}\left(\{RR(k_{min}+1)<\Gamma_{RRT}^{\alpha}(k_{min}+1)\} | \{k_{min}=k_0\}\right) \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathbb{P}(k_{min}=k_0).
\end{array}$$ [ (a) of \[eq:inconsistency2\] follows from the union bound $\mathbb{P}(A \cup B)\geq \mathbb{P}(A)$ and (b) follows from the intersection bound $\mathbb{P}(A\cap B)\leq \mathbb{P}(A)$.]{} Following the proof of Theorem 1 in [@icml], we know that conditional on $k_{min}=j$, for each $k>j$, $RR(k)<Z_k$ where $Z_k \sim \mathbb{B}(\frac{n-k}{2},\frac{1}{2})$. Applying this distributional result in $\mathbb{P}\left(\{RR(k_{min}+1)<\Gamma_{RRT}^{\alpha}(k_{min}+1)\}\right)$ gives $$\label{eq:inconsistency3}
\begin{array}{ll}
\mathbb{P}\left(\{RR(k_{min}+1)<\Gamma_{RRT}^{\alpha}(k_{min}+1)\}\right) \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \geq \mathbb{P}\big(\{Z_{k_0+1} < \Gamma_{RRT}^{\alpha}(k_0+1)\}\big)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ =F_{\frac{n-k_0-1}{2},\frac{1}{2}}\left(\Gamma_{RRT}^{\alpha}(k_0+1)\right)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ =F_{\frac{n-k_0-1}{2},\frac{1}{2}}\left(F^{-1}_{\frac{n-k_0-1}{2},\frac{1}{2}}\left(\frac{\alpha}{k_{max}(p-k_0)}\right)\right)\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\alpha}{k_{max}(p-k_0)}
\end{array}$$ Using Lemma \[lemma:latest\_omp\], we know that $k_{min}=k_0$ once $\|{\bf w}\|_2\leq\epsilon_{omp}$. Hence, $\mathbb{P}(k_{min}=k_0)\geq \mathbb{P}(\|{\bf w}\|_2\leq \epsilon_{omp})$. Since ${\bf w} \overset{P}{\rightarrow} {\bf 0}_n$ as $\sigma^2\rightarrow 0$ for ${\bf w}\sim \mathcal{N}({\bf 0}_n,\sigma^2{\bf I}_n)$, we have $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\|{\bf w}\|_2\leq \epsilon_{omp})=1$ and $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(k_{min}=k_0)=1$. Substituting $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(k_{min}=k_0)=1$ and (\[eq:inconsistency3\]) in (\[eq:inconsistency2\]) gives $\underset{\sigma^2 \rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{RRT}\supset \mathcal{S})\geq \frac{\alpha}{k_{max}(p-k_0)}$.
Appendix B: Proof of Lemma \[lemma:kless\] {#appendix-b-proof-of-lemma-lemmakless .unnumbered}
==========================================
By Lemma \[lemma:latest\_omp\], we have $k_{min}=k_0$ and $\mathcal{S}_{k_0}=\mathcal{S}$ once $\|{\bf w}\|_2\leq \epsilon_{omp}$. This along with the monotonicity of $\mathcal{S}_k$ implies that $\mathcal{S}_k\subset \mathcal{S}$ for each $k<k_0$. We analyse $RR(k)$ assuming that $\|{\bf w}\|_2\leq \epsilon_{omp}$. Applying the triangle inequality $\|{\bf a}+{\bf b}\|_2\leq \|{\bf a}\|_2+\|{\bf b}\|_2$, the reverse triangle inequality $\|{\bf a}+{\bf b}\|_2\geq \|{\bf a}\|_2-\|{\bf b}\|_2$ and the bound $\|({\bf I}_n-{\bf P}_{k}){\bf w}\|_2\leq \|{\bf w}\|_2 $ to $\|{\bf r}^{k}\|_2=\|({\bf I}_n-{\bf P}_{k}){\bf X}\boldsymbol{\beta}+({\bf I}_n-{\bf P}_{k}){\bf w}\|_2$ gives $$\label{aaa1}
\|({\bf I}_n-{\bf P}_{k}){\bf X}\boldsymbol{\beta}\|_2-\|{\bf w}\|_2\leq \|{\bf r}^{k}\|_2\leq \|({\bf I}_n-{\bf P}_{k}){\bf X}\boldsymbol{\beta}\|_2+\|{\bf w}\|_2.$$ Let ${u^k}=\mathcal{S}/\mathcal{S}_k$ denotes the indices in $\mathcal{S}$ that are not selected after the $k^{th}$ iteration. Note that ${\bf X}\boldsymbol{\beta}={\bf X}_{\mathcal{S}}\boldsymbol{\beta}_{\mathcal{S}}={\bf X}_{\mathcal{S}_k}\boldsymbol{\beta}_{\mathcal{S}_k}+{\bf X}_{u^k}\boldsymbol{\beta}_{u^k}$. Since, ${\bf I}_n-{\bf P}_k$ projects orthogonal to the column space $span({\bf X}_{\mathcal{S}_k})$, $({\bf I}_n-{\bf P}_{k}){\bf X}_{\mathcal{S}_k}\boldsymbol{\beta}_{\mathcal{S}_k}={\bf 0}_n$. Hence, $({\bf I}_n-{\bf P}_{k}){\bf X}\boldsymbol{\beta}=({\bf I}_n-{\bf P}_{k}){\bf X}_{u^k}\boldsymbol{\beta}_{u^k}$. Further, $\mathcal{S}_k\subset \mathcal{S}$ implies that $card(\mathcal{S}_k)+card(u^k)= k_0$ and $\mathcal{S}_k\cap u^k=\phi$. Hence, by Lemma 2 of [@latest_omp], $$\label{aaa2}
\sqrt{1-\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2\leq \|({\bf I}_n-{\bf P}_{k}){\bf X}_{u^k}\boldsymbol{\beta}_{u^k}\|_2\leq \sqrt{1+\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2.$$ Substituting (\[aaa2\]) in (\[aaa1\]) gives $$\label{Caibound}
\sqrt{1-\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2-\|{\bf w}\|_2\leq\|{\bf r}^{k}\|_2\leq \sqrt{1+\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2+\|{\bf w}\|_2.$$ Note that $\boldsymbol{\beta}_{u^{k-1}}=\boldsymbol{\beta}_{u^{k}}+\boldsymbol{\beta}_{u^{k-1}/u^{k}}$ after appending enough zeros in appropriate locations. $\boldsymbol{\beta}_{u^{k-1}/u^{k}}$ has only one non zero entry. Hence, $\|\boldsymbol{\beta}_{u^{k-1}/u^{k}}\|_2\leq \boldsymbol{\beta}_{max}$. Applying triangle inequality to $\boldsymbol{\beta}_{u^{k-1}}=\boldsymbol{\beta}_{u^{k}}+\boldsymbol{\beta}_{u^{k-1}/u^{k}}$ gives the bound $$\label{temp_bound}
\|\boldsymbol{\beta}_{u^{k-1}}\|_2\leq \|\boldsymbol{\beta}_{u^{k}}\|_2+\|\boldsymbol{\beta}_{u^{k-1}/u^{k}}\|_2 \leq \|\boldsymbol{\beta}_{u^{k}}\|_2+ \boldsymbol{\beta}_{max}$$ Applying (\[temp\_bound\]) and (\[Caibound\]) in $RR(k)$ for $k<k_0$ gives $$\label{A1bound}
\begin{array}{ll}
RR(k)=\dfrac{\|{\bf r}^{k}\|_2}{\|{\bf r}^{k-1}\|_2} &\geq \dfrac{\sqrt{1-\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}\|\boldsymbol{\beta}_{u^{k-1}}\|_2+\|{\bf w}\|_2}\\
&\geq \dfrac{\sqrt{1-\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}\left[\|\boldsymbol{\beta}_{u^{k}}\|_2+\boldsymbol{\beta}_{max}\right]+\|{\bf w}\|_2}\\
\end{array}$$ whenever $\|{\bf w}\|_2\leq \epsilon_{omp}$. The R.H.S of (\[A1bound\]) can be rewritten as $$\label{A1bound2}
\begin{array}{ll}
\dfrac{\sqrt{1-\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}
\left[\|\boldsymbol{\beta}_{u^{k}}\|_2+\boldsymbol{\beta}_{max}\right]+\|{\bf w}\|_2}=\dfrac{\sqrt{1-\delta_{k_0}}}{\sqrt{1+\delta_{k_0}}} \\ \ \ \ \ \ \ \ \ \ \ \
\left(1-\dfrac{\dfrac{\|{\bf w}\|_2}{\sqrt{1-\delta_{k_0}}}+\dfrac{\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}}+\boldsymbol{\beta}_{max}}{\|\boldsymbol{\beta}_{u^{k}}\|_2+\boldsymbol{\beta}_{max}+\dfrac{\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}}}\right)
\end{array}$$ From (\[A1bound2\]) it is clear that the R.H.S of (\[A1bound\]) decreases with decreasing $\|\boldsymbol{\beta}_{u^k}\|_2$. Note that the minimum value of $\|\boldsymbol{\beta}_{u^{k}}\|_2$ is $\boldsymbol{\beta}_{min}$ itself. Hence, substituting $\|\boldsymbol{\beta}_{u^{k}}\|_2\geq \boldsymbol{\beta}_{min}$ in (\[A1bound\]) gives $$\label{lb_on_klessk0_temp}
RR(k)\geq \dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})+\|{\bf w}\|_2},$$ $\forall k<k_0$ whenever $\|{\bf w}\|_2<\epsilon_{omp}$. This along with the fact $RR(k)\geq 0$ implies that $$RR(k)\geq \dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})+\|{\bf w}\|_2} \mathcal{I}_{\{\|{\bf w}\|_2\leq \epsilon_{omp}\}},$$ where $\mathcal{I}_{\{\mathcal{E}\}}$ is the indicator function returning one when the event $\mathcal{E}$ occurs and zero otherwise. Note that $\|{\bf w}\|_2\overset{P}{\rightarrow }0$ as $\sigma^2 \rightarrow 0$. This implies that $\dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})+\|{\bf w}\|_2} \overset{P}{\rightarrow } \dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})}$ and $\mathcal{I}_{\{\|{\bf w}\|_2\leq \epsilon_{omp}\}} \overset{P}{\rightarrow}1$. Substituting these bounds in (\[lb\_on\_klessk0\_temp\]) one can obtain $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}\left(RR(k)>\dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})}\right)=1$.
Appendix C: Proof of Theorem \[thm:rrm\_finite\] {#appendix-c-proof-of-theorem-thmrrm_finite .unnumbered}
=================================================
For RRM support estimate $\mathcal{S}_{RRM}=\mathcal{S}_{k_{RRM}}$ where $k_{RRM}=\underset{k}{\arg\min}RR(k)$ to satisfy $\mathcal{S}_{RRM}=\mathcal{S}$, it is sufficient that the following four events $\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{A}_3$ and $\mathcal{A}_4$ occur simultaneously.\
$\mathcal{A}_1=\{k_{min}=k_{0}\}$.\
$\mathcal{A}_2=\{RR(k)>RR(k_0),\forall k<k_{0}\}$.\
$\mathcal{A}_3=\{RR(k_{0})<\underset{j=1,\dotsc,k_{max}}{\min}\Gamma_{RRT}^{\alpha}(j)\}$.\
$\mathcal{A}_4=\{RR(k)>\underset{j=1,\dotsc,k_{max}}{\min}\Gamma_{RRT}^{\alpha}(j),\forall k>k_{min} \}$.
This is explained as follows. Event $\mathcal{A}_1$ true implies that $\mathcal{S}_{k_{0}}=\mathcal{S}$ and $k_{min}=k_{0}$. $\mathcal{A}_1 \cap \mathcal{A}_2$ is true implies that $k_{RRM}\geq k_{min}$, i.e., $k_{RRM}$ will not underestimate $k_{min}$. Event $\mathcal{A}_3\cap \mathcal{A}_4$ implies that $RR(k_{min})>RR(k)$ for all $k>k_{min}$ which ensures that $k_{RRM}\leq k_{min}$, i.e., $k_{RRM}$ will not overestimate $k_{min}$. Hence, $\mathcal{A}_2\cap \mathcal{A}_3\cap \mathcal{A}_4$ implies that $k_{RRM}=k_{min}$. This together with $\mathcal{A}_1$ implies that $k_{RRM}=k_0$ and $\mathcal{S}_{RRM}=\mathcal{S}$. Hence, $\mathbb{P}(\mathcal{S}_{RRM}=\mathcal{S})\geq \P\left(\mathcal{A}_1\cap \mathcal{A}_2\cap \mathcal{A}_3\cap \mathcal{A}_4\right)$.
By Lemma \[lemma:latest\_omp\], it is true that $\mathcal{A}_1=\{k_{min}=k_{0}\}$ is true once $\|{\bf w}\|_2\leq \epsilon_{omp}$. [ Using the bound $\sqrt{1-\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2-\|{\bf w}\|_2\leq\|{\bf r}^{k}\|_2\leq \sqrt{1+\delta_{k_0}}\|\boldsymbol{\beta}_{u^k}\|_2+\|{\bf w}\|_2$ from (\[Caibound\]) in the proof of Lemma \[lemma:kless\] and the fact that $u_{k_0}=\emptyset$, we have $\|{\bf r}^{k_0}\|_2\leq \|{\bf w}\|_2$ and $ \|{\bf r}^{k_0-1}\|_2 \geq \sqrt{1-\delta_{k_0}}\|\boldsymbol{\beta}_{u^{k_0-1}}\|_2-\|{\bf w}\|_2 \geq \sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2 $. ]{} Substituting these bounds in $RR(k_0)=\frac{\|{\bf r}^{k_0}\|_2}{\|{\bf r}^{k_0-1}\|_2}$ gives $$RR(k_{0})=RR(k_{min})\leq \dfrac{\|{\bf w}\|_2}{\sqrt{1-\delta_{k_{0}}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2},$$ once $\|{\bf w}\|_2 \leq \epsilon_{omp}$. Hence the event $\mathcal{A}_3$, i.e., $RR(k_{0})<\underset{j=1,\dotsc,k_{max}}{\min}\Gamma_{RRT}^{\alpha}(j)$ is true once $$\dfrac{\|{\bf w}\|_2}{\sqrt{1-\delta_{k_{0}}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2}<\underset{j=1,\dotsc,k_{max}}{\min}\Gamma_{RRT}^{\alpha}(j)$$ which in turn is true once $\|{\bf w}\|_2\leq \min(\epsilon_{omp},\tilde{\epsilon}_{rrt})$.
Next we consider the event $\mathcal{A}_2$. From (\[lb\_on\_klessk0\_temp\]), we have $RR(k)\geq \dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})+\|{\bf w}\|_2}, \ \forall k<k_0$ whenever $\|{\bf w}\|_2<\epsilon_{omp}$. Hence, $RR(k_{0})<\underset{k<k_{0}}{\min}RR(k)$ is true once the lower bound $\dfrac{\sqrt{1-\delta_{k_0}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2}{\sqrt{1+\delta_{k_0}}(\boldsymbol{\beta}_{max}+\boldsymbol{\beta}_{min})+\|{\bf w}\|_2}$ on $RR(k)$ for $k<k_0$ is higher than the upper bound $\dfrac{\|{\bf w}\|_2}{\sqrt{1-\delta_{k_{0}}}\boldsymbol{\beta}_{min}-\|{\bf w}\|_2}$ on $RR(k_0)$. This is true once $\|{\bf w}\|_2\leq \min(\epsilon_{omp},\epsilon_{rrm})$. Consequently, events $\mathcal{A}_1\cap\mathcal{A}_2\cap\mathcal{A}_3$ occur simultaneously once $\|{\bf w}\|_2\leq \min(\epsilon_{omp},\tilde{\epsilon_{rrt}},\epsilon_{rrm})$. Since $\epsilon_{\sigma}=\sigma\sqrt{n+2\sqrt{n\log(n)}}$ satisfies $\P(\|{\bf w}\|_2\leq \epsilon_{\sigma})\geq 1-1/n$, it is true that $\P(\mathcal{A}_1\cap\mathcal{A}_2\cap\mathcal{A}_3)\geq 1-1/n$ once $\epsilon_{\sigma}\leq \min(\epsilon_{omp},\tilde{\epsilon_{rrt}},\epsilon_{rrm})$.
Next we consider the event $\mathcal{A}_4$. Following Lemma \[lemma:RR\_properties\], it is true that $$\begin{array}{ll}
\mathbb{P}(RR(k)>\underset{j=1,\dotsc,k_{max}}{\min}\Gamma_{RRT}^{\alpha}(j),\forall k>k_{min})\\
\ \ \ \ \ \ \ \ \ \ \ \geq \P(RR(k)>\Gamma_{RRT}^{\alpha}(k),\forall k>k_{min})\geq 1-\alpha,
\end{array}$$ for all $\sigma^2>0$. Hence, the event $\mathcal{A}_4$ occurs with probability atleast $1-\alpha$, $\forall \sigma^2>0$.
Combining all these results give $\mathbb{P}(\mathcal{S}_{RRM}=\mathcal{S})\geq 1-1/n-\alpha$ whenever $\epsilon_{\sigma}\leq \min(\tilde{\epsilon_{rrt}},\epsilon_{rrm},\epsilon_{omp})$.
Appendix D: Proof of Theorem \[thm:rmm\_hsc\] {#appendix-d-proof-of-theorem-thmrmm_hsc .unnumbered}
==============================================
To prove that $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{RRM}=\mathcal{S})=1$, it is sufficient to show that for every fixed $\delta>0$, there exists a $\sigma^2_{\delta}>0$ such that $\mathbb{P}(\mathcal{S}_{RRM}=\mathcal{S})\geq 1-\delta$ for all $\sigma^2<\sigma^2_{\delta}$. Consider the events $\{\mathcal{A}_j\}_{j=1}^4$ with the same definition as in Appendix C. Then $\mathbb{P}(\mathcal{S}_{RRM}=\mathcal{S})\geq \mathbb{P}(\mathcal{A}_1\cap \mathcal{A}_2\cap \mathcal{A}_3\cap \mathcal{A}_4)$. Let $\delta>0$ be any given number. Fix the alpha parameter $\alpha=\frac{\delta}{2}$. Applying Lemma \[lemma:RR\_properties\] with $\alpha=\frac{\delta}{2}$ gives the bound $$\label{eq:rrm_hsc1}
\mathbb{P}(\mathcal{A}_4)=\mathbb{P}(RR(k)>\underset{j=1,\dotsc,k_{max}}{\min}\Gamma_{RRT}^{\frac{\delta}{2}}(j), \forall k>k_{min})\geq 1-\frac{\delta}{2},$$ for all $\sigma^2>0$. Following the proof of Theorem \[thm:rrm\_finite\], we have $$\mathbb{P}(\mathcal{A}_1\cap \mathcal{A}_2\cap \mathcal{A}_3)\geq \mathbb{P}\left(\|{\bf w}\|_2\leq \min(\epsilon_{rrm},\tilde{\epsilon}_{rrt},\epsilon_{omp})\right).$$ [ Note that both $\epsilon_{rrm}>0$ and $\epsilon_{omp}>0$ are both independent of $\alpha$ and hence $\delta$. At the same time, $\tilde{\epsilon}_{rrt}=\dfrac{\underset{1<k\leq k_{max}}{\min}\Gamma_{RRT}^{\alpha}(k)\sqrt{1-\delta_{{k_{0}}}}\boldsymbol{\beta}_{min}}
{1+\underset{1\leq k\leq k_{max}}{\min}\Gamma_{RRT}^{\alpha}(k)}$ is dependent on $\alpha$ and hence $\delta$. Since $\mathbb{B}(a,b)$ is a continuous random variable with support in $(0,1)$, for every $z>0$, $a>0$ and $b>0$ , $F^{-1}_{a,b}(z)>0$. Hence, for each $\delta>0$, $\Gamma_{RRT}^{\frac{\delta}{2}}(k)>0$ which implies that $\tilde{\epsilon}_{rrt}>0$. This inturn implies that $\min(\epsilon_{rrm},\tilde{\epsilon}_{rrt},\epsilon_{omp})>0$.]{} Note that $\|{\bf w}\|_2\overset{P}{\rightarrow }0$ as $\sigma^2\rightarrow 0$. This implies that for every fixed $\delta>0$, $\exists \sigma^2(\delta)>0$ such that $$\label{eq:rrm_hsc2}
\begin{array}{ll}
\mathbb{P}(\mathcal{A}_1\cap \mathcal{A}_2\cap \mathcal{A}_3)&\geq \mathbb{P}\left(\|{\bf w}\|_2\leq \min(\epsilon_{rrm},\tilde{\epsilon}_{rrt},\epsilon_{omp})\right)\\
&\geq 1-\frac{\delta}{2}
\end{array}$$ for all $\sigma^2<\sigma^2(\delta)$. Combining (\[eq:rrm\_hsc1\]) and (\[eq:rrm\_hsc2\]), one can obtain $\mathbb{P}(\mathcal{A}_1\cap \mathcal{A}_2\cap \mathcal{A}_3\cap \mathcal{A}_4)\geq 1-\delta$ for all $\sigma^2<\sigma^2(\delta)$. Since this is true for all $\delta>0$, we have $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(S_{RRM}=\mathcal{S})=1$.
Appendix E: Proof of Theorem \[thm:RRTA\] {#appendix-e-proof-of-theorem-thmrrta .unnumbered}
==========================================
[Define the events $\mathcal{E}_1=\{\mathcal{S}_{k_0}=\mathcal{S}\}=\{k_{min}=k_0\}$, $\mathcal{E}_2=\{RR(k_0)<\Gamma_{RRT}^{\alpha^*}(k_0)\}$ and $\mathcal{E}_3=\{RR(k)>\Gamma_{RRT}^{\alpha^*}(k),\forall k>k_{min} \}$. Event $\mathcal{E}_1 \cap \mathcal{E}_2$ implies that the RRTA estimate $k_{RRTA}=\max\{k:RR(k)<\Gamma_{RRT}^{\alpha^*}(k)\}$ satisfies $k_{RRTA}\geq k_{min}$, whereas, the event $\mathcal{E}_2 \cap \mathcal{E}_3$ implies that the RRTA estimate $k_{RRTA}\leq k_{min}$. Hence, Event $\mathcal{E}_1 \cap \mathcal{E}_2\cap \mathcal{E}_3$ implies that $k_{RRTA}=k_{min}=k_0$ and $\mathcal{S}_{RRTA}=\mathcal{S}$. Hence $\mathbb{P}(\mathcal{S}_{RRTA}=\mathcal{S})\geq \mathbb{P}(\mathcal{E}_1 \cap \mathcal{E}_2\cap \mathcal{E}_3)$. ]{}
By Lemma \[lemma:latest\_omp\], $\mathcal{E}_1$ is true once $\|{\bf w}\|_2\leq \epsilon_{omp}$. This along with $\|{\bf w}\|_2\overset{P}{\rightarrow} 0$ as $\sigma^2\rightarrow 0$ implies that $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{E}_1)=1$. Next we consider $\mathcal{E}_2$. By the definition of $\mathcal{E}_2$ $$\mathbb{P}(\mathcal{E}_2)=\mathbb{P}\left(\dfrac{\Gamma_{RRT}^{\alpha^*}(k)}{\underset{k}{\min}RR(k)}\dfrac{\underset{k}{\min}RR(k)}{RR(k_0)}>1\right)$$ Following Theorem \[thm:rmm\_hsc\] and it’s proof, we know that $\underset{k}{\min}RR(k)\overset{P}{\rightarrow} RR(k_0)$ as $\sigma^2\rightarrow 0$. Hence, $\dfrac{\underset{k}{\min}RR(k)}{RR(k_0)}\overset{P}{\rightarrow }1 $ as $\sigma^2\rightarrow 0$. From Theorem \[thm:rmm\_hsc\], we also know that $\underset{k}{\min}RR(k)\overset{P}{\rightarrow} 0$ as $\sigma^2\rightarrow 0$. Since the function $\alpha^*(x)=\min(PFD_{finite},x^q)$ is continuous around $x=0$ for every $q>0$ and $PFD_{finite}>0$, this implies[^1] that $\alpha^*=\min(PFD_{finite},\underset{k}{\min}RR(k)^q)\overset{P}{\rightarrow }0$ as $\sigma^2\rightarrow 0$.
\[lemma:Gamma\] For any function $f(x)\rightarrow 0$ as $x\rightarrow 0$, $\Gamma_{RRT}^{f(x)}(k_0)/x \rightarrow \infty$ as $x\rightarrow 0$ once $f(x)^{\frac{2}{n-k_0}}/x^2 \rightarrow \infty$ as $x\rightarrow 0$.
Please see Appendix F.
Please note that the function $f(x)=\min(PFD_{finite},x^q)$ satisfies $f(x)^{\frac{2}{n-k_0}}/x^2 \rightarrow \infty$ as $x \rightarrow 0$ once $2q/(n-k_0)<2$ which is true once $n>k_0+q$. Since the function $\alpha^*=f(RR(k))=\min(PFD_{finite},RR(k)^q)$ is continuous around zero and $\underset{k}{\min}RR(k)\overset{P}{\rightarrow }0$ as $\sigma^2 \rightarrow 0$, $\dfrac{\Gamma_{RRT}^{\alpha^*}(k)}{\underset{k}{\min}RR(k)}\overset{P}{\rightarrow} \infty$ as $\sigma^2\rightarrow 0$ once $n>k_0+q$. Since $\dfrac{\Gamma_{RRT}^{\alpha^*}(k)}{\underset{k}{\min}RR(k)}\overset{P}{\rightarrow }\infty$ and $\dfrac{\underset{k}{\min}RR(k)}{RR(k_0)} \overset{P}{\rightarrow} 1$, we have $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}\left(\dfrac{\Gamma_{RRT}^{\alpha^*}(k)}{\underset{k}{\min}RR(k)}\dfrac{\underset{k}{\min}RR(k)}{RR(k_0)}>1\right)=1$ and $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{E}_2)=1$.
Next we consider the event $\mathcal{E}_3=\{RR(k)>\Gamma_{RRT}^{\alpha^*}(k),\forall k>k_{min} \}$. Please note that the bound $\mathbb{P}(RR(k)>\Gamma_{RRT}^{\alpha}(k), \forall k>k_{min})\geq 1-\alpha$ for all $\sigma^2>0$ in Lemma \[lemma:RR\_properties\] is derived assuming that $\alpha$ is a deterministic quantity. However, $\alpha^*=\min(PFD_{finite},\underset{k}{\min} RR(k)^q)$ in RRTA is a stochastic quantity and hence Lemma \[lemma:RR\_properties\] is not directly applicable. Note that for any $\delta>0$, we have $$\begin{array}{ll}
\mathbb{P}(RR(k)>\Gamma_{RRT}^{\alpha^*}(k),\forall k>k_{min} )\\
\ \ \ \ \ \ \ \ \ \ \ \ \overset{(a)}{\geq} \mathbb{P}(\{RR(k)>\Gamma_{RRT}^{\alpha^*}(k),\forall k>k_{min}\} \cap \{\alpha^*\leq \delta\} )\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \overset{(b)}{\geq} \mathbb{P}(\{RR(k)>\Gamma_{RRT}^{\delta}(k),\forall k>k_{min}\} \cap \{\alpha^*\leq \delta\} )
\end{array}$$ (a) follows from the intersection bound $\mathbb{P}(A\cap B)\geq \mathbb{P}(A)$. Note that $F^{-1}_{a,b}(z)$ is a monotonically increasing function of $z$. This implies $\Gamma_{RRT}^{\alpha}(k)<\Gamma_{RRT}^{\delta}(k)$ when $\alpha<\delta$. (b) follows from this.
Note that by Lemma \[lemma:RR\_properties\], we have $\mathbb{P}(\{RR(k)>\Gamma_{RRT}^{\delta}(k),\forall k>k_{min}\})\geq 1-\delta$ for all $\sigma^2>0$. Further, $\alpha^*\overset{P}{\rightarrow} 0$ as $\sigma^2\rightarrow $ implies that $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}({\alpha^*\leq \delta})=1$. These two results together imply $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\{RR(k)>\Gamma_{RRT}^{\alpha^*}(k),\forall k>k_{min}\} \cap \{\alpha^*\leq \delta\} )\geq 1-\delta$. Since this is true for all $\delta>0$, we have $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\{RR(k)>\Gamma_{RRT}^{\alpha^*}(k),\forall k>k_{min}\} \cap \{\alpha^*\leq \delta\} )=1$ which in turn imply $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{E}_3)=\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(RR(k)>\Gamma_{RRT}^{\alpha^*}(k),\forall k>k_{min} )=1$.
Since $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{E}_j)=1$ for $j=1,2$ and $3$, it follows that $\underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{S}_{RRTA}=\mathcal{S})\geq \underset{\sigma^2\rightarrow 0}{\lim}\mathbb{P}(\mathcal{E}_1 \cap \mathcal{E}_2\cap \mathcal{E}_3)=1$.
Appendix F: Proof of Lemma \[lemma:Gamma\] {#appendix-f-proof-of-lemma-lemmagamma .unnumbered}
===========================================
Expanding $F^{-1}_{a,b}(z)$ at $z=0$ using the expansion given in \[http://functions.wolfram.com/06.23.06.0001.01\] gives $$\label{beta_exp}
\begin{array}{ll}
F^{-1}_{a,b}(z)=\rho(n,1)+\dfrac{b-1}{a+1}\rho(n,2) \\
+\dfrac{(b-1)(a^2+3ab-a+5b-4)}{2(a+1)^2(a+2)}\rho(n,3)
+O(z^{(4/a)})
\end{array}$$ for all $a>0$. Here $\rho(n,l)=(az{B}(a,b))^{(l/a)}$. Note that $\Gamma_{RRT}^{f(x)}(k_0)=\sqrt{F^{-1}_{\frac{n-k_0}{2},{\frac{1}{2}}}\left(\dfrac{f(x)}{k_{max}(p-k_0+1)}\right)}$. We associate $a=\frac{n-k_0}{2}$, $b=1/2$ , $z=\dfrac{f(x)}{k_{max}(p-k_0+1)}$ and $\rho(n,l)=(az{B}(a,b))^{(l/a)}=\left(\frac{\left(\frac{n-k_0}{2}\right)f(x){B}\left(\frac{n-k_0}{2},0.5\right)}{k_{max}(p-k_0+1)}\right)^{\frac{2l}{n-k_0}}$ for $l\geq 1$. $\dfrac{\Gamma_{RRT}^{f(x)}(k_0)}{x}=\sqrt{\dfrac{F^{-1}_{\frac{n-k_0}{2},\frac{1}{2}}\left(\dfrac{f(x)}{k_{max}(p-k_0+1)}\right)}{x^2}}$. [ Note that the term $f(x)^{\frac{2l}{n-k_0}}$ is the only term in $\rho(n.l)$ that depends on $x$. Now from the expansion and the fact that $\underset{x \rightarrow 0}{\lim}f(x)^{\frac{2l}{n-k_0}}/f(x)^{\frac{2}{n-k_0}}=0$ for each $l>1$, it is clear that $\sqrt{\dfrac{F^{-1}_{\frac{n-k_0}{2},\frac{1}{2}}\left(\dfrac{f(x)}{k_{max}(p-k_0+1)}\right)}{x^2}} \rightarrow \infty$ as $x \rightarrow 0$ once $f(x)^{\frac{2}{n-k_0}}/x^2\rightarrow \infty$. ]{}
[^1]: Suppose that a R.V $Z \overset{P}{\rightarrow} c$ and $g(x)$ is a function continuous at $x=c$. Then $g(Z)\overset{P}{\rightarrow}g(c)$[@wasserman2013all].
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
(for the STAR Collaboration)\
Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China\
E-mail:
bibliography:
- 'CPOD2013\_Xiaofeng.bib'
title: 'Beam Energy Dependence of Higher Moments of Net-proton Multiplicity Distributions in Heavy-ion Collisions at RHIC'
---
Introduction
============
Exploring the Quantum Chromodynamics (QCD) phase structure is one of the most important goal of Beam Energy Scan (BES) program at Relativistic Heavy-Ion Collider (RHIC) [@bes]. The first principle Lattice QCD calculation predicted that the transition from hadronic to partonic matter at zero is a smooth crossover [@crossover], while at finite region is a first order phase transition [@firstorder]. The end point of the first order phase transition boundary is so called the QCD Critical Point (CP). There are large uncertainties for Lattice QCD calculation in determining the first order phase boundary as well as the QCD critical point [@location] in the QCD phase diagram due to the sign problem at finite region [@methods]. In the first phase of the BES program, the Au+Au colliding energy was tuned from 200 GeV down to 7.7 GeV and the corresponding baryon chemical potential () coverage is from about 20 to 450 MeV [@bes]. This allows us to map a broad region of QCD phase digram (temperature ($T$) versus baryon chemical potential ()). Thus, it provides us a good opportunity to look for the first order phase boundary and search for the CP at RHIC.
In heavy-ion collisions, moments (Variance ($\sigma^2$), Skewness($S$), Kurtosis($\kappa$)) of conserved quantities, such as net-baryon, net-charge and net-strangeness, are predicted to be sensitive to the correlation length of the hot dense matter created in the collisions [@qcp_signal; @ratioCumulant; @Neg_Kurtosis] and connected to the various order susceptibilities computed in the Lattice QCD [@Lattice; @MCheng2009; @science; @Wupp_Lattice] and Hardon Resonance Gas (HRG) [@HRG] model. For instance, the higher order cumulants of multiplicity ($N$) distributions are proportional to the high power of the correlation length ($\xi$) as third order cumulant $C_{3}=S \sigma^{3}=<(\delta N)^3> \sim \xi^{4.5}$ and fourth order cumulant $C_{4}=\kappa \sigma^{4}=<(\delta N)^4> - 3(\delta N)^2> \sim \xi^{7}$ [@qcp_signal], where the $\delta N=N-<N>$ and $<N>$ is the mean multiplicities. The moment products, and , are also related to the ratios of various order susceptibilities, such as ratios of baryon number susceptibilities can be compared with the experimental data as $\kappa
\sigma^2=\chi^{(4)}_{B}$/$\chi^{(2)}_{B}$ and $S
\sigma=\chi^{(3)}_{B}$/$\chi^{(2)}_{B}$. The ratios cancel out the volume effect. Theoretical calculations have shown that the experimentally measurable net-proton number (number of protons minus number of anti-protons) fluctuations may reflect the fluctuations of the net-baryon number at CP [@Hatta]. Thus, the net-proton number fluctuations are measured as the approximation to the net-baryon fluctuations.
The proceedings is organized as follows. In Section 2, we discuss the centrality resolution effect. In section 3, we present different methods of estimating the statistical errors for moment analysis. In section 4, we present STAR preliminary measurements for higher moments of event-by-event net-proton multiplicity distributions from the first phase of the BES program at RHIC. Finally, we present a summary of the work.
Centrality Resolution Effect (CRE)
==================================
The collision centrality and/or the initial collision geometry can be represented by many parameters in heavy-ion collisions, such as impact parameter $b$, number of participant nucleons ($N_{part}$) and number of binary collisions ($N_{coll}$). These initial geometry parameters are not independent but are strongly correlated with each other. Experimentally, the collision centrality is determined from a comparison between experimental measures such as the particle multiplicity and Glauber Monte-Carlo simulations [@Glauber]. Particle multiplicity, not only depends on the physics processes, but also reflects the initial geometry of the heavy-ion collision. This indicates that relation between measured particle multiplicity and initial collision geometry does not have a one-to-one correspondence and there are fluctuations in the particle multiplicity even for a fixed initial geometry. Thus, it could have different initial collision geometry resolution (centrality resolution or volume fluctuation) for different centrality definitions with particle multiplicity. This may affect moments of the event-by-event multiplicity distributions. It is natural to expect that the more particles are used in the centrality determination, the better centrality resolution and smaller fluctuations of the initial geometry (volume fluctuations) we get [@techniques].
![(Color Online) Centrality dependence of the moments products of net-proton multiplicity distributions for Au+Au collisions at [[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}=7.7, 11.5, 19.6, 27, 39, 62.4, 200 GeV in AMPT string melting model. Different symbols represent different collision centrality definition.[]{data-label="fig:res-kv"}](res-sd.eps)
![(Color Online) Centrality dependence of the moments products of net-proton multiplicity distributions for Au+Au collisions at [[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}=7.7, 11.5, 19.6, 27, 39, 62.4, 200 GeV in AMPT string melting model. Different symbols represent different collision centrality definition.[]{data-label="fig:res-kv"}](res-kv.eps)
To verify the CRE in the moment analysis, we use the charged kaon and pion multiplicity (as for the real data analysis to avoid autocorrelation effect) produced in the final state within $|\eta|<0.5, 1, 1.5$ and 2 to define the centrality in the AMPT string melting model calculations with a parton-parton interaction cross section 10mb [@ampt]. The proton and anti-proton are selected within mid-rapidity $|y|<0.5$ and transverse momentum range $0.4<p_{T}<0.8$ GeV/c, which are the same kinematical range as used in the real data analysis. Fig. \[fig:res-sd\] and \[fig:res-kv\] shows the centrality dependence of the moment products ($ S\sigma, \kappa\sigma^2$) of net-proton multiplicity distributions for four different $\eta$ range of charged kaon and pion used to determine the centrality. We observe significant difference for moment products ($S\sigma, \kappa\sigma^2$) for the different $\eta$ range of the centrality definition. The behavior can be understood as due to different contribution from volume fluctuations (increasing centrality resolution) arising from different centrality definition. When we increase the $\eta$ range $(|\eta| < 1,1.5,2)$, the values of $S\sigma$ and $\kappa\sigma^2$ will decreases, which indicates the centrality resolution effect will enhance the moments values of net-proton distributions. On the other hand, the moment products ($S\sigma, \kappa\sigma^2$) are closer to the results with centrality directly determined by number of participant nucleons ($N_{part}$) when the $\eta$ range is larger. It confirms that the centrality resolution effect can be suppressed by having more particles to determine the centrality.
Fig. \[fig:SD\_KV\_Energy\_model\] shows the energy dependence of moment product ($S\sigma, \kappa\sigma^2$) of net-proton multiplicity distributions for three different centralities $(0-5\%, 30-40\%, 70-80\%)$ with four different $\eta$ coverage in Au+Au collisions. We can find that the is more sensitive to the CRE than , and the CRE has a larger effect in the peripheral collision as well as at low energies. Thus, we should use a lager $\eta$ coverage in the centrality definition for the real experimental moment analysis to reduce the centrality resolution effects.
![(Color online) Energy dependence of the moments products $(S\sigma , \kappa\sigma^2)$ of net-proton multiplicity distributions for Au+Au collisions at centralities ($0-5\%,30-40\%, 70-80\%$) and at [[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}=7.7, 11.5, 19.6, 27, 39, 62.4, 200GeV in AMPT string melting model. Different symbols represent different collision centrality definition. []{data-label="fig:SD_KV_Energy_model"}](energy.eps){width="60.00000%"}
Statistical Error Estimation
============================
Since we don’t know exactly the underlying probability distributions for proton and anti-proton, it is not accurate to estimate the statistical error with standard error propagation with respect to the number of proton and anti-proton. Several statistical methods (Delta theorem, Bootstrap [@bootstrap], Sub-group) of error estimation in the moment analysis and their comparisons will be discussed by a Monte Carlo simulation. For simplicity, skellam distribution is used to perform the simulation. If protons and anti-protons multiplicity follow independent Poissonian distributions, the net-proton multiplicity will follow the skellam distribution, which is expressed as: $$P(N) = {(\frac{{{M_p}}}{{{M_{\overline p}}}})^{N/2}}{I_N}(2\sqrt {{M_p}{M_{\overline p}}} )\exp [ - ({M_p} + {M_{\overline p}})],$$ where $I_{N}(x)$ is a modified Bessel function, $M_{p}$ and $M_{\overline p}$ are the measured mean values of protons and anti-protons. If the net-proton follows the skellam distribution, then we have, $S\sigma = {C_3}/{C_2} = ({M_p} - {M_{\overline p }})/({M_p} + {M_{\overline p }})$ and $\kappa {\sigma ^2} = {C_4}/{C_2} = 1$, which then provides the Poisson expectations for the moment products. To perform the simulation, we set the two mean values of the skellam distributions as $M_{p}$ = 4.11, $M_{\overline p}$ = 2.99. Then, we generate random numbers as per the skellam distribution. The details of Delta theorem error estimation method for moment analysis can be found in [@Delta_theory]. The bootstrap method [@bootstrap] is based on the repeatedly sampling with the same statistics of the parent distribution and the statistical errors can be obtained as the root mean square of the observable from each sample. In the sub-group method, we randomly divide the whole sample into several sub-groups with same statistics and the errors are obtained as the root mean square of the observable from each sub-group. In our simulation, we set 200 bootstrap times and 5 sub-groups for bootstrap and sub-group methods, respectively.
![(Color online) $\kappa \sigma^{2}$ for 50 samples that independently and randomly generated from the skellam distribution with different number of events (0.01, 0.1, 1 million). The dashed lines are expectations value 1 for the skellam distribution. []{data-label="fig:error"}](error.eps){width="65.00000%"}
Fig. \[fig:error\] shows the error estimation comparison between Delta theorem, Bootstrap and Sub-group methods for $\kappa \sigma^{2}$ of skellam distribution. For each method, fifty independent samples are sampled from the skellam distribution with statistics 0.01 , 0.1 and 1 million, respectively. The probability for the value staying within $\pm 1\sigma$ around expectation is about $68.3 \%$ and it means error bars of 33 out of 50 points should touch the expected value(dashed line at unity) in Fig. \[fig:error\]. We find that the results from the Delta theorem and Bootstrap method show similar error values and satisfies the above criteria, while the random sub-group method over estimates the statistical errors. It indicates that the Delta theorem and Bootstrap error estimation methods for the moment analysis are reasonable and can reflect the statistical properties of moments.
Results and Discussion
======================
The results presented here are obtained from the Au+Au collisions at [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{} =7.7, 11.5, 19.6, 27, 39, 62.4 and 200 GeV in the first phase of the BES program at RHIC and $p + p$ collisions at [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=62.4 and 200 GeV. The protons and anti-protons are identified at midrapidity ($|y| < 0.5$) and within the transverse momentum range $0.4 < p_{T} < 0.8$ GeV/c by using the ionization energy loss ($dE/dx$) of charged particles measured by the Time Projection Chamber (TPC) of STAR detector. To suppress autocorrelation effects between measured net-proton fluctuations and centrality defined using charged particles, a new method of centrality selection is used in the net-proton moment analysis. The new centrality is determined from the uncorrected charge particle multiplicity by excluding the protons and anti-protons within pseudorapdity $|\eta| < 0.5$.
![(Color Online) Centrality dependence of /Poisson and of net-proton distributions for Au+Au collisions at [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=7.7, 11.5, 19.6, 27, 39, 62.4 and 200 GeV. The error bars are statistical and caps are systematic errors.[]{data-label="fig:SD_KV_Centrality"}](Fig3_QM.eps)
Figure \[fig:SD\_KV\_Centrality\] shows the ratios of the cumulants, which are connected to the moment products as $S\sigma = {C_3}/{C_2}$ and $\kappa {\sigma ^2} = {C_4}/{C_2}$. The various order cumulants ($C_{1}-C_{4}$) can be obtained from the net-proton multiplicity distributions and corrected for the finite centrality bin width effect [@WWND2011]. It is observed that the and the values normalized to Poisson expectations are below unity for [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{} above 11.5 GeV and above unity for 7.7 GeV in Au+Au collisions. The shows larger deviation from Poisson expectations than . The statistical error shown in Fig. \[fig:SD\_KV\_Centrality\] are obtained by the Delta theorem method and the systematical errors are estimated by varying the track quality condition and particle identification criteria. The data presented here may allow us to extract freeze-out conditions in heavy-ion collisions using QCD based approaches [@CPOD2011_FKarsch].
![(Color online) Energy dependence of and for net-proton distributions for four collision centralities (0-5%, 5-10%, 30-40% and 70-80%) measured at STAR. The results are compared to UrQMD model calculations and collisions at [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=62.4 and 200 GeV. The lines in top panel are the Poisson expectations and in the bottom panel shows the normalized to the corresponding Poisson expectations. []{data-label="fig:SD_KV_Energy"}](Fig4_QM.eps){width="60.00000%"}
Figure \[fig:SD\_KV\_Energy\] shows the energy dependence of and of net-proton distributions for four centralities (0-5%, 5-10%, 30-40% and 70-80%) in Au+Au collisions. The bottom panel of Fig.\[fig:SD\_KV\_Energy\] shows values normalized to the corresponding Poisson expectations. The and normalized values are close to the Poisson expectations for Au+Au collisions at [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=39 , 62.4 and 200 GeV. They show deviation from Poisson expectations for the 0-5% central Au+Au collisions below [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=39 GeV. The UrQMD model [@urqmd] results are also shown in the Fig. \[fig:SD\_KV\_Energy\] for 0-5% centrality to understand the non-CP effects, such as baryon number conservation and hadronic scattering. The UrQMD calculations show a monotonic decrease with decreasing beam energy.
For the preliminary experimental results shown here and also in the QM2012 proceedings [@QM2012_Xiaofeng], the collision centralities are determined from the uncorrected charge particle multiplicity by excluding the protons and anti-protons within pseudorapdity $|\eta|< 0.5$. Based on the model simulation results, there could be significant centrality resolution effects in mid-central and peripheral collisions at low energies. The amount of particles used in the centrality determination still can be increased by extending the pseudorapdity coverage to the $|\eta|<1$, the current acceptance limit of the STAR TPC. Thus, to obtain more precise results with less centrality resolution effect, we will use the pseudorapdity coverage $|\eta|<1$ to redefine collision centralities in Au+Au collisions for all energies. This analysis is in progress.
Recently, the STAR inner TPC (iTPC) upgrade plan has been proposed and it can enlarge the pseudorapdity coverage of TPC from $|\eta|<1$ to $|\eta|<1.7$. This upgrade is expected to be completed in the year 2017, which can be used for data taking in the second phase of BES at RHIC. It will allow us to define the centralities with a much wider $\eta$ range to further suppress the centrality resolution effects.
Summary
=======
We have presented the beam energy ([[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=7.7$-$200 GeV) and centrality dependence for the higher moments of net-proton distributions in Au+Au collisions from the first phase of the BES program at RHIC. It is observed that the and values are close to the Poisson expectation for Au+Au collisions at [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=39 , 62.4 and 200 GeV. They show deviation from Poisson expectations in the 0-5% central Au+Au collisions below [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=39 GeV. The UrQMD calculations show a monotonic decrease with decreasing beam energy. We also need more statistics to get precise measurements below 19.6 GeV and additional data at [[[[$\sqrt{s_{_{{NN}}}}$]{}]{}]{}]{}=15 GeV. These are planned for the second phase of the BES program at RHIC. The centrality resolution effect in moment analysis has been pointed out and large $\eta$ coverage will be used in centrality definition to suppress this effect. Three statistic error estimation methods and their comparisons in moment analysis have been discussed through a Monto Carlo simulation.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work was supported in part by the National Natural Science Foundation of China under grant No. 11205067 and 11135011. CCNU-QLPL Innovation Fund(QLPL2011P01) and China Postdoctoral Science Foundation (2012M511237).
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'David R. Soderblom'
title: 'George Howard Herbig, 1920-2013'
---
A great astronomer, George Herbig, passed away in Honolulu on October 12, 2013, at the age of 93. His life and career were long and productive, and consistently dedicated to the careful, thorough research that earned him his reputation. Herbig spent most of his career at Lick Observatory, first as a graduate student, and then from 1949 to 1987 as a staff member, rising through the ranks to Astronomer and becoming Professor as well when the observatory moved to the University of California at Santa Cruz in the late 1960s. Herbig retired from UCSC in 1987 and spent the remainder of his life in Honolulu at the Institute for Astronomy of the University of Hawai‘i.
Herbig was particularly known for his work on newly-formed stars in our Galaxy, but his interests broadly encompassed the life history of the Sun and Solar System and the place of the Sun among the stars. As with many of us, this work had its beginnings in an encounter with a senior astronomer, Alfred Joy of the Mount Wilson Observatory, who was studying the variable star known as T Tauri [@joy45]. Herbig was fascinated and worked on these stars associated with nebulosity in his thesis and then over his entire career. In addition, photographs of the regions in which these stars are found led to the discovery of small, dense clouds of glowing gas. The Mexican astronomer G. Haro was working on these objects (now known as Herbig-Haro objects) at the same time in the early 1950s, and subsequent work by them showed that these clouds often occur in matching pairs on either side of a very young star and that they were ejected by that star and can be seen to move on the sky after just a few years, a demonstration that energetic, dynamic processes were at work on human time-scales. In a similar vein, one of these stars in nebulosity, FU Orionis, had brightened by a factor of about one hundred within just a few months and Herbig realized that type of change might be a typical short-lived phase for young stars, in that case caught through luck. A more systematic search has uncovered more of these FUors.
Patient, comprehensive spectroscopic investigations were his forte, and in the early and mid-1950s he was able to demonstrate that the T Tauri stars and stars like them were the youngest stars in our Galaxy, no more than a few million years old. In so doing he forged a new branch of astronomy, the study of pre-main sequence stars. Herbig was not the only one pursuing these stars, but he dominated their observational study by piecing together the clues that could lead to no other conclusion. For example, the very high density of stars in a cluster like NGC 2264 compared to the region around the Sun meant that stars form in concentration and then disperse, for a dense association could not plausibly arise by chance. Prior to that it was not possible to rule out that the stars associated with nebulosity were just more ordinary stars like our Sun that happened to be passing through the clouds they were found in and being affected by them, leading to their spectroscopic peculiarities.
Herbig’s path into a research career was along a steady trajectory of his own making and yet also had good fortune at critical points. He was born on January 2, 1920, on a small island in the Ohio River, part of Wheeling, West Virginia. His father, a tailor, died when he was six, and a few years later his mother took him to live in Los Angeles, where she had a brother. Circumstances during the Depression were difficult for both of them, but Herbig had enough budding interest in astronomy to join the Amateur Astronomical Society of Los Angeles [@ghh38] and build his own telescope. This gave him exposure to the very active astronomical scene of southern California, centered on the Mount Wilson Observatory, whose offices were in Pasadena, within sight of the telescopes atop the mountain.
In addition to this avid interest in science, Herbig had help in going from LA’s Polytechnic High School, from which he graduated in 1937, to UCLA. Having an excellent public university nearby, let alone one with astronomers, was a resource he had the ability and ambition to take advantage of. At the same time, Jack Preston of the amateur society helped Herbig get started at UCLA, and Frederick Leonard, Chairman of the UCLA Astronomy Department, supported him in many ways, including a recommendation that he be hired at the Griffith Observatory, an outreach-oriented facility that overlooks downtown Los Angeles. The head guide at Griffith, George Bunton, and his wife took Herbig under their wings by providing him a job as a guide, providing resources to help Herbig stay in school during those lean years. Herbig graduated from UCLA in 1943, and his first publication (which led to an article in the L.A. Times) appeared in 1940 [@ghh40].
Herbig went to U.C. Berkeley to seek a Ph.D. in astronomy, despite having been advised at one point that he would most likely be best as a high school teacher. While there he was assigned to the Berkeley Radiation Laboratory on a war-related project. He found that work unsatisfying and was reassigned to the Lick Observatory, on Mount Hamilton, another branch of the University of California. We can be confident that he found that much more to his liking, for he was to spend two decades living on Mount Hamilton and another two associated with Lick when the observatory’s research staff moved to U.C. Santa Cruz.
Herbig completed his doctorate in 1948, supervised by Harold Weaver of U. C. Berkeley. The young Herbig clearly impressed the Lick astronomers, for the Lick Director, Donald Shane, was eager to hire Herbig onto the staff as soon as his thesis was completed, urging the President of the university to make a position available. However, Herbig was offered a one-year fellowship by the National Research Council, and took advantage of that year to go to Yerkes Observatory to work with Otto Struve, arguably the greatest of stellar spectroscopists (see Herbig 1968), and the last of the great Struve astronomical dynasty. Struve was later to come to Berkeley and to there influence a generation of observers. Herbig also spent a portion of that year at the McDonald Observatory, in west Texas, and in Pasadena at the Mount Wilson Observatory, working with Walter Baade. Herbig’s first year as a Lick astronomer, 1949, was to be significant in several ways. He had married in 1943 while at UCLA, and the couple’s first child was born in 1949; three more were to follow. Most important for his future efforts, in 1949 Lick bought the 120-inch mirror blank from Palomar that would enable them to build the world’s second largest optical telescope, now known as the Shane 3-meter telescope. Despite his youth, Herbig was to take charge of the design and operation of the new telescope’s coudé focus, where large optics could be installed to take spectra of high resolution. Such facilities were rare, and so Herbig spent time visiting Mount Wilson and the coudé spectrograph of itÕs then 32-year-old 100-inch telescope, and to work with Mount Wilson astronomer Ira Bowen, who was to make the diffraction gratings for the 120-inch coudé because almost no one else had that capability.
The 120-inch opened in 1959, and its initiation and construction make an important story in itself. That telescope helped make Lick a more significant center of astronomical research, and it also led to growth in astronomy at many of the U.C. campuses. Great facilities inspire great science. That growth in U.C. astronomy led to stresses between Lick and the other campuses, and Herbig led the Mount Hamilton staff in deciding to relocate to the brand new U.C. campus at Santa Cruz, a small seaside resort town that the campus was to transform. Staying on Mount Hamilton was no longer a realistic option, while going to Santa Cruz offered the opportunity of maintaining some independence of Lick from the other U.C. campuses while establishing a new academic department, one that has continued to grow in the decades since.
Before the 120-inch was built, Lick’s primary telescope was the 36-inch Crossley reflector. After Herbig returned to Lick in 1949 he lost no time in vigorously continuing his work on the stars associated with nebulosity, and he built an instrument for the Crossley that would allow him to identify more effectively additional T Tauri stars. A full understanding of their nature and significance required spectra of individual stars, but he also needed to know how many and where these stars were found, and that required a means to pick out the T Tauri stars over a large field of view. T Tauri stars were identified by their having spectra roughly similar to the Sun’s and cooler, as well as prominent emission in the 6563 Å H$\alpha$ line of hydrogen. Photographic emulsions of the time responded much more to blue photons than to the red ones of H$\alpha$, and so acquiring the observations meant designing and building a slitless spectrograph of high efficiency. The emission-line stars were then reobserved spectroscopically at higher resolution.
The life of an observer was one that fit Herbig well. In his graduate student years of the 1940s, American astronomy was still a small-scale effort at a very small number of monastic outposts and a few of the major universities. Nearly all work was photographic, and patience and technical skill were required to get quality results. Excellent physical insight was also necessary to interpret what was obtained, and looking back one is impressed with how well those astronomers prized the secrets of the heavens out of the observations they were able to obtain with problematic detectors (photographic plates were inefficient and non-linear) and small telescopes.
Located on Mount Hamilton, east of San Jose, California, Lick was part of the University of California, but it was not an academic department and had an arrangement with U.C. Berkeley to work with graduate students. During that time Herbig supervised or influenced many students who went on to astronomical careers of their own, including Elizabeth Roemer, Robert Kraft, George Preston, Beverly Lynds, Ann Merchant Boesgaard, and Leonard Kuhi. At UCSC, he supervised students under the auspices of the Board of Studies in Astronomy and Astrophysics, including Robert Zappala, William Alschuler, N. Kameswara Rao, Douglas Duncan, and Geoff Marcy, and he significantly influenced many more through the graduate courses he taught and personal interactions. Herbig would note to his class that in physics finding the gold mostly meant great effort “excavating” in large groups, while in astronomy one could walk around and find the nuggets by kicking over rocks. He never tired of exploring new veins.
I was also one of Herbig’s thesis students and also a research assistant of his for several years. We spent many nights in the 120-inch coudé, where I learned the art and craft of observing and a feel for the night sky and also what a special privilege it is to have great instruments to examine the cosmos. There was enough time, and an interest on Herbig’s part, to observe Comet Kohoutek and Comet West with our eyes on the telescope’s coudé slit and to obtain good spectra. Herbig was responsible for the coudé spectrograph, and we stopped in the middle of one dark, clear night to align optics that needed real starlight to test properly. From all this I could see his style for being a scientist: Plan carefully, get all the data you need for the study, and make no statements not rigorously supported by the observations.
Herbig tended toward solo efforts: Of his 171 papers in refereed journals, nearly 2/3 are single author, and only about 10% have 3 or more authors. Yet his influence was great and well recognized. He was awarded the Helen B. Warner Prize of the American Astronomical Society in 1955 for his work on the T Tauri stars, and in 1975 he received the AASÕ highest recognition, the Henry Russell Norris Lectureship.
Herbig’s scientific career included more than just young stars. He developed instruments and improvements to efficiency and data quality constantly, and was ready to adopt new technology as it became available to push his work further. He did significant studies of the interstellar medium and on the enigmatic diffuse interstellar bands. He obtained an extraordinary collection of high-quality spectra of comets, perhaps in the hope of discovering a connection between them and the interstellar medium. And he hosted what was very likely the first ever workshop on detecting exoplanets, in Santa Cruz in 1976 [@green76].
When Herbig left Santa Cruz in 1987 and went to the University of Hawai‘i he took on a different role. He no longer had direct responsibilities as he did at Lick and instead spent full time on research and working with students (including supervising Scott Dahm). He also became someone that the younger staff and students could find a ready ear in, and he influenced many careers as he continued to study young stars in clusters and associations.
Herbig’s research record will continue to speak for itself, but we will no longer have his voice of counsel. Many of us still work by his example.
I am grateful for assistance received and conversations with Hannelore Herbig, Marilyn Wood, Bo Reipurth, Ann Merchant Boesgaard, George Preston, Joseph Wampler, Steven Vogt, Lynne Hillenbrand, and Geoff Marcy. The Special Collections of the library at U.C. Santa Cruz was helpful in allowing access to the Mary Shane Archives of the Lick Observatory. Roger Griffin kindly supplied a copy of the minutes from the 1976 exoplanets workshop that he and I attended.
Annotated list of selected publications
=======================================
It is not possible to present the entire panoply of Herbig’s work here, so I have chosen a few particularly important research papers and some of the insights and comments he made over the years in connection with scientific symposia. There are more, and I encourage the reader to spend a little time looking.
The history of the star formation field {#the-history-of-the-star-formation-field .unnumbered}
---------------------------------------
Herbig was intimately involved in advancing the field of star formation through his observations, he also personally knew the key players, and he had thorough knowledge of the literature. He has left behind several concise accounts of the state of knowledge of star formation that are enjoyable to read and provide key insights. Because they go back some years, they must be found in bound volumes, not on-line, but are well worth the effort.
In an introductory chapter to the Saas-Fee Advanced Course 29 (“Physics of star formation in galaxies," 1999) Herbig summarizes the history of star formation from about 1900 to the mid-1950s with the benefit of hindsight. There are also two accounts that were written contemporaneously. “On the nature and origin of the T Tauri stars” is the transcript of a presentation at IAU Symposium 3, held in Dublin in 1955. It shows the arguments then being made and the uncertainties that still needed to be resolved. “T Tauri stars, flare stars, and related objects as members of stellar associations,” a Vatican Observatory symposium published in Ricerche Astronomische, v. 5, Specola Vaticana, 1958, tells a similar story a few years later. In addition to again providing insight into the state of thinking at the time, it is remarkable to read through the questions and answers recorded at the end of Herbig’s Vatican presentation and to realize what a extraordinary constellation of great astronomers was assembled to hear from one another, a much more diverse group than would occur today. For additional remarks see @ghh69x in one of the Liège symposia.
Scientific insights into the advancing field {#scientific-insights-into-the-advancing-field .unnumbered}
--------------------------------------------
One of Herbig’s most cited publications was his 1962 review [@ghh62a]: “The properties and problems of T Tauri stars and related objects.” This review remains a must-read for anyone wanting to work in the field of star formation, and the problems described are mostly still with us.
A volume titled “Spectroscopic astrophysics: An assessment of the contributions of Otto Struve” was edited by Herbig and published by the University of California Press in 1970 [@ghh70c]. Struve was a truly remarkable astronomer who contributed fundamentally in many ways. Each of a number of seminal papers by Struve is coupled to a modern review of the subject that Struve’s paper opened up. My copy of this is a personal favorite. Herbig himself contributed a chapter on T Tauri stars.
Research papers {#research-papers .unnumbered}
---------------
In the Struve memorial volume that he edited [@ghh70c], Herbig celebrated the many pioneering studies of Struve. Herbig himself merits such a volume because of the many seminal studies he published (and such a volume is being prepared by B. Reipurth). As an example, Herbig’s most-cited paper is his 1960 study that defined the “Herbig AeBe stars.” The T Tauri stars he had worked on so extensively represented mostly just low-mass stars, and the higher masses were conspicuously absent. In @ghh60b (“The spectra of Be- and Ae-type stars associated with nebulosity”) he reported on his search for higher mass examples of pre-main sequence stars; this has now become a subfield of its own [see, e.g., @ghh94w].
We now know that the Sun and stars like it rotate much more slowly than one would expect from a simple extrapolation of the trend of rotation in higher-mass stars, and that that slow rotation is the result of angular momentum loss throughout the star’s life. But @ghh55d (“Axial rotation and line broadening in stars of spectral types F0-K5”) is the first observational study that established the slow rotation of stars F8 and later, even if only upper limits were seen in most cases. Later studies filled in once higher resolution could be achieved, but this study was the first systematic attempt.
Similarly, @ghh65e (“Lithium abundances in F8-G5 stars”) established the basic facts regarding lithium in solar-type stars. Abundant lithium is a defining characteristic of the T Tauri stars, and it was well known that lithium is virtually absent in integrated sunlight, yet very strong in spectra of sunspots. Herbig showed that solar-type stars exhibit a range of lithium abundances and that that range was probably due to a steady decline in the abundance with time. Decades later we have much more and better observations but still do not understand the behavior of lithium in stars, even though it appears that the decline in surface lithium is an indicator of convective processes. Herbig also studied lithium in some clusters [@ghh65g], and attempted to measure lithium isotope ratios [@ghh64h]. All of this was enabled by high-quality spectra from the 120-inch coudé spectrograph.
Herbig studied the interstellar medium extensively, in part because the ISM feeds star formation. He is particularly known for detailed studies of the diffuse interstellar bands. @ghh75q is a classic instance of his very thorough approach to gathering the best observations he could and then analyzing them in detail. The diffuse interstellar bands remain enigmatic, despite years of effort by many individuals. When a new high-resolution spectrograph became available at the 120-inch coudé he immediately applied it to this problem [@ghh82q]. @ghh95x is a comprehensive review of the diffuse interstellar bands. @ghh68t presents an in-depth look at the interstellar lines in just one star, $\zeta$ Oph, including abundance analyses of atoms and molecules.
@ghh66r (“On the interpretation of FU Orionis”) pointed out the extraordinary behavior of FU Orionis, a T Tauri star that brightened suddenly by a factor of 100. Such dramatic changes on short time-scales in pre-main sequence stars had not seemed possible. That paper led to Herbig’s more systematic and detailed study of the phenomenon [@ghh77q] (“Eruptive phenomena in early stellar evolution").
@ghh77b (“Radial velocities and spectral types of T Tauri stars”) reports on his effort over nearly a decade to assemble the quantity and quality of observational data needed to study the composition and dynamics of star-forming regions. To make these faint stars accessible to spectra of sufficient resolution, he built a device that used an image intensifier that was optimized for the near-infrared, with photographic plates. The image intensifier introduced distortions and Herbig developed fitting methods to extract the radial velocities in a way that kept systematics under control. Together he and I took a thousand or more of these spectra over many, many nights.
In “Spectral classification of faint members of the Hyades and Pleiades and the dating problem in Galactic clusters,” @ghh62w pointed out the apparent discrepancy in ages between the main sequence turn-off at the high-mass end and the fact that low-mass stars were also on the main sequence and not displaced above it. Some of this discrepancy has been removed by increases in the turn-off ages as the interior physics of intermediate- and high-mass stars has become better understood, but the problem remains at some level.
Finally, there are the post-T Tauri stars. @ghh73c pointed out that the expected duration of the T Tauri phase is about 10% of the total time needed for a star to contract to the main sequence. By that time many T Tauri stars were known, but where were the somewhat older objects, the “post-T Tauri stars”? There should be many, yet few are observed. In @ghh78a he discussed the post-T Tauri problem further, and it is a problem that is still with us.
Greenstein, J. 1976, Minutes of “First workshop on extrasolar planetary detection,” personal copy of author. Herbig, G. H. 1938, , 46, 293 Herbig, G. H. 1940, , 52, 327 Herbig, G. H. 1960, , 4, 337 Herbig, G. H. 1962a, Adv. Astron. Astrophys., 1, 47 Herbig, G. H. 1962b, , 135, 736 Herbig, G. H. 1964, , 140, 702 Herbig, G. H. 1965, , 141, 588 Herbig, G. H. 1966, Vistas in Astron., 8, 109 Herbig, G. H. 1968, Z. Ap., 68, 243 Herbig, G. H. 1969, Mem. Soc. Roy. Sci. Liège, v. 19 Herbig, G. H. 1970, editor, [*Spectroscopic astrophysics: An assessment of the contributions of Otto Struve*]{}, Berkeley: University of California Press Herbig, G. H. 1973, , 182, 129 Herbig, G. H. 1975, , 196, 129 Herbig, G. H. 1977a, , 214, 747 Herbig, G. H. 1977b, , 217, 693 Herbig, G. H. 1978, in Problems of physics and evolution of the universe, ed. L. V. Mirzoyan, Yerevan: Armenian Acad. Sci., 171 Herbig, G. H. 1994, in The nature and evolutionary status of Herbig Ae/Be stars, eds. P. The et al., San Francisco: ASP Press Herbig, G. H. 1995, Ann. Rev. Astr. Ap., 33, 19 Herbig, G. H., & Soderblom, D. R. 1982, , 252, 610 Herbig, G. H., & Spalding, J. 1955, , 121, 118 Joy, A. H. 1945, , 102, 168 Wallerstein, G., Herbig, G. H., & Conti, P. S. 1965, , 141, 610
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We explore the relation between classical and quantum states in both open and closed (super)strings discussing the relevance of coherent states as a semiclassical approximation. For the closed string sector a gauge-fixing of the residual world-sheet rigid translation symmetry of the light-cone gauge is needed for the construction to be possible. The circular target-space loop example is worked out explicitly.'
---
[ YITP-SB-07-01 ]{}
0.9cm
**On Semiclassical Limits of String States**
0.7cm
Jose J. Blanco-Pillado$^{a,}$[^1], Alberto Iglesias$^{b,}$[^2] and Warren Siegel$^{c,}$[^3]
.3cm
$^a$*Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155*
$^b$*Department of Physics, University of California, Davis, CA 95616*
$^c$*C. N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794*
1.9cm
Introduction
============
The motivation to investigate the semiclassical limit of fundamental strings is twofold. First, in view of the revived interest in the possibility of producing superstrings of cosmic size in models of brane inflation, it has been suggested that brane annihilation would leave behind a network of lower dimensional extended objects [@Tye; @AlexGia; @CMP; @Polchinski] which would be seen as strings from the four dimensional point of view. This realization opens up the possibility of observing the cosmological consequences of cosmic strings, either from fundamental strings or from (wrapped) D-branes. It is therefore interesting to understand how to reconcile the usual quantum mechanical treatment of fundamental strings with their expected classical behaviour at cosmological scales. Second, the semiclassical limit discussed in this paper may play a role in relation to the possible microscopic counting of states [@Sus; @Sen; @Str; @Dab] associated with known classical supergravity solutions.
It is usually assumed that the description of a semiclassical string state (a string of macroscopic size) is in terms of a coherent superposition of the fundamental string quanta. However, ealier attempts to build this type of state in the covariant gauge quantization face serious difficulties [@calucci]. In the next section we motivate the use of the lightcone gauge coherent state to give an accurate microscopic description of extended open string solutions (spinning string configuration). We contrast the result with the alternative mass eigenstates (“perturbative states”) with the same angular momentum, emphasizing the advantages of the former. In section 3 we focus on the closed string case. We describe the obstacle to the naive extrapolation from the open string case. We then provide the solution, suitable for the case of a circular target space loop, by gauge-fixing the residual rigid $\sigma$ translation symmetry. This is done in three ways: in unitary gauge, through a BRST method and in Gupta-Bleuler like quantization.
Open strings
============
In this section we explore the relation between perturbative quantum states and their classical counterparts for open strings. In particular, we will find the closest quantum state to the leading classical Regge trajectory that corresponds to the following spinning configuration, \[spinning\] X\^0 &=& A ,\
X\^1 &=& A ,\
X\^2 &=& A . In the conventions of [@GSW], the general solution for the open strings in light-cone gauge is given by the following expressions, X\^+ &=&x\^+ + l\^2 p\^+ ,\
X\^i &=& x\^i + l\^2 p\^i + i l \_[n= 0]{} [1n]{} \_n\^i [e]{}\^[-i n ]{} n ,\
X\^-&=&x\^-+l\^2p\^-+il\_[n= 0]{}[1n]{} \_n\^- [e]{}\^[-in]{} n . On the other hand, in order to fulfill the constraints the $\alpha_n^-$ are restricted to be functions of the physical transverse directions, \_n\^- = [1]{} (\_[m=-]{}\^ : \_[n-m]{}\^i \_m\^i : -2a\_n) , where the $a$ coefficcient comes from the normal ordering of the $\alpha$ operators. Furthermore we can also write the mass and the angular momentum in terms of transverse modes in the following way, M\^2&=&[2l\^2]{} (\_[n=1]{}\^\_[-n]{}\^i\_n\^i-a) ,\
J\^[ij]{}&=&-i\_[n=1]{}\^[1 n]{} (\_[-n]{}\^i\_n\^j -\_[-n]{}\^j\_n\^i) .
Classical Regge trajectory in the light-cone gauge
--------------------------------------------------
We start out by finding the description in the lightcone gauge of the classical solution of the open string found in (\[spinning\]). It is clear that this state corresponds to a string spinning around its center of mass, which has been set to the origin of the coordinates, which implies that, x\^+=x\^-=x\^i=p\^i=0 . Also, it is easy to see that the only excited oscillators in this solution are,: \_1\^1 &=& (\_[-1]{}\^1)\^\* = -[[i A]{}]{} ,\
\_1\^2 &=& (\_[-1]{}\^2)\^\* = [[ A]{}]{} . Which in turn implies that, X\^1 &=& il (-\_[-1]{}\^1 e\^[i]{} + \_1\^1 e\^[-i ]{}) = A ,\
X\^2 &=& il (-\_[-1]{}\^2 e\^[i]{} + \_1\^2 e\^[-i ]{}) = A . And for the $(-)$ oscillators we obtain, \_0\^-= l p\^- = [1 ]{} ([A l]{})\^2 , with all the other $\alpha_n^-=0$. Note that the only non-trivial cases are $n = \pm 2$, which in our case are still zero, \_2\^- = [1 ]{} (\_1\^1 \_1\^1 + \_1\^2 \_1\^2 ) = [1 ]{} = 0 , and the same for $\alpha^-_{-2}$. If we want the string to move on the $1-2$ plane only, we have to impose that $X^9=0$, X\^9 = [1 ]{} (X\^+ - X\^- ) = [1 ]{}(l\^2p\^+ -l\^2 p\^-) =0 , in other words, that $p^+=p^-=A/\sqrt{2} l^2$. This also means that, X\^0 = [1 ]{} (X\^+ + X\^- ) = [1 ]{} (l\^2p\^+ + l\^2 p\^-) = A . So, putting all these results together, we finally get a solution of the form (\[spinning\]), X\^0 &=& A ,\
X\^1 &=& A ,\
X\^2 &=& A . We can also use the equations given above to compute, in the classical limit, the observables of this state, [*i.e.*]{}, its mass and angular momentum. M\^2&=&[2l\^2]{} (\_[n=1]{}\^ \_[-n]{}\^i \_n\^i) = [2l\^2]{} ( \_[-1]{}\^1 \_1\^1 + \_[-1]{}\^2 \_1\^2 ) = [A\^2 l\^4]{} ,\
J\^[12]{}&=& - i ( \_[-1]{}\^1 \_1\^2 - \_[-1]{}\^2 \_1\^1 ) = [ A\^2 ]{} . which indeed show that these configurations belong to the classical Regge trajectory of maximum angular momentum per unit mass.
Quantum State
-------------
We will now try to obtain the quantum state for the open bosonic string that resembles the classical case described above. There seem to be two natural possibilities:
### “Perturbative” Regge State (Mass Eigenstates)
The classical calculation suggests that we construct the quantum state for this solution in the following way, |= [1 ]{} (\_[-1]{}\^2 - i \_[-1]{}\^1)\^n |0 . The reason to choose this particular configuration becomes clear when we realize that this state is, in fact, an eigenstate of mass and angular momentum, M\^2 |&=&[2l\^2]{}(\_[-1]{}\^1 \_1\^1 + \_[-1]{}\^2\_1\^2-1)|=[2l\^2]{} (n-1)| ,\
J\^[12]{}|&=&-i ( \_[-1]{}\^1 \_1\^2 - \_[-1]{}\^2 \_1\^1 )|=n| . Using the identification $n = 1+A^2/ 2 l^2$, we see that this state has identical values of mass and angular momentum to the classical configuration in the $n \gg 1$ or $A \gg l$ limit. In fact, they saturate the quantum inequality for mass eigenstates given by,
J\^[12]{} M\^2 +1 On the other hand, this is not an eigenstate of the position of the string. It is easy to show that, in this state, the expectation value of the spatial part of the string position operator is equal to zero for all values of $\sigma$ and $\tau$, namely, |\_n\^i|&=&0 ,\
|\_k\^-|&=& | [1]{} \_[m=-]{}\^ \_[k-m]{}\^i \_m\^i |= 0 , for $k \neq 0$. Finally, |\_0\^-|= [[n-1]{} ]{} , which implies that[^4] |X\^0|&=& A ,\
|X\^i|&=& 0 ,\
|X\^9|&=& 0 .
We notice that while it is true that this state has similar properties to the classical one, it clearly does not resemble the macroscopic string state, in the sense that its spacetime motion is not reproduced at all, not even taking a high excitation number ([*i.e.*]{}, $n\gg 1$ does not produce a semiclassical limit).
### “Coherent” Regge State
On the other hand, it seems more reasonable to try to mimic the classical configuration by constructing a coherent state of the form, |= e\^[v\_[-1]{}\^1-v\^\*\_[1]{}\^1]{} e\^[-iv\_[-1]{}\^2-iv\^\*\_[1]{}\^2]{}|0 , where $v$ is a parameter related to $p^+$, ($p^+=l^{-1}\sqrt{2|v|^2-1}$) and to the amplitude of the spacetime oscillations.
This state has the following expectation values for the energy and angular momentum, | M\^2 |&=& [2l\^2]{}|(\_[n=1]{}\^\_[-n]{}\^i \_n\^i - 1 )|\
&=& [2l\^2]{} |( \_[-1]{}\^1 \_1\^1 + \_[-1]{}\^2 \_1\^2 - 1 ) |=[2l\^2]{} (2|v|\^2-1) ,\
| J\^[12]{}|&=& - i | ( \_[-1]{}\^1 \_1\^2 - \_[-1]{}\^2 \_1\^1 ) |= 2|v|\^2 , which also correspond to the values obtained in the previous section by considering the identification $2|v|^2 = n$. The key point, however, is that this state does have the spacetime position expectation value of an extended string, namely, | X\^0|&=& l = A ,\
| X\^1|&=& 2 l v = ,\
| X\^2|&=& 2 l v = ,\
| X\^9|&=& 0 , which for $A \gg l$ approaches the classical solution discussed above.[^5] This shows that the coherent state is a much closer match to the classical solution than the previously considered construction.
Closed strings
==============
The calculations in the previous section show how one can obtain a semiclassical coherent state for open strings in a very similar way to the simple harmonic oscillator. However, no such construction is available for closed strings, except in the approach of “semiclassical” quantization (in the sense of [@ls], where the constraints are satisfied in mean value) as in [@bpi2]. In this section we will present the obstacle in finding a macroscopic state from perturbative closed string states in the lightcone gauge, and propose a solution.
The problem of a microscopic perturbative description
-----------------------------------------------------
Consider the first order form of of the bosonic string action with world-sheet coordinates $m=(0,1)\equiv(\sigma,\tau)$, \[action\] S=[12\^]{}d\^2(\_m XP\^m+g\_[mn]{} [12]{}P\^mP\^n) , where $g_{mn}=(-h)^{-1/2}h_{mn}$ is the unit determinant part of the world-sheet metric $h_{mn}$ (related to the second order form [@act] using $P^m=(-h)^{1/2}h^{mn}\partial_n X$).
Integrating out $P^\sigma$ via its equation of motion: P\^=-[1g\_[11]{}]{}(X\^+g\_[01]{}P\^) the action becomes \[act2\] S=[12\^]{}d\^2 , where $P^\tau\equiv P$ to simplify notation. The reparametrization symmetry of (\[action\]) can be used to set $g_{11}=1$ and $g_{01}=0$. Further, Weyl symmetry can be used to set $h_{11}=1$ and residual semilocal symmetry to set X\^+= , P\_+=1 ([light-cone gauge]{}) . Let us now look closely at the constraint obtained by varying $g_{01}$, [*i.e.*]{}, $X^\prime\cdot P=0$ that has the following mode decomposition ($\int_0^\pi~d\tau~{\rm e}^{2im\tau}\cdots$): C\_m&=&\_m\^–\_m\^-+\_[n= 0]{} (\_[m-n]{}\^i\_n\^i- \^i\_[m-n]{}\_n\^i) , where C\_0=N , ([recall]{} \^-\_0=\_0\^-) is the generator of rigid $\sigma$ shifts ($\delta X=\epsilon \partial_\sigma X$, with constant $\epsilon$). Upon quantization, if a physical state $|{\rm phys}\rangle$ satisfies the constraint $C_0$: N |[phys]{}=0 , then, it follows that 0|\[N, X\] |[phys]{}=\_X . This shows that the string would appear to be stuck at a fixed point $\langle X \rangle$ for all $\sigma$, making it impossible, in this gauge, to have a macroscopic extended closed string. The reason for this is that we have not fixed the gauge completely so, in practice, we are integrating over all the gauges compatible with the lightcone gauge which, of course, yields the aforementioned center of mass position for the whole string. This makes the operator $ X $ not the right quantity to look at in this gauge if we are interested in evaluating the semiclassical position of the string. The way out of this problem that we suggest in the following section is to fix the gauge completely before evaluating the position of the string.
Gauge-fixing $\sigma$ translations
----------------------------------
In this subsection we will fix the gauge for the residual rigid reparametrization symmetry $\sigma\to
\sigma+\epsilon$ that remains after choosing light-cone gauge for closed strings. We do this by prescribing the value of one of the coordinates modes of the string. This fixes the symmetry in a way similar to the way in which the open string arises from the closed string by removing the modes of one handedness [@s].
Our goal here will be that of describing a circular string. Once we are in the light-cone gauge, $X^+=(X^0+X^9)/\sqrt{2}\propto \tau$, we single out two of the coordinates (that span the plane in which the circular loop lies): $X^1$ and $X^2$. The solutions for the equations of motion of these coordinates have the usual decomposition into left and right moving modes, namely, $X^i=X^i_L+X^i_R$. [^6]
We propose the following additional gauge-fixing condition suitable for the description of the circular string loop states: \[GF\] =[1]{}d\^[-2i]{}(\_-X\^1 -v l [e]{}\^[-2i(-)]{})=0 , where $\partial_-=\partial_\tau-\partial_\sigma$ and the parameter $v$ will be related to the radius of the circle. Note that this is only a condition involving the left-moving part of $X^1$ since $\partial_- X_R\equiv 0$.
To the action (\[act2\]) we add the gauge-fixing term: \[Lgf\] [L]{}\_[gf]{}= , such that $\lambda$ acts as the Lagrange multiplier enforcing the gauge conditions $\Phi=0$ ([i.e.]{}, determining $\alpha_1^1$).
After reaching light-cone gauge, the only remaining piece of the third term in the action (\[act2\]) is given by dg\_[0]{}\_[n]{}( \_[-n]{}\^i() \_n\^i()-\_[-n]{}\^i()\_n\^i()) ,where $g_0$ stands for the zero mode of $g_{01}$, we have used the decomposition $\partial_-X^1=\sum \alpha_n^1(\tau) {\rm
e}^{2ni\sigma}$, $\dots$ etc.
Varying with respect to $\lambda$ and $g_0$ we obtain the gauge-fixing condition and the constraint that can be solved (if $v\not =0$) for $\alpha_1^1(\tau)$ and $\alpha_{-1}^1(\tau)$ respectively. On the solutions, $\alpha_n(\tau)=\alpha_n {\rm e}^{-2in\tau},\dots$, etc. Therefore, we obtain, \_1\^1&=&v ,\[sol1\]\
\_[-1]{}\^1&=&-[1v]{}(\_[m2]{} \_[-m]{}\^1 \_m\^1+\_[n1]{} \_[-n]{}\^j \_n\^j-\_[-n]{}\^i\_n\^i) ,\[sol2\] where $j=2,\cdots,8$. These results show that the idea behind our gauge fixing choice in (\[GF\]) is very much like the one used to solve the constraints in the lightcone gauge expressing $\alpha_n^-$ in terms of the transverse modes.
In this gauge, then, the mode decomposition of $X^1_L$ is different from the usual. It reads, \[x1\] X\^1\_L&=&[12]{}x\^i+[12]{} l\^2 p\^i(-) - [i2]{}l \_[-1]{}\^1[e]{} \^[2i(-)]{} + [i 2]{} v l [e]{} \^[-2i(-)]{}\
&&+ [i2]{}l \_[n>1]{} [1n]{} (\_n\^1 [e]{}\^[- 2in(-)]{} -\_[-n]{}\^1 [e]{} \^[2in(-)]{}) , where $\alpha_{-1}^1$ should be interpreted as the operator on the rhs of (\[sol2\]).
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Alternatively, using the BRST method, the same result can be obtained with a gauge-fixing term linear in $\Phi$. The BRST charge in this case contains an extra term: Q\_[extra]{}=c , with corresponding ghost and anti-ghost $c$ and $b$ satisfying $\{c, b\}=1$.
The gauge-fixing term in this case is \_[gf]{}={Q,} , where $\Lambda= b \Phi$ is the gauge-fixing function. The extra term $ Q_{\rm extra}$ gives the contribution (\[Lgf\]), and there are also Fadeev-Popov terms from $b\{Q,\Phi\}$ with contribution from the term originally present in the BRST charge, $\tilde c \Delta N$. By using the gauge condition this contribution is $v\tilde c b$. Thus, the ghosts decouple in this gauge.
Then, one proceeds as before, solving the gauge condition and constraint.
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Quantization in this gauge can also be achieved in a Gupta-Bleuler approach, by imposing the gauge condition and constraint on physical states. For any pair of physical states $|\chi\rangle$ and $|\phi\rangle$, we require: \[phi\] ||=0 . But (\[phi\]) is satisfied if we impose the following condition on physical states: \[cond0\] |[phys]{}=0 , Therefore, (\[cond0\]) implies, \[cond\] \_1\^1|[phys]{}=v |[phys]{} ,
Using (\[cond\]), the $\Delta N=0$ constraint can be rewritten. Again, at the quantum level, for any pair of physical states $|\chi\rangle$ and $|\phi\rangle$ \[DN\] 0=|N| , should hold. But it is enough to impose: (v \_[-1]{}\^1+\_[m2]{} \_[-m]{}\^1 \_m\^1+\_[n1]{} \_[-n]{}\^j \_n\^j-\_[-n]{}\^i\_n\^i)|[phys]{}=0 ,
The state
---------
Let us now consider a state in the gauge of the previous subsection of the following form |\_0=|\_0\_L|n\_R , where the left-moving factor is a coherent state built on a left vacuum, $|0\rangle_L$, \[state\] |\_0\_L = [e]{}\^[-iv\_[-1]{}\^2-iv\^\*\_1\^2]{}|0\_L , and the right-moving part is an eigenstate of $N_R$ with eigenvalue $2
v^2$. We also set the parameters $x^i$ and $p^i$ to zero, making the center of mass the origin of the coordinate system. Notice that $\alpha^2_1|\phi_0\rangle_L=-iv$ which implies that $\langle\phi_0|\alpha^1_{-1}|\phi_0\rangle_L=v$ upon using equation (\[sol2\]).
We can compute now the expectation value of the string coordinates in the normalized state $|\phi_0\rangle$ to find a stationary circular loop of radius $vl$: X\^0&=& 2l ,\
X\^1&=&[[il]{}2]{}\_0| -\_[-1]{}\^1[e]{}\^[2i(-)]{}+ v [e]{}\^[-2i(-)]{}|\_0 ,\
&=&v l [sin]{} 2(-) ,\
X\^2&=&[[il]{}2]{}\_0| -\_[-1]{}\^2[e]{}\^[2i(-)]{}+ \_[1]{}\^2[e]{}\^[-2i(-)]{}|\_0 ,\
&=&v l [cos]{} 2(-) . There is no contribution from the right-moving bosonic excitations to the expectation value because we are considering that this sector is in a $N_R$ eigenstate.
Before ending this subsection, let us note that the circular loop is just one possible coherent state constructed using this gauge. It is not difficult to see the generalization to other shapes. Take a left-moving component for the state of the form: |\_0= [e]{}\^[A]{}|0\_L , [A]{}=\_[{i,m}={1,1}]{}u\_[mi]{} \_[-m]{}\^i+u\_[mi]{}\^\*\_m\^i , $i=1,\cdots,8$ and $m$ runs over positive integers. If the right-moving part of the state has level $N_R=n$, then $\langle X^1\rangle$ is real provided n-v\^2=\_[{i,m}={1,1}]{}|u\_[mi]{}|\^2 . Choosing the parameters $v$ and $n$ appropriately one can build a loop of arbitrary shape in target space. As a trivial example, consider $n=v^2$, $u_{mi}=0$ to obtain a folded string along the $X^1$ axis.
Conclusions
===========
The use of coherent states allows for the construction of semiclassical states in superstring theory that bear close resemblance to the classical solutions. These are of relevance in both studies of some macroscopic defects expected to arise in string theory descriptions of inflationary cosmology and towards a microscopic entropy counting of certain string configurations that correspond to classical supergravity solutions.
For the open string case we have shown that the coherent state reproduces the classical motion in the target spacetime while other ’perturbative’ excitations have large oscillations (averaging to zero) around the center of mass position of the string. In the closed string case similar properties are found. Also, with the gauge-fixing of section 3 we obtained the microscopic description of a left-moving static loop supported by a right-moving world-sheet current; the analog of a classical superconducting vorton solution [@vortons].
As a final remark let us mention that throughout the text we have used ten dimensional spacetime thinking about the bosonic part of a superstring, the modification to include the fermionic sector being straightforward. In particular, the state of subsection 3.3 could have fermionic right-moving excitations accounting for part or the whole of the level $N_R$ needed. The latter possibility leads to the loop completely stabilized by fermionic excitations [@bpi2] which has no classical gravitational radiation.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Jaume Garriga, Ken Olum, Alex Vilenkin and especially Roberto Emparan and Jorge Russo for illuminating discussions. AI is grateful to the organizers of the Simons Workshop at Stony Brook and Perimeter Institute for their hospitality while this work was in progress. The work of AI was supported by DOE Grant DE-FG03-91ER40674. WS was supported in part by NSF Grant PHY-0354776.
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[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail: [email protected]
[^4]: Where we have used the relations noted before between the different sets of parameters, $p^+ = {A \over {\sqrt{2} l^2}}$ as well as $n = {{A^2}\over {2 l^2}}+1 $.
[^5]: Note that in this limit we also recover the same values for the mass and angular momentum of the classical configuration.
[^6]: For closed strings the standard decomposition into left and right-movers is $$\begin{aligned}
X^i&=&X^i_L+X^i_R~,\\
X^i_L&=&{1\over 2}x^i+{1\over 2} l^2 p^i(\tau-\sigma)+
{i\over 2}l
\sum_{n\not = 0} {1\over n} \alpha_n^i{\rm e} ^{-2in(\tau-\sigma)}~,\\
X^i_R&=&{1\over 2}x^i+{1\over 2} l^2 p^i(\tau+\sigma)+{i\over 2}l
\sum_{n\not = 0} {1\over n} \tilde\alpha_n^i{\rm e}^{-2in(\tau+\sigma)}~,\end{aligned}$$ where $i=1, \cdots, 8$, and $x^i$ and $p^i$ are the center of mass position and momentum of the loop.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle’s result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao’s influential result had shown that the problem is already non-trivial in one dimension [@Yao-1Dlb]: using $m$ units of space, the query time $Q(n)$ must be $\Omega( \alpha(m,n) + \frac{n}{m-n+1})$ where $\alpha(\cdot,\cdot)$ is the inverse Ackermann’s function, a very slowly growing function. In $d$ dimensions, Bernard Chazelle [@Chazelle.LB.II] proved that the query time must be $Q(n) = \Omega( (\log_\beta n)^{d-1})$ where $\beta = 2m/n$. Chazelle’s lower bound is known to be tight for when space consumption is “high” i.e., $m = \Omega(n \log^{d+\varepsilon}n)$.
We have two main results. The first is a lower bound that shows Chazelle’s lower bound was not tight for “low space”: we prove that we must have $m Q(n) = \Omega(n (\log n \log\log n)^{d-1})$. Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions.
author:
- Peyman Afshani
bibliography:
- 'ref.bib'
title: A New Lower Bound for Semigroup Orthogonal Range Searching
---
Introduction
============
The Lower Bound {#sec:lb}
===============
The Upper Bounds {#sec:ub}
================
Conclusions {#sec:conc}
===========
In this paper we considered the semigroup range searching problem from a lower bound point of view. We improved the best previous lower bound trade-off offered by Chazelle by analysing a well-distributed point set for $(2d-1)$-sided queries for an idempotent semigroup. Furthermore, we showed that our analysis is tight which leads us to suspect that we have found an (almost) optimal lower bound for idempotent semigroups as we believe it is unlikely that a more difficult point set exists. Thus, two prominent open problems emerge: (i) Can we improve the known data structures under the [*extra*]{} assumption that the semigroup is idempotent? (ii) Can we improve our lower bound under the [*extra*]{} assumption that the semigroup is not idempotent? Note that the effect of idempotence on other variants of range searching was studied at least once before [@Arya.idem].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We compute the charged pion loop contribution to the light-by-light scattering amplitude for off-shell photons in chiral perturbation theory through next-to-leading order (NLO). We show that NLO contributions are relatively more important due to a fortuitous numerical suppression of the leading-order (LO) terms. Consequently, one expects theoretical predictions for the hadronic light-by-light (HLBL) contribution to the muon anomalous magnetic moment, $a_\mu^\mathrm{HLBL}$, to be sensitive to the choice of model for the higher momentum-dependence of the LBL amplitude. We show that models employed thus far for the charged pion loop contribution to $a_\mu^\mathrm{HLBL}$ are not consistent with low-momentum behavior implied by quantum chromodynamics, having omitted potentially significant contributions from the pion polarizability.'
author:
- 'Kevin T. Engel'
- 'Hiren H. Patel'
- 'Michael J. Ramsey-Musolf'
title: 'Hadronic Light-by-Light and the Pion Polarizability'
---
The anomalous magnetic moment of the muon, $a_\mu=(g_\mu-2)/2$, continues to be a quantity of considerable interest in particle and nuclear physics. The present experimental value, $a_\mu^\mathrm{exp}= 116592089(63)\times 10^{-11}$ obtained by the E821 Collaboration[@Bennett:2006fi; @Bennett:2004pv; @Bennett:2002jb] differs from theoretical expectations by $3.6\sigma$ assuming the Standard Model (SM) of particle physics and state-of-the-art computations of hadronic contributions, including those obtained using data on $\sigma(e^+e^-\to\mathrm{hadrons})$ and dispersion relation methods: $a_\mu^\mathrm{SM}= 116591802(49)\times 10^{-11}$ (for recent reviews, see Ref. [@Passera:2010ev; @Jegerlehner:2009ry] as well as references therein). A deviation of this magnitude can be naturally explained in a number of scenarios for physics beyond the Standard Model (BSM), including (but not limited to) supersymmetry, extra dimensions, or additional neutral gauge bosons [@Stockinger:2006zn; @Hertzog:2007hz; @Czarnecki:2001pv] . A next generation experiment planned for Fermilab would reduce the experimental uncertainty by a factor of four[@FERMILAB-PROPOSAL-0989]. If a corresponding reduction in the theoretical, SM uncertainty were achieved, the muon anomalous moment could provide an even more powerful indirect probe of BSM physics.
The most significant pieces of the error quoted above for $a_\mu^\mathrm{SM}$ are associated with the leading order hadronic vacuum polarization (HVP) and the HLBL contributions: $\delta a_\mu^\mathrm{HVP}(\mathrm{LO}) = \pm 42 \times 10^{-11}$ and $\delta a_\mu^\mathrm{HLBL} = \pm 26 \times 10^{-11}$ [@Prades:2009tw] (other authors give somewhat different error estimates for the latter [@Hayakawa:1997rq; @Hayakawa:1996ki; @Hayakawa:1995ps; @Knecht:2001qg; @Knecht:2001qf; @Melnikov:2003xd; @Bijnens:2007pz; @Nyffeler:2009tw; @RamseyMusolf:2002cy] , but we will refer to these numbers as points of reference; see [@Nyffeler:2010rd] for a review). In recent years, considerable scrutiny has been applied to the determination of $a_\mu^\mathrm{HVP}(\mathrm{LO})$ from data on $\sigma(e^+e^-\to\mathrm{hadrons})$ and hadronic $\tau$ decays. Use of the latter indicating a somewhat smaller discrepancy between the SM and experimental values for $a_\mu$ than quoted above. Clearly, a significant improvement in this determination will be needed if the levels of theoretical and future experimental precision are to be commensurate.
Here, we concentrate on the $a_\mu^\mathrm{HLBL}$, focusing in particular on the contributions from charged pion loops. Subsequent to the first results from the E821 Collaboration, the theoretical community devoted substantial effort to refining the predictions for pseudoscalar pole" contributions, which appear at leading order in the expansion of the number of colors $N_C$ and which are numerically dominant. However, the error quoted for the charged pion loop contributions, which enter at subleading order in $N_C$, is now comparable to the uncertainty associated with the pseudoscalar pole terms. Thus, we are motivated to revisit the former as part of the effort to improve the level of confidence in the theoretical SM prediction for $a_\mu^\mathrm{HLBL}$.
As a first step in that direction, we have computed the HLBL scattering amplitude for off-shell photons to NLO in Chiral Perturbation Theory ($\chi$PT). $\chi$PT is an effective field theory for low-energy interactions of hadrons and photons that incorporates the approximate chiral symmetry of quantum chromodynamics (QCD) for light quarks. Long-distance hadronic effects can be computed order-by-order in an expansion of $p/\Lambda_\chi$, where $p$ is a typical energy scale (such as the pion mass $m_\pi$ or momentum) and $\Lambda_\chi=4\pi F_\pi\sim 1$ GeV is the hadronic scale with $F_\pi=93.4$ MeV being the pion decay constant. At each order in the expansion, presently incalculable strong interaction effects associated with energy scales of order $\Lambda_\chi$ are parameterized by a set of effective operators whose coefficients – low energy constants" (LECs) – are fit to experimental results and then used to predict other low-energy observables.
$\chi$PT has been applied with considerable success to the analysis of a variety of hadronic and electromagnetic processes (for a recent review, see [*e.g.*]{} [@Bijnens:2006zp]), making it an in principle appropriate and model-independent framework for investigating hadronic contributions to $a_\mu$, another low-energy observable. In the $\chi$PT analysis of the pseudoscalar pole contributions to $a_\mu^\mathrm{HLBL}$, however, one encounters a new LEC that cannot be determined independent of the $a_\mu$ measurement itself. Consequently, hadronic modeling is presently unavoidable if one wishes to predict the anomalous moment. Nevertheless, the calculable terms in $\chi$PT can be used to test or constrain model input, as any credible model for the LBL amplitude must reproduce behavior in the low-energy regime that is dictated by QCD. Indeed, the $\chi$PT computation of the leading $\ln^2$ term in the pion pole contribution revealed a critical sign error in earlier numerical computations of the pion pole contribution[@Knecht:2001qg; @Knecht:2001qf]. The sub-leading $\ln$ term can be obtained from a combination of analytic computation[@RamseyMusolf:2002cy] and a determination of the relevant LEC from a determination of the $\pi^0\to e^+e^-$ branching ratio[@Abouzaid:2006kk], and it can be used to further constrain the model input.
In this spirit, we have analyzed the charged pion loop contribution to the LBL amplitude to NLO and have compared with corresponding predictions implied by models used in the computation of $a_\mu^\mathrm{HLBL}$. The leading order (in chiral counting) contribution is fixed entirely by gauge invariance and contains no unknown constants. As we show below, this contribution is fortuitously suppressed. As a result, higher order contributions are likely to be relatively more important than one might expect on general grounds, rendering this quantity more susceptible to model-dependent uncertainties. Thus, it becomes all the more important that any model used for the charged pion contribution to $a_\mu^\mathrm{HLBL}$ respect the requirements of QCD at NLO in the low-momentum regime. In this respect, we find that models utilized to date have omitted a potentially significant contribution associated with the pion polarizability, leading one to question the reliability of the presently-quoted value for $a_\mu^\mathrm{HLBL}$. Below, we provide details of the calculation leading to this conclusion.
We compute the charged pion contributions to the LBL vertex function $\Pi^{\mu\nu\alpha\beta}$ through NLO from the diagrams in Figure 1, expanding the result as a power series in the external (photon) momentum and pion mass. The LO amplitude that corresponds to a pure scalar QED calculation for point-like charged pions follows from Fig. 1(a) and yields a finite result that is free from any LECs. The result contains two $\mathcal{O}(p^4)$ structures that can be expressed in terms of two dimension eight ($d=8$) operators, $32\, \mathcal{O}_1^{(8)}\equiv(F^2)^2\equiv(F_{\mu\nu} F^{\mu\nu})^2$ and $8\,\mathcal{O}_2^{(8)}\equiv F^4=F_{\alpha\beta} F^{\beta\gamma} F_{\gamma\lambda} F^{\lambda\alpha}$, whose coefficients are given in Table \[tab:LO\] (the operators are defined to absorb symmetry factors). Naively, one would expect the magnitude of the coefficients to be set by $1/(4\pi)^2 \times 1/m_\pi^4$ . However, we find that each operator contains an additional suppression factor of $1/9$ and $1/45$, respectively. Thus, we anticipate that the NLO contributions from the graphs of Fig. 1(b-d) will be relatively more important.
The graphs in Figures 1(b-d) correspond respectively to the propagator, vertex, and polarizability corrections. The first two classes are divergent and require the introduction of counterterms from the $\mathcal{O}(p^4)$ chiral Lagrangian. We carry out the calculation using dimensional regularization in $d=4-2\epsilon$ dimensions and define the counterterms to remove the contributions proportional to $1/\epsilon-\gamma+\ln 4\pi +1$ as is the standard convention for $\chi$PT[@Bijnens:2006zp]. We find that the explicit dependence on the counterterms needed for renormalization of the pion propagator is cancelled by charge and mass renormalization, leaving only a dependence on the $\mathcal{O}(p^4)$ operator associated with the charge radius of the pion: \_9 = ie \_9 F\_ (Q) , where $Q=\mathrm{diag}(2/3,-1/3)$ is the electric charge matrix and $\Sigma=\mathrm{exp}(i \tau^a\ \pi^a/F_\pi)$ with $a=1,2,3$ giving the non-linear realization of the spontaneously broken chiral symmetry. After renormalization, one has for the square of the pion charge radius r\_\^2 = \_9\^r()+ where the superscript $r$" indicates the finite component after the subtraction of $1/\epsilon-\gamma+\ln 4\pi+1$ term is performed. Choosing $\mu=m_\rho$ and taking the experimental value for $r_\pi^2$ gives $\alpha_9^r(m_\rho)= (7.0\pm0.2)\times 10^{-3}$ for two-flavor $\chi$PT at $\mathcal{O}(p^4)$. Within error bars, this result is the same as obtained in Ref. [@hep-ph/0203049] for the three-flavor case.
The $\pi\pi\gamma\gamma$ vertex correction shown in Fig. 1(d) is finite, but the polarizability amplitude nevertheless receives an additional finite contribution from $\mathcal{L}_9$ and \_[10]{} = e\^2\_[10]{} F\^2 (QQ\^) . The corresponding combination entering the LBL amplitude is $\alpha_9^r+\alpha_{10}^r$. As the sum of the one-loop polarizability sub-graphs is finite, this combination of LECs is independent of the renormalization scale. An experimental value $(\alpha_9^r+\alpha_{10}^r)_\mathrm{exp} = (1.32\pm 0.14)\times 10^{-3}$ has been obtained from radiative pion decay [@arXiv:0801.2482]. As a cross check on the extraction of these LECs we also consider the determination of $\alpha_{10}^r$ from semileptonic $\tau$-decays given in Ref. [@arXiv:0810.0760]. Converting from three- to two-flavor $\chi$PT we obtain $\alpha_{10}^r(m_\rho)=-(5.19\pm 0.06)\times 10^{-3}$, in reasonable agreement with the determination of $\alpha_9^r(m_\rho)$ from the pion form factor and $(\alpha_9^r+\alpha_{10}^r)$ from pion radiative decay. The resulting prediction for the pion polarizability[@Gasser:2006qa], which we confirm by taking the on-shell photon limit of our off-shell $\pi^+\pi^-\gamma\gamma$ computation, disagrees with the latest experimental determination[@Ahrens:2004mg] by a factor of two.
The final NLO results for the LBL amplitude are summarized in Table I. To lowest order in external momenta, the only change from LO are polarizability corrections which modify the $\mathcal{O}^{(8)}_1$ coefficient. To see the full impact of the (higher momentum) NLO terms, we expand our result to $\mathcal{O}(p^6)$, introducing a complete basis of seven $d=10$ four-photon operators: $$\begin{aligned}
\nonumber
16\, \mathcal{O}_1^{(10)} &=& \partial_\rho F_{\mu\nu}\partial^\rho F^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} \\
\nonumber
8\ \mathcal{O}_2^{(10)}&=&\partial_\rho F_{\mu\nu}F^{\mu\nu}\partial^\rho F_{\alpha\beta}F^{\alpha\beta} \\
\nonumber
2\ \mathcal{O}_3^{(10)}& =&\partial_\rho F_{\alpha\beta}\partial^\rho F^{\beta\gamma}F_{\gamma\delta}F^{\delta\alpha}\\
\nonumber
4\ \mathcal{O}_4^{(10)}&=&\partial_\rho F_{\alpha\beta}F^{\beta\gamma}\partial^\rho F_{\gamma\delta}F^{\delta\alpha} \\
\nonumber
4\ \mathcal{O}_5^{(10)}&=&\partial^\mu F_{\mu\nu}F^{\alpha\nu}\partial_\alpha F_{\beta\gamma}F^{\beta\gamma} \\
\nonumber
4\ \mathcal{O}_6^{(10)}&=&F_{\mu\nu}F^{\alpha\nu}\partial^\mu F_{\beta\gamma}\partial_\alpha F^{\beta\gamma} \\
\nonumber
2\ \mathcal{O}_7^{(10)}&=& F_{\mu\nu}\partial^\mu F_{\alpha\beta}\partial^\nu F^{\beta\gamma}F_{\gamma\alpha} \end{aligned}$$ The coefficients of these operators are given in Table II. At this order, both vertex and polarizability corrections modify the LO result.
To obtain a sense of the numerical impact of the two-loop corrections, including those involving $\alpha_9^r+\alpha_{10}^r$, we utilize the values of the LECs discussed above. In the case of $\mathcal{O}_1^{(8)}$, the NLO (two-loop) contribution represents a $\sim 20\%$ correction to the LO term, substantially larger than the $\sim m_\pi^2/\Lambda_\chi^2\sim 0.01$ magnitude one might expect from power counting arguments. In the case of the $d=10$ operators, the NLO corrections range from a few to $\sim 30\%$. The largest impact of the charge radius corrections is on $\mathcal{O}_1^{(10)}$ ($\sim 30\%$) while the most important effect of the polarizability is on $\mathcal{O}_2^{(10)}$ ($\sim 10\%$). As we discuss below, the numerical impact of the various NLO contributions on the low-momentum HLBL amplitude – while illustrating their relative importance due to the LO suppression – may not be indicative of their impact on the $a_\mu^\mathrm{HLBL}$. Indeed, previous experience with the inclusion of the pion form factor in earlier work [@Hayakawa:1995ps; @Bijnens:1995cc; @Bijnens:1995xf; @Hayakawa:1996ki] suggests that the effect on $a_\mu^\mathrm{HLBL}$ may be even more pronounced than implied by these low-momentum comparisons.
Operator 1 loop $\chi$PT 2 loop VMD
----------------------- ----------------- ----------------------------------------------------------------- -----
$\mathcal{O}_1^{(8)}$ $1/9$ $\frac{m_\pi^2}{F_\pi^2}\frac{16}{3}(\alpha_9^r+\alpha_{10}^r)$ 0
$\mathcal{O}_2^{(8)}$ $1/45$ 0 0
: Coefficients of lowest dimension ($d=8$) operators contributing to the HLBL amplitude, scaled by $(4\pi)^2 m_\pi^4/e^4$. Second and third columns give LO and NLO contributions in $\chi$PT, while final column indicates the VMD result [@Bijnens:1995xf]. []{data-label="tab:LO"}
$n$ 1 loop 2 loop VMD
----- ----------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------
$1$ $\frac{1}{45}$ $\frac{1}{3} \bigl\{\frac{1}{9}(m_\pi r_\pi)^2+\frac{4}{5}(\frac{m_\pi}{F_\pi})^2(\alpha_9^r+\alpha_{10}^r) \bigr\}$ $ \frac{2}{9}\frac{m_\pi^2}{M_V^2}$
$2$ $\frac{2}{45}$ $\frac{1}{9}\bigl\{\frac{1}{3} (m_\pi r_\pi)^2+\frac{1}{2}\frac{m_\pi^2}{\Lambda_\chi^2}+\frac{44}{5} (\frac{m_\pi}{F_\pi})^2 (\alpha_9^r+\alpha_{10}^r)\bigr\}$ $\frac{2}{9}\frac{m_\pi^2}{M_V^2}$
$3$ $\frac{2}{315}$ $ \frac{1}{135} (m_\pi r_\pi)^2 $ $\frac{2}{45}\frac{m_\pi^2}{M_V^2}$
$4$ $\frac{1}{189}$ $ \frac{1}{135} (m_\pi r_\pi)^2 $ $\frac{2}{45}\frac{m_\pi^2}{M_V^2}$
$5$ $\frac{1}{135}$ $\frac{4}{45} (\frac{m_\pi}{F_\pi})^2 (\alpha_9^r+\alpha_{10}^r)$ 0
$6$ $\frac{1}{315}$ 0 0
$7$ $\frac{1}{945}$ 0 0
: Coefficients of $d=10$ operators $\mathcal{O}_n^{(10)}$ contributing to the HLBL amplitude, scaled by $(4\pi)^2 m_\pi^6/e^4$. First column denotes operator index $n$. Second and third columns give LO and NLO contributions in $\chi$PT, while final column indicates VMD result. Identifying $r_\pi^2=6/M_V^2$ (see text) implies agreement between the two-loop $\chi$PT and VMD predictions for the charge radius contribution. []{data-label="tab:NLO"}
We now compare the explicit NLO results in $\chi$PT with the corresponding expectations for the operators in Tables I and II derived from models used to compute the charged pion loop contribution to $a_\mu^\mathrm{HLBL}$. For concreteness, we focus on the extended Nambu-Jona-Lasinio (ENJL) model adopted in Ref. [@Bijnens:1995xf]. In that work, the point-like contributions to the LBL vertex function $\Pi^{\mu\nu\alpha\beta}$ are modified by the inclusion of vector meson dominance (VMD) type propagator functions $
V_{\mu\lambda}(k^2) = ( g_{\mu\lambda} M_V^2-p_\mu p_\lambda)/( M_V^2-p^2)
$ as \[eq:vmd1\] \^V\_(p\_1) V\_(p\_2) V\_(p\_3) V\_(p\_4) \^ , with the vector meson mass" $M_V$ in general a function of the photon momentum $p_j^2$. The Ward identities imply that the $p_\mu p_\lambda$ terms do not contribute to the overall LBL vertex function; hence, the replacement of Eq. (\[eq:vmd1\]) is equivalent introducing a VMD form factor for each photon when $M_V$ is taken to be a constant. The corresponding prediction for the charge radius is $(r_\pi^2)_\mathrm{VMD} = 6/{M_V^2}$. For $M_V=m_\rho$, one obtains a value for $r_\pi^2$ in good agreement with experiment. An analogous treatment using a Hidden Local Symmetry approach [@Hayakawa:1995ps; @Hayakawa:1996ki] agrees with the ENJL prescription to $\mathcal{O}(p^6)$.
Expanding the right hand side of Eq. (\[eq:vmd1\]) to first order in $p^2/M_V^2$ we obtain the model prediction for the NLO operator coefficients given in the last column of Table I. Identifying $6/M_V^2$ with the corresponding quantity that gives the pion charge radius , we observe that the VMD model reproduces some but not all of the physics that one expects at NLO for the LBL amplitude. In particular, the polarizability contributions to $\mathcal{O}_1^{(8)}$ as well as $\mathcal{O}_{1,2,5}^{(10)}$ are absent from the VMD prescription. As a point of principle, the results of this comparison imply that the VMD-type models employed for $a_\mu^\mathrm{HLBL}$ are not fully consistent with the strictures of QCD for the low-momentum behavior of $\Pi^{\mu\nu\alpha\beta}$ and that use of a more consistent model prescription is warranted.
On a practical level, given the relative magnitudes of the $\alpha_9^r+\alpha_{10}^r$ and $\alpha_9^r$, one has reason to suspect that the omission of the polarizability contribution could have numerically significant implications for $a_\mu^\mathrm{HLBL}$. As discussed earlier, a comparison of the low-momentum LO and NLO contributions to the low-momentum HLBL amplitude indicates that the both the charge radius and polarizability contributions that appear at NLO can generate substantially larger corrections than one might expect based on power counting, due to the fortuitous numerical suppression of the LO terms. Moreover, the charge radius and polarizability contributions can have comparable magnitudes in the case of some operators, while for others, one or the other dominates.
At this point, one may only speculate as to the effect on $a_\mu^\mathrm{HLBL}$ of the previously neglected polarizability contribution. Nevertheless, it is instructive to refer to existing model computations that introduce a pion form factor at the $\pi^+\pi^-\gamma$ vertices. In the original computation of Ref. [@CLNS-84-606], inclusion of the form factor via a VMD prescription reduced the magnitude of the charged pion loop contribution to $a_\mu^\mathrm{HLBL}$ by a factor of three from the scalar QED/point-like pion result. The subsequent computation using the HLS procedure yielded an even stronger suppression (a factor of ten)[@Hayakawa:1995ps; @Hayakawa:1996ki]. The ENJL calculation of Ref. [@Bijnens:1995xf] leads to a result that is about four times larger than the HLS computation, but still strongly suppressed compared to the point-like pion/scalar QED limit. In all cases, the use of a VMD type procedure that matches onto the $r_\pi^2$ terms for the HLBL amplitude at low-momentum has a much more significant numerical impact on $a_\mu^\mathrm{HLBL}$ than the low-momentum comparisons would suggest. Given that the latter already indicate a substantial contribution from the pion polarizability, it appears important to include the corresponding physics in modeling the charged pion contribution to $a_\mu^\mathrm{HLBL}$. An effort to do so will be reported in forthcoming work.
[*Acknowledgements*]{} We thank J. Bijnens, E. de Rafael, and A. Vainshtein for useful conversations and Mark B. Wise for discussions in the formulation of this project. The work is partially supported by U.S. Department of Energy contracts DE-NNN (KTE) and DE-FG02-08ER41531 (HHP and MJRM) and the Wisconsin Alumni Research Foundation (HHP and MJRM).
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We propose a scheme for probabilistic teleportation of an unknown two-particle state of general formation in ion trap. It is shown that one can realize experimentally this teleportation protocol of two-particle state with presently available techniques.'
author:
- 'LIAN Shi-Man, YAN Feng-Li'
date: 'December 28, 2009'
title: 'Probabilistic Teleportation of Two-Particle State of General Formation in Ion Trap'
---
[^1]
Entanglement is the most peculiar feature of quantum physics and lies at the heart of quantum information [@NielsenChuang]. One of the most striking application of entanglement is quantum teleportation [@s1], which says that an unknown state can be transmitted from a sender Alice to a spatially distant receiver Bob with the aid of classical communication and a previously shared entanglement. Quantum teleportation has been widely used in the development of quantum computation and quantum communication [@s1; @s2; @s3; @s4; @s5; @s6; @BDEFMS; @DINGShengChaoJINZhi; @zhangguohuayanfengli].
In the original protocol of Bennett et al. [@s1], the sender Alice and a spatially distant receiver Bob are sharing an Einstein-Podolsky-Rosen (EPR) pair first. Then Alice makes the Bell state measurement and transmits her measurement outcome to Bob via a classical channel. According to Alice’s measurement result, Bob performs a corresponding unitary transformation. After that the quantum state has been teleported successfully. Here Alice does not know either the state to be teleported or the location of the intended receiver, Bob. Bennett et al [@s1] also gave a protocol involving teleporting an unknown state of a qudit via a maximally entangled state in $d\times d$ dimensional Hilbert space and by transmitting $2{\rm log}_2d$ bits of classical information.
Since the seminal work of Bennett et al. [@s1], a lot of work has been done in the field of quantum teleportation [@Braunstein; @VaidmanPRA1994; @BKprl1998; @GRpra; @KB; @s7; @GYLScienceinChina2; @WangYan; @APpla; @PAjob; @DLLZWpra2005; @Zhangzhanjun; @TIANDongPingTAOYingJuanQINMeng; @ZHANGXinHuaYANGZhiYongXUPeiPei; @LiXHandDengFG; @ZHANZHANGQunWANGMA; @ZHANGLIUZUOZHANGZHANG; @QIANFANG; @ZHOUYANGLUCAO; @DONGTENG] and quantum teleportation has been experimentally demonstrated by several groups [@BPMEWZ; @FSBFKP; @NKL; @Boschi]. Furthermore, quantum teleportation has also been generalized to more general situations, for example, where two parties may share not with a set of pure entangled states, but with a noisy quantum channel. In this situation they can use an error correcting code [@Gottesman], or they can share the entanglement through this noisy channel and then use teleportation [@BBP96].
It has been demonstrated that for infinite dimensional Hilbert spaces, quantum teleportation is also possible [@Braunstein; @VaidmanPRA1994; @BKprl1998].
Karlsson and Bourennane proposed the so-called controlled quantum teleportation protocol [@KB; @DLLZWpra2005; @Zhangzhanjun; @s7; @GYLScienceinChina2; @LiXHandDengFG]. In the protocol, one can perfectly transport an unknown state from one place to another via a previously shared Greenberger-Horne-Zeilinger (GHZ) state using local operations and classical communications under the control of a third party. The unknown quantum state can not be teleported unless the third party permits them to transmit the state.
In the situation where the shared entanglement between the sender Alice and the receiver Bob is not in a maximally entangled state then they cannot teleport a qubit with both unit fidelity and unit probability. However, it is possible to have unit fidelity teleportation but with a probability less than unity by using a non-maximally entangled state. This is so called probabilistic quantum teleportation [@APpla; @PAjob], and has been shown to be possible using a non-maximally entangled basis as a measurement basis. Furthermore, this probabilistic scheme has been generalized in two ways: (i) to enable teleportation of $N$ qubits [@GRpra] and (ii) controlled teleportation [@s7; @GYLScienceinChina2; @LiXHandDengFG].
So far, many schemes have been proposed about the teleportation of two-particle state [@ShiJiangGuo; @LuGuo; @DaiChenLi; @yantan]. But most of them settled at mathematic logic level and did not present a concretely physics system. Solano et al. [@SolanoCesarMatosFilho] propose a method for implementing a reliable teleportation protocol of an arbitrary internal state in trapped ions, and Home et al. [@HomeStean] proposed a scheme for implementing optimal probabilistic teleportation between two separated trapped ions. In this Letter we propose a scheme for probabilistically teleporting an unknown two-particle state in ion trap.
We suppose that ion 1 and ion 2 in trap $A$ are in an arbitrary electronic state $$|\phi\rangle_{12}=\alpha
|ee\rangle_{12}+\beta|eg\rangle_{12}+\gamma|ge\rangle_{12}+\delta|gg\rangle_{12},$$ where $|\alpha|^2+|\beta|^2+|\gamma|^2+|\delta|^2=1$; $|g\rangle$ and $|e\rangle$ denote ground state and excited state of the ion, respectively. The ions 3, 4, 5 and 6 are in trap $B$. The ions 3 and 4 is in the following state $$|\psi\rangle_{34}=a|ee\rangle_{34}+b|gg\rangle_{34},$$ and the state of ions 5 and 6 is $$|\psi\rangle_{56}=c|ee\rangle_{56}+d|gg\rangle_{56}.$$ Here $|a|^2+|b|^2=1$, $|a|\geq |b|$, $|c|^2+|d|^2=1$, $|c|\geq
|d|$. The overall state of the six ions is $$\label{3}
\begin{array}{lll}
|\psi\rangle_{\rm total} & = &|\phi\rangle_{12}\otimes
|\psi\rangle_{34}\otimes|\psi\rangle_{56}\\
& = & \alpha ac|eeeeee\rangle_{123456}+\alpha
ad|eeeegg\rangle_{123456}\\
&&+\alpha bc|eeggee\rangle_{123456}+\alpha
bd|eegggg\rangle_{123456}\\
& & +\beta ac|egeeee\rangle_{123456}+\beta
ad|egeegg\rangle_{123456}\\
&&+\beta bc|egggee\rangle_{123456}+\beta
bd|eggggg\rangle_{123456}\\& & +\gamma
ac|geeeee\rangle_{123456}+\gamma
ad|geeegg\rangle_{123456}\\
&&+\gamma bc|geggee\rangle_{123456}+\gamma
bd|gegggg\rangle_{123456}\\& & +\delta
ac|ggeeee\rangle_{123456}+\delta
ad|ggeegg\rangle_{123456}\\
&&+\delta bc|ggggee\rangle_{123456}+\delta
bd|gggggg\rangle_{123456}.\\\end{array}$$
One transports adiabatically the ions 3 and 5 to the trap $A$. The purpose of the teleportation protocol is to transmit the arbitrary state $|\phi\rangle$ of the ions 1 and 2, mentioned in Eq.(1), to the ions 4 and 6.
In order to make ion 3 to undergo entanglement with ion 1, Alice performs a C-Not operation between the ion 1 and the ion 3. Similarly, the same C-Not operation has been implemented between the ion 2 and ion 5. After that Alice makes the Hadamard transformations on ion 1 and ion 2, respectively. So the state $|\psi\rangle_{\rm total}$ becomes $$\label{4}
\begin{array}{ll}
|\psi'\rangle =&\frac {1}{2}|ee\rangle_{13}[|ee\rangle_{25}(\alpha
ac|ee\rangle_{46}+\beta ad|eg\rangle_{46}\\
& +\gamma
bc|ge\rangle_{46}+\delta bd|gg\rangle_{46})\\
&+|eg\rangle_{25}(\alpha
ad|eg\rangle_{46}+\beta ac|ee\rangle_{46}\\
& +\gamma bd|gg\rangle_{46}+\delta bc|ge\rangle_{46})\\
&+|ge\rangle_{25}(\alpha
ac|ee\rangle_{46}-\beta ad|eg\rangle_{46}\\
& +\gamma bc|ge\rangle_{46}-\delta bd|gg\rangle_{46})\\
&+|gg\rangle_{25}(\alpha
ad|eg\rangle_{46}-\beta ac|ee\rangle_{46}\\
& +\gamma bd|gg\rangle_{46}-\delta bc|ge\rangle_{46})]\\
&\frac {1}{2}|eg\rangle_{13}[|ee\rangle_{25}(\alpha
bc|ge\rangle_{46}+\beta bd|gg\rangle_{46}\\
& +\gamma
ac|ee\rangle_{46}+\delta ad|eg\rangle_{46})\\
&+|eg\rangle_{25}(\alpha
bd|gg\rangle_{46}+\beta bc|ge\rangle_{46}\\
& +\gamma ad|eg\rangle_{46}+\delta ac|ee\rangle_{46})\\
&+|ge\rangle_{25}(\alpha
bc|ge\rangle_{46}-\beta bd|gg\rangle_{46}\\
& +\gamma ac|ee\rangle_{46}-\delta ad|eg\rangle_{46})\\
&+|gg\rangle_{25}(\alpha
bd|gg\rangle_{46}-\beta bc|ge\rangle_{46}\\
& +\gamma ad|eg\rangle_{46}-\delta ac|ee\rangle_{46})]\\
&\frac {1}{2}|ge\rangle_{13}[|ee\rangle_{25}(\alpha
ac|ee\rangle_{46}+\beta ad|eg\rangle_{46}\\
& -\gamma
bc|ge\rangle_{46}-\delta bd|gg\rangle_{46})\\
&+|eg\rangle_{25}(\alpha
ad|eg\rangle_{46}+\beta ac|ee\rangle_{46}\\
& -\gamma bd|gg\rangle_{46}-\delta bc|ge\rangle_{46})\\
&+|ge\rangle_{25}(\alpha
ac|ee\rangle_{46}-\beta ad|eg\rangle_{46}\\
& -\gamma bc|ge\rangle_{46}+\delta bd|gg\rangle_{46})\\
&+|gg\rangle_{25}(\alpha
ad|eg\rangle_{46}-\beta ac|ee\rangle_{46}\\
& -\gamma bd|gg\rangle_{46}+\delta bc|ge\rangle_{46})]\\
&\frac {1}{2}|gg\rangle_{13}[|ee\rangle_{25}(\alpha
bc|ge\rangle_{46}+\beta bd|gg\rangle_{46}\\
& -\gamma
ac|ee\rangle_{46}-\delta ad|eg\rangle_{46})\\
&+|eg\rangle_{25}(\alpha
bd|gg\rangle_{46}+\beta bc|ge\rangle_{46}\\
& -\gamma ad|eg\rangle_{46}-\delta ac|ee\rangle_{46})\\
&+|ge\rangle_{25}(\alpha
bc|ge\rangle_{46}-\beta bd|gg\rangle_{46}\\
& -\gamma ac|ee\rangle_{46}+\delta ad|eg\rangle_{46})\\
&+|gg\rangle_{25}(\alpha
bd|gg\rangle_{46}-\beta bc|ge\rangle_{46}\\
& -\gamma ad|eg\rangle_{46}+\delta ac|ee\rangle_{46})].\\
\end{array}$$
For the purpose of teleportation, some measurements on ions 1 and 3 and ions 2 and 5 have to be made by Alice in trap $A$. After that Alice transmits the measurement outcomes to Bob, who controls the trap $B$, via a classical channel. For example, if Alice tells Bob that her measurement outcomes are $|ee\rangle_{13}$ and $|ee\rangle_{25}$, then Bob can make conclusion that the state of ions 4 and 6 is $$\alpha ac|ee\rangle_{46}+\beta ad|eg\rangle_{46} +\gamma
bc|ge\rangle_{46}+\delta bd|gg\rangle_{46}.$$
Assume that the center-of-mass vibration mode in trap $B$ is initially prepared in the vacuum state $|0\rangle$, so the state of the system in trap $B$ becomes $$|\psi(0)\rangle= (\alpha ac|ee\rangle_{46}+\beta ad|eg\rangle_{46}
+\gamma bc|ge\rangle_{46}+\delta bd|gg\rangle_{46})|0\rangle.$$
In the trap B, when a laser standing wave tuned to the first lower vibrational sideband was applied to the ion 6 and the Lamb-Dicke criterion is satisfied, the Hamiltonian in an interaction picture reads [@CiracZoller; @ZhengSB] $$H_I=g(c\sigma_+{\rm e}^{-{\rm i}\phi}+c^+\sigma_-{\rm e}^{{\rm
i}\phi}),$$ where $c^+$ and $c$ are the creation and annihilation operators of phonon, $g$ is the Rabi frequency, $\sigma_+=|e\rangle_6\langle g|$ and $\sigma_-=|g\rangle_6\langle e|$, $\phi$ is the phase of this laser field. As the Hamiltonian for the quantum system is not a time-varying, hence if the laser beam is on for a certain time $t$, the evolution of the system will be described by the unitary operator $$\begin{array}{ll}
U(t)&={\rm e}^{-\mathrm{i}Ht}\\
&= \cos(gt\sqrt {1+c^+c})|e\rangle_6\langle e|\\
&~~ -{\rm i}\mathrm{e}^{-\mathrm{i}\phi}\frac {\sin(gt\sqrt
{1+c^+c})}{\sqrt
{1+c^+c}}c|e\rangle_6\langle g|\\
&~~-{\rm i}\mathrm{e}^{\mathrm{i}\phi}\frac {\sin(gt\sqrt
{c^+c})}{\sqrt {c^+c}}c^+|g\rangle_6\langle e|\\
&~~+ \cos(gt\sqrt {c^+c})|g\rangle_6\langle g|.\end{array}$$ Therefore, after a certain time $t$, we have $$\begin{array}{l}
|e\rangle_6|0\rangle \rightarrow \cos(gt)|e\rangle_6|0\rangle -{\rm
i e}^{{\rm
i}\phi}\sin(gt)|g\rangle_6|1\rangle,\\
|g\rangle_6|0\rangle \rightarrow |g\rangle_6|0\rangle.
\end{array}$$ So, when the laser beam is applied on the ion 6 for the time interval $t_1$, the state $|\psi(0)\rangle$ evolves into $$\begin{array}{ll}
|\psi(t_1)\rangle=&[\alpha ac \cos(gt_1)|ee\rangle_{46}+\beta ad
|eg\rangle_{46}\\
&+\gamma bc \cos(gt_1)|ge\rangle_{46}+\delta bd
|gg\rangle_{46}]|0\rangle\\
&-{\rm i e}^{{\rm i}\phi}\sin(gt_1)(\alpha ac|eg\rangle_{46}+\gamma
bc|gg\rangle_{46})|1\rangle.\\
\end{array}$$
Let us choose $t_1=\frac {1}{g}\arccos |\frac {d}{c}|$, then we have $$\begin{array}{ll}
|\psi(t_1)\rangle'=&[\mathrm{e}^{\mathrm{i}(\theta_1+\theta_3)}|ad|
\alpha
|ee\rangle_{46}+\mathrm{e}^{\mathrm{i}(\theta_1+\theta_4)}|ad|
\beta|eg\rangle_{46}\\
&+\mathrm{e}^{\mathrm{i}(\theta_2+\theta_3)} |bd|\gamma
|ge\rangle_{46}+ \mathrm{e}^{\mathrm{i}(\theta_2+\theta_4)}|bd|
\delta|gg\rangle_{46}]|0\rangle\\
&-{ \mathrm{i e}}^{{\mathrm{ i}}\phi}\sin(gt_1)(\alpha
ac|eg\rangle_{46}+\gamma
bc|gg\rangle_{46})|1\rangle,\\
\end{array}$$ where $a=|a|e^{\rm i\theta_1}$, $b=|b|e^{\rm i\theta_2}$, $c=|c|e^{\rm i\theta_3}$, $d=|d|e^{\rm i\theta_4}$. Then Bob makes a measurement on phonon. If the result $|1\rangle$ is obtained, then the teleportation fails. When the measurement result is $|0\rangle$, the state of the ions 4 and 6 can be written as $$\begin{array}{ll}
|\psi\rangle'=&[\mathrm{e}^{\mathrm{i}(\theta_1+\theta_3)}|ad|
\alpha
|ee\rangle_{46}+\mathrm{e}^{\mathrm{i}(\theta_1+\theta_4)}|ad|
\beta|eg\rangle_{46}\\
&+\mathrm{e}^{\mathrm{i}(\theta_2+\theta_3)} |bd|\gamma
|ge\rangle_{46}+ \mathrm{e}^{\mathrm{i}(\theta_2+\theta_4)}|bd|
\delta|gg\rangle_{46}].\\
\end{array}$$
After that, Bob applies a laser standing wave tuned to the first lower vibrational sideband on the ion 4. By the same argument used to ion 6, after an interaction time $t_2$, we have $$\begin{array}{l}
|e\rangle_4|0\rangle \rightarrow \cos(gt_2)|e\rangle_4|0\rangle
-{\rm i e}^{{\rm i}\phi}\sin(gt
_2)|g\rangle_4|1\rangle,\\
|g\rangle_4|0\rangle \rightarrow |g\rangle_4|0\rangle.
\end{array}$$ Hence, the state stated in Eq.(13) becomes $$\begin{array}{ll}
|\psi(t_2)\rangle=&[\mathrm{e}^{\mathrm{i}(\theta_1+\theta_3)}|ad|
\alpha
\cos(gt_2)|ee\rangle_{46}\\&+\mathrm{e}^{\mathrm{i}(\theta_1+\theta_4)}|ad|
\beta\cos(gt_2)|eg\rangle_{46}\\
&+\mathrm{e}^{\mathrm{i}(\theta_2+\theta_3)} |bd|\gamma
|ge\rangle_{46}
\\&+ \mathrm{e}^{\mathrm{i}(\theta_2+\theta_4)}|bd|
\delta|gg\rangle_{46}]|0\rangle\\
&-{ \mathrm{i e}}^{{\mathrm{
i}}\phi}\sin(gt_2)[\mathrm{e}^{\mathrm{i}(\theta_1+\theta_3)}
|ad|\alpha|ge\rangle_{46}\\
&+\mathrm{e}^{\mathrm{i}(\theta_1+\theta_4)}|ad|\beta|gg\rangle_{46}]|1\rangle.\\
\end{array}$$ With the choice $t_2=\frac {1}{g}\arccos |\frac {b}{a}|$, we have $$\begin{array}{ll}
|\psi(t_2)\rangle'=&\mathrm{e}^{\mathrm{i}(\theta_1+\theta_3)}|bd|[
\alpha
|ee\rangle_{46}\\&+\mathrm{e}^{\mathrm{i}(\theta_4-\theta_3)}\beta
|eg\rangle_{46}\\
&+\mathrm{e}^{\mathrm{i}(\theta_2-\theta_1)} \gamma |ge\rangle_{46}
\\&+ \mathrm{e}^{\mathrm{i}(\theta_2+\theta_4-\theta_1-\theta_3)}
\delta|gg\rangle_{46}]|0\rangle\\
&-{ \mathrm{i e}}^{{\mathrm{
i}}\phi}\sin(gt_2)[\mathrm{e}^{\mathrm{i}(\theta_1+\theta_3)}
|ad|\alpha|ge\rangle_{46}\\
&+\mathrm{e}^{\mathrm{i}(\theta_1+\theta_4)}|ad|\beta|gg\rangle_{46}]|1\rangle.\\
\end{array}$$
Now Bob makes a measurement on phonon again. If the measurement result is $|1\rangle$, it means that the teleportation fails. When the measurement outcome $|0\rangle$ is obtained, the state of the ions 4 and 6 can be written as $$\begin{array}{l}\alpha
|ee\rangle_{46}+\mathrm{e}^{\mathrm{i}(\theta_4-\theta_3)}\beta
|eg\rangle_{46} +\mathrm{e}^{\mathrm{i}(\theta_2-\theta_1)} \gamma
|ge\rangle_{46}
\\+ \mathrm{e}^{\mathrm{i}(\theta_2+\theta_4-\theta_1-\theta_3)}
\delta|gg\rangle_{46}. \end{array}$$
Under the basis $\{|ee\rangle_{46}, |eg\rangle_{46},
|ge\rangle_{46}, |gg\rangle_{46}\}$, a collective unitary transformation $$U=\left (\begin{array}{cccc} 1&0&0&0\\0&e^{-i\phi_1}&0&0\\0&0&e^{-i\phi_2}&0\\0&0&0&e^{-i(\phi_1+\phi_2)}\\
\end{array}\right )$$ is made, then the state in Eq.(17) becomes $$\alpha
|ee\rangle_{46}+\beta
|eg\rangle_{46} +\gamma |ge\rangle_{46} + \delta|gg\rangle_{46}.$$ Here $\phi_1=\theta_4-\theta_3$, $\phi_2=\theta_2-\theta_1$. The state in Eq.(19) is just the state which we want to teleport.
Without any difficult one can show that the successful probability of the teleportation protocol is $4|bd|^2$. When $|a|=|b|$, $|c|=|d|$, the probability of successful teleportation is 1.
In summary, we propose a scheme for probabilistic teleportation of an unknown two-particle state of general formation in ion trap. We hope that this teleportation protocol of two-particle state can be realized experimentally with presently available techniques in the future.
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[^1]: To whom correspondence should be addressed. Email: [email protected]\
Supported by the National Natural Science Foundation of China under Grant No: 10971247, Hebei Natural Science Foundation of China under Grant No: F2009000311.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. In this paper, we numerically demonstrate that the normalized localization length $\beta$ of the eigenfunctions of multilayer random networks follows a simple scaling law given by $\beta=x^*/(1+x^*)$, with $x^*=\gamma(b_{\mbox{\tiny eff}}^2/L)^\delta$, $\gamma,\delta\sim 1$ and $b_{\mbox{\tiny eff}}$ being the effective bandwidth of the adjacency matrix of the network, whose size is $L=M\times N$. The reported scaling law for $\beta$ might help to better understand criticality in multilayer networks as well as used as to predict the eigenfunction localization properties of them.'
author:
- 'J. A. Méndez-Bermúdez,$^1$ Guilherme Ferraz de Arruda,$^{2,3}$ Francisco A. Rodrigues,$^2$ and Yamir Moreno$^{3,4,5}$'
title: Scaling Properties of Multilayer Random Networks
---
Real systems are naturally structured in levels or interconnected substructures, which in turn can be made of nodes organized in its own non-trivial manner [@BoccalettiPR2014; @Kivela2014]. For instance, individuals are connected according to friendship, working and family relations, defining different social circles, each of which can be thought of as a network. People and goods are transported through different mobility modes, such as airlines, roads and ships. All the previous systems made up what is nowadays referred to as multilayer networks [@Kivela2014], which can also be found in biological and technological systems. Mathematically, the multilayer organization of real systems can be represented in different ways, being the matrix representation [@Kivela2014; @Cozzo2015] the most popular.
On the other hand, it has also been shown that many critical properties of several phenomena that take place on top of complex networks are determined by the topology of them, and specifically by the spectral and eigenfunction properties of the adjacency and the Laplacian matrices of the networks. One particularly suitable approach to address the relation between the structure and the dynamics of a networked system is given by Random Matrix Theory (RMT), see for instance [@Bandyopadhyay; @MAM15]. Random Matrix Theory has numerous applications in many different fields, from condensed matter physics to financial markets [@RMT]. In the case of complex networks, the use of RMT techniques might reveal universal properties, i.e., the nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks follow Gaussian Orthogonal Ensemble (GOE) statistics [@Bandyopadhyay]. For instance, the analysis of Erdös-Rényi networks shows that the level spacing distribution and the entropic eigenfunction localization length of the adjacency matrices are universal for fixed average degrees [@MAM15].
Universal properties are always of interest, as they allow to reduce the set of parameters describing the system and provide relations that allow to deduce its behavior from those few global parameters, without the need of having precise details of the system. In this paper, we study whether there are universal scaling properties in multilayer systems. To this end, we perform a scaling analysis of the eigenfunction localization properties of multilayer networks using RMT models and techniques. We explore multilayer networks whose networks of layers are of two types: (i) a line and (ii) a complete graph (node-aligned multiplex networks). In the first case, we study weighted layers coupled by weighted matrices, whereas in the latter case, weighted and binary layers coupled by identity matrices are considered. We demonstrate that the normalized localization length of the eigenfunctions of multilayer random networks exhibits a well defined scaling function, being the scaling law robust for all the aforementioned networks. Our results can be used to predict or design the localization features of the eigenfunctions of multilayer random networks and to better understand critical properties that depend on eigenfunction properties.
A multilayer network is formed by $M$ undirected random layers with corresponding adjacency matrices $A^{(m)}$ having $N_m$ nodes each. The respective adjacency matrix of the whole network is expressed by ${\bf A} = \bigoplus_{m=1}^M A^{(m)} + {\it p} \bf{C}$, where $\bigoplus$ represents the Kronecker product, $p$ is a parameter that defines the strength of the inter-layer edges and $\bf{C}$ is the interlayer coupling matrix, whose elements represent the relations between nodes in different layers. Observe that the coupling matrix has implicitly the information of a network of layers. On such network, nodes represent layers of the multilayer network, i.e., there is an edge if at least one node is connected on both layers. Moreover, the network of layers can be extracted in different manners, as can be seen in [@Garcia2014]. Examples of multilayers are shown in Fig. \[fig:schematic\]. Observe that the spectra of the adjacency matrix [**A**]{} is a function of the parameter $p$. As a consequence, eigenvalue crossings, structural transitions [@Radicchi2013], near crossings [@Arruda2015] or localization problems [@Goltsev2012; @Arruda2015] are inherent to the network spectra, depending on $p$ for multilayer networks. Regarding dynamical processes, such parameter plays a fundamental role. For instance on diffusion processes it can drive the network to what is called super-diffusion [@Gomez2013], which means that the time scale of the multiplex is faster than the time scale observed when the layers are separated. Another important example is the case of contagion dynamics, for which $p$ can be interpreted as the ratio of intra and inter-layer spreading rates, giving raise to the existence of both localized and delocalized states [@Arruda2015]. Here we restrict ourselves to $p=1$: for $p \ll 1$ the layers can be considered as uncoupled, while for $p \gg 1$ the topology of the network of layers dominates the spectral properties [@Garcia2014; @Cozzo2016]. In this way, $p = 1$ represents a suitable intermediary case (multilayer phase).
We define two ensembles of multilayer random networks as adjacency matrices. As the first model we consider a network of layers on a line, see Fig. \[fig:schematic\](a), whose adjacency matrix ${\bf A}$ has the form $${\bf A} =
\left(
\begin{array}{ccccc}
A^{(1)} & C^{(1,2)} & 0 & \cdots & 0 \\
C^{(2,1)} & A^{(2)} & C^{(2,3)} & \ & 0 \\
0 & C^{(3,2)} & A^{(3)} & \ & 0 \\
\vdots & \ & \ & \ddots & C^{(M-1,M)} \\
0 & 0 & 0 & C^{(M,M-1)} & A^{(M)}
\end{array}
\right) \ ,
\label{eq:A_line}$$ where $\left(C^{(m,m')}\right)_{i,j}=\left(C^{(m,m')}\right)^{\mbox{\tiny T}}_{j,i}$ are real rectangular matrices of size $N_m \times N_{m'}$ and 0 represents null matrices. Furthermore, we consider a special class of matrices $A^{(m)}$ and $C^{(m,m')}$ which are characterized by the sparsities $\alpha_A$ and $\alpha_C$, respectively. In other words, since with a probability $\alpha_*$ their elements can be removed, these matrices represent Erdös-Rényi–type random networks. Notice that when the $N_m$ are all the same $N_m=\mbox{constant}\equiv N$, which is the case we explore here. Also, the adjacency matrix ${\bf A}$ has the structure of a block-banded matrix of size $L=M\times N$. In addition, we consider this model as a model of weighted networks; i.e., the non-vanishing elements ${\bf A}_{i,j}$ are independent Gaussian variables with zero mean and variance $1+\delta_{i,j}$. We justify the addition of self-loops and random weights to edges by recognizing that in real-world networks the nodes and the interactions between them are in general non-equivalent. Moreover, with this prescription we retrieve well known random matrices [@Metha] in the appropriate limits: According to this definition a diagonal random matrix is obtained for $\alpha_A=\alpha_C=0$ (Poisson case), whereas the GOE is recovered when $\alpha_A=\alpha_C=1$ and $M=2$. For simplicity, and without loss of generality, in this work we consider the case where $\alpha\equiv\alpha_A=\alpha_C$. As an example, this network model can be applied to transportation networks, where the inter-layer edges represent connections between two different means of transport. An obvious constraint is that no layer can be connected to more than two layers. In addition to the above configuration, we are also interested on the node-aligned multiplex case, whose network of layers is a complete graph [@SM], see Fig. \[fig:schematic\](b).
![(color online) Illustration of the two types of multilayer networks studied here. The network of layers are (a) a line and (b) a complete network. Here, each network is composed by $M=3$ layers having $N=5$ nodes.[]{data-label="fig:schematic"}](Fig1.pdf){width="\columnwidth"}
There is a well known RMT model known as the banded random matrix (BRM) model which was originally introduced to emulate quasi-one-dimensional disordered wires of length $L$ and width $b$ (with $b\ll L$). The BRM ensemble is defined as the set of $L\times L$ real symmetric matrices whose entries are independent Gaussian random variables with zero mean and variance $1+\delta_{i,j}$ if $|i-j|<b$ and zero otherwise. Therefore $b$ is the number of nonzero elements in the first matrix row which equals 1 for diagonal, 2 for tridiagonal, and $L$ for matrices of the GOE. There are several numerical and theoretical studies available for this model, see for example Refs. [@CMI90; @EE90; @FM91; @CIM91; @FM92; @MF93; @FM93; @FM94; @I95; @MF96; @CGM97; @S97; @KPI98; @KIP99; @W02]. In particular, outstandingly, it has been found [@CMI90; @EE90; @FM91; @FM92; @MF93; @FM93; @FM94; @I95] that the eigenfunction properties of the BRM model, characterized by the [*scaled localization length*]{} $\beta$ (to be defined below), are [*universal*]{} for the fixed ratio $x = b^2/L$. More specifically, it was numerically and theoretically shown that the scaling function $$\beta = \gamma x/(1+\gamma x) \ ,
\label{betascaling0}$$ with $\gamma\sim 1$, holds for the eigenfunctions of the BRM model [^1].
Admittedly, the ensemble of adjacency matrices of the multilayer network with layers on a line, see Eq. (\[eq:A\_line\]), can be considered as a [*non-homogeneous diluted version*]{} of the BRM model. Therefore, motivated by the similarity between these two complex matrix models, we propose the study of eigenfunction properties of the adjacency matrices of multilayer and multiplex random networks as a function of the parameter $$x = b_{\mbox{\tiny eff}}^2/L \ ,
\label{x}$$ where $b_{\mbox{\tiny eff}}\equiv b_{\mbox{\tiny eff}}(N,\alpha)$ is the adjacency matrix effective bandwidth and $L=M\times N$.
A commonly accepted tool to characterize quantitatively the complexity of the eigenfunctions of random matrices (and of Hamiltonians corresponding to disordered and quantized chaotic systems) is the information or Shannon entropy $S$. This measure provides the number of principal components of an eigenfunction in a given basis. In fact, $S$ has been already used to characterize the eigenfunctions of the adjacency matrices of random network models; see some examples in Refs. [@ZYYL08; @GT06; @JSVL10; @PS12; @MRP14; @MAM15]. The Shannon entropy for the eigenfunction $\Psi^l$ is given as $S = -\sum_{n=1}^L (\Psi^l_n)^2 \ln (\Psi^l_n)^2$ and allows computing the scaled localization length as [@I90] $$\label{beta}
\beta = \exp\left( {\left\langle}S {\right\rangle}- S_{{\mbox{\tiny GOE}}} \right) \ ,$$ where $S_{{\mbox{\tiny GOE}}}\approx\ln(L/2.07)$, which is used as a reference, is the entropy of a random eigenfunction with Gaussian distributed amplitudes (i.e., an eigenfunction of the GOE). With this definition for $\beta$ and in the case of the multilayer network with layers on a line, when $\alpha=0$ (i.e., when all vertices in the network are isolated), since the eigenfunctions of the adjacency matrices of Eq. (\[eq:A\_line\]) have only one non-vanishing component with magnitude equal to one, ${\left\langle}S {\right\rangle}=0$ and $\beta\approx 2.07/L$. On the other hand, when all nodes in this multilayer network are fully connected we recover the GOE and ${\left\langle}S {\right\rangle}=S_{{\mbox{\tiny GOE}}}$. Thus, the [*fully chaotic*]{} eigenfunctions extend over the $L$ available vertices in the network and $\beta=1$. Therefore, $\beta$ can take values in the range $(0,1]$.
Here, as well as in BRM model studies, we look for the scaling properties of the eigenfunctions of our multilayer random network models through $\beta$. Below we use exact numerical diagonalization to obtain eigenfunctions $\Psi^l$ ($l=1\ldots L$) of the adjacency matrices of large ensembles of multilayer random networks characterized by $M$, $N$, and $\alpha$. We perform the average ${\left\langle}S {\right\rangle}$ taking half of the eigenfunctions, around the band center, of each adjacency matrix.
![(Color online) (a) Scaled localization length $\beta$ as a function of $x=b_{\mbox{\tiny eff}}^2/L$ for ensembles of multilayer networks characterized by the sparsity $\alpha$. The black dashed line close to the data for $\alpha=0.8$ corresponds to Eq. (\[betascaling0\]) with $\gamma=1.4$. Horizontal black dot-dashed lines at $\beta\approx 0.12$ and 0.88 are shown as a reference, see the text. (b) Logarithm of $\beta/(1-\beta)$ as a function of $\ln(x)$. Upper inset: Power $\delta$, from the fittings of the data with Eq. (\[betascaling2\]), as a function of $\alpha$. Lower inset: Enlargement in the range $\ln[\beta/(1-\beta)]=[-2,4]$ including data for $\alpha=0.6$, 0.8, and 1. Lines are fittings of the data using Eq. (\[betascaling2\]).[]{data-label="fig:BBRM"}](Fig2.pdf){width="\columnwidth"}
![(Color online) (a) $\beta$ as a function of $x^*$ \[as defined in Eq. (\[x\*\])\] for ensembles of multilayer networks with $\alpha\in [0.2,1]$ in steps of $0.05$. Inset: Data for $\alpha\in [0.5,1]$ in steps of $0.05$. Orange dashed lines in main panel and inset are Eq. (\[betax\*\]). (a) Logarithm of $\beta/(1-\beta)$ as a function of $\ln(x^*)$ for $\alpha\in [0.2,1]$ in steps of $0.05$. Inset: Enlargement in the range $\ln[\beta/(1-\beta)]=[-2,2]$ including curves for $\alpha\in [0.5,1]$ in steps of $0.05$. Orange dashed lines in main panel and inset are Eq. (\[betascaling3\]).[]{data-label="fig:BBRM_collapsed"}](Fig3.pdf){width="\columnwidth"}
We now analyze in detail the multilayer network model with adjacency matrix given by Eq. (\[eq:A\_line\]). In Fig. \[fig:BBRM\](a), we present $\beta$ as a function of $x$, see Eq. (\[x\]), for ensembles of networks characterized by the sparsity $\alpha$. We have defined $b_{\mbox{\tiny eff}}$ as the average number of non-vanishing elements per adjacency-matrix row, i.e., $$b_{\mbox{\tiny eff}} = 2N\alpha \ .
\label{beff}$$
We observe that the curves of $\beta$ vs. $x$ in Fig. \[fig:BBRM\](a) have a functional form similar to that for the BRM model. To show this we are including Eq. (\[betascaling0\]) (black dashed line) with $\gamma=1.4$ (the value of $\gamma$ reported in Ref. [@CMI90] for the BRM model) which is very close to our data for $\alpha=0.8$. In addition, in Fig. \[fig:BBRM\](b) the logarithm of $\beta/(1-\beta)$ as a function of $\ln(x)$ is presented. The quantity $\beta/(1-\beta)$ was useful in the study of the scaling properties of the BRM model [@CMI90; @FM92] because $\beta/(1-\beta) = \gamma x$, which is equivalent to scaling (\[betascaling0\]), implies that a plot of $\ln[\beta/(1-\beta)]$ vs. $\ln(x)$ is a straight line with unit slope. Even though, this statement is valid for the BRM model in a wide range of parameters (i.e., for $\ln[\beta/(1-\beta)]<2$) it does not apply to our multilayer random network model; see Fig. \[fig:BBRM\](b). In fact, from this figure we observe that plots of $\ln[\beta/(1-\beta)]$ vs. $\ln(x)$ are straight lines (in a wide range of $x$) with a slope that depends on the sparsity $\alpha$. Therefore, we propose the scaling law $$\beta/(1-\beta) = \gamma x^\delta \ ,
\label{betascaling2}$$ where both $\gamma$ and $\delta$ depend on $\alpha$. Indeed, Eq. (\[betascaling2\]) describes well our data, mainly in the range $\ln[\beta/(1-\beta)]=[-2,2]$, as can be seen in the inset of Fig. \[fig:BBRM\](b) where we show the numerical data for $\alpha=0.6$, 0.8 and 1 and include fittings through Eq. (\[betascaling2\]). We stress that the range $\ln[\beta/(1-\beta)]=[-2,2]$ corresponds to a reasonable large range of $\beta$ values, $\beta\approx[0.12,0.88]$, whose bounds are indicated with horizontal dot-dashed lines in Fig. \[fig:BBRM\](a). Finally, we notice that the power $\delta$, obtained from the fittings of the data using Eq. (\[betascaling2\]), is very close to unity for all the sparsity values we consider here (see the upper inset of Fig. \[fig:BBRM\](b)).
Therefore, from the analysis of the data in Fig. \[fig:BBRM\], we are able to write down a [*universal scaling function*]{} for the scaled localization length $\beta$ of the eigenfunctions of our multilayer random network model as $$\beta/(1-\beta) = x^* \ ,
\label{betascaling3}$$ where the scaling parameter $x^*=\gamma x^\delta$, as a function of the multilayer network parameters, is given by $$x^* \equiv \gamma \left( 4N\alpha^2/M \right)^\delta \ .
\label{x*}$$ To validate Eq. (\[betascaling3\]) in Fig. \[fig:BBRM\_collapsed\](b) we present again the data for $\ln[\beta/(1-\beta)]$ shown in Fig. \[fig:BBRM\](b) but now as a function of $\ln(x^*)$. We do observe that curves for different values of $\alpha$ fall on top of Eq. (\[betascaling3\]) for a wide range of the variable $x^*$. Moreover, the collapse of the numerical data is excellent in the range $\ln[\beta/(1-\beta)]=[-2,2]$ for $\alpha\ge 0.5$, as shown in the inset of Fig. \[fig:BBRM\_collapsed\](b).
Finally, we rewrite Eq. (\[betascaling3\]) into the equivalent, but explicit, scaling function for $\beta$: $$\beta = x^*/(1+x^*) \ .
\label{betax*}$$ In Fig. \[fig:BBRM\_collapsed\](a) we confirm the validity of Eq. (\[betax\*\]). We emphasize that the universal scaling given in Eq. (\[betax\*\]) extends outside the range $\beta\approx[0.12,0.88]$, for which Eq. (\[betascaling2\]) was shown to be valid, see the main panel of Fig. \[fig:BBRM\_collapsed\](a). Furthermore, the collapse of the numerical data following Eq. (\[betax\*\]) is remarkably good for $\alpha\ge 0.5$, as shown in the inset of Fig. \[fig:BBRM\_collapsed\](a). Additionally, we have verified [@SM] that the scaling (\[betax\*\]) is also applicable to node-aligned multiplex networks $-$which are relevant for certain applications$-$, once the effective bandwidth $b_{\mbox{\tiny eff}}$ is properly defined.
Summarizing, in this study we have demonstrated that the normalized localization length $\beta$ of the eigenfunctions of multilayer random networks scales as $x^*/(1+x^*)$. Here, $x^*=\gamma(b_{\mbox{\tiny eff}}^2/L)^\delta$; where $b_{\mbox{\tiny eff}}$ is the effective bandwidth of the network’s adjacency matrix, $L$ is the adjacency matrix size, and $\gamma,\delta\sim 1$. We showed that such scaling law is robust covering weighted multilayer and both weighted and unweighted node-aligned multiplex networks [@SM]. Our results might shed additional light on the critical properties and structural organization of multilayer systems. Interestingly enough, our findings might be used to either predict or design (e.g, tune), by means of Eq. (\[betax\*\]), the localization properties of the eigenfunctions of multilayer random networks. For instance, we anticipate the following cases: (i) Due to the banded nature of the adjacency matrices of the network models considered here, $b_{\mbox{\tiny eff}}<L$, it is unlikely to observe fully delocalized eigenfuctions unless the value of $x^*$ is driven to large values, for example, by increasing the size of the subnetworks $N$ and/or their sparsity $\alpha$ for a fixed value of $M$; (ii) For a fixed subnetwork size $N$ and sparsity $\alpha$, the eigenfunctions of the multilayer network become more localized when increasing the number of subnetworks $M$; and (iii) the procedure of adding/removing subnetworks in our network models may be used to tune their conduction properties [^2], since $M$ could drive the network from a regime of delocalized eigenfunctions (metallic regime), $x^*\gg 1$, to a regime of localized eigenfunctions (insulating regime), $x^*\ll 1$. We hope our results motivate further numerical and theoretical studies.\
[*Acknowledgments*]{}.– This work was partially supported by VIEP-BUAP (Grant No. MEBJ-EXC16-I), Fondo Institucional PIFCA (Grant No. BUAP-CA-169), and CONACyT (Grant Nos. I0010-2014/246246 and CB-2013/220624). FAR acknowledges CNPq (Grant No. 305940/2010-4), FAPESP (Grant No. 2011/50761-2 and 2013/26416-9), and NAP eScience - PRP - USP for financial support. GFA would like to acknowledge FAPESP (grants 2012/25219-2 and 2015/07463-1) for the scholarship provided. Y. M. acknowledges support from the Government of Aragón, Spain through a grant to the group FENOL, by MINECO and FEDER funds (grant FIS2014-55867-P) and by the European Commission FET-Proactive Project Multiplex (grant 317532).
Scaling analysis of Multiplex Networks
======================================
In the node-aligned multiplex case, whose network of layers is a complete graph, the coupling matrices are restricted to identity matrices and all layers have the same number of nodes, see Fig. \[fig:schematic\](b). The adjacency matrix of a node-aligned multiplex is given as $${\bf A} =
\left(
\begin{array}{ccccc}
A^{(1)} & I & I & \cdots & I \\
I & A^{(2)} & I & \ & I \\
I & I & A^{(3)} & \ & I \\
\vdots & \ & \ & \ddots & I \\
I & I & I & I & A^{(M)}
\end{array}
\right) \ .
\label{eq:A_mux}$$ Similarly to the multilayer model of Eq. (\[eq:A\_line\]), this configuration is characterized by the sparsity $\alpha$ which we choose to be constant for all the $M$ matrices $A^{(m)}$ of size $N\times N$ composing the adjacency matrix ${\bf A}$ of size $L=M\times N$. Additionally, the configuration (\[eq:A\_mux\]) is considered in two different setups: weighted and unweighted multiplex without self-loops. In the weighted case, the non-vanishing elements of the matrices $A^{(m)}$ are chosen as independent Gaussian variables with zero mean and variance $1+\delta_{i,j}$. A realistic example of this configuration are online social systems, where each layer represents a different online network (e.g., Facebook, Twitter and Google+, etc). In the unweighted case, the non-vanishing elements of $A^{(m)}$ are equal to unity. In (\[eq:A\_mux\]), $I$ are identity matrices of size $N\times N$.
![(Color online) (a) Scaled localization length $\beta$ as a function of $x=b_{\mbox{\tiny eff}}^2/L$ for ensembles of weighted multiplex networks characterized by the sparsity $\alpha$. The black dashed line close to the data for $\alpha=0.6$ corresponds to Eq. (\[betascaling0\]) with $\gamma=1.4$. Horizontal black dot-dashed lines at $\beta\approx 0.12$ and 0.88 are shown as a reference, see the text. (b) Logarithm of $\beta/(1-\beta)$ as a function of $\ln(x)$. Upper inset: Power $\delta$, from the fittings of the data with Eq. (\[betascaling2\]), as a function of $\alpha$. Lower inset: Enlargement in the range $\ln[\beta/(1-\beta)]=[-2,2]$ including data for $\alpha=0.6$, 0.8, and 1. Lines are fittings of the data with Eq. (\[betascaling2\]).[]{data-label="fig:W_Mux"}](Fig4.pdf){width="\columnwidth"}
![(Color online) (a) $\beta$ as a function of $x^*$ \[as defined in Eq. (\[x\*\])\] for ensembles of weighted multiplex networks with $\alpha\in [0.2,1]$ in steps of $0.05$. Inset: Data for $\alpha\in [0.5,1]$ in steps of $0.05$. Orange dashed lines in main panel and inset are Eq. (\[betax\*\]). (a) Logarithm of $\beta/(1-\beta)$ as a function of $\ln(x^*)$ for $\alpha\in [0.2,1]$ in steps of $0.05$. Inset: Enlargement in the range $\ln[\beta/(1-\beta)]=[-2,2]$ including curves for $\alpha\in [0.5,1]$ in steps of $0.05$. Orange dashed lines in main panel and inset are Eq. (\[betascaling3\]).[]{data-label="fig:W_Mux_collapsed"}](Fig5.pdf){width="\columnwidth"}
Weigthed Multiplex
------------------
![(Color online) (a) Scaled localization length $\beta$ as a function of $x=b_{\mbox{\tiny eff}}^2/L$ for ensembles of unweighted multiplex networks characterized by the sparsity $\alpha$. The black dashed line corresponds to Eq. (\[betascaling0\]) with $\gamma=1.4$. Horizontal black dot-dashed lines at $\beta\approx 0.5$ and 0.98 are shown as a reference, see the text. (b) Logarithm of $\beta/(1-\beta)$ as a function of $\ln(x)$. Upper inset: Power $\delta$, from the fittings of the data with Eq. (\[betascaling2\]), as a function of $\alpha$. Lower inset: Enlargement in the range $\ln[\beta/(1-\beta)]=[-2,4]$ including data for $\alpha=0.6$, 0.7, and 0.85. Lines are fittings of the data with Eq. (\[betascaling2\]).[]{data-label="fig:A_Mux"}](Fig6.pdf){width="\columnwidth"}
![(Color online) (a) $\beta$ as a function of $x^*$ \[as defined in Eq. (\[x\*\])\] for ensembles of unweighted multiplex networks with $\alpha\in [0.2,1]$ in steps of $0.05$. Inset: Data for $\alpha\in [0.5,1]$ in steps of $0.05$. Orange dashed lines in main panel and inset are Eq. (\[betax\*\]). (a) Logarithm of $\beta/(1-\beta)$ as a function of $\ln(x^*)$ for $\alpha\in [0.2,1]$ in steps of $0.05$. Inset: Enlargement in the range $\ln[\beta/(1-\beta)]=[-1,4]$ including curves for $\alpha\in [0.5,1]$ in steps of $0.05$. Orange dashed lines in main panel and inset are Eq. (\[betascaling3\]).[]{data-label="fig:A_Mux_collapsed"}](Fig7.pdf){width="\columnwidth"}
Now we consider weighted multiplex networks (i.e., where the non-vanishing elements of the adjacency matrices $A^{(m)}$ in (\[eq:A\_mux\]) are chosen as independent Gaussian variables with zero mean and variance $1+\delta_{i,j}$). We follow the same methodology as in the multilayer case. Thus, in Fig. \[fig:W\_Mux\](a) we first present curves of $\beta$ vs. $x$ as given in Eqs. (\[beta\]) and (\[x\]), respectively; however, we redefine $b_{\mbox{\tiny eff}}$ as $$b_{\mbox{\tiny eff}} = N\alpha \ ,
\label{beff2}$$ which is the average number of non-vanishing elements per row inside the adjacency-matrix band in the multiplex setup. From Fig. \[fig:W\_Mux\](a) we observe that the curves of $\beta$ vs. $x$ have functional forms similar to those for the multilayer model (compare with Fig. \[fig:BBRM\](a)); however, with larger values of $\beta$ for given values of $x$. As a reference we also include Eq. (\[betascaling0\]) (black-dashed line) with $\gamma=1.4$, corresponding to the BRM model, which is even below the data for $\alpha=1$. Moreover, in Fig. \[fig:W\_Mux\](b) we show the logarithm of $\beta/(1-\beta)$ as a function of $\ln(x)$. As in the multilayer case, here we observe that plots of $\ln[\beta/(1-\beta)]$ vs. $\ln(x)$ are straight lines mainly in the range $\ln[\beta/(1-\beta)]=[-2,2]$ with a slope that depends on the sparsity $\alpha$. We indicate the bounds of this range with horizontal dot-dashed lines in Fig. \[fig:W\_Mux\](a). Therefore, the scaling law of Eq. (\[betascaling2\]) is also valid here. Indeed, in the upper inset of Fig. \[fig:W\_Mux\](b) we report the power $\delta$ obtained from fittings of the data with Eq. (\[betascaling2\]).
In order to validate the scaling hypothesis of Eq. (\[betascaling2\]) for the node-aligned multiplex setup, in Fig. \[fig:W\_Mux\_collapsed\](b) we present the data for $\ln[\beta/(1-\beta)]$ shown in Fig. \[fig:W\_Mux\](b), but now as a function of $\ln(x^*)$. We observe that curves for different values of $\alpha$ fall on top of Eq. (\[betascaling3\]) for a wide range of the variable $x^*$. Moreover, the collapse of the numerical data on top of Eq. (\[betascaling3\]) is excellent in the range $\ln[\beta/(1-\beta)]=[-2,2]$ for $\alpha\ge 0.5$, as shown in the inset of Fig. \[fig:W\_Mux\_collapsed\](b). Finally, in Fig. \[fig:W\_Mux\_collapsed\](a) we confirm the validity of Eq. (\[betax\*\]) which is as good here as for the multilayer case. We emphasize that the collapse of the numerical data on top of Eq. (\[betax\*\]) is remarkably good for $\alpha\ge 0.5$, as shown in the inset of Fig. \[fig:W\_Mux\_collapsed\](a).
Unweighted Multiplex
--------------------
The last analyzed scenario is the binary multiplex case. We recall that, in contrast to the two previous random network models, this model does not include weighted self-loops. Therefore the Poisson limit is not recovered when $\alpha\to 0$ and $\beta$ is not well defined there. Thus, we will compute $\beta$ for values of $x$ as smaller as the adjacency-matrix diagonalization produces meaningful results. Also, as for the weighted multiplex, we use here the effective bandwidth given in Eq. (\[beff2\]).
The conducted experiments are similar to the previous ones. Then, in Figs. \[fig:A\_Mux\](a) and \[fig:A\_Mux\](b) we present curves of $\beta$ vs. $x$ and $\ln[\beta/(1-\beta)]$ vs. $\ln(x)$, respectively. Here, due to the absence of self-loops we observe important differences with respect to the previous cases: In particular, the curves $\beta$ vs. $x$ present minima at given small values of $x$. This feature can be seen clearer in Fig. \[fig:A\_Mux\](b) since it is magnified there. Also, from Fig. \[fig:A\_Mux\](b) we can notice that the range where $\ln[\beta/(1-\beta)]$ is a linear function of $\ln(x)$ has been shifted upwards for all the values of $\alpha$ considered. Therefore, we perform fittings to the curves $\ln[\beta/(1-\beta)]$ vs. $\ln(x)$ with Eq. (\[betascaling2\]) in the interval $\ln[\beta/(1-\beta)]=[0,3.75]$; the bounds of this interval are marked as dot-dashed lines in Fig. \[fig:A\_Mux\](a). The corresponding values of $\delta$ are reported in the upper inset of Fig. \[fig:A\_Mux\](b).
Now, under the above conditions, we validate our scaling hypothesis by plotting $\beta$ vs. $x^*$ and $\ln[\beta/(1-\beta)]$ vs. $\ln(x^*)$, see Figs. \[fig:A\_Mux\_collapsed\](a) and \[fig:A\_Mux\_collapsed\](b), respectively. Remarkably, we observe a clear scaling behaviour also in the unweighted multiplex case (despite the minima in the curves $\beta$ vs. $x^*$ for small $x^*$).
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[^1]: It is relevant to mention that the scaling (\[betascaling0\]) was also shown to be valid for the kicked-rotator model [@CGIS90] (a [*quantum-chaotic*]{} system characterized by a random-like banded Hamiltonian matrix), the one-dimensional Anderson model, and the Lloyd model [@CGIFM92].
[^2]: In order to study transport properties one may consider to open the multilayer random network model by attaching conducting leads to chosen vertices, see for example [@MAM13; @M16].
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{
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**Is the $\nu_\mu \to \nu_{s}$ oscillation solution to the atmospheric neutrino anomaly**
**excluded by the superKamiokande data?**
1.1cm
R. Foot[^1]
.7cm
[*School of Physics*]{}
[*Research Centre for High Energy Physics*]{}
[*The University of Melbourne*]{}
[*Victoria 3010 Australia*]{}
Abstract
Recently the SuperKamiokande collaboration have claimed that their data exclude the $\nu_\mu \to \nu_s$ solution to the atmospheric neutrino anomaly at more than 99% C.L. We critically examine this claim.
Something mysterious with neutrinos is a foot. It is clear that about half of the upward going atmospheric $\nu_\mu 's$ are mising[@sk; @atmos]. Furthermore, about half of the solar $\nu_e 's$ have also disappeared[@solar]. There is also strong evidence that $\nu_e \leftrightarrow \nu_\mu$ oscillations take place with small mixing angles from the LSND experiment[@lsnd]. An elegant explanation of these facts is that each neutrino oscillates maximally with an approximately sterile partner, with small angles between generations[@fv]. For the status of the maximal $\nu_e \to \nu_s$ solution to the solar neutrino problem, see Ref.[@smok]. The status of the maximal $\nu_\mu \to \nu_s$ solution to the atmospheric neutrino problem is the subject of this paper.
As was pointed out sometime ago[@2], both $\nu_\mu \to \nu_s$ and $\nu_\mu \to \nu_\tau$ oscillations are able to explain the sub-GeV and multi-GeV superKamiokande single ring events (while 2 flavour $\nu_\mu \to \nu_e$ oscillations cannot because there is no observed anomaly with the electron events[@fvy]). Recently, however, the SuperKamiokande Collaboration have argued that the $\nu_\mu \to \nu_s$ oscillation explanation of the observed deficit of atmospheric neutrinos is disfavoured at more than 99% C.L.[@3], while the interpretation interms of $\nu_\mu \to \nu_\tau$ oscillations fits all of their data extremely well. This conclusion relies on an analysis of the upward through going muon data (UTM), the partially contained events with $E_{visible} > 5\ GeV$(PC) and the neutral current enriched multi-ring events (NC). These data sets lead to slightly different expectations for the $\nu_\mu \to \nu_\tau$ vs $\nu_\mu \to \nu_s$ oscillations because of earth matter effects for $\nu_\mu \to \nu_s$ oscillations which are important for UTM[@lip] and PC events[@lip98; @2] while neutral current interactions in the detector are utilized for the NC events[@hall]. These three data sets, obtained from Ref.[@3] (for 1100 live days), are shown in Figures 1a,b,c. Also shown is the theoretically expected result for maximal $\nu_\mu \to \nu_s$ oscillations with $\delta m^2 = 3\times 10^{-3}\ eV^2$ also obtained from Ref.[@3]. SuperKamiokande analyse the data by taking particular ratios and have not as yet provided detailed justification of the systematic uncertainties in the theoretically expected rates.
Let us discuss each of the three data sets in turn: 0.2cm a) Upward through going muons: The overall normalization of the through going muon fluxes have an estimated 20 % uncertainty, however the uncertainty in the expected shape of the zenith angle distribution is significantly lower (for some discussion of these uncertainties, see Ref.[@5; @6]). A recent estimate[@5] of the uncertainty in the vertical/horizontal ratio due to the uncertainties in the atmospheric fluxes is of order 4%. This systematic uncertainty is dominated by the uncertainty in the ratio $K/\pi$ produced in the atmosphere from the interactions of cosmic rays[@5]. In addition there will be other uncertainties in the shape of the zenith angle distribution due to the uncertainty in the energy dependence of the neutrino - nucleon cross section [^2] and from cosmic ray muons masquerading as neutrino induced muons. The latter uncertainty, while mainly affecting the most horizontal bin ($-0.1 < \cos\Theta < 0$) may be very important, as we will show. 0.2cm b) Partially contained events (with $E_{visible} > 5\ GeV$). The systematic uncertainty in the expected normalization of these events is quite large, again of order 20%[@5; @6]. The systematic uncertainty on the expected shape of the zenith angle distribution of these events should be relatively small ($\stackrel{<}{\sim}
\ few \%$ in the up/down ratio). 0.2cm c) Neutral current enriched multi-ring events. The systematic uncertainty in the expected normalization is again quite large, of order 20-40% due to the highly uncertain cross sections (as well as the uncertain atmospheric fluxes). The uncertainty in the expected shape of the zenith angle distribution will of course be much smaller, but may be significant (i.e. of order 5% in the up/down ratio). The uncertainty is due in part to the uncertainty in the relative contributions due to $\nu_e$ interactions (which are expected to be approximately up/down symmetric) and $\nu_\mu$ interactions (which are up/down asymmetric due to the oscillations affecting the upward going $\nu_\mu$’s). The relative contributions due to the neutral current weak interactions and the charged current weak interactions will also be uncertain. In addition to the cross section uncertainties there are also the uncertainties in the scattering angle distribution between the angles of the multi-ring events and the incident neutrino.
Note that the normalization uncertainties between the three data sets will be largely uncorrelated because of the different energy ranges for the atmospheric neutrino fluxes and also because of the different cross sections involved. Nevertheless, some weak correlation between UTM and PC events may be expected. While analysing the data using ratio’s does eliminate the normalization uncertainty, the remaining uncertainties will be important for the data sets a) and c). Furthermore, a conclusion based on particular ratios could only be robust if it agreed with a $\chi^2$ fit of the binned data points. SuperKamiokande are in the best position to do this for their data, and we hope that they will do this at some point.
In the meantime we will do this using the superKamiokande theoretical Monte-Carlo results for their given test point of maximal mixing with $\delta m^2 = 3 \times 10^{-3}\
eV^2$ (which is not expected to be the best fit for $\nu_\mu \to \nu_s $ oscillations). We define the $\chi^2$ by: $$\chi^2_{total} = \chi^2_{UTM} + \chi^2_{PC} + \chi^2_{NC},$$ with $$\chi^2_{y} = \sum^{10}_{i=1} \left({data_y (i) - f_y \times theory_y (i) \over
\delta data_y (i)}\right)^2 +
\left({f_y - 1\over \delta f}\right)^2,
\label{5}$$ where $y = UTM, PC, NC$ and the sum runs over the 10 zenith angle bins, and $\delta data_y$ is the statistical uncertainty in the data, $data_y(i)$. The normalization factor, $f_y$ parameterizes the overall normalization uncertainty in the theoretical expected value, $theory_y (i)$, and $\delta f_y$ is the expected normalization uncertainty, and we take $\delta f_y = 0.2$ for $y=UTM,PC,NC$. It is understood that $\chi^2_y$ is minimized with respect to $f_y$.
Doing this exercise (using the superKamiokande experimental data, $data_y (i), \delta data_y (i)$ and also the superKamiokande theoretically expected results $theory_y(i)$ for maximal mixing with $\delta m^2 = 3\times 10^{-3}\ eV^2$), we find that $\chi^2_y$ is minimized when $f_{UTM} \simeq 0.90$, $f_{PC} \simeq 0.87$ and $f_{NC} \simeq 1.07$. In Figures 2a,b,c we compare the data with $f_y theory_y(i)$, which is the theoretical prediction for $\delta m^2 = 3\times
10^{-3} \ eV^2$ (neglecting systematic uncertainties in the shape). We obtain the following $\chi^2_y$ values: $$\begin{aligned}
\chi^2_{UTM} = 17.0 \ \ for \ 10 \ degrees \ of \ freedom,\nonumber \\
\chi^2_{PC} = 13.4 \ \ for \ 10 \ degrees \ of \ freedom,\nonumber \\
\chi^2_{NC} = 16.0 \ \ for \ 10 \ degrees \ of \ freedom.\end{aligned}$$ Thus we obtain $\chi^2_{total} \simeq 46 $ for $30$ degrees of freedom which corresponds to an allowed C. L. of about 3%. While this allowed C.L. is low, it is only a [*lower limit*]{} because we haven’t varied $\delta m^2$ or incorporated the systematic uncertainties in the shape of $theory_y (i)$, which we now discuss.
Varying $\delta m^2$ within the allowed region identified from a fit to the contained events should improve $\chi^2_{PC}$ somewhat as well as slightly improving $\chi^2_{UTM,NC}$. For example, for $\delta m^2 = 5\times 10^{-3}\ eV^2$ using our code developed in Ref.[@2] we find that $\chi^2_{PC} \simeq 12$ (c.f. $13.4$ for $\delta m^2 = 3\times 10^{-3}\ eV^2$).
With regard to the UTM and NC events the effect of systematic uncertainties on $\chi^2$ can be very dramatic. We illustrate this by introducing a slope factor s(i) defined by $$s(i) = 0.95 + 0.01*i,
\label{xxu}$$ where $i=1,...,10$ (with $i=1$ the vertical upward going bin). In Eq.(\[5\]) we replace $f_{UTM}(i) \to s(i)*f_{UTM}(i)$, which is roughly within the estimated 1-sigma systematic uncertainty for the UTM events. In fact, this would be roughly equivalent to reducing the atmospheric $K/\pi$ ratio by about $30-40\%$ to be compared with the estimated $25\%$ uncertainty for the $K/\pi$ ratio[@5]. While the uncertainty in the $K/\pi$ ratio may be the largest single contribution to the uncertainty in the shape of the zenith angle distribution of UTM events, the total systematic uncertainty in the shape of the zenith angle distribution gets many contributions [^3] which is why the slope factor in Eq.(\[xxu\]) might be expected to be roughly within the 1-sigma systematic uncertainty. With the above slope factor, we find $\chi^2_{UTM} \simeq 13$, which represents a significantly improved fit. From our ealier discussion, the systematic uncertainties in the shape of the zenith angle distribution of the events for UTM and NC events are expected to be completely uncorrelated. This means that the best fit for the NC events can have a slope factor with a slope of a different sign, and this is needed to improve the fit. To illustrate the effect then, for NC we replace $f_{NC}(i) \to f_{NC}(i)/s(i)$ and find $\chi^2_{NC} \simeq 12$. This demonstrates that a $\chi^2$ fit to the three data sets incorporating the systematic uncertainties and varying $\delta m^2$ would be expected to reduce $\chi^2_{total}$ by at least 9 leading to a $\chi^2_{total}$ of about 37 or less. This corresponds to an allowed C.L. of 15% or more. Of course a global fit of all the superKamiokande data gives a much larger allowed C.L. because of the excellent fit of the $\nu_\mu \to \nu_s$ oscillations to the lower energy contained events (both sub-GeV and multi-GeV)[@val]. The results obtained for UTM and NC events using the slope factor $s(i)$ are given by the dotted lines in Figure 2a,c.
We would also like to emphasise that the poor $\chi^2$ fit for UTM events is due largely to the most horizontal bin ($-0.1 < \cos\Theta < 0$). Excluding this bin we find that $$\chi^2_{UTM} = 12.5 \ for \ 9 \ bins$$ [*excluding*]{} any systematic uncertainty in the shape of the zenith angle distribution (i.e. with $s(i)=1$). The reason for questioning the horizontal bin is clear: It is expected that the systematic uncertainty for the most horizontal bin should be relatively large. This is because atmospheric muons can contribute. (In fact the Kamiokande collaboration[@kam] made the cut $\cos\Theta < -0.04$ and incorporated large systematic errors for this bin). SuperKamiokande, in their published analysis of 537 days [@superk] included the whole horizontal bin, and made an estimate of the contamination of atmospheric muons in this bin (of order 4%) and subtracted it off. This is based on an extrapolation from $\cos\Theta > 0$ where the background falls off exponentially. This exponential assumption is not discussed in any detail, and needs to be justified if it can be. In fact, from their Figure 1[@superk], which compares the distribution of through-going muons near the horizon observed by superKamiokande for regions with thick and thin rock overburden, it seems possible that the atmospheric muon background could be higher by a factor of two or three or even more. This is rather important. For example, if a background of $10\%$ is assumed (which means that we must lower the superKamiokande data value by $6\%$ for this bin), then we obtain a $\chi^2_{UTM} \simeq 14$ for 10 degrees of freedom (excluding the effects of the systematic uncertainties in the shape of the zenith angle distribution, i.e. $s(i) = 1$) or $\chi^2_{UTM} \simeq 11$ including the modest slope factor in Eq.(\[xxu\]). Unless the level of contamination of atmospheric muons in the horizontal bin can be rigorously justified, it is probably safest to exclude the horizontal bin altogether because the systematic uncertainties may be so large as to make it too uncertain to be useful[^4].
Thus, we have shown that a $\chi^2$ analysis of the recent upward through going muon binned data, partially contained events with $E_{visible} > 5\ GeV$ and neutral current enriched multi-ring events does not exclude maximal $\nu_\mu \to \nu_s$ oscillation solution to the atmospheric neutrino problem with any significant confidence level. This is not inconflict with the superKamiokande results since they fit three particular ratio’s rather than the binned data. However it does show that the conclusion that the $\nu_\mu \to \nu_s$ osillations are disfavoured does depend on how one analyses the data. Furthermore, the overall fit (i.e. including also the lower energy single ring events) of the $\nu_\mu \to
\nu_s$ oscillations to the superKamiokande data is good. Fortunately future data will eventually decide the issue. In the meantime, important work needs to be done on carefully estimating and checking the possible systematic uncertainties.
0.4cm [**Acknowledgement**]{} 0.4cm The author would like to thank Paolo Lipari and Ray Volkas for many related discussions/communications and for comments on a preliminary version of the paper. The author is an Australian Research Fellow.
0.6cm [**Figure Captions**]{} 0.4cm Figure 1: SuperKamiokande data for the upward through going muons (Fig.1a), partially contained events with $E_{visible} > 5\ GeV$ (Fig.1b) and neutral current enriched multi-ring events (Fig.1c), all obtained from Ref.[@3]. Also shown are the superKamiokande expected results for maximal $\nu_\mu \to \nu_s$ oscillations with $\delta m^2 = 3 \times 10^{-3}\ eV^2$, also obtained from Ref.[@3]. 0.3cm Figure 2: Same as Figure 1 except that the theoretical expectation for maximal $\nu_\mu \to \nu_s$ oscillations with $\delta m^2 = 3 \times 10^{-3}\ eV^2$, are renormalized by an overall scale factor (as discussed in the text). In Figures 2a and 2c, the dotted line includes the effect of a modest correction to the expected shape of the zenith angle distribution given by Eq.(\[xxu\]), as discussed in the text.
[99]{}
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[^1]: Email address: [email protected]
[^2]: Uncertainties in the energy dependence of the cross section leads to uncertainties in the expected shape of the zenith angle distribution of UTM events, because the zenith angle dependence of the atmospheric neutrino flux is energy dependent.
[^3]: Due to e.g uncertainty in the energy dependence of the neutrino nucleon cross section, uncertainty in the interaction length of the cosmic rays in the upper atmosphere, modelling of the atmosphere, uncertainty in the primary cosmic ray energy spectrum and composition of cosmic rays etc.
[^4]: In terms of analysis with ratio’s we suggest that the vertical be defined as $-1 < \cos\Theta < -0.5$ and the horizontal as $-0.5 < \cos\Theta < -0.9$.
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---
author:
- Fabien André
- 'Anne-Marie Kermarrec'
- Nicolas Le Scouarnec
bibliography:
- 'references.bib'
title: 'Quicker ADC : Unlocking the hidden potential of Product Quantization with SIMD'
---
=1
Introduction
============
Background
==========
Quicker ADC
===========
Evaluation
==========
Related Work
============
Conclusion
==========
|
{
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}
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---
abstract: 'We report on the emergence of robust multi-clustered chimera states in a dissipative-driven system of symmetrically and locally coupled identical SQUID oscillators. The “snake-like” resonance curve of the single SQUID (Superconducting QUantum Interference Device) is the key to the formation of the chimera states and is responsible for the extreme multistability exhibited by the coupled system that leads to attractor crowding at the geometrical resonance (inductive-capacitive) frequency. Until now, chimera states were mostly believed to exist for nonlocal coupling. Our findings provide theoretical evidence that nearest neighbor interactions are indeed capable of supporting such states in a wide parameter range. SQUID metamaterials are the subject of intense experimental investigations and we are highly confident that the complex dynamics demonstrated in this manuscript can be confirmed in the laboratory.'
author:
- 'J. Hizanidis, N. Lazarides, and G. P. Tsironis'
title: Robust chimera states in SQUID metamaterials with local interactions
---
Since the first report on chimera states [@KUR02a], the number of works dedicated to this phenomenon of coexisting synchronous and desynchronous oscillatory behavior has grown immensely (see [@panaggio:2015] and references within). The counterintuitive nature of chimeras inspired Abrams and Strogatz [@ABR04] to name them after the mythological hybrid creature Chimera (Greek: Qímaira) which has a lion’s head, a goat’s body and a snake’s tail. The latest studies on chimera states focus on their stabilization and manipulation through various control techniques [@SIE14; @BIC15; @ISE15; @OME16] and their experimental verification [@tinsley:2012; @hagerstrom:2012; @wickramasinghe:2013; @martens:2013; @Rosin2014; @schmidt:2014; @Gambuzza2014; @Kapitaniak2014].
Chimera states have mostly been found for nonlocal coupling between the oscillators [@OME13; @ZAK14; @OME15]. This fact has given rise to a general notion that nonlocal coupling is an essential ingredient for their existence. However, recently, it has been demonstrated that chimeras can be achieved for global coupling too [@schmidt:2014; @SET14; @Yeldesbay2014; @BOE15]. The case of local coupling (i.e. nearest-neighbor interactions) has been studied less: In [@LAI15] chimera states were found in locally coupled networks, but the oscillators in the investigated systems were not completely identical. Very recently, the emergence of single- and double-headed (i. e. with one and two (in)coherent regions, respectively) chimera states in neural oscillator networks with local coupling has been reported [@BER16]. That system, however, is known to exhibit high metastability, which renders the chimera state non-stationary when tracked in long time intervals [@HIZ16]. Here we demonstrate numerically the emergence of multi-clustered robust chimera states in SQUID metamaterials described in the local coupling approximation, in a relevant parameter region which has been determined experimentally in [@Trepanier2013; @Zhang2015].
Superconducting metamaterials comprising SQUIDs have been realized in both one and two dimensions [@Trepanier2013; @Zhang2015; @Butz2013; @Jung2014a; @Jung2014b; @Ustinov2015] and possess extraordinary properties such as negative magnetic permeability, dynamic multistability, broadband tunability, switching between different magnetic permeability states, as well as a unique form of transparency whose development can be manipulated through multiple parametric dependences. Some of these observed properties had been theoretically predicted both for the quantum [@Du2006] and the classical regime [@Lazarides2007; @Lazarides2013]. SQUID metamaterials are richly nonlinear effective media modeled by discrete phenomenological equations of coupled individual SQUID oscillators, which introduce qualitatively new macroscopic quantum effects into both the metamaterials and the coupled oscillator networks communities, i.e., magnetic flux quantization and the Josephson effect [@Josephson1962].
![(Color online) Schematic of a SQUID in a magnetic field ${\bf H}(t)$ (a), and equivalent electrical circuit (b). The real Josephson junction is represented by the circuit elements in the brown-dashed box. (c) Schematic of a one-dimensional SQUID metamaterial. \[fig1\] ](fig1ab.png "fig:"){width="0.9\linewidth"} ![(Color online) Schematic of a SQUID in a magnetic field ${\bf H}(t)$ (a), and equivalent electrical circuit (b). The real Josephson junction is represented by the circuit elements in the brown-dashed box. (c) Schematic of a one-dimensional SQUID metamaterial. \[fig1\] ](fig1c.png "fig:"){width=".9\linewidth"}
A SQUID consists of a superconducting ring interrupted by a Josephson junction (JJ) as shown schematically along with its electrical equivalent circuit in Figs. \[fig1\](a) and \[fig1\](b), respectively; it is a highly nonlinear oscillator with a resonant response to an applied alternating (ac) magnetic field. When a periodic arrangement of $N$ identical SQUIDs is driven by a spatially uniform, ac field \[Fig. \[fig1\](c)\], its elements are coupled together through magnetic dipole-dipole forces which decay as the inverse cube of the distance. In the following, it is considered that each SQUID in the array is coupled only to its nearest neighbors, neglecting further-neighbor interactions. Then, the magnetic flux $\Phi_n$ threading the loop of the $n$th SQUID is $$\begin{aligned}
\label{01}
\Phi_n =\Phi_{ext} +L\, I_n +M\, ( I_{n-1} +I_{n+1} ) ,\end{aligned}$$ where $\Phi_{ext}$ is the external flux to each SQUID, $L$ is the self-inductance of the individual SQUID, $M$ is the mutual inductance between neighboring SQUIDs, and $$\begin{aligned}
\label{02}
I_n =-C\frac{d^2\Phi_n}{dt^2} -\frac{1}{R} \frac{d\Phi_n}{dt}
-I_c\, \sin\left(2\pi\frac{\Phi_n}{\Phi_0}\right), \end{aligned}$$ is the current in the $n$th SQUID as provided by the resistively and capacitively shunted junction (RCSJ) model of the JJ [@Likharev1986], and $\Phi_0$ is the flux quantum. Within the RCSJ framework, $R$, $C$, and $I_c$ are the resistance, capacitance, and critical current of the JJ, respectively. Combining Eqs. (\[01\]) and (\[02\]), while neglecting all terms proportional to higher than the first power of the dimensionless coupling coefficient $\lambda=M/L$ [@Lazarides2013], gives $$\begin{aligned}
\label{05}
\ddot{\phi}_n +\gamma \dot{\phi}_n +\phi_n +\beta \sin\left( 2\pi \phi_n \right) =
\lambda ( \phi_{n-1} +\phi_{n+1} )
\nonumber \\
+(1 -2\lambda) \phi_{ac} \cos(\Omega \tau) , \end{aligned}$$ in which a sinusoidal external flux is considered. The flux through the $n$th SQUID loop $\phi_n$ and the amplitude of the external flux $\phi_{ac}$ are normalized to $\Phi_0$, the driving frequency $\Omega$ and the temporal variable $\tau$ (the overdots denote derivation with respect to $\tau$) are normalized to the geometrical (inductive-capacitive) resonance frequency of the SQUID $\omega_{LC} =1 / \sqrt{L C}$ and its inverse $\omega_{LC}^{-1}$, and $\beta=\frac{I_c L}{\Phi_0} =\frac{\beta_L}{2\pi}$, $\gamma=\frac{1}{R} \sqrt{ \frac{L}{C} }$ is the SQUID parameter and loss coefficient, respectively.
The corresponding equation for a single SQUID is obtained from Eq. (\[05\]) by setting $\lambda=0$ and $\phi_n =\phi$. Then, by linearization of that equation and by neglecting dissipation and forcing, the SQUID resonance frequency can be obtained as $\Omega_{SQ} =\sqrt{1 +\beta_L}$ in units of $\omega_{LC}$. The single SQUID equation for a certain range of parameters exhibits a “snake-like” resonance curve in which multiple stable and unstable periodic orbits coexist and vanish through saddle-node bifurcations of limit cycles. This dynamical behavior bears a big resemblance to the snaking bifurcation curves of localized structures reported in the Swift-Hohenberg equation [@Kozyreff2006].
![(Color online) The “snake-like” resonance curve of a single SQUID for $T=6.24$ ($\Omega\simeq 1.007$), $\beta_L = 0.86$, $\gamma=0.024$, and external ac flux $\phi_{ac}=0.06$. Solid blue and dashed lines correspond to branches of stable and unstable periodic solutions, respectively. Saddle-node bifurcations points are denoted as “SN”. Thick gray line corresponds to $\Omega=1.007$ and turquoise lines are obtained from Eq. (\[07\]). Inset: Enlargement around the maximum multistability frequency. Phase portraits on the right show the corresponding periodic orbits of points $A-D$ on the resonance curve. \[fig2\]](fig2.png){width="\columnwidth"}
The parameters which are responsible for the SQUID multistability are the loss coefficient $\gamma$ and the external ac flux $\phi_{ac}$. As $\gamma$ decreases and $\phi_{ac}$ increases, the snaking curve becomes more winding, achieving higher flux values and adding, thus, to the multistability. A typical such curve is shown in the left panel of Fig. \[fig2\] in which the amplitude of the flux variable $\phi_{\text{max}}$ is plotted against the driving frequency $\Omega$. The blue solid lines correspond to the branches of stable periodic solutions while the dashed lines mark the unstable orbits. At each turning point where stable and unstable branches merge, a saddle node (“SN”) bifurcation of limit cycles takes place. The inset figure shows a blow-up around $\Omega=1$ where the multistability is more prominent. This is illustrated by the intersections of the gray line with the snaking curve marked by the letters $A-D$. For this value of the driving frequency, we can distinguish $K=5$ coexisting periodic states of increasing amplitude; the corresponding orbits are shown in the phase portraits on the right.
An approximate resonance curve can be actually obtained from the single-SQUID equation using a truncated series expansion for $\sin(2\pi \phi)$ with a trial solution $\phi=\phi_m (\tau)\, \cos[\Omega \tau +\theta (\tau)]$, where $\phi_m (\tau)$ and $\theta (\tau)$ are the slowly varying amplitude and phase, respectively. Then, by applying the rotating wave approximation (RWA) in which only the terms at the fundamental frequency $\Omega$ are retained, neglecting terms $\propto \ddot{\phi_m}$, $\ddot{\theta}$, $\dot{\theta}^2$, $\dot{\phi_m}$, $\dot{\theta}$, etc., and seeking for steady state solutions of the resulting algebraic system for $\phi_m (\tau)$ and $\theta (\tau)$, we get $$\label{07}
\Omega^2 =\Omega_{SQ}^2 -\beta_L \phi_m^2
\{ a_1 -\phi_m^2 [a_2 -\phi_m^2 ( a_3 -a_4 \phi_m^2)]\}
\pm \frac{\phi_{ac}}{\phi_m} ,$$ where $a_1=\pi^2/2$, $a_2=\pi^4/12$, $a_3=\pi^6/144$, $a_4=\pi^8/2880$, which implicitly provides the sought $\phi_m(\Omega)$ relation, in which the first four terms in the series expansion are kept. The curves obtained from the earlier equation are shown in Fig. \[fig2\] in turquoise color, and reproduce the resonance curve up to $\phi_m \sim 0.6$ that includes the first saddle-node bifurcation.
In a metamaterial of $N$ weakly coupled SQUIDs there is a multiplicity of possible collective states that the system can reach, the number of which is of the order of $K^N$ or higher. The complexity in analyzing the behavior of such a system becomes clear by coupling together just two SQUIDs; the corresponding resonance curve maintains its “snake-like” form but with a thicker contour due to the additional (un)stable branches that are created (not shown here). Apart from the new periodic solutions, a number of coexisting chaotic attractors are also to be found. Figure \[fig3\](a) shows a scan in $\Omega$ for values around the maximum multistability point, for two coupled SQUIDs. The different colors correspond to solution branches for different initial conditions and it is clear that around $\Omega=1.007$ the magnetic flux exhibits chaotic behavior. For $N=256$ coupled SQUIDs the complexity of the dynamics is even higher: Figure \[fig3\](b) shows the stroboscopic maps corresponding to individual oscillators of a single configuration of the full system. The numbers next to the orbits denote the oscillator indices. This huge multiplicity of attractors is known as [*attractor crowding*]{} and has been observed before in coupled nonlinear oscillator systems [@Wiesenfeld1989].
![(Color online) (a) Solution branches for two coupled SQUIDs for an $\Omega$ scan around the maximum multistability point. At least ten (10) SQUID states are visible at this frequency. (b) $N=256$ coupled SQUIDs: Stroboscopic maps of some individual oscillators. Numbers denote the index of the respective SQUID oscillator. Coupling strength $\lambda=-0.025$ in both figures. All other parameters as in Fig. \[fig2\]. \[fig3\] ](fig3a.png "fig:"){width="0.5\columnwidth"} ![(Color online) (a) Solution branches for two coupled SQUIDs for an $\Omega$ scan around the maximum multistability point. At least ten (10) SQUID states are visible at this frequency. (b) $N=256$ coupled SQUIDs: Stroboscopic maps of some individual oscillators. Numbers denote the index of the respective SQUID oscillator. Coupling strength $\lambda=-0.025$ in both figures. All other parameters as in Fig. \[fig2\]. \[fig3\] ](fig3b.png "fig:"){width="0.46\columnwidth"}
{width=".44\textwidth"} {width=".44\textwidth"}
Equations (\[05\]) are integrated numerically in time with a fourth-order Runge-Kutta algorithm with constant time-step and periodic boundary conditions, i. e., $\phi_n(\tau)=\phi_{N+n}(\tau)$ for all $n$. The parameters used in the simulations are close to the design parameters of the SQUID meta-atoms that make two-dimensional SQUID metamaterials [@Zhang2015], i.e., $L=60~pH$, $C=0.42~pF$, $I_c=4.7~\mu A$, and subgap resistance $R=500~\Omega$, which give $\beta_L =0.86$ and $\gamma=0.024$ according to their definitions, while the value of the coupling coefficient between neighboring SQUIDs is $\lambda=-0.025$. The amplitude of the ac field is selected to be $\phi_{ac}=0.06$, within the experimentally accessible range $0.001 -0.1$ [@Trepanier2013]. The selected values of $\gamma$ and $\phi_{ac}$ bring the SQUID metamaterial in the strongly nonlinear regime.
Figure \[fig4\] shows time-snapshots of the magnetic fluxes $\phi_n$ for different initial conditions and for two different values of the loss coefficient $\gamma$. The left panel is for $\gamma=0.024$ which is the value corresponding to the resonance curve of Fig. \[fig2\]. The initial “sine wave” magnetic flux distribution for each simulation is shown by the gray solid line. The SQUIDs that are prepared at lower values form the coherent clusters of the chimera state, while those that are initially set at higher magnetic flux values oscillate incoherently. Moreover, as the “wave-length” of the initial magnetic flux distribution increases, so does the chimera state multiplicity (number of (in)coherent regions highlighted by the colored areas). Similar behavior is observed for lower values of the loss coefficient ($\gamma=0.0024$) as shown in the right panel of Fig.\[fig4\]. Here, the incoherent clusters are better illustrated since they are approximately of equal size and do not contain oscillators that “escape” from the incoherent cluster abiding around low magnetic flux values, something which is visible in the left panel. Furthermore, the coherent clusters (emphasized by the blue solid lines) are fixed around $\phi=0$, unlike in the left panel where additional clusters located at slightly higher values also form. Here we must recall that for low values of $\gamma$ (right panel) the winding of the “snake-like” resonance curve increases significantly creating, thus, new branches of stable (and equally unstable) periodic solutions. These branches are larger in number and smaller in size compared to those of higher $\gamma$ values (left panel). The lower amplitude branches which are the longer ones attract the SQUIDs that eventually form the coherent clusters. The other oscillators have a plethora of higher states to choose from and, therefore, create a more chaotic incoherent cluster than in the case of higher $\gamma$ values.
-(d). Right panel: Map of dynamic regimes in the $(\gamma,\lambda)$ parameter space for the initial conditions of Figs. \[fig4\](a) and (b). Numbers in brackets denote the multiplicity of the chimera state while “synch” stands for (route to) synchronization. All other parameters as in Fig. \[fig2\]. \[fig5\] ](fig5left.png "fig:"){width="0.47\columnwidth"} -(d). Right panel: Map of dynamic regimes in the $(\gamma,\lambda)$ parameter space for the initial conditions of Figs. \[fig4\](a) and (b). Numbers in brackets denote the multiplicity of the chimera state while “synch” stands for (route to) synchronization. All other parameters as in Fig. \[fig2\]. \[fig5\] ](fig5right.png "fig:"){width="0.51\columnwidth"}
The observed chimera states can be quantified through the Kuramoto local order parameter [@HIZ16a] which is a measure for local synchronization: $$Z_n=\left | \frac{1}{2\delta} \sum_{|j-n| \le \delta} e^{i\phi_j} \right |, \quad n=1,\dots, N.$$ We use a spatial average with a window size of $\delta=5$ elements. A $Z_n$ value close to unity indicates that the $n$th SQUID belongs to the coherent cluster of the chimera state, while $Z_n$ is closer to $0$ in the incoherent parts. In the left panel of Fig. \[fig5\] the space-time plots of the local order parameter corresponding to the chimera states of Fig. \[fig4\](a)-(d) are shown. The number of (in)coherent regions increases according to the initial conditions and the size and location of the clusters is constant in time. Previous works on SQUID metamaterials showed that for nonlocal coupling single and double-headed chimera states coexist with solitary states [@Jaros2015] and metastable states of drifting (in)coherence [@LAZ15; @HIZ16a]. Note that in these studies the focus was on a different dynamical area with the external driving frequency lying outside the multistability regime. For a suitable choice of $\Omega$, stable chimera states can be achieved for nonlocal coupling also. However, they exist only for low coupling strengths $\lambda$; the threshold value of the coupling strength in the case of local coupling is much higher. Local coupling is therefore crucial for the emergence of *robust* chimera states, both in structure and in lifetime, for a wide range of parameters.
Previously we stressed the importance of multistability and the impact of the loss coefficient $\gamma$ in the formation of chimeras in our system. In addition to that, it is important to note the role of the network topology which is defined through the local nature of interactions and the coupling strength $\lambda$. As already shown in Fig. \[fig4\], our system exhibits a variety of coexisting multi-clustered chimera states. A systematic study in the $(\lambda, \gamma)$ parameter space is depicted in the right panel of Fig. \[fig5\], where the observed patterns for two different sets of initial conditions (namely those of Fig. \[fig4\](a,a’) and (b,b’)) are mapped out. The numbers in the brackets correspond to the multiplicity of the respective chimeras and “synch” denotes the synchronized states. The black and white asterisk mark the $(\lambda, \gamma)$ values used in the left and right panel of Fig. \[fig4\], respectively. We see that for low values of the loss coefficient and for low and medium (in absolute value) coupling strengths, single- and four-headed chimera states coexist (yellow area). As $\gamma$ increases, the effect of multistability diminishes and the system enters the synchronized state (red area) either directly or through a region where three-headed chimeras coexist with single-headed ones (green area). For stronger couplings (blue area), double-headed chimeras coexist with single chimeras. The latter persist also for high $\gamma$ values where the synchronized state is achieved. For initial conditions with a larger modification in space (like in Fig. \[fig4\](c,c’) and (d,d’)) chimera states with higher multiplicity can be found, but the mechanism towards synchronization is the same: the fully coherent state is reached through the appearance of solitary states [@Jaros2015; @HIZ16a].
In conclusion, the model equations for a SQUID metamaterial truncated to only nearest-neighbor coupling were integrated in time with properly chosen initial conditions that allow the system to reach chimera states. These novel states emerge due to the extreme multistability that leads to attractor crowding at the geometrical resonance frequency. Typical chimera states are presented and characterized with respect to their local synchronization level. A systematic study in the relevant parameter space reveals the coexistence of multi-headed chimeras and the oscillators for a metamaterial in a chimera state elucidate a number of different trajectories, some of which are chaotic. Since chimera states have been intimately connected with nonlocal coupling, the present results point towards the need to revise the general consensus on the essential conditions for their existence.
Acknowledgement {#acknowledgement .unnumbered}
---------------
This work was partially supported by the European Union Seventh Framework Programme (FP7-REGPOT-2012-2013-1) under grant agreement n$^o$ 316165, and the Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of NUST “MISiS” (No. K2-2015-007). J. H. would like to thank Thomas Isele and Yuri Maistrenko for valuable discussions.
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Supplemental video SM1 Supplemental video SM2
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We prove spectral analogues of the classical strong multiplicity one theorem for newforms. Let $\Gamma_1$ and $\Gamma_2$ be uniform lattices in a semisimple group $G$. Suppose all but finitely many irreducible unitary representations (resp. spherical) of $G$ occur with equal multiplicities in $L^{2}(\Gamma_1 \backslash G)$ and $L^{2}(\Gamma_2 \backslash G)$. Then $L^{2}(\Gamma_1 \backslash G)
\cong L^{2}(\Gamma_2 \backslash G)$ as $G$ - modules (resp. the spherical spectra of $L^{2}(\Gamma_1 \backslash G)$ and $L^{2}(\Gamma_2 \backslash G)$ are equal).
address: |
Tata Institute of Fundamental Research\
Homi Bhabha Road\
Mumbai 400005, India.
author:
- Chandrasheel Bhagwat
- 'C. S. Rajan'
title: |
On a spectral analogue of the strong\
multiplicity one theorem
---
**[Introduction]{}** {#intro}
====================
The beginnings of the analogy between the spectrum and arithmetic of Riemannian locally symmetric spaces can be attributed to Maass, who defined non-analytic modular forms as eigenfunctions of the Laplacian satisfying suitable modularity and growth conditions. From the viewpoint of Gelfand, the theory of Maass forms can be re-interpreted in terms of the representation theory of $PSL(2,{{\mathbb R}})$ on $L^{2}(\Gamma
\backslash PSL(2,{{\mathbb R}}))$ for a lattice $\Gamma$ in $PSL(2,{{\mathbb R}})$. Subsequently, the analogy between the spectrum and arithmetic has been extended by the work of A. Selberg, P. Sarnak, M. F. Vigneras and T. Sunada amongst others.
In this paper, our aim is to establish an analogue in the spectral context of the classical strong multiplicity one theorem for cusp forms. Suppose $f$ and $g$ are newforms for some Hecke congruence subgroup $\Gamma_{0}(N)$ such that the eigenvalues of the Hecke operator at a prime $p$ are equal for all but finitely many primes $p$. Then the strong multiplicity one theorem of Atkin and Lehner states that $f$ and $g$ are equal (*cf*. [@La p.125]).
Now, let $G$ be a semisimple Lie group and $\Gamma$ be a uniform lattice (a discrete cocompact subgroup) in $G$. Let $R_{\Gamma}$ be the right regular representation of $G$ on $L^2(\Gamma \backslash G)$:
$$R_{\Gamma}(g)(\phi)(y) = \phi(yg)\quad \forall\ g,~y \in G\ \text{and}\ \phi \in\ L^2(\Gamma \backslash G)$$
This defines a unitary representation of $G$. It is known ([@GGP p.23]) that $R_{\Gamma}$ decomposes discretely as a direct sum of irreducible unitary representations of $G$ occurring with finite multiplicities. Let $\widehat{G}$ be the set of equivalence classes of irreducible unitary representations of $G$. For $\pi \in
\widehat{G}$, let $m(\pi, \Gamma )$ be the multiplicity of $\pi$ in $R_{\Gamma}$.
Let $\Gamma_1$ and $\Gamma_2$ be uniform lattices in $G$. The lattices $\Gamma_1$ and $\Gamma_2$ are said to be representation equivalent in $G$ if $$L^{2}(\Gamma_1 \backslash G) \cong L^{2}(\Gamma_2 \backslash G)$$ as representations of $G$, i.e. for every $\pi\ \in\ \widehat{G}$, $$m(\pi, \Gamma_1 ) = m(\pi, \Gamma_2)$$
In this article we prove the following result :
\[thm1\] Let $\Gamma_1 $ and $\Gamma_2 $ be uniform lattices in a semisimple Lie group $G$. Suppose all but finitely many irreducible unitary representations of $G$ occur with equal multiplicities in $L^{2}(\Gamma_1 \backslash G)$ and $L^{2}
(\Gamma_2 \backslash G)$. Then the lattices $\Gamma_1$ and $\Gamma_2$ are representation equivalent in $G$.
The proof of Theorem \[thm1\] uses the Selberg trace formula and fundamental results of Harish Chandra on the character distributions of irreducible unitary representations of $G$; in particular, we make crucial use of a deep and difficult result of Harish Chandra that the character distribution of an irreducible unitary representation of $G$ is given by a locally integrable function on $G$.
We now consider an analogue of Theorem \[thm1\] for the spherical spectrum of uniform lattices. Let $K$ be a maximal compact subgroup of $G$. An irreducible unitary representation $\pi$ of $G$ is said to be spherical if there exists a non-zero vector $v \in \pi$ such that $$\pi(k) v = v\quad \forall \ k \in K.$$ The spherical spectrum $\widehat{G}_s$ of $G$ is the subset of $\widehat{G}$ consisting of equivalence classes of irreducible unitary spherical representations of $G$.
\[prin\] Let $G$ be a connected, semisimple Lie group. Suppose $\Gamma_1$, $\Gamma_2$ are uniform torsion-free lattices in $G$ such that $$m(\pi, \Gamma_1 ) = m(\pi, \Gamma_2 )$$ for all but finitely many representations $\pi$ in $\widehat{G}_s$. Then $$m(\pi, \Gamma_1 ) = m(\pi, \Gamma_2 )$$ for all representations $\pi$ in $\widehat{G}_s$.
The proof of this theorem follows the broad outline of Theorem \[thm1\], but requires a more delicate control of $K\times
K$-saturation of a conjugacy class of an element of $\Gamma$ (see Proposition \[B\]). This is achieved by looking at the behaviour of the conjugacy classes of elements of $\Gamma$ in a neighbourhood of identity.
We now relate the spherical spectrum with the spectrum of $G$-invariant differential operators on the associated symmetric space $X = G/K$. For a torsion-free uniform lattice $\Gamma$ in $G$, let $X_{\Gamma} = \Gamma \backslash G / K$ be the associated compact Riemannian locally symmetric space. The space of smooth functions on $X_{\Gamma}$ can be considered as the space of smooth functions on $X$ invariant under the action of $\Gamma$. Let $D(G / K)$ be the algebra of $G$-invariant differential operators on $X$. For a character $\lambda$ of $D(G/K)$ (i.e. an algebra homomorphism of $D(G/K)$ into ${{\mathbb C}}$), consider the eigenspace of $\lambda$, $$\label{V} V(\lambda,\Gamma) = \left\{f \in
C^{\infty}(X_{\Gamma})\ :\ D(f) = \lambda(D)(f)\quad \forall\ D \in
D(G / K) \right\}.$$ It is known that the space $V(\lambda,\Gamma)$ is of finite dimension (see Section \[prooflaplacian\]).
Let $\Gamma_1$ and $\Gamma_2$ be torsion-free uniform lattices in $G$. The locally symmetric spaces $X_{\Gamma_1} =
\Gamma_1 \backslash G / K$ and $X_{\Gamma_2} = \Gamma_2 \backslash G /
K$ are said to be *compatibly isospectral* if $$\text{dim} ( V (\lambda, \Gamma_1) ) = \text{dim}
\ ( V (\lambda, \Gamma_2) )$$ for every character $\lambda$ of $D(G / K)$.
From the generalized Sunada criterion proved by Berard [@Be p.566] and DeTurck - Gordon [@DG], it can be seen that if two uniform lattices in $G$ are representation equivalent, then the associated compact locally symmetric Riemannian spaces $X_{\Gamma_1}$ and $X_{\Gamma_2}$ are compatibly isospectral.
We prove the following result in Section \[prinlapla\] :
\[laplacian\] Let $G$ be a connected, semisimple Lie group. Suppose $\Gamma_1$, $\Gamma_2$ are uniform torsion-free lattices in $G$. Suppose $$\text{dim}\ ( V (\lambda, \Gamma_1 )) = \text{dim}
( V (\lambda, \Gamma_2 ) )$$ for all but finitely many characters $\lambda$, then $X_{\Gamma_1}$ and $X_{\Gamma_2} $ are compatibly isospectral.
If $X$ is of rank one, the algebra $D(G / K)$ is the polynomial algebra in the Laplace-Beltrami operator $\Delta$ on $G/K$ (see [@He p.397]). Hence the eigenvalues of $\Delta$ determine the characters of $D(G / K)$. Consequently we get :
\[lapla\] Let $X_1$ and $X_2$ be two locally symmetric Riemannian spaces of rank one and $\Delta_1$, $\Delta_2$ be the Laplace-Beltrami operators acting on the space of smooth functions on $X_1$ and $X_2$ respectively. If all but finitely many eigenvalues occur with equal multiplicities in the spectra of $\Delta_1$ and $\Delta_2$, then the spaces are isospectral with respect to the Laplace-Beltrami operators.
Using an analytic version of the Selberg Trace formula, J. Elstrodt, F. Grunewald, and J. Mennicke (on a suggestion of M. F. Vigneras) proved Corollary \[lapla\] for $G = PSL(2,{{\mathbb R}})$ and $G = PSL(2,{{\mathbb C}})$ ([@EGM Theorem 3.3, p.203]).
When $G=PSL(2,{{\mathbb R}})$, it can be seen that the spherical spectrum of determines the full spectrum $L^2(\Gamma\backslash G)$ ([@Pe]). One can raise the question whether such a result will be true in general. This fits in with the conjectures linking spectrum and arithmetic in the context of automorphic forms (see [@Ra Conjecture 3]).
We thank S. Kudla for raising the question of proving an analogue of the strong multiplicity one theorem in the spectral case during the conference on ‘Modular forms’ held at Schiermonikoog, Netherlands in October 2006. The second author thanks the organizers of the conference for the invitation and warm hospitality.\
Preliminaries {#Prelim}
=============
Representations of semisimple groups {#repgrp}
-------------------------------------
We recall some facts about representations of semisimple groups. Let $G$ be a semisimple Lie group with a Haar measure $\mu$. Let $\pi$ be a unitary representation of $G$ on a Hilbert space $V$. For a compactly supported smooth function $f$ on $G$, define the convolution operator $\pi(f)$ on $V$ as follows :
$$\pi(f)(v) = \int\limits_G{f(g)\ \pi(g)v\ d\mu(g)}$$
This defines a bounded linear operator on $V$. We recall the following result from [@Kn Theorem 10.2; p.334] :
\[trace\] Let $G$ be a semisimple group and $\pi$ be an irreducible unitary representation of $G$. Then the convolution operator $\pi(f)$ is of trace class for every compactly supported smooth function $f$ on $G$.
Let $\pi$ be an irreducible unitary representation of $G$. Let $C_{c}^{\infty}(G) $ be the space of compactly supported smooth functions on $G$. Define the character distribution $\chi_{\pi}$ by, $$\chi_{\pi}(f) = \text{trace}~(\pi(f)) \quad \forall f \in C_{c}^{\infty}(G).$$
Some results of Harish Chandra {#HC}
------------------------------
We recall some results of Harish Chandra on the characters of irreducible unitary representations of $G$.
\[LinInd\] [@Kn Theorem 10.6; p.336] Let $\left\{\pi_i\right\}$ be a finite collection of mutually inequivalent irreducible unitary representations of $G$. Then their characters $\left\{\chi_{\pi_i}\right\}$ are linearly independent distributions on $C_{c}^{\infty}(G)$.
Let $L^{1}_{loc}(G)$ be the space of all complex valued measurable functions $f$ on $G$ such that $$\int\limits_{C} |f(g)|\ d\mu (g)\ < \infty\ \ \ \text{for all compact subset}\ C\ \text{of}\ G.$$
The following deep result of Harish Chandra will be crucially used in the proof of Theorem \[thm1\].
\[locint\] [@Kn Theorem 10.25; p.356] Let $\pi$ be an irreducible unitary representation of $G$. The distribution character $ \chi_{\pi}$ is given by a locally integrable function $h$ on $G$. i.e. there exists $ h \in L^{1}_{loc}(G)$ such that $$\chi_{\pi}(f) = \int \limits_{G}\ f(g)\ h(g)\ d\mu(g)
\quad \forall f \in C^{\infty}_c(G).$$
Selberg trace formula for compact quotient {#STF}
------------------------------------------
We recall the Selberg trace formula for compact quotient. (For details, see Wallach [@Wa p.171-172]). Let $\mu'$ be the normalized $G$-invariant measure on the quotient space $\Gamma \backslash G$. For a compactly supported smooth function $f$ on $G$, the convolution operator $R_{\Gamma}(f)$ on $L^2(\Gamma \backslash G)$ is given by :
$$\begin{split} R_{\Gamma}(f)(\phi)(y)\ \ \ = \int \limits_{G}f(x)\
\phi(yx)\ d\mu(x) \ \ \ \ \ \ \forall\ \phi \in L^2(\Gamma \backslash
G)\ \text{and}\ y \in G.
\end{split}$$
$$\begin{split} = \int \limits_{\Gamma \backslash G}\left[
\sum\limits_{\gamma\ \in\ \Gamma}f(y^{-1} \gamma x) \right]\ \phi(x)\
d\mu'(x).
\end{split}$$
Since $f$ is a smooth and compactly supported function on $G$ and $\Gamma$ is uniform lattice, the sum $K_{f}(y,x) =
\sum\limits_{\gamma\ \in\ \Gamma}f(y^{-1} \gamma x)$ is a finite sum, and hence it follows that the operator $R_{\Gamma}(f)$ is of Hilbert-Schmidt class. The trace of $R_{\Gamma}(f)$ is defined and it is given by integrating the kernel function $K_{f}(y,x)$,
$$\begin{split} \text{tr}(R_{\Gamma}(f)) = \int \limits_{\Gamma
\backslash G} \left[\sum_{\gamma\ \in\ \Gamma} f(x^{-1}\gamma
x)\right] d\mu'(x)
\end{split}$$
Let $[\gamma]_{G}$ (resp. $[\gamma]_{\Gamma}$) be the conjugacy class of $\gamma$ in $G$ (resp. in $\Gamma$). Let $\left[\Gamma\right]$ (resp. $\left[\Gamma \right]_G$) be the set of conjugacy classes in $\Gamma$ (resp. the $G$-conjugacy classes of elements in $\Gamma$). For $\gamma \in \Gamma$, let $G_{\gamma}$ be the centralizer of $\gamma$ in $G$. Put $\Gamma_{\gamma} = \Gamma \cap G_{\gamma}$. It can be seen that $\Gamma_{\gamma}$ is a lattice in $G_{\gamma}$ and the quotient $\Gamma_{\gamma} \backslash G_{\gamma}$ is compact. Since $G_{\gamma}$ is unimodular, there exists a $G$-invariant measure on $G_{\gamma} \backslash G$, denoted by $d_{\gamma}x$. After normalizing the measures on $G_{\gamma}$ and $G_{\gamma} \backslash G$ appropriately and rearranging the terms on the right hand side of above equation, we get :
$$\label{1} \text{tr}(R_{\Gamma}(f)) \ \ \ \ \ \ \ =
\sum \limits_{[\gamma]\ \in \ \left[\Gamma \right]} \text{vol}
(\Gamma_{\gamma} \backslash G_{\gamma}) \ \int\limits_{G_{\gamma}
\backslash G} f(x^{-1} \gamma x)\ d_{\gamma}x$$
$$\label{geom} = \sum \limits_{[\gamma]\ \in\
[\Gamma]_G}a(\gamma,\Gamma)\ O_{\gamma}(f)$$
where $O_{\gamma}(f)$ is the orbital integral of $f$ at $\gamma$ defined by, $$O_{\gamma}(f) = \int\limits_{G_{\gamma} \backslash G} f(x^{-1}
\gamma x)\ d_{\gamma}x.$$ Here $$a(\gamma, \Gamma) = \sum
\limits_{[\gamma']_\Gamma \ \subseteq\ [\gamma]_G}\ \text{vol} \
(\Gamma_{\gamma'} \backslash G_{\gamma'}).$$ If $\gamma$ is not conjugate to an element in $\Gamma$, we define $a(\gamma,
\Gamma)=0$. On the other hand, the trace of $R_{\Gamma}(f)$ on the spectral side can be written as an absolutely convergent series as, $$\label{rep} \text{tr}(R_{\Gamma}(f)) = \sum
\limits_{\pi\ \in \ \widehat{G}} m(\pi,\Gamma)\chi_{\pi}(f)$$
Hence from (\[1\]) and (\[rep\]), we obtain the Selberg trace formula:
$$\label{STF1} \sum \limits_{\pi\ \in \ \widehat{G}}
m(\pi,\Gamma) \chi_{\pi}(f) = \sum \limits_{[\gamma]\ \in\
[\Gamma]_G}a(\gamma,\Gamma)\ O_{\gamma}(f).$$
**[Proof of Theorem \[thm1\]]{}**
==================================
Some preliminary lemmas
-----------------------
We first recall some known results about the geometry of conjugacy classes in $G$.
\[conj\] Let $\Gamma$ be a uniform lattice in $G$. Let $\gamma \in \Gamma$. Then the $G$-conjugacy class $[\gamma]_G$ is a closed subset of measure zero in $G$.
Let $\left\{g_{n}^{-1} \gamma g_{n}\right\}_{n=1}^{\infty}$ be a sequence of points in $[\gamma]_G$ which converges to $h$ in $G$. Since $\Gamma \backslash G$ is compact, there exists a relatively compact set $D$ of $G$ such that $G = \Gamma D$. Write $
g_{n} = \gamma_{n}\ d_{n}$ where $\gamma_{n}\ \in\ \Gamma$ and $d_{n}\
\in\ D$. Hence, $$g_{n}^{-1}\ \gamma\ g_{n} = d_{n}^{-1}\ \gamma_{n}^{-1}\ \gamma\
\gamma_{n}\ d_{n}.$$ Since $D$ is relatively compact, there is a convergent subsequence of $\left\{ d_{n} \right\}_{n=1}^{\infty}$, which converges to some element $d$ of $G$. Hence we get, $$\lim \limits_{n \rightarrow \infty} \gamma_{n}^{-1}\ \gamma\
\gamma_{n} = d^{-1}hd.$$ Since $\Gamma$ is discrete, for large $n$, $$\gamma_{n}^{-1}\ \gamma\ \gamma_{n} = d^{-1}hd.$$ Hence $h\ \in\
[\gamma]_G$. Thus $[\gamma]_G$ is closed in $G$.
The conjugacy class $[\gamma]_G$ is homeomorphic to the homogeneous space $G_{\gamma}\backslash G$. Hence there exists a natural structure of a smooth manifold on it such that it is a submanifold of $G$. Since $G_{\gamma}$ is non trivial (it contains a Cartan subgroup of $G$), it is of lower dimension than $G$ and hence of measure zero with respect to the Haar measure $\mu$ on $G$.
\[berard\] Let $\Omega$ be a relatively compact subset of $G$. Then the set $$A_{\Omega} = \left\{\ [\gamma]_G\ :\ \gamma \in \Gamma\
\text{and}\ [\gamma]_G\ \cap\ \Omega\ \neq\ \emptyset\ \right\}$$ is finite.
Let $x \in G$ be such that $x^{-1} \gamma x \in \Omega$. As in Lemma \[conj\], write $x = \gamma_1. \delta$ where $\gamma_1
\in \Gamma$ and $\delta \in D$. Hence $\gamma_1^{-1} \gamma \gamma_1
\in D \Omega D^{-1}$ which is relatively compact in $G$. Hence $\gamma_1^{-1} \gamma \gamma_1 \in D \Omega D^{-1}\ \cap\ \Gamma$ which is a finite set.
\[E\] Let $E$ be the union of the conjugacy classes $\ [\gamma]_G\ \text{such that}\ \gamma \in {\Gamma_1} \cup
{\Gamma_2}$. Then $E$ is a closed subset of measure zero in $G$.
By using above two lemmas, it follows that $E \cap C$ is finite for every compact subset $C \subseteq G$. Hence $E$ is closed in $G$. It is of measure zero since it is a countable union of sets of measure zero.
Proof of Theorem \[thm1\]
--------------------------
For $\pi \in \widehat{G}$, let $t_\pi = m(\pi,\Gamma_1) - m(\pi,\Gamma_2)$. Let $f \in
C^{\infty}_{c}(G)$. Since the series in equation (\[STF1\]) converges absolutely, by comparing equation (\[STF1\]) for $\Gamma_1$ and $\Gamma_2$, we obtain:
$$\label{trf1} \sum \limits_{\pi\ \in \ \widehat{G}}
t_\pi\ \chi_{\pi}(f) = \sum \limits_{\substack{[\gamma]\ \in \
[\Gamma_1]_G\ \cup\ [\Gamma_2]_G}} (a(\gamma,\Gamma_1) -
a(\gamma,\Gamma_2)) \ O_{\gamma}(f).$$
By hypothesis, $t_\pi = 0$ for all but finitely many $\pi \in \hat{G}$. Hence there exists a finite subset $S$ of $\widehat{G}$ such that,
$$\label{trf3}
\begin{split} \sum \limits_{\pi\ \in\ S} \ t_{\pi}\chi_{\pi}(f) =
\sum \limits_{\substack{[\gamma]\ \in \ [\Gamma_1]_G\ \cup\
[\Gamma_2]_G}} (a(\gamma,\Gamma_1) - a(\gamma,\Gamma_2))\
O_{\gamma}(f).
\end{split}$$
Since $S$ is a finite set, by Harish Chandra’s Theorem \[locint\], there exists a function $\phi \in L^{1}_{loc}(G)$ such that
$$\label{loc} \sum \limits_{\pi\ \in\ S}
t_{\pi}. \chi_{\pi}(f) = \int \limits_{G} f(g)\ \phi(g)\ d\mu(g) \quad
\forall\ f \in C^{\infty}_c(G).$$
Let $E$ be as in Corollary \[E\] above. Let $g \in G$ be any point outside $E$. Since $E$ is closed in $G$, there exists a relatively compact neighborhood $U$ of $g$ such that $U \cap E =
\emptyset $. Hence, if $ f\ \in\ C^{\infty}_c(G) $ is supported on $U$, we have $$O_{\gamma}(f) = 0 \quad \quad \forall\ \gamma \in \Gamma_1 \cup
\Gamma_2.$$ Hence from equations (\[trf3\]) and (\[loc\]) above, we get : $$\int \limits_{G} f(g)\ \phi(g)\ d\mu(g) = 0,$$ for all smooth compactly supported functions $f$ supported in $U$.
But this means that $ \phi(g)$ is essentially $0$ on $U$. Since $U$ was a neighborhood of an arbitrary point $g$ outside $E$, and $E$ is a closed subset of measure zero, we conclude that $ \phi(g)$ is essentially $0$ on $G$. By equation (\[loc\]) above : $$\sum \limits_{\pi\ \in\ S} t_{\pi} \chi_{\pi}(f)\ =\ 0 \quad
\forall\ f\ \in C^{\infty}_c(G).$$ From the linear independence of characters (Theorem \[LinInd\]), we get that $ t_{\pi} = 0 $ for any $\pi \in S$. Hence, $$m(\pi, \Gamma_{1}) = m(\pi, \Gamma_{2}) \quad \forall\ \pi \in
\widehat{G}.$$ i.e., the lattices $\Gamma_1$ and $\Gamma_2$ are representation equivalent in $G$.
Proof of Theorem \[prin\] {#prinlapla}
==========================
In this section we give a proof of Theorem \[prin\], following the broad outline of the proof of Theorem \[thm1\]. Since the analogue of Corollary \[E\] does not seem available to us, we need to establish a more delicate proposition concerning the $K\times K$-saturation $KC_{\gamma}K$ of the conjugacy class of elements $\gamma\in
\Gamma$. Corresponding to use of Harish Chandra’s theorem on the local integrability of the character of an irreducible unitary representation of $G$, we instead use the analyticity of the spherical functions on $G$.
[@GV p.399] A complex valued function $\phi$ on $G$ is called a spherical function if
1. $\phi(e) = 1$.
2. $\phi(k_{1} x k_{2}) = \phi(x)\quad \forall\ k_{1},\ k_{2}
\in K\ \text{and}\ x \in G$.
3. $\phi$ is a common eigenfunction for all $D$ in the space $D(G/K)$ of $G$-invariant differential operators on $G / K$ with eigenvalue $\lambda(D)$: $$D\phi = \lambda(D)\phi \ \ \forall\ D \in D(G/K).$$
The map $D \rightarrow \lambda(D)$ defines a algebra homomorphism of $D(G/K)$ into ${{\mathbb C}}$. Denote by $C^{\infty}_{c}(G//K)$ the space of smooth and compactly supported bi-$K$-invariant functions on $G$.
Let $\pi$ be a spherical unitary representation of $G$. The space $\pi^{K}$ of $K$-fixed vectors is one dimensional (*cf*. Helgason [@He p.416]). Let $\phi_{\pi}$ be the associated elementary spherical function defined by $$\phi_{\pi}(x) = \left\langle\ \pi(x)\ e_{\pi}, e_{\pi}\
\right\rangle,$$ where $e_{\pi}$ is a $K$-fixed vector of the representation space of $\pi$ such that $\|e_{\pi}\| = 1$.
We have the following proposition:
\[warn\] Let $\pi$ be an irreducible unitary spherical representation of $G$. Then the following hold:
(i) The associated elementary spherical functions $\phi_{\pi}$ are analytic on $G$.
(ii) The relationship of the elementary spherical function $\phi_{\pi}$ to character $\chi_{\pi}$ is given by the following equation: $$\chi_{\pi}(f) = \int \limits_{G} f(g)\
\phi_{\pi}(g)\ dg \quad f\in C^{\infty}_{c}(G//K).$$
(iii) Let $\left\{\pi_{j} : 1 \leq j \leq k \right\} $ be a finite collection of mutually inequivalent irreducible spherical representations of $G$. The associated elementary spherical functions $\left\{\phi_{\pi_{j}} \ : 1 \leq j \leq k \right\} $ are linearly independent.
<!-- -->
(i) Since the algebra $D(G/K)$ contains the Laplace-Beltrami operator which is an elliptic, essentially self adjoint differential operator, it follows that the elementary spherical functions $\phi_{\pi}$ are analytic on $G$.
(ii) Let $V$ be the space underlying the representation $\pi$. Given $f \in$ $ C^{\infty}_{c}(G//K)$, the image $\pi(f)(V)$ of the convolution operator $\pi(f)$ lands in the space $V^K$ of $K$-invariants. Hence the trace is given by, $$\chi_{\pi}(f) = \text{trace}~(\pi(f)) =\left\langle \pi(f)
e_{\pi}, e_{\pi} \right\rangle = \int \limits_{G} f(g)\
\phi_{\pi}(g)\ dg.$$
(iii) The function $\phi_{\pi_{j}}$ is an eigenvector for the character $\lambda_{\pi_{j}}$. Since the representations $\pi_{j}$ are mutually inequivalent, the homomorphisms $\lambda_{\pi_{j}}$ are distinct and hence the corresponding eigenvectors are linearly independent.
Now we turn to the geometric aspects of the Selberg trace formula. Let $G//K$ denote the collection of orbits under the action of $K\times K$ acting on $G$ by, $$\label{biKaction} (k, l)g=k^{-1}gl \quad \quad k, l\in
K, ~g\in G.$$
\[sph\] The space $C^{\infty}_{c}(G//K)$ consisting of bi-$K$-invariant compactly supported smooth functions on $G$ separate points on $G//K$.
The orbits of $K\times K$ being compact are closed subsets of $G$. Given two orbits $KxK, ~KyK$ choose a compactly supported, smooth function which is positive on $KxK$ and vanishes on $KyK$. Then, $$F(g) =\int_{K\times K} f(kgl)dk dl,$$ is a bi-$K$-invariant compactly supported smooth function on $G$ which separates the two orbits.
\[gamma\] Let $\Gamma$ be a torsion-free uniform lattice in $G$. For a non-trivial element $\gamma \in \Gamma$, the conjugacy class $C_{\gamma}$ is disjoint from $K$.
The group $x^{-1}\gamma x \cap\ K$ is discrete and contained in the compact group $K$, hence finite. Consequently, $x^{-1}\gamma x$ is of finite order in $G$ and hence $\gamma$ is of finite order. Since $\Gamma$ is torsion-free, $\gamma$ is the identity element of $G$.
\[e\] If $\gamma \neq e$, then $e \notin
KC_{\gamma}K$.
Let $x \in G$ and $k,l \in K$ be such that $ k x^{-1}
\gamma x l = e$. Then $x^{-1} \gamma x \in K$, which is not possible by Lemma \[gamma\].
\[B\] There exists an open set $B$ in $G$ such that $C_{\gamma} \cap B$ is empty for all $\gamma \in \Gamma_1
\cup \Gamma_2$ and $B$ is stable under $K\times K$ action on $G$ given by equation \[biKaction\].
Let $U'$ be a relatively compact open neighborhood of $e$ in $G$. Let $U = KU'K$. Then $U$ is relatively compact and hence it intersects atmost finitely many conjugacy classes $C_{\gamma}$. Since the map $G\to G//K$ is proper and the conjugacy class $C_{\gamma}$ is closed, the set $KC_{\gamma}K$ is closed in $G$. Since $U$ is $K$-stable, $KC_{\gamma}K \cap\ U$ is non-empty if and only if $C_{\gamma}\cap U$ is non-empty. Hence, the set $E = \bigcup
\limits_{\gamma \neq e}[KC_{\gamma}K]\ \cap U$, being a finite union of closed sets, is a $K\times K$-stable closed subset of $U$. By Lemma \[e\], the identity element $e$ does not belong to $E$. Choose an open set $V \subseteq U$ containing $e$ such that $E \cap V =
\emptyset$. Let $B = KVK \cap K^{c}$, where $K^c$ is the complement of $K$ in $G$. It can be seen that B satisfies the desired property.
Now we give the proof of Theorem \[prin\].
By hypothesis of Theorem \[prin\], there exists a finite subset $S$ of $\widehat{G}_{s}$ such that $$m(\pi, \Gamma_1) = m(\pi, \Gamma_2)\ \ \forall\ \pi \notin S.$$ Let $f \in C^{\infty}_{c}(G//K)$. Since $f$ is bi-$K$-invariant, $\chi_{\pi}(f) = 0$ if $\pi \notin \widehat{G}_{s} $. Using the Selberg trace formula for $f$ , we get : $$\label{trf5}
\begin{split} \sum \limits_{\pi\ \in\ S} \ t_{\pi} \chi_{\pi}(f) =
\sum \limits_{\substack{[\gamma]\ \in \ [\Gamma_{1}]_{G}\ \cup\
[\Gamma_{2}]_{G}}} (a(\gamma,\Gamma_1) - a(\gamma,\Gamma_2))\
O_{\gamma}(f)
\end{split}$$ Let $\phi = \sum \limits_{\alpha\ \in\ S} t_{\pi}
\phi_{\pi}$. By using proposition \[warn\], we get : $$\int \limits_{G} f(g)\ \phi(g)\ d\mu(g) = \sum
\limits_{\substack{[\gamma]\ \in \ [\Gamma_{1}]_{G}\ \cup\
[\Gamma_{2}]_{G} }} (a(\gamma,\Gamma_1) - a(\gamma,\Gamma_2))\
O_{\gamma}(f).$$ Let $B$ be as in the proof of Proposition \[B\]. The term on right hand side in above equation vanishes for every function $f$ in $C^{\infty}_{c}(G//K)$ which is supported on $B$. Hence for such functions $f$, $$\int \limits_{G} f(g)\ \phi(g)\ d\mu(g) = 0.$$ By Lemma \[sph\], the functions $f$ separate points on $B$. Hence $\phi$ must vanish on the open subset $B$ of $G$. Since $\phi$ is analytic, it vanishes on all of G. By the linear independence of functions $\phi_{\pi}$ (Proposition \[warn\]), we conclude that $$m(\pi, \Gamma_1) = m(\pi, \Gamma_2)\ \ \forall\ \pi \in
\widehat{G_s}.$$
**[Proof of Theorem \[laplacian\]]{}** {#prooflaplacian}
=======================================
We now proceed to derive Theorem \[laplacian\] from Theorem \[prin\]. We follow the notation given in the introduction. Let $\pi$ be an irreducible, unitary, spherical representation of $G$. Let $e_{\pi}$ be a $K$-fixed vector of unit length in $\pi$. The associated spherical function $\phi_{\pi}$ is an eigenfunction of $D(G/K)$ with eigencharacter $\lambda_{\pi}$: $$D(\phi_{\pi})=\lambda_{\pi}(D)\phi_{\pi} \quad D\in D(G/K).$$ The main observation is the following proposition.
\[pro1\] Let $\Gamma$ be a torsion-free uniform lattice in $G$. Let $\pi$ be an irreducible, unitary spherical representation of $G$. Then $$m(\pi, \Gamma ) = \text{dim}\ (V(\lambda_{\pi},\Gamma)).$$ In particular, $V (\lambda_{\pi},\Gamma)$ is finite dimensional.
Conversely, if $\lambda$ is a character of $D(G/K)$ and the dimension of $V (\lambda_{\pi},\Gamma)$ is positive, then $\lambda =
\lambda_{\pi}$ for some spherical representation $\pi$ of $G$.
When $G = PSL(2,{{\mathbb R}})$ this is the duality theorem proved by Gelfand, Graev and Pyatetskii-Shapiro ([@GGP p.50]), relating the spectrum of the Laplace-Beltrami on the upper half plane and the multiplicities of spherical representations of $PSL(2,{{\mathbb R}})$ occurring in $L^2(\Gamma\backslash PSL(2, {{\mathbb R}}))$. We follow their proof. The above fact is probably well known to the experts but we have included a proof for sake of completeness. The proof also indicates that Theorems \[prin\] and Theorem \[laplacian\] are equivalent.
Let $ \mathfrak{g}$ be the complexification of Lie algebra of $G$ consisting of left invariant vector fields on $G$. Let $\mathfrak{U}(\mathfrak{g})^{K}$ be the $K$-invariant subspace of the universal enveloping algebra $\mathfrak{U}(\mathfrak{g})$ under the right action of $K$ on $\mathfrak{U}(\mathfrak{g})$. We consider the right action of $G$ on itself. This gives raise to a surjective map from $\mathfrak{U}(\mathfrak{g})^{K}$ to $D(G/K)$ ([@GV page 52, Proposition 1.7.5]). Hence it can be seen that $D(e_{\pi})$ is a $K$-fixed vector for each $D \in D(G/K)$. Since the dimension of the space of $K$-fixed vectors of $\pi$ is one, it follows that $e_{\pi}$ is an eigenvector of $D(G/K)$ with respect to the eigencharacter $\lambda_{\pi}$ i.e. it lies in the eigenspace in $V(\lambda_{\pi},
\Gamma)$. Therefore, we conclude that $ m(\pi, \Gamma ) \leq
\text{dim}\ (V(\lambda_{\pi},\Gamma))$.
Conversely let $f \in C^{\infty}(X)$ be an eigenvector of some character $\lambda$ of $D(G/K)$. Since $$L^{2}(\Gamma \backslash G) = \bigoplus_{\pi\ \in\ \widehat{G}} m
\left(\pi, \Gamma \right)\ \pi$$ we write $$\label{fdecomp} f = \sum \limits_{\pi\ \in\
\widehat{G}} a_{\pi}\ v_{\pi}$$ such that $v_{\pi} \in \pi$ is a vector of unit length. Let $W$ be the space of $K$-invariants of $L^{2}(\Gamma
\backslash G)$. Let $P_W$ be the orthogonal projection of $L^{2}(\Gamma \backslash G)$ onto $W$. Since $f$ is right invariant under $K$, $P_{W}(f) = f$. Hence we get : $$f = \sum \limits_{\pi\ \in\ \widehat{G}} a_{\pi}\ P_{W}(v_{\pi})$$
The algebra $D(G/K)$ is generated by essentially self-adjoint differential operators. Hence, if the character $\lambda_{\pi}$ is distinct from $\lambda$, there exists an essentially self-adjoint $D\in D(G/K)$ such that $\lambda_{\pi}(D)\neq \lambda(D)$. Hence the eigenvectors $v_{\pi}$ and $f$ are orthogonal. If $\pi$ is not a spherical representation, $P_{W}(v_{\pi}) = 0$. Hence the indexing set in equation (\[fdecomp\]) is restricted to those irreducible unitary spherical representations with character $\lambda_{\pi}$ equal to $\lambda$.
Since the associated spherical functions to inequivalent representations are linearly independent, the characters are distinct. Hence we conclude that there is an unique irreducible unitary spherical representation $\pi$ of $G$ such that $\lambda =
\lambda_{\pi}$. Hence, $$m(\pi, \Gamma ) = \text{dim}\ (V (\lambda_{\pi},\Gamma)).$$
Now we give the proof of Theorem \[laplacian\]. Let $T$ be a finite subset of characters of $D(G/K)$ such that $$\text{dim}\ ( V
(\lambda,\Gamma_1) ) = \text{dim}\ (V(\lambda,\Gamma_2))$$ for all characters $\lambda \notin T$. By above Proposition \[pro1\], we get that : $$m(\pi,\Gamma_{1}) = m(\pi, \Gamma_{2})$$ for all but finitely many irreducible, unitary spherical representations of $G$. Hence using Theorem \[prin\] and Proposition \[pro1\], we get a proof of Theorem \[laplacian\].
[amsalpha]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
RICHARD G. CLEGG\
\
*Dept. Of Mathematics, University of York, YO10 5DD\
[email protected]*
bibliography:
- 'rgc\_ijs.bib'
date: |
*\
This paper describes, in detail, techniques for measuring the Hurst parameter. Measurements are given on artificial data both in a raw form and corrupted in various ways to check the robustness of the tools in question. Measurements are also given on real data, both new data sets and well-studied data sets. All data and tools used are freely available for download along with simple “recipes” which any researcher can follow to replicate these measurements.*
title: A PRACTICAL GUIDE TO MEASURING THE HURST PARAMETER
---
INTRODUCTION AND BACKGROUND
===========================
Long-Range Dependence (LRD) is a statistical phenomenon which has received much attention in the field of telecommunications in the last ten years. A time-series is described as possessing LRD if it has correlations which persist over all time scales. A good guide to LRD is given by [@beran1994] and a summary in the context of telecommunications is given by [@clegg2004 chapter one] (from which some of the material in this paper is taken). In the early nineties, LRD was measured in time-series derived from Internet traffic [@leland1993]. The importance of this is that LRD can impact heavily on queuing. LRD is characterised by the parameter $H$, the Hurst parameter, (named for a hydrologist who pioneered the field in the fifties [@hurst1951]) where $H \in (1/2,1)$ indicates the presence of LRD. There are a number of different statistics which can be used to estimate the Hurst parameter and several papers have been written comparing these estimators both in theory and practice [@taqqu1995; @taqqu1997; @bardet2003]. The aim of this paper is not to make a rigorous comparison of the estimators but, instead, to present a simple and readable guide to what a researcher can expect from attempting to assess whether LRD is absent or present in a data set. All the tools used are available online using free software. Software can be downloaded from:\
[www.richardclegg.org/\
lrdsources/software/]{}
Long-Range Dependence in Telecommunications
-------------------------------------------
In their classic paper, Leland et al [@leland1993] measure traffic past a point on an Ethernet Local Area Network. They conclude that “In the case of Ethernet LAN traffic, self-similarity is manifested in the absence of a natural length of a ‘burst’; at every time scale ranging from a few milliseconds to minutes and hours, bursts consist of bursty sub-periods separated by less burst sub-periods. We also show that the degree of self-similarity (defined via the Hurst parameter) typically depends on the utilisation level of the Ethernet and can be used to measure ‘burstiness’ of LAN traffic.” Since then, a number of authors have replicated these experiments on a variety of measurments of Internet traffic and the majority found evidence of LRD or related multi-fractal behaviour. Summaries are given in [@sahinoglu1999; @willinger2003]. The reason for the interest in the area is that LRD can, in some circumstances, negatively impact network performance. The exact details of the scale and nature of the effect are uncertain and depend on the particular LRD process being considered.
A Brief Introduction to Long-Range Dependence
---------------------------------------------
Let $\{X_t: t \in {\mathbb{N}}\}$ be a time-series which is weakly stationary (that is it has a finite mean and the covariance depends only on the separation or “lag” between two points in the series). Let $\rho(k)$ be the auto-correlation function (ACF) of $X_t$.
The ACF, $\rho(k)$ for a weakly-stationary time series, $\{X_t: t \in {\mathbb{N}}\}$ is given by $$\rho(k)= \frac{{{\text{E}}\left[(X_t - \mu) (X_{t+k} - \mu)\right]}} {\sigma^2},$$ where ${{\text{E}}\left[X_t\right]}$ is the expectation of $X_t$, $\mu$ is the mean and $\sigma^2$ is the variance.
There are a number of different definitions of LRD in use in the literature. A commonly used definition is given below.
The time-series $X_t$ is said to be [*long-range dependent*]{} if $\sum_{k=-\infty}^{\infty} \rho(k)$ diverges. \[defn:lrd\_weak\]
Often the specific functional form $$\rho(k) \sim C_\rho k^{-\alpha},
\label{eqn:lrd}$$ is assumed where $C_\rho > 0$ and $\alpha \in (0,1)$. Note that the symbol $\sim$ is used here and throughout this paper to mean [*asymptotically equal to*]{} or $f(x) \sim g(x) \Rightarrow f(x)/g(x) = 1$ as $x \rightarrow \infty$ or, where indicated, as $x \rightarrow 0$. The parameter $\alpha$ is related to the Hurst parameter via the equation $\alpha = 2 - 2H$.
If holds then a similar definition can be shown to hold in the frequency domain.
The [*spectral density*]{} $f(\lambda)$ of a function with ACF $\rho(k)$ and variance $\sigma^2$ can be defined as $$f(\lambda)= \frac{\sigma^2}{2 \pi} \sum_{k= -\infty}^{\infty}
\rho(k) e^{ik\lambda},
$$ where $\lambda$ is the frequency, $\sigma^2$ is the variance and $i = \sqrt{-1}$. \[defn:spectral\]
Note that this definition of spectral density comes from the Wiener-Kninchine theorem [@wiener1930].
The weakly-stationary time-series $X_t$ is said to be [*long-range dependent*]{} if its spectral density obeys $$f(\lambda) \sim C_f |\lambda|^{-\beta},
$$ as $\lambda \rightarrow 0$, for some $C_f > 0$ and some real $\beta \in (0,1)$. \[defn:lrd\_freq\]
The parameter $\beta$ is related to the Hurst parameter by $H = (1 + \beta)/2$.
LRD relates to a number of other areas of statistics, notably the presence of statistical self-similarity. Self-similarity can be characterised by a self-similarity parameter $H$. If a self-similar process has stationary increments and $H \in (1/2,1)$ then its increments themselves, taken as a process, form an LRD process with Hurst parameter $H$. Indeed analysis of telecommunications traffic is often described in terms of self-similarity and not long-range dependence. (Sometimes the phrase “asymptotic second-order self-similarity" is used. This refers to self-similarity in the data when it is aggregated and is synonymous with LRD.)
In summary, LRD can be thought of in two ways. In the time domain it manifests as a high degree of correlation between distantly separated data points. In the frequency domain it manifests as a significant level of power at frequencies near zero. LRD is, in many ways, a difficult statistical property to work with. In the time-domain it is measured only at high lags (strictly at infinite lags) of the ACF — those very lags where only a few samples are available and where the measurement errors are largest. In the frequency domain it is measured at frequencies near zero, again where it is hardest to make measurements. Time series with LRD converge slowly to their mean. While the Hurst parameter is perfectly well-defined mathematically, it will be shown that it is, in fact, a very difficult property to measure in real life.
MEASURING THE HURST PARAMETER
=============================
While the Hurst parameter is perfectly well-defined mathematically, measuring it is problematic. The data must be measured at high lags/low frequencies where fewer readings are available. Early estimators were biased and converged only slowly as the amount of available data increased. All estimators are vulnerable to trends in the data, periodicity in the data and other sources of corruption. Many estimators assume specific functional forms for the underlying model and perform poorly if this is misspecified. The techniques in this paper are chosen for a variety of reasons. The R/S parameter, aggregated variance and periodogram are well-known techniques which have been used for some time in measurements of the Hurst parameter. The local Whittle and wavelet techniques are newer techniques which generally fare well in comparative studies. All the techniques chosen have freely available code which can be used with free software to estimate the Hurst parameter.
The problems with real-life data are worse than those faced when measuring artificial data. Real life data is likely to have periodicity (due to, for example, daily usage patterns), trends and perhaps quantisation effects if readings are taken to a given precision. The naive researcher taking a data set and running it through an off-the-shelf method for estimating the Hurst parameter is likely to end up with a misleading answer or possibly several different misleading answers.
Data sets to be studied
-----------------------
A large number of methods are used for generating data exhibiting LRD. A review of some of the better known methods are given in [@bardet2003b]. In this paper trial data sets with LRD and a known Hurst parameter are generated using fractional auto-regressive integrated moving average (FARIMA) modelling and fractional Gaussian noise (FGN). The software used to generate the data is included with at the web address previously mentioned.
A FARIMA model is a well-known time series modelling technique. It is a modification of the standard time series ARIMA ($p,d,q$) model. An ARIMA model is defined by $$(1 - \sum_{j=1}^p \phi_j \mathbf{B}^j)(1-\mathbf{B})^dX_i=
(1 - \sum_{j=1}^q \theta_j \mathbf{B}^j) \varepsilon_i,$$ where $p$ is the order of the AR part of the model, the $\phi_i$ are the AR parameters, $p$ is the order of the MA part of the model, the $\theta_j$ are the MA parameters, $d \in {\mathbb{Z}}$ is the order of differencing, the $\varepsilon_i$ are i.i.d. noise (usually normally distributed with zero mean) and $\mathbf{B}$ is the backshift operator defined by $\mathbf{B}(X_t) = X_{t-1}$. If, instead of being an integer, the model is changed so that $d \in (0,1/2)$ then the model is a FARIMA model. If the $\phi_i$ and $\theta_i$ are chosen so that the model is stationary and $d \in(0,1/2)$ then the model will be LRD with $H= d+ 1/2$. FARIMA processes were proposed by [@granger1980] and a description in the context of LRD can be found in [@beran1994 pages 59–66].
Fractional Brownian Motion is a process $B_H(t)$ for $t \geq 0$ obeying,
- $B_H(0) = 0$ almost surely,
- $B_H(t)$ is a continuous function of $t$,
- The distribution of $B_H(t)$ obeys $$\begin{aligned}
{{\mathbb{P}}\left[B_H(t+k) - B_H(t) \leq x\right]} = \\ (2 \pi) ^{-\frac{1}{2} }k ^{-H}
\int \limits_{-\infty}^{x} \exp\left(\frac{-u^2}{2k^{2H}}\right)du,\end{aligned}$$
where $H \in (1/2,1)$ is the Hurst parameter. The process $B_H(t)$ is known as fractional Brownian motion (FBM) and its increments are known as fractional Gaussian noise (FGN). FBM is a self-similar process with self-similarity parameter $H$ and, when $H \in (1/2,1)$, FGN exhibits long-range dependence with Hurst parameter $H$. When $H = 1/2$ in the above, then the process is the well known Weiner process (Brownian motion) and the increments are independent (Gaussian noise). A number of authors have described computationally efficient methods for generating FGN and FBM. The one used in this paper is due to [@paxson1997].
Data generated from these models will be tested using the various measurement techniques and then the same data set will be corrupted in several ways to see how this disrupts measurements:
- Addition of zero mean AR(1) model with a high degree of short-range correlation ($X_t= 0.9 X_{t-1} + \varepsilon_t$). This simulates a process with very high local correlations which might be mistaken for a long-range dependence.
- Addition of periodic function (sine wave) — ten complete cycles of a sin wave are added to the signal. This simulates a seasonal effect in the data, for example, a daily usage pattern.
- Addition of linear trend. This simulates growth in the data, for example the data might be a sample of network traffic at a time of day when the network is growing busier as time continues.
The noise signals are normalised so the standard deviation of the corrupting signal is identical to the standard deviation of the original LRD signal to which it is being added. Note that strictly speaking, while the addition of an AR(1) model does not change the LRD in the model and theoretically will leave the Hurst parameter unchanged, techincally the addition of a trend or of periodic noise makes the time-series non-stationary and hence the time-series produces are, strictly speaking, not really LRD.
In addition, some real-life traffic traces are studied to provide insight into how well different measurements agree across data sets with and without various transforms being applied to clean the data. The data sets used are listed below.
- The famous (and much-studied) Bellcore data [@leland1991] which was collected in 1989 and has been used for a large number of studies since. Note that, unfortunately, the exact traces used in [@leland1993] are not available for download. Data from the same sites collected at a similar time is available online at:\
[ita.ee.lbl.gov/\
html/contrib/BC.html]{}
- A data set collected at the University of York in 2001 which consists of a tcpdump trace of 67 minutes of incoming and outgoing data from the external link to the university from the rest of the Internet.
Three techniques (listed below) were tried to filter real-life traces in addition to making measurements purely on the raw data. These methods have been selected from the literature as techniques commonly used by researchers in the field. Often a high pass filter would be used to remove periodicity and trends. However, since LRD measurements are most important at low-frequency, that is an obviously inappropriate technique.
- Transform to log of original data (only appropriate if data is positive).
- Removal of mean and linear trend (that is, subtract the best fit line $Y= at + b$ for constant $a$ and $b$).
- Removal of high order best-fit polynomial of degree ten (the degree ten was chosen after higher degrees showed evidence of overfitting).
Note that the “transform to log” option is not available if the data contains zeros. In practice some rule of thumb could be considered for replacing zeros with a minimal value but this substitution was not done here and this pre-processing technique has not been used where the data contains zeros.
Measurement techniques
----------------------
The measurement techniques used in this paper can only be described briefly but references to fuller descriptions with mathematical details are given. The techniques used here are chosen for various reasons. The R/S statistic, aggregated variance and periodogram are well-known techniques with a considerable history of use in estimating long-range dependence. The wavelet analysis technique and local Whittle estimator are newer techniques which perform well in comparative studies and have strong theoretical backing.
The R/S statistic is a well-known technique for estimating the Hurst parameter. It is discussed in [@mandelbrot1969] and also [@beran1994 pages 83–87]. Let $R(n)$ be the range of the data aggregated (by simple summation) over blocks of length $n$ and $S^2(n)$ be the sample variance of the data aggregated at the same scale. For FGN or FARIMA series the ratio $R/S(n)$ follows $${{\text{E}}\left[R/S(n)\right]} \sim C_Hn^H,$$ where $C_H$ is a positive, finite constant independent of $n$. Hence a log-log plot of $R/S(n)$ versus $n$ should have a constant slope as $n$ becomes large. A problem with this technique which is common to many Hurst parameter estimators is knowing which values of $n$ to consider. For small $n$ short term correlations dominate and the readings are not valid. For large $n$ then there are few samples and the value of $R/S(n)$ will not be accurate. Similar problems occur for most of the estimators described here.
The aggregated variance technique is described in [@beran1994 page 92]. It considers ${\text{var}\left(X^{(m)}\right)}$ where $X^{(m)}_t$ is a time series derrived from $X_t$ by aggregating it over blocks of size $m$. The sample variance ${\text{var}\left(X^{(m)}\right)}$ should be asymptotically proportional to $m^{2H-2}$ for large $N/m$ and $m$.
The periodogram, described by [@geweke1983] is defined by $$I(\lambda) = \frac{1}{2 \pi N} \left|
\sum_{j=1}^N X_je^{ij\lambda}\right|^2,$$ where $\lambda$ is the frequency. For a series with finite variance, $I(\lambda)$ is an estimate of the spectral density of the series. From Definition \[defn:lrd\_freq\] then, a log-log plot of $I(\lambda)$ should have a slope of $1 - 2H$ close to the origin.
Whittle’s estimator is a Maxmimum Likelihood Estimator which assumes a functional form for $I(\lambda)$ and seeks to minimise parameters based upon this assumption. A slight issue with the Whittle estimator is that the user must specify the functional form expected, typically either FGN or FARIMA (with the order specified). If the user misspecifies the underlying model then errors may occur. Local Whittle is a semi-parametric version of this which only assumes a functional form for the spectral density at frequencies near zero [@robinson1995].
Wavelet analysis has been used with success both to measure the Hurst parameter and also to simulate data [@riedi2003]. Wavelets can be thought of as akin to Fourier series but using waveforms other than sine waves. The estimator used here fits a straight line to a frequency spectrum derived using wavelets. A 95% confidence interval is given, however, this should be interpreted only as a confidence interval on the fitted line and, as will be seen, not as a confidence inteval on the fitted Hurst parameter. This is an important distinction — it is tempting to consider the confidence intervals given by some estimators as literally confidence intervals on the measurement of $H$. Often this is not the case (as in the case of Wavelet analysis) or is only the case if certain conditions are met.
RESULTS
=======
Results here are in two sections. Firstly, results are given for simulated data. In these cases the expected “correct” answer is known and therefore it can be seen how well the estimators have performed. The data is then corrupted by the addition of noise with the same standard deviation as the original data sets. Three types of noise are considered as described previously.
In the second section results are given for real data. The York data is analysed as a time series of bytes per unit time for two different time units. The Bellcore data is analysed both in terms of interarrival times and in terms of bytes per unit time. Note that, strictly speaking, the interarrival times do not consititute a proper “time-series” since the time units between readings are not constant.
Results on Simulated Data
-------------------------
For each of the simulation methods chosen, traces have been generated. Each trace is 100,000 points of data. Hurst parameters of 0.7 and 0.9 have been chosen to represent a low and a high level of long-range dependence in data. The errors on the wavelet estimator are a 95% confidence interval on the fitted regression line (not, as might be thought, the Hurst parameter measured).
[|l| l l l l l|]{} Added & R/S Plot & Aggreg. & Period. & Wavelet & Local\
Noise & & Variance & ogram & Estimate & Whittle\
\
None &0.66 & 0.668 & 0.686 & 0.707 $\pm$ 0.013 & 0.72\
AR(1) &0.767 & 0.657 & 0.794 & 0.888 $\pm$ 0.034 & 0.904\
Sin &0.667 & 0.969 & 0.692 & 0.707 $\pm$ 0.013 & 0.787\
Trend &0.66 & 0.968 & 0.777 & 0.707 $\pm$ 0.013 & 0.766\
\
None &0.641 & 0.692 & 0.7 & 0.694 $\pm$ 0.007 & 0.721\
AR(1) &0.775 & 0.671 & 0.795 & 0.882 $\pm$ 0.036 & 0.902\
Sin &0.66 & 0.97 & 0.705 & 0.694 $\pm$ 0.007 & 0.788\
Trend &0.641 & 0.968 & 0.769 & 0.694 $\pm$ 0.007 & 0.765\
\
None &0.636 & 0.69 & 0.704 & 0.708 $\pm$ 0.009 & 0.723\
AR(1) &0.734 & 0.654 & 0.79 & 0.876 $\pm$ 0.038 & 0.905\
Sin &0.64 & 0.969 & 0.709 & 0.708 $\pm$ 0.009 & 0.787\
Trend &0.636 & 0.971 & 0.783 & 0.708 $\pm$ 0.009 & 0.77\
\
None &0.782 & 0.864 & 0.905 & 0.901 $\pm$ 0.009 & 0.934\
AR(1) &0.805 & 0.784 & 0.88 & 0.969 $\pm$ 0.042 & 1.066\
Sin &0.772 & 0.961 & 0.907 & 0.901 $\pm$ 0.009 & 0.945\
Trend &0.782 & 0.958 & 0.928 & 0.901 $\pm$ 0.009 & 0.939\
\
None &0.862 & 0.837 & 0.891 & 0.902 $\pm$ 0.003 & 0.933\
AR(1) &0.856 & 0.76 & 0.877 & 0.969 $\pm$ 0.038 & 1.062\
Sin &0.858 & 0.955 & 0.894 & 0.902 $\pm$ 0.003 & 0.943\
Trend &0.862 & 0.954 & 0.921 & 0.902 $\pm$ 0.003 & 0.938\
\
None &0.793 & 0.884 & 0.907 & 0.904 $\pm$ 0.007 & 0.93\
AR(1) &0.818 & 0.802 & 0.871 & 0.972 $\pm$ 0.041 & 1.066\
Sin &0.8 & 0.967 & 0.91 & 0.904 $\pm$ 0.007 & 0.943\
Trend &0.794 & 0.959 & 0.924 & 0.904 $\pm$ 0.007 & 0.936\
Table \[tab:fgn\] shows results for various FGN models. Three runs each are done with a Hurst parameter of 0.7 and then 0.9. Firstly it should be noted that, in all cases, for H=0.7 all estimators are relatively close when no noise is applied. The R/S method performs worst, as it consistently underestimates the Hurst parameter. The addition of AR(1) noise confuses all the methods with the Local Whittle performing particularly poorly. The correct answer is well outside the confidence intervals of the Wavelet estimate after this addition (although, as previously stated, the confidence interval should not be taken literally). Addition of a sine wave or a trend causes trouble for the aggregated variance method but the frequency domain methods (wavelets and local Whittle) do not seem greatly affected.
When considering runs with Hurst parameter H=0.9, the R/S method gets a considerable underestimate even with no corrupting noise. Note also that the R/S and aggregated variance method actually produce quite different estimates for the three runs. Most methods seem to perform badly with the AR(1) noise corruption. Again the frequency domain methods seem to be able to cope with the sine wave and with the addition of a trend. The aggregated variance method seems to perform particularly badly in the presence of a corrupting sin wave and a corrupting trend (perhaps not surprising as such a series is no longer weakly stationary).
[|l| l l l l l|]{} Added & R/S Plot & Aggreg. & Period. & Wavelet & Local\
Noise & & Variance & ogram & Estimate & Whittle\
\
None & 0.663 & 0.692 & 0.699 & 0.696 $\pm$ 0.004 & 0.681\
AR(1) & 0.823 & 0.673 & 0.792 & 0.896 $\pm$ 0.033 & 0.876\
Sin & 0.665 & 0.972 & 0.704 & 0.696 $\pm$ 0.004 & 0.765\
Trend & 0.662 & 0.973 & 0.786 & 0.696 $\pm$ 0.004 & 0.746\
\
None & 0.706 & 0.701 & 0.71 & 0.702 $\pm$ 0.007 & 0.679\
AR(1) & 0.837 & 0.673 & 0.791 & 0.891 $\pm$ 0.034 & 0.873\
Sin & 0.714 & 0.972 & 0.714 & 0.702 $\pm$ 0.007 & 0.764\
Trend & 0.706 & 0.972 & 0.782 & 0.702 $\pm$ 0.007 & 0.742\
\
None & 0.718 & 0.684 & 0.696 & 0.687 $\pm$ 0.005 & 0.679\
AR(1) & 0.827 & 0.667 & 0.776 & 0.868 $\pm$ 0.044 & 0.872\
Sin & 0.723 & 0.973 & 0.701 & 0.687 $\pm$ 0.005 & 0.765\
Trend & 0.718 & 0.972 & 0.778 & 0.687 $\pm$ 0.005 & 0.743\
\
None & 0.684 & 0.693 & 0.706 & 0.697 $\pm$ 0.006 & 0.68\
AR(1) & 0.818 & 0.656 & 0.774 & 0.88 $\pm$ 0.041 & 0.878\
Sin & 0.689 & 0.973 & 0.71 & 0.697 $\pm$ 0.006 & 0.766\
Trend & 0.684 & 0.972 & 0.786 & 0.697 $\pm$ 0.006 & 0.743\
\
None & 0.757 & 0.882 & 0.91 & 0.886 $\pm$ 0.004 & 0.861\
AR(1) & 0.804 & 0.789 & 0.873 & 0.969 $\pm$ 0.036 & 1.011\
Sin & 0.764 & 0.967 & 0.913 & 0.886 $\pm$ 0.004 & 0.883\
Trend & 0.757 & 0.974 & 0.933 & 0.886 $\pm$ 0.004 & 0.875\
\
None & 0.856 & 0.854 & 0.881 & 0.887 $\pm$ 0.006 & 0.858\
AR(1) & 0.888 & 0.773 & 0.874 & 0.959 $\pm$ 0.04 & 1.001\
Sin & 0.86 & 0.963 & 0.885 & 0.887 $\pm$ 0.006 & 0.879\
Trend & 0.856 & 0.968 & 0.92 & 0.887 $\pm$ 0.006 & 0.872\
\
None & 0.807 & 0.74 & 0.817 & 0.966 $\pm$ 0.048 & 1.05\
AR(1) & 0.814 & 0.691 & 0.822 & 1.007 $\pm$ 0.059 & 1.136\
Sin & 0.8 & 0.94 & 0.821 & 0.966 $\pm$ 0.048 & 1.052\
Trend & 0.807 & 0.939 & 0.856 & 0.966 $\pm$ 0.048 & 1.051\
Table \[tab:farima\] shows a variety of results for FARIMA models. The first three runs are for a FARIMA $(0,d,0)$ model (that is one with no AR or MA components) and with a Hurst parameter $H = 0.7$. In this case, all methods peform adequately with no noise (although the R/S plot perhaps underestimates the answer). Addition of AR(1) noise causes problems for the R/S plot, wavelet and local Whittle methods and to a lesser extent the periodogram. The addition of a sin wave and a trend causes problems for the aggregated variance.
For a FARIMA $(1,d,1)$ model with $H = 0.7$ and with the AR parameter $\phi_1 = 0.5$ and the MA parameter $\theta_1 = 0.5$ (implying a moderate degree of short range correlation) all estimators provide a reasonable result for the uncorrupted series. As before, the wavelet and local Whittle method seem relatively robust to the addition of a trend. The AR(1) noise again causes problems for most of the methods.
For a FARIMA $(0,d,0)$ model with $H = 0.9$ the R/S method under predicts the Hurst parameter but all others perform well in the absence of noise. The AR(1) noise causes problems for the local Whittle and wavelet methods and the sine wave and trend cause problems for the aggregated variance.
For a FARIMA $(1,d,1)$ model with $H = 0.9$ and with the AR parameter $\phi_1 = 0.5$ and the MA parameter $\theta_1 = 0.5$ (implying, as before, a moderate degree of short range correlation) all estimators do relatively well initially. The corruption produces the same problems with the same estimators — that is to say, wavelets and local Whittle do not cope with the AR(1) noise and Aggregated variance reacts badly to the sine wave and local trend.
For a FARIMA $(2,d,1)$ model with $H = 0.9$ and with the AR parameters $\phi_1 = 0.5$, $\phi_2 = 0.2$ and the MA parameter $\theta_1 = 0.1$ indicating quite strong short-range correlations, none of the estimators perform particularly well. Presented with these results, a researcher would certainly not know the Hurst parameter of the underlying model from looking at the results given by the estimators. In the case of the AR(1) corrupted data the measurement from the Wavelet estimator is outside of the usual range for the Hurst parameter. In fact it is not unusual for Hurst parameter estimators to produce estimates outside the range $(1/2,1)$. All five estimators are producing different results in most cases (there is some aggreement between the R/S plot and periodogram but it would be hard to put this down to anything more than coincidence and, in any case, they are agreeing on an incorrect value for the Hurst parameter). It is interesting that, even in this relatively simple case where the theoretical correct result is known, five well-known estimators of the Hurst parameter all fail to get the correct answer.
Autocorrelations for the Artificial Data
----------------------------------------
It’s instructive to look at the ACF of these data sets to understand why the various methods fail or succeed with the data sets. Figure \[fig:fgn-acf\] shows the ACF up to lag 1000 for a data set of 100,000 points of FGN data with $H=0.7$. For this data, it is possible to fit “by eye” a straight line to the log-log plot of the ACF and obtain an estimate of the Hurst parameter. From Table \[tab:fgn\] it can be seen that all the estimators performed well on this data set. Note also that in the log-log plot it can be seen that at the higher lags the error on the ACF estimate is large.
When the time series is corrupted by the addition of AR(1) noise as described earlier in the paper then the ACF changes markedly. The ACF is then shown in Figure \[fig:fgn-acf-noise\]. The degree to which the ACF has changed is only really clear in the log-log plot. It can be seen that, for low lags, the ACF remains much higher than in the noise-free data of the previous series. It would be difficult indeed to make a convincing case for fitting a straight line to this data. As for the higher lags, the ACF estimate certainly does not seem to produce anything like a straight line in the log-log plot for lags over fifty. Two things can be clearly seen from this picture, firstly that it is impossible to get a good estimate of LRD simply by fitting a straight line to the ACF and secondly that the addition of highly correlated short range dependent data can vastly change the nature of the estimation problem. From considering this ACF it may be no surprise that the estimators mainly performed so badly at removing AR(1) noise.
Finally, the ACF is shown for the a data set which is FARIMA $(2,d,1)$ with $H= 0.7$, $\phi_1 = 0.5, \phi_2 = 0.2, \theta_1 = 0.1$. This was the data which proved hardest to estimate in Table \[tab:farima\]. Some of the difficulties of this estimation can be seen by looking at the ACF in Figure \[fig:far-acf\]. Even before the addition of noise, it can be seen that this data looks as hard to find a single best fit line as it was in Figure \[fig:fgn-acf-noise\]. It is, again, unsurprising, that the estimators performed badly with this data set even without the addition of noise.
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Results on Real Data
--------------------
In analysing the real data it is hard to know where to begin. Since the genuine answer (if, indeed, it can be really said that there is a genuine answer) is not known it cannot be said that one result is more “right” than another. The suggested methods for preprocessing data (taking logs, removing a linear trend and removing a best fit polynomial — in this case of order ten) have all been found in the literature on measuring the Hurst parameter.
[|l| l l l l l|]{} Filter & R/S Plot & Aggreg. & Period. & Wavelet & Local\
Type & & Variance & ogram & Estimate & Whittle\
\
None & 0.749 & 0.88 & 1.186 & 0.912 $\pm$ 0.052 & 0.981\
Log & 0.758 & 0.894 & 1.105 & 0.921 $\pm$ 0.039 & 0.932\
Trend & 0.749 & 0.873 & 1.212 & 0.912 $\pm$ 0.052 & 0.981\
Poly & 0.756 & 0.723 & 0.732 & 0.895 $\pm$ 0.04 & 0.972\
\
None & 0.826 & 0.924 & 0.928 & 0.909 $\pm$ 0.012 & 0.881\
Trend & 0.826 & 0.923 & 0.932 & 0.909 $\pm$ 0.012 & 0.881\
Poly & 0.827 & 0.892 & 0.863 & 0.909 $\pm$ 0.012 & 0.878\
Table \[tab:yorkdata\] shows analysis of data collected at the University of York. The same data set is analysed firstly as a series of bytes/second and then as bytes/tenth of a second. While theoretically the results should be the same, in practice this is not the case. Obviously there are only one tenth as many points in the data set when seconds are used rather than tenths of seconds. Firstly, looking at the data aggregated over a time period of one second, there is no good agreement between estimators. The periodogram estimate is hopelessly out of the correct range. The other estimators, while in the range $(1/2,1)$ show no particular agreement. Of the suggested filtering techniques, little changes between them except that removal of a polynomial greatly reduces the estimate found by the periodogram and slightly reduces the estimate found by aggregated variance. No conclusion can realistically be drawn about the data from these results.
Considering the data aggregated into tenths of a second time units the picture is somewhat clearer. Taking a log of data was impossible at this time scale due to presence of zeros. The estimators, with the exception of the R/S plot are all relatively near $H = 0.9$. While it seems somewhat arbitrary to ignore the results of the R/S plot it should be remembered that this technique performed poorly with high Hurst parameter measurements on theoretical data and underestimated badly in those cases. No great difference is observed from any of the suggested filtering techniques except, perhaps, a slight reduction in the aggregated variance and periodogram results from removal of a polynomial. A tentative conclusion from this data would be that $0.85 < H < 0.95$ and that the R/S plot is inaccurate for this trace.
[|l| l l l l l|]{} Filter & R/S Plot & Aggreg. & Period. & Wavelet & Local\
Type & & Variance & ogram & Estimate & Whittle\
\
None & 0.73 & 0.742 & 0.762 & 0.73 $\pm$ 0.018 & 0.661\
Log & 0.722 & 0.806 & 0.797 & 0.77 $\pm$ 0.02 & 0.652\
Trend & 0.73 & 0.74 & 0.762 & 0.73 $\pm$ 0.018 & 0.661\
Poly & 0.73 & 0.733 & 0.751 & 0.73 $\pm$ 0.018 & 0.66\
\
None & 0.709 & 0.703 & 0.742 & 0.746 $\pm$ 0.025 & 0.655\
Log & 0.721 & 0.795 & 0.779 & 0.778 $\pm$ 0.011 & 0.673\
Trend & 0.709 & 0.703 & 0.742 & 0.746 $\pm$ 0.025 & 0.655\
Poly & 0.709 & 0.691 & 0.732 & 0.746 $\pm$ 0.025 & 0.654\
\
None & 0.707 & 0.8 & 0.817 & 0.786 $\pm$ 0.017 & 0.822\
Trend & 0.707 & 0.797 & 0.815 & 0.786 $\pm$ 0.017 & 0.822\
Poly & 0.707 & 0.789 & 0.787 & 0.786 $\pm$ 0.017 & 0.822\
\
None & 0.62 & 0.802 & 0.808 & 0.762 $\pm$ 0.012 & 0.825\
Trend & 0.62 & 0.802 & 0.808 & 0.762 $\pm$ 0.012 & 0.825\
Poly & 0.618 & 0.786 & 0.777 & 0.762 $\pm$ 0.012 & 0.824\
In the case of the Bellcore measurements, the data has been split into two sections and analysed seperately for interarrival times and for bytes per unit time. Considering first the interarrival times, all estimators seem to have a result which is not too distant from $H = 0.7$ in both cases. The various filtering techniques tried do little to change this. It is hard to come to a really robust conclusion since the estimators are as high as $0.806$ (aggregated variance after taking logs) and as low as $0.652$ (local Whittle after taking logs).
When the bytes per unit time are considered, the log technique cannot be used due to zeros in the data. The most comfortable conclusion abou this data might be that the Hurst parameter is somewhere around $H = 0.8$ with the R/S plot underestimating again. As before, it is hard to reach a strong conclusion on the exact Hurst parameter. Certainly it would be foolish to take the confidence intervals on the wavelet estimator at face value. The various filters tried seem to make little difference except perhaps a slight reduction in the answer given by some estimators after the polynomial is removed. A tentative conclusion might be that $0.75 < H < 0.85$ for this data with the R/S plot being in error.
CONCLUSION
==========
This paper has looked at measuring the Hurst parameter, firstly in the case of artificial data contaminated by various types of noise and secondly in the case of real data with various filters to try to improve the performance of the estimators used.
The most striking conclusion of this paper is that measuring the Hurst parameter, even in artificial data, is very hit and miss. In the artificial data with no corrupting noise, some estimators performed very poorly indeed. Confidence intervals given should certainly not be taken at face value (indeed should be considered as next to worthless).
Corrupting noise can affect the measurements badly and different estimators are affected in by different types of noise. In particular, frequency domain estimators (as might be expected) are robust to the addition of sinusoidal noise or a trend. All estimators had problems in some circumstances with the addition of a heavy degree of short-range dependence even though this, in theory, does not change the long-range dependence of the time series.
When considering real data, researchers are advised to use extreme caution. A researcher relying on the results of any single estimator for the Hurst parameter is likely to be drawing false conclusions, no matter how sound the theoretical backing for the estimator in question. While simple filtering techniques are suggested in the literature for improving the performance of Hurst parameter estimation, they had little or no effect on the data analysed in this paper.
All the data and tools used in this paper are available for download from the web and can be found at:\
[www.richardclegg.org/\
lrdsources/software/]{}
BIOGRAPHY {#biography .unnumbered}
=========
**Richard G. Clegg** is a Research Assistant in the Department of Mathematics at the University of York. He obtained his PhD in “The Statistics of Dynamic Networks” in May 1994 and works on applied mathematics in networks, mainly road networks and traffic networks. His main research interests are equilibrium and driver route choice in road networks and long-range dependence in computer networks.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The linear polarization produced by scattering processes in the hydrogen Ly$\alpha$ line of the solar disk radiation is a key observable for probing the chromosphere-corona transition region (TR) and the underlying chromospheric plasma. While the line-center signal encodes information on the magnetic field and [the]{} three-dimensional structure of the TR, the sizable scattering polarization signals that the joint action of partial frequency redistribution and $J$-state interference produce in the Ly$\alpha$ wings have generally been thought to be sensitive only to the thermal structure of the solar atmosphere. Here we show that the wings of the $Q/I$ and $U/I$ scattering polarization profiles of this line are actually sensitive to the presence of chromospheric magnetic fields, with strengths similar to those that produce the Hanle effect in the line core [(i.e., [between 5 and]{} 100 gauss, approximately).]{} In spite of the fact that the Zeeman splitting induced by such weak fields is very small compared to the total width of the line, the magneto-optical effects that couple the transfer equations for Stokes $Q$ and $U$ are actually able to produce sizable changes in the $Q/I$ and $U/I$ wings. [We find that magnetic fields with longitudinal components [larger than]{} $100$ G produce an almost complete depolarization of the wings of the Ly$\alpha$ $Q/I$ profiles within a ${\pm}5$ Å spectral range around line center, while stronger fields are required for the $U/I$ wing signals to be depolarized to a similar extent. ]{} The theoretical results presented here further expand the diagnostic content of the unprecedented spectropolarimetric observations provided by the Chromospheric Lyman-Alpha Spectropolarimeter (CLASP).'
author:
- 'E. Alsina Ballester, L. Belluzzi, and J. Trujillo Bueno'
title: |
Magnetic sensitivity in the wing scattering polarization signals\
of the hydrogen Lyman-$\alpha$ line of the solar disk radiation
---
Introduction {#Sect:Introd}
============
The linear polarization produced by scattering processes in ultraviolet (UV) resonance lines of the solar disk radiation encodes key information on the plasma of the upper solar chromosphere and transition region (TR). For example, it is known that the line-center scattering polarization signals are sensitive to magnetic fields via the Hanle effect [e.g., @LandiLandolfi04 hereafter LL04]. Of particular interest is the hydrogen Ly$\alpha$ resonance line at $121.6$ nm, the strongest emission line in the solar UV spectrum. A few years ago, the Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) sounding rocket experiment, motivated by theoretical predictions based on radiative transfer (RT) calculations [@TrujilloBueno+11; @Belluzzi+12; @Stepan+15], discovered conspicuous scattering polarization signals in Ly$\alpha$ [see @Kano+17]. Theoretical modeling of the observed Stokes $Q/I$ and $U/I$ line-center signals recently allowed us to constrain the magnetic field strength and geometrical complexity of the corrugated surface that delineates the chromosphere-corona TR [@TrujilloBueno+18].
While the line-center photons of the hydrogen Ly$\alpha$ line stem mainly from the TR, the wing photons encode information on the underlying chromospheric layers (e.g., at $\Delta{\lambda}={\pm}1$ Å from the line center, the height in the solar atmosphere where the optical depth is unity lies a few hundred kilometers below the TR). Unlike the $Q/I$ and $U/I$ line-center signals, which are sensitive to the presence of magnetic fields in the TR via the Hanle effect, the wing signals have always been thought to be sensitive only to the thermal structure of the solar atmosphere [e.g., @Belluzzi+12]. The main aim of the present [paper]{} is to show that the wings of the $Q/I$ and $U/I$ profiles of the hydrogen Ly$\alpha$ line are sensitive to the presence of magnetic fields in the solar chromosphere, with strengths similar to those that [characterize]{} the onset of the Hanle effect in the line core. The physical mechanism at the origin of this magnetic sensitivity is as follows.
In some resonance lines for which the effects of partial frequency redistribution (PRD) produce large $Q/I$ wing signals, the $\rho_V\,U$ and $\rho_V\,Q$ magneto-optical (MO) terms of the transfer equations for Stokes $Q$ and $U$, respectively, can [induce]{} a significant magnetic sensitivity in the line’s scattering polarization wings. Given that in the line wings $\rho_V$ is significant already for relatively weak magnetic fields, the above-mentioned $\rho_V\,Q$ term introduces sizable, magnetically sensitive, $U/I$ wing signals. In turn, such large $U/I$ wing signals allow the $\rho_V\,U$ term to introduce a magnetic sensitivity in the $Q/I$ wing signals. [This mechanism causes both a rotation of the plane of linear polarization as the radiation travels through the solar atmosphere [see @AlsinaBallester+17] and an effective decrease of the degree of total linear polarization [see @AlsinaBallester+18].]{} [Recent]{} RT investigations have indicated that such MO effects should play an important role in the wings of many strong chromospheric lines, such as the Mg [ii]{} k line [@AlsinaBallester+16], the Mg [ii]{} h & k lines [@delPinoAleman+16], the Sr [ii]{} $407.8$ nm line [@AlsinaBallester+17], and the Ca [i]{} $422.7$ nm line [@AlsinaBallester+18].
Although the physical mechanism that introduces magnetic sensitivity in the Ly$\alpha$ scattering polarization wings is therefore not new, it is remarkable that it is capable of producing measurable effects even in a far UV line like hydrogen Ly$\alpha$. This is because, [at line-wing wavelengths]{}, the $\rho_V$ coefficient takes sizeable values relative to the absorption coefficient already when the Zeeman splitting becomes comparable to the radiative and collisional line broadening. In contrast, the signals produced by the familiar Zeeman effect depend on the ratio of the magnetic splitting over the Doppler width of the line and therefore scale with the wavelength of the spectral line under consideration.
Formulation of the problem
==========================
We present the results of non-local thermodynamic equilibrium (NLTE) RT calculations of the intensity and linear polarization of the hydrogen Ly$\alpha$ line, considering the semi-empirical model C of [@Fontenla+93], hereafter FAL-C. The use of this static one-dimensional (1D) solar atmospheric model allows us to isolate the influence of the magnetic field (although neglecting its possible horizontal fluctuations) from other possible symmetry-breaking mechanisms. The magnetic fields we have imposed in this model atmosphere are deterministic. [Hereafter, we specify their direction by their inclination and azimuth, defined as illustrated in]{} Figure 1 of [@AlsinaBallester+18]. The lines of sight (LOSs) for the considered Stokes profiles are specified by $\mu = \cos\theta$, where $\theta$ is the heliocentric angle. The positive direction for Stokes $Q$ has been taken along the $Y$ axis (i.e., parallel to the limb for all LOSs with $\mu<1$). In the calculations presented below, the line-broadening effect of both elastic and inelastic collisions [is]{} taken into account according to the rates presented in LL04 and [@PrzybillaButler04], respectively. The depolarizing effect of the former has not been taken into account, after having verified numerically that its impact is negligible for this very strong chromospheric line.
[The Ly$\alpha$ line is produced by the transition between the hydrogen levels $n=1$ and $n=2$. Taking the fine structure (FS) of hydrogen into account, and neglecting the contribution from forbidden transitions (under the electric dipole approximation), this line [receives contributions]{} from two FS transitions, namely those between the $^2$P$_{1/2}$ and $^2$P$_{3/2}$ FS levels of the $^2$P upper term and the $^2$S$_{1/2}$ FS level of the $^2$S lower term (i.e., the ground state).]{} It has been established from previous theoretical investigations in the unmagnetized case (Belluzzi et al. 2012) that reliable calculations of the wing linear polarization of the hydrogen Ly$\alpha$ line must account for quantum interference between the $^2$P$_{1/2}$ and $^2$P$_{3/2}$ [upper levels]{} (i.e., $J$-state interference), in addition to PRD effects. [An atomic model accounting for the various FS transitions between two terms, as well as for the quantum interference between different FS $J$-levels belonging to the same term, is generally referred to as a two-term atom (see LL04). A]{} correct modeling of the wing scattering polarization of the Ly$\alpha$ line thus requires considering at least a two-term ($^2$S – $^2$P) model atom.
On the other hand, observing that the FS components are very close to each other, it can be shown that, far from the line center, this line behaves in resonance scattering as a spinless two-level $0$ – $1$ transition, in compliance with the principle of spectroscopic stability (PSS)[^1]. [The good agreement between the modeling that accounts for FS and the one that neglects it can]{} be clearly seen in the left panel of Figure \[Fig1\], in which the scattering polarization profiles obtained by considering [both a two-term (${}^2$S – $^2$[P]{}) and a two-level ($0$ – $1$) model are compared]{}, in the absence of magnetic fields. Unless otherwise noted, an LOS with $\mu = 0.3$ is considered in the figures presented [in this work]{}. The expected discrepancy in the line core is a clear manifestation of the depolarizing effect of the FS (e.g., LL04). Indeed, the gray area across the line-core region, [appearing in several of the figures presented in this paper,]{} indicates the spectral interval where the approximation of neglecting FS is not justified. The very small deviations found outside the line-core region are due to the approximate treatment of elastic collisions in the two-term atom calculations [see @BelluzziTrujilloBueno14].
At spectral distances from the line center that are much greater than both the Doppler width of the line and the magnetic splitting of the energy levels, the line emissivity is insensitive to both the Hanle and Zeeman effects, provided that the collisional broadening is significantly smaller than the natural width of the line [see LL04; also Appendix B of @AlsinaBallester+18]. [For illustrative purposes, throughout this work we will focus on]{} [a]{} wing wavelength [around]{} which the linear polarization maximizes.[^2] [More precisely, we consider the wavelength at $360$ mÅ to the blue of the line center (hereafter $\lambda_m$) and we point out that]{} this spectral separation is much greater than the magnetic splitting of the energy levels, even in the presence of magnetic fields of a few kilogauss. It is also considerably larger than the Doppler width [corresponding to]{} the atmospheric [regions]{} where most of the radiation at wavelength $\lambda_m$ [comes]{} from. [Indeed, considering the FAL-C model, the Doppler width is approximately $55$ mÅ at $z_m = 1998.5$ km; at this height the optical depth at wavelength $\lambda_m$ is close to unity for an LOS with $\mu = 0.3$. ]{}
The magnetic sensitivity of the scattering polarization in the wings of this line is instead governed by the MO effects quantified by the RT coefficient $\rho_V$. [It is important to note that the]{} impact [of such effects]{} is only appreciable if another physical mechanism, such as scattering processes subject to PRD [phenomena]{}, produces sizable linear polarization signals outside the Dopper core [see @AlsinaBallester+17]. Using the two-level atomic model, we have verified that, when artificially setting $\rho_V$ to zero, magnetic fields with strengths up to $5$ kG have no impact on the line’s wing linear polarization. In the right panel of Figure \[Fig1\], we compare the ratio of $\rho_V$ over the absorption coefficient $\eta_I$ obtained from the two-term atom equations to that found for a $0$ – $1$ two-level atom, in the presence of a horizontal magnetic field of 50 G[^3]. The [results of the]{} two [calculations, carried out at height $z_m$ in the FAL-C model, present]{} an excellent agreement, confirming the suitability of neglecting FS when modeling the magnetic sensitivity of this line’s wing scattering polarization signals. [As shown in Appendix \[App2L\], the far-wing value of $\rho_V$ is proportional to the spectral distance between the centers of gravity of the $\sigma_b$ and $\sigma_r$ components of the line. Thus, the above-mentioned agreement is ultimately related to the fact that - in accordance with the PSS - the frequency shifts of the centers of gravity of the $\sigma_b$, $\pi$, and $\sigma_r$ components for a two-term atom, obtained accounting for the incomplete Paschen-Back (IPB) effect, coincide with those of a normal Zeeman triplet, i.e., of a spinless two-level atomic transition (e.g., Section 3.4 of LL04).]{}
The impact of magneto-optical effects {#Sect:MO}
=====================================
[In this]{} section, we present the results of illustrative RT calculations of the Ly$\alpha$ wing scattering polarization signals, considering a spinless two-level model atom and accounting for the joint impact of PRD and of magnetic fields through the Hanle, Zeeman, and MO effects. Details on the theoretical and numerical framework can be found in [@AlsinaBallester+17]. [We]{} consider magnetic fields with a constant strength and orientation throughout the FAL-C model atmosphere, paying particular attention to [vertical and horizontal (as well as nearly vertical and [nearly]{} horizontal)]{} magnetic fields.
Linear Polarization Profiles for Deterministic Magnetic Fields
--------------------------------------------------------------
{width="\textwidth"}
[The top panels of]{} Figure \[Fig2\] [show]{} the linear polarization profiles [at an LOS with $\mu = 0.3$, in the presence of]{} [horizontal ($\theta_B = 90^\circ$) magnetic]{} fields [of]{} various strengths with [azimuth $\chi_B = 0^\circ$]{} [(this choice of azimuth maximizes the longitudinal component of the magnetic field).]{} Outside the Doppler core, the MO effects induced by such magnetic fields produce a $U/I$ signal and a depolarization in $Q/I$. [The influence of such effects is controlled by the ratio of $\rho_V$ over $\eta_I$, which depends on the longitudinal component of the magnetic field. Interestingly, this ratio scales with the same parameters that characterize the efficacy of the Hanle effect (see Appendix \[App2L\]). Indeed, such MO effects are expected to noticeably impact the wings of the linear polarization signals when the magnetic field strength is comparable to the Hanle critical field. Moreover, in the presence of increasingly strong magnetic fields, the impact of such MO effects is appreciable in the wings of both $Q/I$ and $U/I$ at greater spectral distances from the line center. As pointed out in [@AlsinaBallester+18], the relative contribution of continuum processes to $\eta_I$ is greater farther into the line wings, implying that stronger magnetic fields are required in order for the $\rho_V/\eta_I$ ratio to be significant. Note also that, in]{} addition to their amplitude, also the sign of the $U/I$ wing signals is sensitive to the [orientation]{} of the magnetic field. [For instance, comparing horizontal magnetic fields with $\chi_B = 0^\circ$ and $\chi_B = 180^\circ$, which have longitudinal components of the same magnitude but point in the opposite direction, we have verified that the depolarization of $Q/I$ is the same, while the resulting $U/I$ wing signal is identical in absolute value but with opposite sign.]{} [We point out that, because the wing $Q/I$ scattering polarization signals are negative in the unmagnetized case, the MO effects induced by a magnetic field with a positive (negative) longitudinal component give rise to negative (positive) $U/I$ signals.]{}
[We have also considered the case of vertical magnetic fields ($\theta_B = 0^\circ$) of increasing strength, for an LOS with $\mu = 0.3$. As seen in the bottom left panel of Figure \[Fig2\], for $B = 300$ G the wings of $Q/I$ are almost completely depolarized within a ${\pm}5$ Å spectral range around line center, and a significant depolarization is also appreciable much farther into the wings. These profiles have a strong resemblance to those obtained in the presence of a horizontal magnetic field of $100$ G discussed above. This can be easily understood by observing that the longitudinal components of the two aforementioned field configurations are very similar (around $90$ G). Interestingly, in the presence of magnetic fields with such longitudinal components, the near wings of the $U/I$ profiles still have a considerable amplitude (see the right panels of Figure \[Fig2\]), and stronger magnetic fields are required in order for them to be considerably depolarized. Indeed, we have checked that the absolute value of the $U/I$ wing signal at $\lambda_m$ does not fall below $0.1$% until magnetic fields with longitudinal components larger than $900$ G are considered. Finally, just as in the case of a horizontal magnetic field, we have also verified that if the vertical magnetic field is oriented in the opposite direction (i.e., $\theta_B = 180^\circ$), the resulting depolarization of $Q/I$ is the same and the $U/I$ profile has the opposite sign, again as a consequence of the sign reversal of the LOS projection of the magnetic field.]{}
[The previously discussed signatures of the MO effects, namely the depolarization of $Q/I$ together with the appearance of a $U/I$ signal whose sign depends on the orientation of the magnetic field, offer]{} a new tool for inferring the longitudinal component of the magnetic fields in the chromospheric regions where the Ly$\alpha$ wings originate. The magnetic sensitivity of this line’s wing scattering polarization signals can be expected to be well above the noise level, [even]{} in quiet regions of the solar atmosphere [where]{} the circular polarization signals produced by the Zeeman effect would be extremely weak. Moreover, the scattering polarization signal is clearly appreciable very far into the line wings, thus encoding information on the magnetic activity in deeper chromospheric layers.
Center-to-limb Variation for Determinstic Magnetic Fields
---------------------------------------------------------
[Figures \[Fig4a\] and \[Fig4b\] show]{} the center-to-limb variation (CLV) of the $Q/I$ and $U/I$ signals at $\lambda_m$, calculated for
nearly vertical ($\theta_B = 20^\circ$; Figure \[Fig4a\]) and nearly horizontal ($\theta_B = 70^\circ$; Figure \[Fig4b\]) magnetic fields of various strengths up to $300$ G, both for $\chi_B = 0^\circ$ and $\chi_B = 180^\circ$.
The CLV for $Q/I$ and $U/I$ found in the presence of magnetic fields with $\theta_B = 20^\circ$ and $\chi_B = 0^\circ$ can be explained in a relatively straightforward manner. The projection of the magnetic field along the LOS – and therefore the value of $\rho_V/\eta_I$ – has the same sign for all $\mu$ between $0$ and $1$ and increases monotonically up to $\mu \approx 0.94$. As the magnetic field strength increases, one finds a progressively greater departure
from the $(1 - \mu^2)$ trend for $Q/I$, theoretically predicted in the unmagnetized case. The amplitude of the $U/I$ signals, produced by the same MO effects, is found to decrease monotonically with $\mu$, because it depends on both the longitudinal component of the magnetic field and the amplitude of the $Q/I$ signals. [The $U/I$ signals increase in amplitude with the field strength up to roughly $50$ G, but for even stronger fields they begin to decrease, as the MO effects produce a net reduction of the total fraction of linear polarization [see Appendix A of @AlsinaBallester+18]]{}.
[The situation is substantially different in the presence of magnetic fields with $\theta_B = 20^\circ$ and $\chi_B = 180^\circ$ (see the bottom panels of Figure \[Fig4a\]). In this case, the magnetic field points away from the observer for LOSs with small $\mu$ values, it becomes completely transversal at $\mu \approx 0.34$, and its longitudinal component becomes positive and increases as one continues approaching $\mu = 1$. Compared to the case in which $\chi_B = 0^\circ$, the longitudinal component is smaller when considering LOSs with small $\mu$ values, resulting in a much more modest depolarization in $Q/I$, especially around the LOS at which the magnetic field is transversal. It is interesting to note that, even at this LOS, the magnetic field still produces some depolarization, despite the fact that $\rho_V$ is zero in this direction. This can be explained because the pumping radiation field is nevertheless modified by MO effects, thereby impacting the linear polarization [emitted in]{} this direction [e.g., @AlsinaBallester+16; @AlsinaBallester+18]. There are also clear qualitative differences with respect to the previous case in the CLV for $U/I$; in this case their signals are positive for LOS with large inclinations and become negative when directions closer to the vertical are considered. The sign inversion occurs around the LOS for which such fields are transversal, although the exact $\mu$ value changes with the field strength because of modification of the pumping radiation field induced by MO effects.]{}
[On the other hand, when considering nearly horizontal ($\theta_B = 70^\circ$) magnetic fields (see Figure \[Fig4b\]), the CLV obtained in the presence of fields with $\chi_B = 0^\circ$ and $\chi_B = 180^\circ$ are qualitatively very similar to each other. Nevertheless, is worth noting that, for LOSs with small $\mu$ values, the longitudinal component of the magnetic field – and thus the depolarization of $Q/I$ – is slightly greater for the former case than for the latter. For small $\mu$, such nearly horizontal magnetic fields give rise to a considerably stronger depolarization than those with an inclination of $\theta_B = 20^\circ$, also in this case due to their larger longitudinal components. ]{}
A Look at Observational Data: CLASP
-----------------------------------
Recently, CLASP successfully measured the linear polarization signals of the Ly$\alpha$ line emerging from quiet regions of the Sun, spanning from off-limb positions to close to the disk center [see @Kano+17]. [In the wings of the $Q/I$ and $U/I$ profiles, considerable fluctuations along the spatial direction of the radially oriented slit were found. [The]{} amplitude of the wing $Q/I$ signal was found to decrease with $\mu$ (in agreement with our theoretical expectations), while no [serious]{} CLV was observed in the amplitude of $U/I$]{}. [We are confident that the observed lack of CLV in the $U/I$ wing signals can be explained by accounting for]{} horizontal variations in the longitudinal component of the magnetic field, [which could substantially modify]{} the net amplitude of the signals resulting from MO effects, [and/or ]{} by the axial asymmetries in other thermodynamical properties of the solar atmosphere, which may produce $U/I$ signals of non-magnetic origin. An accurate modeling of the [scattering]{} polarization signals observed [in strong resonance lines such as H [i]{} Ly$\alpha$]{} must [therefore account]{} for the full three-dimensional complexity of the solar atmosphere, as well as the joint action of scattering polarization with PRD phenomena and the Hanle, Zeeman, and MO effects.
[In spite of the simplification that the FAL-C semi-empirical model implies, it is worthwhile to note that the results of our radiative transfer calculations in this 1D model of the solar atmosphere can be invoked to qualitatively explain one of the other spectacular observational results provided by CLASP. In addition to the wavelength variation of the linear polarization profiles, CLASP provided Stokes $I$ and Stokes $Q/I$ broadband images over a large field of view [see @Kano+17]. Within this field of view there was a bright plage and a multitude of network and inter-network features. Interestingly, the bright plage region and some of the network features that can be distinguished in the Stokes $I$ image show nearly zero linear polarization in the Stokes $Q/I$ image, while the surrounding quiet regions instead show very significant polarization signals. In a forthcoming publication we will investigate whether this can be explained on the basis of the results reported above, by noting that the broadband $Q/I$ signals observed by CLASP are dominated by the linear polarization in the Ly$\alpha$ wings [see @Belluzzi+12] and by bearing in mind that plages and the network have stronger magnetic fields than the surrounding quieter regions.]{}
Unresolved Magnetic Fields
--------------------------
Further insights into the magnetic sensitivity of the linear polarization in the wings can be gained by studying its behavior on the $Q/I-U/I$ plane. The closed curves in the polarization diagrams shown in Figure \[Fig5\] indicate how the fractional linear polarization signals obtained at $\lambda_m$ change with $\chi_B$ in the presence of $50$ G magnetic fields with a fixed inclination. The diagram is symmetric around the $U/I = 0$ axis for $\theta_B = 90^\circ$ (left panel), but this is not the case for arbitrary inclinations, implying the following [see @AlsinaBallester+18]. If one measures the net $U/I$ to be zero in a given spatially unresolved region of the solar atmosphere, this is an indication that the magnetic field therein is [transversal, or otherwise]{} has a distribution such that the averaged longitudinal component is zero[^4].
[We]{} also point out that magnetic field [distributions]{} that do not fulfil the aforementioned condition [are]{} capable of producing a net $U/I$ signal [even if]{} their orientations change at scales below the [mean free path of the line’s photons]{} [(i.e., micro-structured magnetic fields)]{}. [Indeed, for such a field configuration, in which the inclination is fixed and the azimuth changes randomly, the $\rho_V$ is generally not zero [see Equations (6b) and (50a) of @AlsinaBallester+17], implying that a $U/I$ signal may be produced in the wings. By contrast, for the same field configuration the Hanle effect may modify the line-core $Q/I$ amplitude of the scattered radiation, but it produces no $U/I$ signal (see Eqs. (11), (21), (22), and (50b) of the same paper). Such qualitative differences can be understood by realizing that the MO effects quantified by $\rho_V$ depend only on the net longitudinal component of the magnetic field, which is only zero for all LOSs if $\theta_B = 90^\circ$. On the other hand, the Hanle effect also depends on the angle between the magnetic field and the symmetry axis of the pumping radiation field, which in a 1D unmagnetized atmospheric model is parallel to the local vertical. The field configuration discussed here is symmetric around this axis and, as a result, the Hanle effect does not cause a rotation of the plane of linear polarization of the scattered radiation, although it may decrease the degree of linear polarization.]{}
Concluding comments {#Sect:Concl}
===================
Motivated by the recent theoretical discovery that the wing scattering polarization of some strong resonance lines is highly sensitive to the MO effects quantified by the $\rho_V$ terms of the transfer equations for Stokes $Q$ and $U$, we have conducted an RT investigation on [the]{} wing linear polarization signals [of]{} the hydrogen Ly$\alpha$ line. [We have]{} modeled [this line considering]{} a spinless two-level atom [(i.e., the impact of FS has been neglected), [having]{} shown that this approximation is suitable outside the Doppler core.]{} We [have found]{} that the wing scattering polarization signals of this far UV line are in fact sensitive to longitudinal magnetic fields, even [when]{} they are considerably weaker than the Hanle critical field.
Such signals extend far into the line wings, potentially offering a method to simultaneously infer the LOS components of the magnetic fields present in [a wide range of depths throughout]{} the solar chromosphere. The sign of such components can be determined from that of $U/I$, while the combined amplitude of $Q/I$ and $U/I$ are indicative of their magnitude. From symmetry [considerations applied]{} to the polarization diagrams, we conclude that measuring a nonzero $U/I$ wing signal may be a signature of an asymmetry of the distribution of the LOS component of the magnetic field[ within the considered spatial resolution element]{}.
This investigation reveals that relatively weak magnetic fields may strongly impact [the]{} wing scattering polarization signals [of the Ly$\alpha$ line]{} via MO effects. [Interestingly, the broadband $Q/I$ images provided by the CLASP suborbital rocket experiment revealed [linear polarization]{} signals [close to zero]{} in the regions of the field of view corresponding to a plage and to some of the network features, in contrast to the much less magnetized surrounding regions. As we shall show in detail in a forthcoming publication, these observations can potentially be explained on the basis of the depolarization that MO effects produce in the wings of the Ly$\alpha$ line.]{}
[Finally, we emphasize that]{} an accurate RT modeling of the [scattering polarization in the]{} hydrogen Ly$\alpha$ line requires accounting for the 3D structure of the solar atmosphere, in addition to the joint action of resonance scattering with PRD and the Hanle, Zeeman, and MO effects.
E.A.B. and L.B. gratefully acknowledge financial support by the Swiss National Science Foundation (SNSF) through Grant 200021\_175997. J.T.B. acknowledges the funding received from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Advanced Grant agreement No. 742265).
A. The far-wing limit of the elements of the propagation matrix {#AppFarProp}
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Here we present an analytical study of the magnetic dependence of the elements of the line contribution to the so-called propagation matrix (see LL04), focusing on spectral regions far beyond the Doppler core. We consider a two-term atomic model [without hyperfine structure]{}, in the presence of magnetic fields of arbitrary strength. In order to [determine]{} the various eigenstates of an atomic system in the presence of an external magnetic field, one must diagonalize the total Hamiltonian $H = H_0 + H_B$, in which $H_0$ is the Hamiltonian of the unperturbed atomic system and $H_B$ is the magnetic Hamiltonian [see @CondonShortley35]. Taking the quantization axis of total angular momentum $J$ (i.e., the $z$-axis) parallel to the magnetic field, the magnetic Hamiltonian obeys the following commutation rules, $$[H_B, J_z] = 0 \, , \quad [H_B, J_x] \ne 0 \, , \quad [H_B, J_y] \ne 0 \, .$$ Therefore, in the presence of a magnetic field the quantum number $J$ generally loses [the]{} property of being a “good” quantum number, while this property is preserved for the quantum number $M$. When the magnetic energy is much smaller than the energy intervals of $H_0$ the effect of $H_B$ can be computed through a perturbative approach to first order [e.g., @LandiDeglInnocenti14], which implies its diagonalization over the degenerate eigenvectors of $H_0$. The matrix $\bra{\beta L S J M}\! |H_B|\!\ket{\beta L S J M^\prime}$ is diagonal and the magnetic field produces an energy splitting of the magnetic sublevels that scales linearly with the field strength. This approach is commonly known as the linear Zeeman splitting approximation (LZS). In the more general case, commonly referred to as the IPB effect regime, when performing the diagonalization of the total Hamiltonian on the basis $\ket{\beta L S J M}$, one finds that the magnetic field produces a mixing of the various $J$-levels. The ensuing eigenvectors are characterized by quantum number $M$ as well as by the label $j$: $$\begin{aligned}
& H \, \ket{\beta_u L_u S j_u M_u} = E_{j_u}(\beta_u L_u S, M_u) \, \ket{\beta_u L_u S j_u M_u} \, ; \quad \quad
\ket{\beta_u L_u S j_u M_u} = \sum_{J_u} C^{j_u}_{J_u}(\beta_u L_u S, M_u) \, \ket{\beta_u L_u S J_u M_u} \, , \\
& H \, \ket{\beta_\ell L_\ell S j_\ell M_\ell} = E_{j_\ell}(\beta_\ell L_\ell S, M_\ell) \, \ket{\beta_\ell L_\ell S j_\ell M_\ell} \, ;
\quad \quad \; \;
\ket{\beta_\ell L_\ell S j_\ell M_\ell} = \sum_{J_\ell} C^{j_\ell}_{J_\ell}(\beta_\ell L_\ell S, M_\ell) \, \ket{\beta_\ell L_\ell S J_\ell M_\ell} \, ,
\end{aligned}$$ where the $u$ and $\ell$ subscripts refer to the states of the upper and lower term, respectively. $E_j (\beta L S, M)$ is the energy for each eigenstate and the $C^j_J(\beta L S, M)$ coefficients describe the coupling between such states and the $\ket{\beta L S J M}$ basis eigenvectors. Given that the sum of the eigenvalues of a Hamiltonian are equal to its trace, it can be shown that for each term $$\sum_{j M} E_j(\beta L S, M) = n \, E(\beta L S) \, ,$$ where $n$ is the number of different eigenstates belonging to the considered term and $E(\beta L S)$ is the energy of the term. Each of the (electric-dipole) radiative transitions between the various states of the upper term $\ket{\beta_u L_u S j_u M_u}$ and those of the lower term $\ket{\beta_\ell L_\ell S j_\ell M_\ell}$ are characterized by their frequencies $$\nu_{j_u M_u, j_\ell M_\ell} = \bigl[E_{j_u}(\beta_u L_u S, M_u) - E_{j_\ell}(\beta_\ell L_\ell S, M_\ell)\bigr]/h \, ,
\label{Freq}$$ where $h$ is the Planck constant. [These frequencies can also be expressed as shifts with respect to the reference frequency of the multiplet $\nu_0 = \bigl[E(\beta_u L_u S) - E(\beta_\ell L_\ell S)\bigr]/h$, in units of the Doppler width $\Delta\nu_D$, as ]{} $$x_{j_u M_u, j_\ell M_\ell} = \frac{\nu_{j_u M_u, j_\ell M_\ell} - \nu_{0}}{\Delta\nu_D} \, .
\label{FreqShiftTrans}$$ [Introducing also the reduced frequency]{} $$x = \frac{\nu_0 - \nu}{\Delta\nu_D} \, ,$$ [we note that $x_{j_u M_u, j_\ell M_\ell} + x = (\nu_{j_u M_u, j_\ell M_\ell} - \nu)/\Delta \nu_D$. Moreover, ]{} it can easily be shown that $$\sum_{j_u M_u j_\ell M_\ell} x_{j_u M_u, j_\ell M_\ell} = 0 \, .$$ The various transitions between the upper and lower term can be divided into three groups according to $\Delta M \equiv (M_u - M_\ell) = (\pm 1,0)$. Following the terminology generally used in the literature, we refer to the groups with $q = - \Delta M = (-1,0,1)$ as the $\sigma_r$, $\pi$, and $\sigma_b$ components, respectively. The strength of each transition is given by (see LL04) $$\begin{aligned}
& S^{j_u M_u, j_\ell M_\ell}_q = \frac{3}{2 S + 1} %\notag \\
\sum_{J_u J_u^\prime} C^{j_u}_{J_u}(\beta_u L_u S, M_u) \, C^{j_u}_{J_u^\prime}(\beta_u L_u S, M_u)
\sum_{J_\ell J_\ell^\prime} C^{j_\ell}_{J_\ell}(\beta_\ell L_\ell S, M_\ell) \, C^{j_\ell}_{J_\ell^\prime}(\beta_\ell L_\ell S, M_\ell) \notag \\
& \times \sqrt{(2 J_u + 1)(2 J_u^\prime + 1)(2 J_\ell + 1)(2 J_\ell^\prime + 1)} %\notag \\
\left\{\begin{array}{c c c}
J_u & J_\ell & 1 \\
L_\ell & L_u & S
\end{array} \right\}
\left\{\begin{array}{c c c}
J_u^\prime & J_\ell^\prime & 1 \\
L_u & L_\ell & S
\end{array} \right\} % \notag \\
\left(\begin{array}{c c c}
J_u & J_\ell & 1 \\
-M_u & M_\ell & -q
\end{array} \right)
\left(\begin{array}{c c c}
J_u^\prime & J_\ell^\prime & 1 \\
-M_u & M_\ell & -q
\end{array} \right) \, ,
\label{TransStr}\end{aligned}$$ which fulfil the following normalization condition $$\sum_{j_u M_u j_\ell M_\ell} S^{j_u M_u, j_\ell M_\ell}_q = 1 \, , \; \quad q = (-1,0,1) \, .$$
![Left panels: reduced frequency shifts (see Equation ) for the various transitions between the upper ($^2$P) and lower ($^2$S) term of the Ly$\alpha$ line, as a function of magnetic field strength. The Doppler width has been taken at a height of $1998.5$ km in the FAL-C atmospheric model, corresponding to $54.4$ mÅ. Right panels: normalized strengths for the same transitions (see Equation ), as a function of magnetic field strength. The top (bottom) panels illustrate the transitions whose upper state has total angular momentum $J_u = 3/2$ ($J_u = 1/2$) in the absence of magnetic field. The black solid curves represent the results of the calculation accounting for the incomplete Paschen-Back effect, while those represented by the red dashed-dotted curves are obtained under the linear Zeeman splitting approximation. [Note that several of the curves corresponding to the strengths of different transitions may overlap.]{}[]{data-label="FigAppendix1"}](FigAppendix1_240419.pdf){width="\textwidth"}
As discussed in this paper, under the electric-dipole approximation the H [i]{} Lyman-$\alpha$ line can be modeled as a two-term atom whose upper term has two FS levels, $^2$P$_{1/2}$ and $^2$P$_{3/2}$. Relative to the ground state, their energies are $82258.919$ cm$^{-1}$ and $82259.285$ cm$^{-1}$, respectively. Considering field strengths of up to $500$ G, we have verified that the $x_{j_u M_u, j_\ell M_\ell}$ frequency shifts calculated making the LZS approximation present a very good agreement with those obtained in the general IPB effect regime, as is shown in the left panels of Figure \[FigAppendix1\]. The quality of this agreement should not be surprising, because the energy separation between the FS levels of the upper term is more than one order of magnitude larger than the splitting between $M$-levels induced by a magnetic field of such strength.
On the other hand, we note that when the LZS approximation is made, the $C^{j}_{J}(\beta L S, M)$ coefficients reduce to $\delta_{J, j}$ and the magnetic dependence of the transition strengths given in Equation completely vanishes. This contrasts with the results obtained in the IPB effect regime, in which the transition strengths are appreciably modified by such weak magnetic fields, as is shown in the right panels of Figure \[FigAppendix1\].
The frequencies of the centers of gravity of the $\sigma_b$, $\pi$, and $\sigma_r$ groups, relative to $\nu_0$ and in units of Doppler width are defined as $$\bar{x}_q \, = \sum_{j_u j_\ell M_u M_\ell} S^{j_u M_u, j_\ell M_\ell}_q x_{j_u M_u, j_\ell M_\ell} \, .
\label{FrequencyShifts}$$ For the discussions below, it will also be useful to introduce $\bar{\nu}_q \equiv \bar{x}_q \, \Delta\nu_D$. It can be shown that, for a two-term atomic model in the IPB effect regime, the frequency shifts of the centers of gravity of the three groups scale linearly with the strength of the magnetic field (see LL04), according to $$\bar{x}_q = -q \, \frac{\nu_L}{\Delta\nu_D} \, ,
\label{FreqShiftNZT}$$ in which we have introduced the Larmor frequency $\nu_L = \mu_0 B/h$, where $\mu_0$ is the Bohr magneton. Such frequency shifts coincide with those for a spinless two-level atomic model.
![Spectral positions of the centers of gravity (see Equation ), taking a two-term atomic model for the H [i]{} Ly$\alpha$ line. The Doppler width has been taken at a height of $1998.5$ km in the FAL-C atmospheric model, corresponding to $54.4$ mÅ. The blue solid lines represent the spectral positions obtained in the incomplete Paschen-Back effect regime, the red dashed-dotted lines represent the same values obtained under the linear Zeeman splitting approximation, and the black dashed lines follow Equation . []{data-label="FigAppendix2"}](FigAppendix2.pdf){width="68.00000%"}
On the other hand, such shifts are considerably overestimated when the LZS approximation is made (see Figure \[FigAppendix2\]), as a consequence of neglecting the magnetic dependence of the strengths of the various transitions. The necessity of fully accounting for the IPB effect in order to correctly determine the spectral positions of the centers of gravity, also in the presence of magnetic fields weak enough that the splitting they induce is much smaller than separation between FS levels, was already pointed out by [@SocasNavarro+04].
The explicit expressions for the elements of the propagation matrix for a two-term atom with an unpolarized lower term, in the presence of a magnetic field of arbitrary strength, can be obtained as a particular case of those for a multi-term atom given in Section 7.6 of LL04. For the the purposes of this work, it is convenient to write such coefficients (defined taking the quantization axis parallel to the magnetic field), for a given frequency $\nu$ and direction $\boldsymbol{\Omega}$ as $$\begin{aligned}
\eta_i(\nu,\boldsymbol{\Omega}) & = k_M \sum_{K} \sqrt{\frac{2 K + 1}{3}} \, {\mathcal T}^K_0(i,\boldsymbol{\Omega}) \sum_q (-1)^{1+q}
\left(\begin{array}{c c c}
1 & 1 & K \\
q & -q & 0
\end{array} \right) \phi_q(\nu) \, , \quad \quad (i = 0, 1, 2, 3) \label{EtaMatQuant}\\
\rho_i(\nu,\boldsymbol{\Omega}) & = k_M \sum_{K} \sqrt{\frac{2 K + 1}{3}} \, {\mathcal T}^K_0(i,\boldsymbol{\Omega}) \sum_q (-1)^{1+q}
\left(\begin{array}{c c c}
1 & 1 & K \\
q & -q & 0
\end{array} \right) \psi_q(\nu) \, , \quad \quad (i = 1, 2, 3)
\label{RhoMatQuant}
\end{aligned}$$ where $k_M$ is the so-called frequency-integrated absorption coefficient and ${\mathcal T}^K_0(i,\boldsymbol{\Omega})$ are the polarization tensors introduced in [@LandiDeglInnocenti83]. The $\phi_q$ and $\psi_q$ profiles appearing in the previous expression are given by $$\phi_q(\nu) \, = \sum_{j_u M_u j_\ell M_\ell} S^{j_u M_u, j_\ell M_\ell}_q \, \mbox{Re}\bigl\{\Phi(\nu_{j_u M_u, j_\ell M_\ell} - \nu) \bigr\} \, ; \quad \quad
\psi_q(\nu) \, = \sum_{j_u M_u j_\ell M_\ell} S^{j_u M_u, j_\ell M_\ell}_q \, \mbox{Im}\bigl\{\Phi(\nu_{j_u M_u, j_\ell M_\ell} - \nu) \bigr\} \, .
\label{PhiPsiProfs}$$ In the proofs presented hereafter, we consider the observer’s reference frame, making the assumption that the distribution of atomic velocities is Maxwellian. In terms of reduced frequencies, the complex absorption profiles $\Phi(\nu_{j_u M_u, j_\ell M_\ell} - \nu)$ introduced above can be given as $$\Phi\bigl(\nu_{j_u M_u, j_\ell M_\ell} - \nu \bigr) = \frac{1}{\sqrt{\pi} \Delta\nu_D} \biggl(H\bigl(x + x_{j_u M_u, j_\ell M_\ell}, a\bigr)
+ \mathrm{i} \, L\bigl(x + x_{j_u M_u, j_\ell M_\ell},a\bigr) \biggr) \, ,
\label{ProfRelIm}$$ where $H$ and $L$ are the Voigt profile and the associated dispersion profile, respectively (see LL04 for their explicit expressions). They contain the damping parameter $a = \Gamma/(4 \pi \Delta\nu_D)$, where $\Gamma$ is the line-broadening parameter. Note that $\Gamma = \Gamma_R + \Gamma_E + \Gamma_I$, where $\Gamma_R$ is the radiative de-excitation rate, which corresponds to the Einstein coefficient for spontaneous emission $A(\beta_u L_u S \rightarrow \beta_\ell L_\ell S)$, and $\Gamma_I$ and $\Gamma_E$ are the de-excitation rates due to inelastic and elastic collisions, respectively. The discussion presented below concerns frequencies far from line center, for which the condition $x^2 + a^2 \gg 1$ is fulfilled and so one can take the asymptotic expansion (see LL04) for the $H$ and $L$ to the lowest order in $x$, $$H(x,a) \sim \frac{1}{\sqrt{\pi}} \frac{a}{x^2 + a^2} \, , \quad \quad \; L(x,a) \sim \frac{1}{\sqrt{\pi}} \frac{x}{x^2 + a^2} \, .
\label{AsympExpan}$$
B. Particular case: The two-level atom for a $0 - 1$ transition {#App2L}
===============================================================
Before considering a two-term atomic model with arbitrary values for $L_u$, $L_\ell$, and $S$, let us first consider the particular case in which $S = 0$ so that $J_u = L_u$ and $J_\ell = L_\ell$ (corresponding to the case of a two-level atomic model). In a reference frame such that the quantization axis is taken along the direction of the magnetic field, the elements of the propagation matrix given in Eqs. and can be written in a more compact form by introducing the generalized profile $\Phi^{K, K^\prime}_Q$ and the generalized dispersion profile $\Psi^{K, K^\prime}_Q$ [e.g., @LandiDeglInnocenti+91a], yielding $$\eta_i(x,\boldsymbol{\Omega}) =
k_M \sum_{K} {\mathcal T}^K_0(i,\boldsymbol{\Omega}) \, \Phi^{0,K}_0(J_\ell, J_u, x) \, , \quad \quad
\rho_i(x,\boldsymbol{\Omega}) =
k_M \sum_{K} {\mathcal T}^K_0(i,\boldsymbol{\Omega}) \Psi^{0,K}_0(J_\ell, J_u, x) \, .
\label{2lev}$$ By selecting the reference direction for positive Stokes $Q$ so that the $\eta_U$ and $\rho_U$ coefficients are zero, the previous expression can be given explicitly in terms of the angle $\alpha$ between the direction of propagation and the magnetic field as $$\begin{aligned}
& \eta_I(x,\boldsymbol{\Omega}) = k_M \, \Bigl(\Phi^{0,0}_0(J_\ell,J_u;x) + \frac{\sqrt{2}}{4} (3\cos^2\alpha - 1) \; \Phi^{0,2}_0(J_\ell,J_u;x)
\Bigr) \, , \notag \\
& \eta_Q(x,\boldsymbol{\Omega}) = k_M \frac{3\sqrt{2}}{4} \sin^2\alpha \; \Phi^{\,0,\,2}_0(J_\ell,J_u;x) \, ,
\quad \quad \quad \rho_Q(x, \boldsymbol{\Omega}) = k_M \frac{3 \sqrt{2}}{4} \,\sin^2\alpha \; \Psi^{\,0,\,2}_0(J_\ell,J_u;x) \, . \notag \\
& \eta_V(x,\boldsymbol{\Omega}) = k_M \frac{\sqrt{6}}{2} \cos\alpha \; \Phi^{\,0,\,1}_0(J_\ell,J_u;x) \, ,
\; \quad \quad \quad \rho_V(x, \boldsymbol{\Omega}) = k_M \frac{\sqrt{6}}{2} \cos\alpha \; \Psi^{\,0,\,1}_0(J_\ell,J_u;x) \, .
\label{Prop2lev}\end{aligned}$$ Taking also $J_u = 1$ and $J_\ell = 0$, as in the two-level model considered in previous sections, the generalized profiles and generalized dispersion profiles can be written as $$\begin{aligned}
& \Phi^{\,0,\,0}_0(0,1;x) = \frac{1}{3} \, \Bigl[\phi_1 + \phi_0 + \phi_{-1} \Bigr] \, ,
& \Phi^{\,0,\,1}_0(0,1;x) = \frac{\sqrt{6}}{6} \, \Bigl[\phi_1 - \phi_{-1} \Bigr] \, , \quad \quad
& \Phi^{\,0,\,2}_0(0,1;x) = \frac{\sqrt{2}}{6} \, \Bigl[\phi_1 - 2 \phi_0 + \phi_{-1} \Bigr] \notag \\
& \Psi^{\,0,\,0}_0(0,1;x) = \frac{1}{3} \, \Bigl[\psi_1 + \psi_0 + \psi_{-1} \Bigr] \, ,
& \Psi^{\,0,\,1}_0(0,1;x) = \frac{\sqrt{6}}{6} \, \Bigl[\psi_1 - \psi_{-1} \Bigr] \, , \quad \quad
& \Psi^{\,0,\,2}_0(0,1;x) = \frac{\sqrt{2}}{6} \, \Bigl[\psi_1 - 2 \psi_0 + \psi_{-1} \Bigr] \, .\end{aligned}$$ We note that, for such a two-level atomic model, the transition strengths introduced in Equation can simply be written as $$\begin{aligned}
S^{M_u, M_\ell}_q = 3 \left(\begin{array}{c c c}
J_u & J_\ell & 1 \\
- M_u & M_\ell & -q
\end{array} \right)^2 \, .
\label{2LevTRS}\end{aligned}$$ Therefore, one can easily see that, in the case that $J_u = 1$ and $J_\ell = 0$, the profiles $\phi_q$ and $\psi_q$ given in Equation can also be given in the following, more compact, form: $$\phi_q = \frac{1}{\sqrt{\pi}\Delta\nu_D} H\bigl(x + \bar{x}_{q}, a \bigr) \, , \quad \quad
\psi_q = \frac{1}{\sqrt{\pi}\Delta\nu_D} L\bigl(x + \bar{x}_{q}, a \bigr) \, .$$ Using these expressions for the $\phi_q$ and $\psi_q$ profiles together with Equation , valid when considering spectral regions far from the line core, one can write the generalized profiles and generalized dispersion profiles as a sum of fractions of polynomials as $$\begin{aligned}
& \Phi^{0,0}_0(0,1;x) \sim \frac{a}{3 \pi \Delta\nu_D} \left[\frac{1}{a^2 + \bigl(x + \bar{x}_{1} \bigr)^2} + \frac{1}{a^2 + x^2}
+ \frac{1}{a^2 + \bigl(x + \bar{x}_{-1} \bigr)^2} \right] \, , \\
& \Phi^{0,1}_0(0,1;x) \sim \frac{\sqrt{6} \,a}{6\pi\Delta\nu_D} \left[\frac{1}{a^2 + \bigl(x + \bar{x}_{1}\bigr)^2} - \frac{1}{a^2 + \bigl(x + \bar{x}_{-1}\bigr)^2} \right] \, , \\
& \Phi^{0,2}_0(0,1;x) \sim \frac{\sqrt{2} \,a}{6\pi\Delta\nu_D} \left[\frac{1}{a^2 + \bigl(x + \bar{x}_{1}\bigr)^2} - \frac{2}{a^2 + x^2}
+ \frac{1}{a^2 + \bigl(x + \bar{x}_{-1}\bigr)^2} \right] \, , \\
& \Psi^{0,0}_0(0,1;x) \sim \frac{1}{3 \pi \Delta\nu_D} \left[\frac{x + \bar{x}_{1}}{a^2 + \bigl(x + \bar{x}_{1}\bigr)^2} + \frac{x}{a^2 + x^2}
+ \frac{x + \bar{x}_{-1}}{a^2 + \bigl(x + \bar{x}_{-1}\bigr)^2} \right] \, , \\
& \Psi^{0,1}_0(0,1;x) \sim \frac{\sqrt{6}}{6\pi\Delta\nu_D} \left[\frac{x + \bar{x}_{1}}{a^2 + \bigl(x + \bar{x}_{1}\bigr)^2} - \frac{x + \bar{x}_{-1}}{a^2 + \bigl(x + \bar{x}_{-1}\bigr)^2} \right] \, , \\
& \Psi^{0,2}_0(0,1;x) \sim \frac{\sqrt{2}}{6\pi\Delta\nu_D} \left[\frac{x + \bar{x}_{1}}{a^2 + \bigl(x + \bar{x}_{1}\bigr)^2} - \frac{2 x}{a^2 + x^2}
+ \frac{x + \bar{x}_{-1}}{a^2 + \bigl(x + \bar{x}_{-1}\bigr)^2} \right] \, . \end{aligned}$$ Summing the various terms in the square parenthesis, each of the previous profiles can be expressed as a single ratio of polynomials. Taking the leading order in $x$ in the numerator and denominator, one reaches the following limits for their ratios over $\Phi^{0,0}_0(0,1;x)$, $$\begin{aligned}
\frac{\Phi^{0, 0}_0(0,1;x)}{\Phi^{0,0}_0(0,1;x)} = 1 \, ,
& \quad \quad \quad \quad \quad \lim_{x \to \infty} \frac{\Phi^{0, 1}_0(0,1;x)}{\Phi^{0,0}_0(0,1;x)} \to 0 \, ,
& \lim_{x \to \infty} \frac{\Phi^{0, 2}_0(0,1;x)}{\Phi^{0,0}_0(0,1;x)} \to 0 \, , \\
\lim_{x \to \infty} \frac{\Psi^{0, 0}_0(0,1;x)}{\Phi^{0,0}_0(0,1;x)} \to \infty \, ,
& \quad \quad \quad \quad \quad \lim_{x \to \infty} \frac{\Psi^{0, 1}_0(0,1;x)}{\Phi^{0,0}_0(0,1;x)} \to \frac{\sqrt{6}}{3} \frac{\nu_L}{a \Delta\nu_D} \, ,
& \lim_{x \to \infty} \frac{\Psi^{0, 2}_0(0,1;x)}{\Phi^{0,0}_0(0,1;x)} \to 0 \, .\end{aligned}$$ Thus, the only off-diagonal element of the propagation matrix [that]{}, divided by $\eta_I$, does not eventually fall to zero as $x$ increases is $\rho_V/\eta_I$. This ratio instead reaches the constant value $$\lim_{x\to\infty} \frac{\rho_V(x,\boldsymbol{\Omega})}{\eta_I(x,\boldsymbol{\Omega})} \to \frac{4 \pi \nu_L \cos\alpha}{\Gamma} \, .
\label{FarWingRVEI}$$ One immediate conclusion is that, far enough from the line center, the $\rho_V/\eta_I$ ratio is independent of the Doppler width of the line and it scales linearly with the magnetic field strength. In the absence of collisions, $\Gamma$ simply becomes the Einstein coefficient for spontaneous emission $A(\beta_u L_u S J_u \rightarrow \beta_\ell L_\ell S J_\ell)$. Interestingly, [the]{} onset of the Hanle effect is [likewise]{} determined by [the ratio of the Larmor frequency associated to the ambient magnetic field $\nu_L$ over the line-broadening parameter $\Gamma$]{}. [For a two-level atom the efficacy of the Hanle effect is characterized by the parameter]{} $H_u = (2 \pi \nu_L g_u)/A(\beta_u L_u S J_u \rightarrow \beta_\ell L_\ell S J_\ell)$, [where $g_u$ is the Landé factor of the upper level]{}, [and for]{} which the role played by collisions has also been neglected. This illustrates why one should expect the modification of the scattering polarization signatures in the line core (due to the Hanle effect) and in the line wings (produced by magneto-optical effects) to become significant at similar magnetic field strengths. Furthermore, the relation between the magnetic field strength (through the Larmor frequency) and $\bar{x}_q$, given in Equation , implies that $\nu_L = \bigl(\bar{\nu}_{-1} - \bar{\nu_1} \bigr)/2$. Thus, the far-wing limit given in Equation can be directly related to the frequency separation between the centers of gravity of the $\sigma_b$ and $\sigma_r$ components as $$\lim_{x\to\infty} \frac{\rho_V(x,\boldsymbol{\Omega})}{\eta_I(x,\boldsymbol{\Omega})} \to
\frac{2 \pi \, \bigl(\bar{\nu}_{-1} - \bar{\nu}_{1} \bigr) \cos\alpha}{\Gamma} \, .
\label{RhoLimIllust}$$
C. The two-term atom in the incomplete Paschen-Back regime {#AppGeneral2T}
==========================================================
We can now generalize the results presented in Appendix. \[App2L\] to the case of a two-term atom with arbitrary values of $S$, $L_u$, and $L_\ell$, accounting for the IPB effect. Taking a reference frame for which the quantization axis is along the magnetic field direction, the elements of the propagation matrix given in Eqs. and can be rewritten as
& \_I = k\_M ,\
& \_V = k\_M \_q (-1)\^[1+q]{} (
[c c c]{} 1 & 1 & 1\
q & -q & 0
) \_q(x) , \_Q = k\_M \^2 \_q (-1)\^[1+q]{} (
[c c c]{} 1 & 1 & 2\
q & -q & 0
) \_q(x) ,\
& \_V = k\_M \_q (-1)\^[1+q]{} (
[c c c]{} 1 & 1 & 1\
q & -q & 0
) \_q(x) , \_Q = k\_M \^2 \_q (-1)\^[1+q]{} (
[c c c]{} 1 & 1 & 2\
q & -q & 0
) \_q(x) .
Considering a frequency far [enough]{} from the line center [that]{} the asymptotic expansion in Equation can be applied to the absorption profiles, the $\phi_q$ and $\psi_q$ profiles become $$\phi_q(x) = \sum_{r = 1}^N \frac{a}{\pi \Delta\nu_D} \, S^{r}_q \frac{1}{a^2 + \bigl(x + x_r \bigr)^2} \, , \quad \quad
\psi_q(x) = \sum_{r = 1}^N \frac{1}{\pi \Delta\nu_D} \, S^{r}_q \frac{x + x_r}{a^2 + \bigl(x + x_r \bigr)^2 } \, . \label{FarQB}$$ The label $r$ stands for the set of quantum numbers ($j_u$, $M_u$, $j_\ell$, $M_\ell$) [that correspond]{} to the transition between states $\ket{\beta_u L_u S j_u M_u}$ and $\ket{\beta_\ell L_\ell S j_\ell M_\ell}$ and $N$ is the total number of distinct transitions between the two terms. As in the derivation presented in the previous section, the ratios of polynomials appearing in the profiles can be summed into a single ratio. In order to obtain the expressions for the elements of the propagation matrix presented below, which are valid where $x \gg 1$, we have used the identities $$\begin{aligned}
\sum_{r=1}^N S^r_q = 1 \, , \;\quad \sum_{r=1}^N S^r_q \, x_r \equiv \bar{x}_q = -q \,\bigl(\nu_L/\Delta\nu_D\bigr) \, , \, \quad \sum_{s\ne r} x_s = - x_r \, , \end{aligned}$$ We recall that the last equality in the second identity holds in the IPB effect regime, while the spectral shifts $\bar{x}_q$ are instead overestimated when the LZS approximation is made. We have also used the following useful relations for the Racah algebra $3j$ symbols: $$\begin{aligned}
& \sum_q (-1)^{1+q}
\left(\begin{array}{c c c}
1 & 1 & 0 \\
q & -q & 0
\end{array}\right)
= \sqrt{3} \, ,
\quad \quad
\sum_q (-1)^{1+q}
\left(\begin{array}{c c c}
1 & 1 & 1 \\
q & -q & 0
\end{array}\right)
= 0 \, ,
\quad \quad \quad \;\;\;
\sum_q (-1)^{1+q}
\left(\begin{array}{c c c}
1 & 1 & 2 \\
q & -q & 0
\end{array}\right)
= 0 \, , \\
& \sum_q (-1)^{1+q} \, q
\left(\begin{array}{c c c}
1 & 1 & 0 \\
q & -q & 0
\end{array}\right)
= 0 \, ,
\quad \quad
\sum_q (-1)^{1+q} \, q
\left(\begin{array}{c c c}
1 & 1 & 1 \\
q & -q & 0
\end{array}\right)
= \frac{\sqrt{6}}{3} \, ,
\quad \quad
\sum_q (-1)^{1+q} \, q
\left(\begin{array}{c c c}
1 & 1 & 2 \\
q & -q & 0
\end{array}\right)
= 0 \, .
\end{aligned}$$ Taking only the leading orders in $x$ for both the numerator and denominator, after some tedious algebra one reaches the following expressions for the elements of the propagation matrix
$$\begin{aligned}
& \eta_I(x,\boldsymbol{\Omega}) \approx k_M \frac{1}{\pi \Delta\nu_D} \frac{a}{x^2} \, , \\
& \eta_V(x,\boldsymbol{\Omega}) \approx k_M \frac{2\, a}{\pi \Delta\nu_D} \, \frac{1}{x^3} \, \frac{\nu_L}{\Delta\nu_D} \cos\alpha \, ,
\quad \quad \; \eta_Q(x,\boldsymbol{\Omega}) \approx k_M \frac{\sqrt{30}}{4 \pi \Delta\nu_D} \, \frac{a}{x^4} \sin^2\alpha \sum_q (-1)^{1+q}
\left(\begin{array}{c c c}
1 & 1 & 2 \\
q & -q & 0
\end{array} \right) \mathrm{v}_q \, , \\
& \rho_V(x,\boldsymbol{\Omega}) \approx k_M \frac{1} {\pi \Delta\nu_D} \, \frac{1}{x^2} \, \frac{\nu_L}{\Delta\nu_D} \cos\alpha \, ,
\quad \quad \; \rho_Q(\nu,\boldsymbol{\Omega}) \approx k_M \frac{\sqrt{30}}{4 \pi \Delta\nu_D} \, \frac{1}{x^3} \sin^2\alpha \sum_q (-1)^{1+q}
\left(\begin{array}{c c c}
1 & 1 & 2 \\
q & -q & 0
\end{array} \right) \mathrm{w}_q \, ,
\end{aligned}$$
\[PropMatElemLeading\]
where $$\mathrm{v}_q = \sum_{r=1}^N S^r_q \, \biggl(\sum_{s\ne r} x^2_s + 4 \sum_{s\ne r} \, x_s \sum_{\substack{t>s \\ t\ne r}} \, x_t \biggr) \, , \quad \quad \quad
\mathrm{w}_q = \sum_{r=1}^N S^r_q \, \biggl(\sum_{s\ne r} x^2_s + 4 \sum_{s\ne r} \, x_s \sum_{\substack{t>s \\ t\ne r}} \, x_t - 2 \, x^2_r \biggr) \, .$$ It is immediate to realize that, also for a two-term atom with arbitrary values of $L_u$, $L_\ell$, and $S$, the only coefficient in the propagation matrix whose ratio over $\eta_I$ does not fall to zero when $x \to \infty$ is $\rho_V$. Moreover, the expressions relating such ratio to the Larmor frequency and to the spectral distance between the centers of gravity of the $\sigma_b$ and $\sigma_r$ components are also recovered exactly as given in Eqs. and , respectively. It [should be emphasized]{} that this proof is based on the relation $\bar{x}_q = -q \, \nu_L/\Delta \nu_D$, which is strictly valid in the IPB effect regime. In contrast, making the LZS approximation may introduce significant errors in the determination of the far-wing value of the $\rho_V/\eta_I$ relation, even in the presence of relatively weak magnetic fields.
[17]{} natexlab\#1[\#1]{}
, E., [Belluzzi]{}, L., & [Trujillo Bueno]{}, J. 2016, , 831, L15
—. 2017, , 836, 6
—. 2018, , 854, 150
, L., & [Trujillo Bueno]{}, J. 2014, , 564, A16
, L., [Trujillo Bueno]{}, J., & [[Š]{}t[ě]{}p[á]{}n]{}, J. 2012, , 755, L2
, E. U., & [Shortley]{}, G. H. 1935, The Theory of Atomic Spectra (Cambridge: Cambridge University Press)
, T., [Casini]{}, R., & [Manso Sainz]{}, R. 2016, , 830, L24
, J. M., [Avrett]{}, E. H., & [Loeser]{}, R. 1993, , 406, 319
, R., [Trujillo Bueno]{}, J., [Winebarger]{}, A., [et al.]{} 2017, , 839, L10
, E. 1983, , 85, 3
—. 2014, [Atomic Spectroscopy and Radiative Processes]{} (Verlag Mailand: Springer)
, E., [Bommier]{}, V., & [Sahal-Brechot]{}, S. 1991, , 244, 391
, E., & [Landolfi]{}, M. 2004, Polarization in Spectral Lines (Dordrecht: Kluwer Academic Publishers)
, N., & [Butler]{}, K. 2004, , 609, 1181
, H., [Trujillo Bueno]{}, J., & [Landi Degl’Innocenti]{}, E. 2004, , 612, 1175
, J., [[Š]{}t[ě]{}p[á]{}n]{}, J., & [Casini]{}, R. 2011, , 738, L11
, J., [[Š]{}t[ě]{}p[á]{}n]{}, J., [Belluzzi]{}, L., [et al.]{} 2018, , 866, L15
, J., [Trujillo Bueno]{}, J., [Leenaarts]{}, J., & [Carlsson]{}, M. 2015, , 803, 65
[^1]: [The principle of spectroscopic stability is often stated as follows (see Sect 10.17 of LL04): “If two different descriptions are used to characterize a quantum system – a detailed description which takes an inner quantum number into account and a simplified description which disregards it – the predicted results must be the same in all physical experiments where the structure described by the inner quantum number is unimportant.”]{}
[^2]: The exact spectral position of the maximum of the linear polarization fraction has a slight dependence on the LOS and on the magnetic field under consideration.
[^3]: The Hanle critical field of the hydrogen Ly$\alpha$ line, i.e., the magnetic field strength at which the Zeeman splitting of the level with $J = 3/2$ is equal its natural width, is [approximately]{} $53$ G.
[^4]: From symmetry considerations it can be seen that the two following possible scenarios fulfil this condition: (a) a magnetic field distribution with axial symmetry around a given axis that is perpendicular to the LOS, and (b) a distribution with axial symmetry around any given axis, having also reflective symmetry with respect to the plane normal to the same axis. The configuration presented in the left panel of Figure \[Fig5\] is a particular case of the latter.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
[Wei Gao$^{1}$, Weifan Wang$^{2}$, Juan L.G. Guirao$^{3}$[^1]]{}\
[1. School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China]{}\
[2. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China]{}\
[3. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, ]{}\
[Hospital de Marina, 30203-Cartagena, Región de Murcia, Spain]{}\
title: '[**The Extension Degree Conditions for Fractional Factor**]{}'
---
\[theorem\][Definition]{} \[theorem\][Proposition]{}
=0.5cm
[**Abstract:**]{} In Gao’s previous work, the authors determined several graph degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if $b=f(x)=g(x)=a$ for all vertices $x$ in $G$. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference $\Delta$ between $g(x)$ and $f(x)$ for every vertex $x$ in $G$. These obtained new degree conditions reformulate Gao’s previous conclusions, and show how $\Delta$ acts in the results. Furthermore, counterexamples are structured to reveal the sharpness of degree conditions in the setting $f(x)=g(x)+\Delta$.
[**]{} [fractional factor, degree condition, independent set]{}
[**]{} [05C70.]{}
Introduction
============
In many engineering applications, their mathematical models can be expressed as a (direct or undirect) graph. For example, we look upon the network as a graph. Some correspondences are given here: the site matches with a vertex and the channel matches with an edge in the graph. In conventional network, the mission of data transmission is based on the selection of the shortest way between vertices. However, the computation of network flow in software definition network determines the data transmission. It chooses the path that is least congested at present. In this view, the pattern of data transmission problem in SDN setting is just the existence of fractional factor in the corresponding graph.
Graph $G=(V,E)$ mentioned here are all simple graph with its edge set $E(G)$ and its vertex set $V(G)$. Throughout this paper, we set $n=|V(G)|$ as the order of graph. For a vertex $x$ in $G$, $N_{G}(x)$ and $d_{G}(x)$ are used to denote the neighborhood and the degree of $x$ in $G$, respectively. Let $N_{G}[x]=N_{G}(x)\cup \{x\}$. To simplify, we use $N(x)$,$d(x)$ and $N[x]$ to express $d_{G}(x)$, $N_{G}(x)$ and $N_{G}[x]$, respectively. Set $\delta(G)$ as the minimum degree of $G$. We set $G[S]$ as the sub-graph of $G$ deduced from $S\subseteq V(G)$, and $G-S=G[V(G) \backslash
S]$. Set $e_{G}(S_{1},S_{2})=|\{e=v_{1}v_{2}|v_{1}\in S_{1},v_{2}\in S_{2}\}|$ for any $S_{1},S_{2}\subseteq V(G)$ with $S_{1}\cap S_{2}=\emptyset$. Denote $\sigma_{2}(G)=\min\{d_{G}(u)+d_{G}(v)|uv\notin E(G)\}$. The other terms used without clear definitions here can be refered to classic graph theory book [@Bondy08].
Functions $f$ and $g$ are two integer-valued defined on $V(G)$ satisfying $f(x)\ge g(x)\ge0$ for all vertices $x$ in $G$. A [*fractional $(g,f)$-factor*]{} is regarded as a score function $h$ which maps to every element in $E(G)$ a real number belongs to \[0,1\]. As a result, for every vertex $x$ we get $g(x)\le d_{G}^{h}(x)\le f(x)$, and $\sum\limits_{e\in E(x)}h(e)=d_{G}^{h}(x)$ where $E(x)=\{y|yx\in E(G)\}$. Fractional $f$-factor is regarded as a special case of fractional $(g,f)$-factor if the values of two functions are equal for any vertex $x$ in $G$. Fractional $[a,b]$-factor is another special case of fractional $(g,f)$-factor if $f(x)=b$, $g(x)=a$ for any vertex $x$ in $G$. In addition, if the value of both $f$ and $g$ equal to $k\in\Bbb N$ for any vertex $x$ in $G$, then it’s a fractional $k$-factor.
A fractional $(g,f,m)$-deleted graph and a fractional $(g,f,n')$-critical graph imply the existence of fractional factor in special setting when delete $m$ edges and $n'$ vertices, respectively. As the combination of the above two concepts, Gao [@Gao12] introduced fractional $(g,f,n',m)$-critical deleted graph to denote a graph to be fractional $(g,f,m)$-deleted after removing any $n'$ vertices. When functions $g$ and $f$ take special value for all vertices, the fractional $(g,f,n',m)$-critical deleted graph becomes various names which are presented in Table \[table1\].
setting (for any $v\in V(G)$) name
------------------------------- ------------------------------------------------
$g(x)=f(x)$ fractional $(f,n',m)$-critical deleted graph
$f(x)=b$ and $g(x)=a$ fractional $(a,b,n',m)$-critical deleted graph
$f(x)=g(x)=k$ fractional $(k,n',m)$-critical deleted graph
: Special cases of fractional $(g,f,n',m)$-critical deleted graph[]{data-label="table1"}
Several recent contributions in this topic were presented in Zhou et al. [@5], [@4], [@7], [@6] and [@8], and Gao et al. [@add1], [@add2], [@add4] and [@add3], Knor et al. [@AMNS1], and Liu et al. [@AMNS2].
In Zhou [@Zhou15] and Zhou et al. [@zhouetalinpress], the setting was different from the previous situations in which there is a difference $\Delta$ between $g(x)$ and $f(x)$ for every vertex $x$ in $G$, i.e., $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every $x$ in $G$. We observe that if $\Delta=0$, then binding number (minimum $\frac{|N(X)|}{|X|}$ where $\emptyset\ne X\subset V(G)$) condition for ID-$(g,f)$-factor-critical graph (this concept will be explain later) is $$bind(G)>\frac{(n-1)(b+2a-1)}{an-(b+a-2)}.$$ After adding the variable $\Delta$ (i.e., $a\le g(x)\le
f(x)-\Delta\le b-\Delta$), by the conclusion obtained by Zhou et al. [@zhouetalinpress], the binding number condition becomes $$bind(G)>\frac{(n-1)(b+2a-1+\Delta)}{(a+\Delta)n-(b+a-2)}.$$ This fact reveals that if the setting changes, the lower bound of binding number for ID-$(g,f)$-factor-critical graph is changed as well, and the new binding number heavily depends on $\Delta$. There is one thing we must emphasize here is that all the results in this paper are independent from the maximum degree of the graph, and $\Delta$ is only used to represent the difference between $g$ and $f$ throughout the article.
In our article, we consider the degree condition for the existence of fractional factors in the setting that $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$. Intuitively, in our new setting, the new degree conditions should be relied on the variable $\Delta$, or at least the new degree conditions are different from the previous ones. Thus, it inspired us to strictly study it theoretically.
In the following context, we first present the major results of part one in fractional $(g,f,n',m)$-critical deleted setting and prove it in details which extended Theorem 1-3 raised in Gao et. al. [@Gaowei4], perspectively.
\[theorem1\] Assume $G$ is a graph with $n$ vertices, and set $b,a,n',m$, and $\Delta$ as non-negative integers meeting $b-\Delta\ge a\ge2$ and $n>
\frac{(b+a+2m-2)(b+a)}{\Delta+a}+n'$. Functions $g,f$ are integer-valued on its vertex set and $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$. Then $G$ is fractional $(g,f,n',m)$-critical deleted if $\delta(G)\ge\frac{(b-\Delta)n+(\Delta+a)n'}{b+a}$.
\[theorem2\] Assume $G$ is a graph with $n$ vertices, and set $b,a, n',m$, and $\Delta$ as non-negative integers meeting $b-\Delta\ge a\ge2$, $\delta(G)\ge m+n'+\frac{b(b-\Delta)}{\Delta+a}$ and $n>
\frac{(b+a+2m-1)(a+b)}{\Delta+a}+n'$. Functions $g,f$ as integer-valued on its vertex set and meet $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$ in $G$. Then $G$ is fractional $(g,f,n',m)$-critical deleted if for any $xy\ne E(G)$, we have $$\max\{d_{G}(x),d_{G}(y)\}\ge\frac{(b-\Delta)n+(\Delta+a)n'}{b+a}.$$
\[theorem3\] Assume $G$ is a graph with $n$ vertices, and set $b,a, n',m$, and $\Delta$ as non-negative integers meeting $b-\Delta\ge a\ge2$, $\delta(G)\ge\frac{b(b-\Delta)}{\Delta+a}+m+n'$ and $n>
\frac{(b+a+2m-2)(a+b)}{\Delta+a}+n'$. Functions $g,f$ are integer-valued defined on the vertex set so that $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$ in $G$. Then $G$ is fractional $(g,f,n',m)$-critical deleted if $\sigma_{2}(G)\ge\frac{2(n(b-\Delta)+n'(\Delta+a))}{b+a}$.
The above three theorems manifest conditions for a graph to be fractional $(g,f,n',m)$-critical deleted from different aspects. The corollaries on fractional $(g,f,m)$-deleted graphs can be stated in Table \[table2\].
------------------------------------------------------------------------------------------------------------------------------------------
order of graph degree condition additional condition
----------------------------------- ------------------------------------------------------- ----------------------------------------------
$n> $\delta(G)\ge\frac{(b-\Delta)n}{b+a}$
\frac{(b+a+2m-2)(a+b)}{\Delta+a}$
$n> $\max\{d_{G}(x),d_{G}(y)\}\ge\frac{(b-\Delta)n}{b+a}$ $\delta(G)\ge\frac{b(b-\Delta)}{\Delta+a}+m$
\frac{(b+a+2m-1)(b+a)}{\Delta+a}$
$n> $\sigma_{2}(G)\ge\frac{2(b-\Delta)n}{b+a}$ $\delta(G)\ge\frac{b(b-\Delta)}{\Delta+a}+m$
\frac{(b+a+2m-2)(b+a)}{\Delta+a}$
------------------------------------------------------------------------------------------------------------------------------------------
: Three degree conditions of fractional $(g,f,m)$-deleted graph by setting $n'=0$ in above three theorems[]{data-label="table2"}
The data in Table \[table2\] can be regarded as the extension of Corollary 1, Corollary 2 and Corollary 3 in Gao et al. [@Gaowei4], respectively. Furthermore, we will further to discuss the relevant conditions in setting both $f$ and $g$ are constant functions in subsection \[subsection2.4\].
Set $d_{H}(T)=\sum_{x\in T}d_{H}(x)$ and $f(S)=\sum_{x\in S}f(x)$. The lemma below will be used in the demonstration process of our Theorem \[theorem1\]-\[theorem3\].
\[lemma1\][(Gao [@Gao12])]{} Assume $G$ is a graph, functions $f$ and $g$ are integer-valued on its vertex set meeting $f(x)\ge g(x)$ for every $x$ in $G$. Set $n'$, $m\in\Bbb N^{+}\cup\{0\}$. Then $G$ is fractional $(g,f,n',m)$-critical deleted iff $$\begin{aligned}
\label{1}
&\quad&f(S)-g(T)+d_{G-S}(T)\\\nonumber
&\ge&\max_{U\subseteq S, H\subseteq
E(G-U), |U|=n', |H|=m}\{f(U)+\sum_{x\in T}d_{H}(x)-e_{H}(T,S)\}\end{aligned}$$ for any subsets $S,T$ of $V(G)$ with $S\cap T=\emptyset$ and $|S|\ge n'$.
In very special circumstances, $n'$ vertices consist an independent set, then it comes to [*fractional ID-$(g,f,m)$-deleted graph*]{}. Analogously, when functions $g$ and $f$ take special value for all vertices, it becomes different names which are presented in Table \[table3\].
setting (for any $v\in V(G)$) name
------------------------------- ---------------------------------------------
$g=f$ fractional ID-$(f,m)$-deleted graph
$f(x)=b$ and $g(x)=a$ fractional ID-$(a,b,m)$-deleted graph
$m=0$ fractional ID-$(g,f)$-factor-critical graph
: Special cases of fractional ID-$(g,f,m)$-deleted graphs[]{data-label="table3"}
The following results in fractional ID-$(g,f,m)$-deleted setting as second part of main conclusions which are the extension of Theorem 4, Theorem 5 and Theorem 6 showed in Gao et al. [@Gaowei4], respectively.
\[theorem4\] Assume $G$ is a graph with $n$ vertices, and $b,a,m,\Delta$ are non-negative integers meeting $b-\Delta\ge a\ge2$ and $n>
\frac{(b+\Delta+2a)(b+2m+a-2)}{\Delta+a}$. Functions $g,f$ are integer-valued on its vertex set satisfy $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$. Then $G$ is fractional ID-$(g,f,m)$-deleted if $\delta(G)\ge\frac{(b+a)n}{b+2a+\Delta}$.
\[theorem5\] Assume $G$ is a graph with $n$ vertices, and $b,a,m,\Delta$ are non-negative integers meeting $b-\Delta\ge a\ge2$, $\delta(G)\ge\frac{(\Delta+a)n}{b+2a+\Delta}+\frac{b(b-\Delta)}{\Delta+a}+m$ and $n>
\frac{(b+a+2m-1)(b+2a+\Delta)}{\Delta+a}$. Functions $g,f$ as integer-valued on its vertex set satisfy $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$ in $G$. Then $G$ is fractional ID-$(g,f,m)$-deleted if for any $xy\ne E(G)$, we have $$\max\{d_{G}(y),d_{G}(x)\}\ge\frac{(b+a)n}{b+2a+\Delta}.$$
\[theorem6\] Assume $G$ is a graph with $n$ vertices, and $b,\Delta,a,m$ as non-negative integers meeting $b-\Delta\ge a\ge2$, $\delta(G)\ge\frac{(\Delta+a)n}{b+2a+\Delta}+\frac{b(b-\Delta)}{a+\Delta}+m$ and $n>
\frac{(b+2m+a-2)(b+\Delta+2a)}{\Delta+a}$. Functions $g,f$ are integer-valued on its vertex set satisfy $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$ in $G$. Then $G$ is fractional ID-$(g,f,m)$-deleted if $\sigma_{2}(G)\ge\frac{2(b+a)n}{b+2a+\Delta}$.
Proof of first part results: Theorem \[theorem1\]-\[theorem3\]
==============================================================
By observing, we find that $\delta(G)\ge\frac{(b-\Delta)n+(a+\Delta)n'}{a+b}$ in Theorem \[theorem1\] implies $\sigma_{2}(G)\ge\frac{2((b-\Delta)n+(a+\Delta)n')}{a+b}$ and $\delta(G)\ge n'+m+\frac{(b-\Delta)b}{\Delta+a}$ in Theorem \[theorem3\]. Hence, it’s sufficient to make Theorem \[theorem2\] and \[theorem3\] proved.
We deduce the conclusion on graph without non-adjacent vertices below.
\[lemma2\] Assume $G$ is a complete graph with $n$ vertices, and $b,\Delta,a, n',m$ as non-negative integers meeting $b-\Delta\ge a\ge 2$ and $n>\frac{(b+a+2m-2)(b+a)}{a+\Delta}+n'$. Functions $g,f$ as integer-valued on its vertex set satisfy $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$ in $G$. Then $G$ is fractional $(g,f,n',m)$-critical deleted.
[**Proof.**]{} Assume $G$ meets the conditions of Lemma \[lemma2\] without being fractional $(g,f,n',m)$-critical deleted. Clearly, $T\ne\emptyset$. According to Lemma \[lemma1\] and the fact that $\sum_{x\in
T}d_{H}(x)-e_{H}(T,S)$ at most $2m$, subsets $T$ and $S$ of $V(G)$ with $T\cap S\emptyset$ exist to satisfy $$f(S)+d_{G-S}(T)-g(T)\le \max_{U\subseteq S,|U|=n'}f(U)-1+2m$$ or $$\label{2}
f(S-U)-g(T)+d_{G-S}(T)\le2m-1,$$ in which $|S|\ge n'$. Choose $T$ and $S$ with minimum $|T|$. Hence, for every $x\in T$, we derive $b-1-\Delta\ge g(x)-1\ge d_{G-S}(x)$.
Note that $G-S$ is also complete for each vertex subset $S$. Thus, for disjoint subsets $T,S\subseteq V(G)$, we deduce $$\begin{aligned}
&\quad& f(S-U)-g(T)-2m+d_{G-S}(T)\\
&\ge&(|S|-n')(\Delta+a)+\sum_{x\in T}d_{G-S}(x)-(b-\Delta)|T|-2m\\
&\ge&(|S|-n')(\Delta+a)-(n-|S|)(b-n-\Delta+1+|S|)-2m\\
&=&(b+a-2n+1)|S|+|S|^{2}-(b-\Delta)n+n^{2}-n-(a+\Delta)n'-2m.\end{aligned}$$ Regarding it as the function of $|S|$, we look into the following cases in view of the fact that $|S|$ is an integer.
[**Case 1.**]{} $a+b$ is even. Since $n>\frac{(b+a+2m-2)(b+a)}{\Delta+a}+n'$ and $b+a\ge4$, we have $$\begin{aligned}
&\quad& (b+a+1-2n)|S|+|S|^{2}-(b-\Delta)n+n^{2}-2m-n-(a+\Delta)n'\\
&\ge&(n-\frac{a+b}{2})(b+a+1-2n)+(n-\frac{b+a}{2})^{2}-n-(b-\Delta)n+n^{2}-2m-(a+\Delta)n'\\
&=&(\Delta+a)n-\frac{b+a}{2}-(\frac{b+a}{2})^{2}-2m-(\Delta+a)n'\\
&>&(\frac{(b+a+2m-2)(b+a)}{\Delta+a}+n')(\Delta+a)-2m-\frac{b+a}{2}-(\frac{b+a}{2})^{2}-(\Delta+a)n'\\
&=&\frac{3}{4}(b+a)^{2}+(b+a-1)2m-\frac{5}{2}(b+a)\\
&\ge&\frac{3}{4}\cdot16-\frac{5}{2}\cdot4>0,\end{aligned}$$ which contradicts (\[2\]).
[**Case 2.**]{} $b-a\equiv 1$ (mod 2). By $n>\frac{(b+a+2m-2)(b+a)}{\Delta+a}+n'$ and $b+a\ge5$, we get $$\begin{aligned}
&\quad& (b+a+1-2n)|S|+|S|^{2}-(b-\Delta)n+n^{2}-n-(\Delta+a)n'-2m\\
&\ge& (n-\frac{a+b+1}{2})(b+a-2n+1)+(n-\frac{b+a+1}{2})^{2}\\
&\quad&-(b-\Delta)n+n^{2}-2m-n-(a+\Delta)n'\\
&=&(\Delta+a)n-2m-(\frac{b+a+1}{2})^{2}-(a+\Delta)n'\\
&>&(\Delta+a)(\frac{(b+a)(b+a+2m-2)}{\Delta+a}+n')-(\frac{b+a+1}{2})^{2}-(\Delta+a)n'-2m\\
&=&(b+a-1)2m+\frac{3}{4}(b+a)^{2}-\frac{1}{4}-\frac{5}{2}(b+a)\\
&\ge&\frac{3}{4}\cdot25-\frac{5}{2}\cdot5-\frac{1}{4}>0,\end{aligned}$$ a contradiction.
The proof of complete graph setting is done. [$\Box$ ]{}
Clearly, Lemma \[lemma2\] is the extension of previous conclusion on the complete graph marked in Lemma 2 of Gao et al. [@Gaowei4]. By setting $n'=0$ in Lemma \[lemma2\], the corollary present below will be employed in Section \[secondpartsection\].
\[corollary4\] Assume $G$ is a complete graph having $n$ vertices, and $b,\Delta,a,m$ as non-negative integers meeting $b-\Delta\ge a\ge2$ and $n>\frac{(b+a)(b+a+2m-2)}{a+\Delta}$. Functions $g,f$ are integer-valued on its vertex set satisfy $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for every vertex $x$ in $G$. Therefore, $G$ is fractional $(g,f,m)$-deleted.
Graph is supposed to be non-complete in what follows. From this point of view, the degree condition $\max\{d_{G}(y),d_{G}(x)\}\ge\frac{(b-\Delta)n+(a+\Delta)n'}{a+b}$ for every $xy\ne E(G)$ in Theorem \[theorem2\] and $\sigma_{2}(G)\ge\frac{2((b-\Delta)n+(a+\Delta)n')}{a+b}$ in Theorem \[theorem3\] are meaningful.
Correctness of Theorem \[theorem2\] {#subsection2.1}
-----------------------------------
Assume $G$ meets all the assumptions of Theorem \[theorem2\] without being fractional $(g,f,n',m)$-critical deleted. It can be inferred $|T|\ge1$, and there exist disjoint subsets $T,S\subseteq V(G)$ satisfies (\[2\]) with $|S|\ge n'$. We have $b-1-\Delta\ge g(x)-1\ge d_{G-S}(x)$ for all vertex $x$ in $T$ by means of selecting $S$ and $T$ with minimum $|T|$.
Let $d_{1}=\min\{d_{G-S}(x):x\in T\}$. We deduce $b-1-\Delta\ge d_{1}\ge0$ and $$f(S-U)+d_{G-S}(T)-g(T)\ge d_{1}|T|+(\Delta+a)(|S|-n')-(b-\Delta)|T|.$$ This implies $$\label{3}
2m-1\ge (|S|-n')(\Delta+a)-(b-\Delta-d_{1})|T|.$$ We choose vertex $x_{1}$ in $T$ to meet $d_{G-S}(x_{1})=d_{1}$.
If $|T|\le b-\Delta$, in terms of (\[3\]) and $|S|+d_{1}\ge
d_{G}(x_{1})\ge\delta(G)\ge\frac{b(b-\Delta)}{a+\Delta}+n'+m$, we verify $$\begin{aligned}
&\quad&2m-1\\
&\ge& (|S|-n')(\Delta+a)+|T|(\Delta+d_{1}-b)\\
&\ge& (a+\Delta)(\frac{b(b-\Delta)}{a+\Delta}+n'+m-d_{1}-n')+(\Delta+d_{1}-b)(b-\Delta)\\
&=&(b-a-\Delta)d_{1}+am+\Delta(m-d_{1}-\Delta+b)\\
&\ge&2m,\end{aligned}$$ which gets contradicted. Thus, $|T|\ge
b+1-\Delta\ge 1+a$.
On the condition that $T-N_{T}[x_{1}]\ne\emptyset$, set $d_{2}=\min\{d_{G-S}(x):x\in
T-N_{T}[x_{1}]\}$ and take vertex $x_{2}$ belongs to $T-N_{T}[x_{1}]$ such that $d_{G-S}(x_{2})=d_{2}$. Hence, $d_{1}\le d_{2}\le b-\Delta-1$. Since $b-\Delta-1\ge d_{G-S}(x)$ for any vertex $x$ in $T$ and $|T|\ge
b-\Delta+1$, $T-N_{T}[x_{1}]\ne\emptyset$, thus $x_{1}$, $x_{2}$ must be existed. Considering the non-adjacent vertices assumption, we deduce $$\frac{n'(a+\Delta)+n(b-\Delta)}{b+a}\le\max\{d_{G}(x_{1}),d_{G}(x_{2}))\}\le|S|+d_{2},$$ which reveals $$\label{4}
|S|\ge\frac{n'(a+\Delta)+n(b-\Delta)}{b+a}-d_{2}.$$
In view of $b-\Delta-d_{2}>0$ and $n-|S|-|T|\ge0$, we infer $$\begin{aligned}
&\quad&(n-|T|-|S|)(b-\Delta-d_{2})\\
&\ge&(\Delta+a)(|S|-n')+\sum_{x\in T}(d_{G-S}(x)-b+\Delta)+1-2m\\
&\ge&(d_{1}+\Delta-b)|N_{T}[x_{1}]|-2m+(\Delta+a)|S|+1\\
&\quad&+(d_{2}-b+\Delta)(|T|-|N_{T}[x_{1}]|)-(a+\Delta)n'\\
&=&(a+\Delta)|S|+(d_{1}-d_{2})|N_{T}[x_{1}]|+(d_{2}+\Delta-b)|T|-(a+\Delta)n'-2m+1\\
&\ge&(d_{1}-d_{2})(1+d_{1})+(a+\Delta)|S|+(\Delta+d_{2}-b)|T|-(a+\Delta)n'-2m+1.\end{aligned}$$
It follows that $$\label{5}
0\le
n(b-d_{2}-\Delta)-(b+a-d_{2})|S|+2m+(1+d_{1})(d_{2}-d_{1})+(a+\Delta)n'-1.$$
Using (\[4\]), (\[5\]), $n>
\frac{(a+b)(a+b+2m-1)}{a+\Delta}+n'$ and $d_{1}\le d_{2}\le b-1-\Delta$, we obtain $$\begin{aligned}
0&\le&(b-d_{2}-\Delta)n-(b+a-d_{2})(\frac{(a+\Delta)n'+(b-\Delta)n}{b+a}-d_{2})+(d_{1}+1)(d_{2}-d_{1})\\
&\quad&+(a+\Delta)n'-1+2m\\
&=&-nd_{2}\frac{a+\Delta}{b+a}+d_{2}\frac{(a+\Delta)n'}{a+b}+(b+a)d_{2}-d_{1}^{2}-d_{2}^{2}+d_{1}d_{2}+d_{2}-d_{1}+2m-1\\
&<&-d_{1}^{2}-d_{2}^{2}+d_{1}d_{2}+2d_{2}-d_{1}+2m(1-d_{2})-1.\\\end{aligned}$$
If $d_{2}=0$, then we have $d_{1}=d_{2}=0$. According to (\[4\]), we have $|S|\ge\frac{(b-\Delta)n+(a+\Delta)n'}{a+b}$ and $|T|\le n-|S|\le
\frac{(a+\Delta)n-(a+\Delta)n'}{a+b}$. By $\sum_{x\in
T}d_{H}(x)-e_{G}(T,S)\le d_{G-S}(T)$, we yield $$\begin{aligned}
&\quad&f(S-U)-g(T)+d_{G-S}(T)-(\sum_{x\in T}d_{H}(x)-e_{G}(T,S))\\
&\ge&(a+\Delta)\cdot(\frac{(b-\Delta)n+(\Delta+a)n'}{b+a}-n')-(b-\Delta)\cdot\frac{(\Delta+a)n-n'(\Delta+a)}{b+a}\\
&\quad&+e_{G}(T,S)+d_{G-S}(T)-\sum_{x\in T}d_{H}(x)\\
&\ge&0,\end{aligned}$$ a contradiction.
If $d_{2}\ge1$, we infer $$0<-d_{1}^{2}-d_{2}^{2}+d_{1}d_{2}+2d_{2}-d_{1}+2m(1-d_{2})-1\le-d_{2}^{2}+(d_{1}+2)d_{2}-d_{1}^{2}-d_{1}-1.$$ Let $$h_{1}(d_{2})=-d_{2}^{2}+(d_{1}+2)d_{2}-d_{1}^{2}-d_{1}-1.$$ This implies, $$\max\{h_{1}(d_{2})\}=h_{1}(\frac{d_{1}+2}{2})=-\frac{3}{4}d_{1}^{2}\le0,$$ which is a contradiction. Thus, we complete the derivation for the correctness.[$\Box$ ]{}
Correctness of Theorem \[theorem3\] {#subsection2.2}
-----------------------------------
Assume $G$ meets all the assumptions of Theorem \[theorem3\] without being fractional $(g,f,n',m)$-critical deleted. Apparently, $|T|\ge1$ and there exist $T,S\subseteq V(G)$ with $T\cap S=\emptyset$ satisfies (\[2\]) with $|S|\ge n'$. Selecting $T$ and $S$ with minimum $|T|$, we obtain $b-1-\Delta\ge g(x)-1-\Delta\ge d_{G-S}(x)$ for any vertex $x$ in $T$.
Set $d_{1}$, $d_{2}$, $x_{1}$ and $x_{2}$ as defined before. Similarly as discussed in Section 2.1, we yield $d_{1}\le d_{2}\le
b-\Delta-1$, $|T|\ge b-\Delta+1$ and $x_{1}$, $x_{2}$ must be existed.
By means of degree assumption, we arrive $$\frac{2(n(b-\Delta)+n'(\Delta+a))}{b+a}\le\sigma_{2}(G)\le2|S|+d_{2}+d_{1},$$ which reveals $$\label{6}
|S|\ge\frac{(a+\Delta)n'+(b-\Delta)n}{a+b}-\frac{d_{2}+d_{1}}{2}.$$
Using the consideration in Subsection \[subsection2.1\], (\[5\]) holds as well. In light of (\[5\]), (\[6\]), $n>
\frac{(b+a-2+2m)(b+a)}{\Delta+a}+n'$ and $d_{1}\le d_{2}\le b-1-\Delta$, we derive $$\begin{aligned}
0&\le&(d_{1}+1)(d_{2}-d_{1})+n(b-d_{2}-\Delta)-(b+a-d_{2})(\frac{n'(\Delta+a)+n(b-\Delta)}{b+a}-\frac{d_{1}+d_{2}}{2})\\
&\quad&+(\Delta+a)n'-1+2m\\
&=&d_{2}\frac{(\Delta+a)n'}{b+a}-nd_{2}\frac{\Delta+a}{b+a}+(b+a)\frac{d_{1}+d_{2}}{2}-d_{1}^{2}-\frac{d_{2}^{2}}{2}+\frac{d_{1}d_{2}}{2}+d_{2}-d_{1}+2m-1\\
&<&-d_{2}(a+b-3)+\frac{a+b}{2}(d_{1}+d_{2})-d_{1}^{2}-\frac{d_{2}^{2}}{2}+\frac{d_{1}d_{2}}{2}-d_{1}+2m(1-d_{2})-1.\\\end{aligned}$$
The case for $d_{2}=0$ can be proved in the similar way as Subsection \[subsection2.1\].
If $d_{2}\ge1$, then we verify $$\begin{aligned}
0&<&-d_{2}(a+b-3)+\frac{a+b}{2}(d_{1}+d_{2})-d_{1}^{2}-\frac{d_{2}^{2}}{2}+\frac{d_{1}d_{2}}{2}-d_{1}+2m(1-d_{2})-1\\
&\le&-\frac{d_{2}^{2}}{2}-d_{2}(\frac{a+b}{2}-3-\frac{d_{1}}{2})-d_{1}^{2}+(\frac{a+b}{2}-1)d_{1}-1.\\\end{aligned}$$ Let $$h_{2}(d_{2})=-\frac{d_{2}^{2}}{2}-d_{2}(\frac{a+b}{2}-3-\frac{d_{1}}{2})-d_{1}^{2}+(\frac{a+b}{2}-1)d_{1}-1.$$ If $d_{2}$ can reach to $3+\frac{d_{1}}{2}-\frac{a+b}{2}$ (i.e., $3+\frac{d_{1}}{2}-\frac{a+b}{2}\ge1$), then $$\max\{h_{2}(d_{2})\}=h_{2}(3+\frac{d_{1}}{2}-\frac{a+b}{2}),$$ and $d_{2}\le 1$ in terms of $b\ge a\ge2$ and $d_{1}\le b-1$. Hence, $(d_{1}, d_{2})=(0,1)$ or $d_{1}=d_{2}=1$. By $b\ge a\ge
2$, we get $h_{2}(d_{2})\le0$ for both $(d_{1},d_{2})=(0,1)$ and $(d_{1},d_{2})=(1,1)$, a contradiction.
If $d_{2}$ can’t take $3+\frac{d_{1}}{2}-\frac{a+b}{2}-\frac{1}{a+b}$ as its value, then we have $$\begin{aligned}
0&<&-\frac{d_{2}^{2}}{2}-d_{2}(\frac{a+b}{2}-3-\frac{d_{1}}{2})-d_{1}^{2}+(\frac{a+b}{2}-1)d_{1}-1\\
&\le&-\frac{d_{1}^{2}}{2}-d_{1}(\frac{a+b}{2}-3-\frac{d_{1}}{2})-d_{1}^{2}+(\frac{a+b}{2}-1)d_{1}-1\\
&=&-d_{1}^{2}+2d_{1}-1\le 0,\end{aligned}$$ which also gets contradicted. In result, Theorem \[theorem3\] is proven.[$\Box$ ]{}
Sharpness
---------
In this part, we give an example to prove the sharpness of the degree conditions in Theorem \[theorem1\]-\[theorem3\] in some sense. That is to say, the minimal condition $\delta(G)\ge\frac{(b-\Delta)n+(a+\Delta)n'}{a+b}$ can’t be changed to $\delta(G)\ge\frac{(b-\Delta)n+(a+\Delta)n'}{a+b}-1$; we can’t replace $\max\{d_{G}(y),d_{G}(x)\}\ge\frac{n(b-\Delta)+n'(a+\Delta)}{a+b}$ by $\max\{d_{G}(y),d_{G}(x)\}\ge\frac{n(b-\Delta)+n'(a+\Delta)}{a+b}-1$ in Theorem \[theorem2\]; and the degree sum condition $\sigma_{2}(G)\ge\frac{2((b-\Delta)n+(a+\Delta)n')}{a+b}$ in Theorem \[theorem3\] can’t be transferred to $\sigma_{2}(G)\ge\frac{2((b-\Delta)n+(a+\Delta)n')}{a+b}-1$.
Let $b=a+\Delta$, $G_{1}=K_{at+n'}$ be a complete graph, $G_{2}=(bt+1)K_{1}$, and $G=G_{1}\vee
G_{2}$, where $t\in\Bbb N$ is a large number which ensures the graph to meet $\delta(G)\ge m+n'+\frac{b(b-\Delta)}{\Delta+a}$ and $n>
\frac{(b+a-2+2m)(b+a)}{\Delta+a}+n'$), so $n=|G_{1}|+|G_{2}|=(a+b)t+1+n'$. Let $a=g(x)$ and $b=a+\Delta=f(x)$ for every vertex $x$ in $G$. We have $$\frac{n'(a+\Delta)+n(b-\Delta)}{b+a}
>\delta(G)=n'+at>\frac{n'(a+\Delta)+n(b-\Delta)}{b+a}-1,$$ $$\frac{n'(a+\Delta)+n(b-\Delta)}{b+a}
>\max\{d_{G}(y),d_{G}(x)\}
=n'+at>\frac{n'(a+\Delta)+n(b-\Delta)}{b+a}-1,$$ $$\frac{2(n'(a+\Delta)+n(b-\Delta))}{b+a}>\sigma_{2}(G)=2(at+n')
\ge\frac{2(n'(a+\Delta)+n(b-\Delta))}{b+a}-1.$$ Let $T=V(G_{2})$ and $S=V(G_{1})$, we get $$\begin{aligned}
&\quad&f(S)-g(T)+d_{G-S}(T)-\max_{U\subseteq S, |U|=n', H\subseteq
E(G-U), |H|=m}\{f(U)-e_{H}(T,S)+\sum_{x\in
T}d_{H}(x)\}\\
&=&b|S|-a|T|-(a+\Delta)n'=-a<0.\end{aligned}$$ According to Lemma \[lemma1\], $G$ isn’t fractional ($g,f,n',m$)-critical deleted.
Specific case in setting $(g,f)=(a,b)$ {#subsection2.4}
--------------------------------------
According to the techniques in the proof of Lemma \[lemma2\], we infer a likely conclusion for a graph without non-adjacent vertices.
\[lemma3\] Assume $G$ is a complete graph having $n$ vertices, and $b,n',a,
m,\Delta$ are non-negative integers meeting $n>\frac{(b+a-2+2m)(b+a)}{\Delta+a}+n'$ where $b-\Delta\ge a\ge2$. Then $G$ is fractional $(a,b,n',m)$-critical deleted.
We arrive the corollary below by setting $n'=0$ in Lemma \[lemma3\], which is a sufficient condition for a fractional $(a,b,m)$-deleted complete graph.
\[corollary5\] Assume complete graph $G$ having $n$ vertices, and $b,a,m,\Delta$ are non-negative integers meeting $n>\frac{(b+a-2+2m)(b+a)}{\Delta+a}$ where $b-\Delta\ge a\ge2$. Then $G$ is fractional $(a,b,m)$-deleted.
Note that Lemma \[lemma3\] and Corollary \[corollary5\] here are the extension results for the corresponding conclusions in Gao et al. [@Gaowei4].
Set $f(x)=b$, $g(x)=a$ for arbitrary vertex $x$ in $G$. The necessary and sufficient condition is achieved from Lemma \[lemma1\].
\[lemma4\] Assume $G$ is a graph, $b$, $a$, $n'$, and $m$ are non-negative integers meeting $b\ge a$. Therefore, $G$ is fractional $(a,b,n',m)$-critical deleted iff for any disjoint subsets $T,S\subseteq V(G)$ with $|S|\ge n'$, we have $$\label{7}
b|S|+d_{G-S}(T)-a|T|\ge\max_{ |H|=m}\{(a+\Delta)n'+\sum_{x\in
T}d_{H}(x)-e_{H}(T,S)\}.$$
Using Lemma \[lemma3\] and Lemma \[lemma4\], in view of the approaches used in Subsection \[subsection2.1\] and Subsection \[subsection2.2\], we deduce the degree conditions depicted in Table \[table4\] in fractional $(a,b,n',m)$-critical deleted setting, which are corresponding to Theorem \[theorem1\]-\[theorem3\]. We omit the detailed proof.
--------------------------------------------------------------------------------------------------------------------------------------------------------------
order of graph degree condition additional condition
-------------------------------------- -------------------------------------------------------------------- --------------------------------------------------
$n> $\delta(G)\ge\frac{(a+\Delta)n'+(b-\Delta)n}{b+a}$
\frac{(b+a-2+2m)(b+a)}{\Delta+a}+n'$
$n> $\max\{d_{G}(x),d_{G}(y)\}\ge\frac{(\Delta+a)n'+(b-\Delta)n}{b+a}$ $\frac{b(b-\Delta)}{\Delta+a}+m+n'\le \delta(G)$
\frac{(b+a-2+2m)(b+a)}{\Delta+a}+n'$
$n> $\sigma_{2}(G)\ge\frac{2(n'(\Delta+a)+n(b-\Delta))}{b+a}$ $\frac{b(b-\Delta)}{\Delta+a}+m+n'\le \delta(G)$
\frac{(b+a-2+2m)(b+a)}{\Delta+a}+n'$
--------------------------------------------------------------------------------------------------------------------------------------------------------------
: Degree conditions in fractional $(a,b,n',m)$-critical deleted setting []{data-label="table4"}
Again, three theorems above present the new extension versions of Theorem 7-9 in Gao et al. [@Gaowei4], respectively. Moreover, the example in Subsection \[sharpness\] shows that these degree conditions in Table \[table4\] are tight.
In particular, by taking $n'=0$ in Table \[table4\], the corresponding degree conditions in fractional $(a,b,m)$-deleted setting are obtained in Table \[table5\].
-----------------------------------------------------------------------------------------------------------------------------
order of graph degree condition additional condition
----------------------------------- ------------------------------------------------------- ---------------------------------
$n> $\delta(G)\ge\frac{(b-\Delta)n}{b+a}$
\frac{(b+a-2+2m)(b+a)}{\Delta+a}$
$n> $\max\{d_{G}(x),d_{G}(y)\}\ge\frac{(b-\Delta)n}{b+a}$ $\delta(G)\ge
\frac{(b+a-2+2m)(b+a)}{\Delta+a}$ \frac{b(b-\Delta)}{\Delta+a}+m$
$n> $\sigma_{2}(G)\ge\frac{2n(b-\Delta)}{a+b}$ $\delta(G)\ge
\frac{(b+a-2+2m)(b+a)}{\Delta+a}$ \frac{b(b-\Delta)}{\Delta+a}+m$
-----------------------------------------------------------------------------------------------------------------------------
: Degree conditions in fractional $(a,b,m)$-deleted setting[]{data-label="table5"}
Proof of second part results: Theorem \[theorem4\]-\[theorem6\] {#secondpartsection}
===============================================================
Since $\delta(G)\ge\frac{n(a+b)}{2a+\Delta+b}$ in Theorem \[theorem4\] implies $\delta(G)\ge\frac{(\Delta+a)n}{b+2a+\Delta}+m+\frac{(b-\Delta)b}{\Delta+a}$ and $\sigma_{2}(G)\ge\frac{2n(b+a)}{2a+\Delta+b}$ in Theorem \[theorem6\], it is sufficient for the proof of Theorem \[theorem5\]-\[theorem6\].
Correctness of Theorem \[theorem5\]-\[theorem6\] {#subsection3.1}
------------------------------------------------
Here, first let’s prove Theorem \[theorem5\]. Let $G'=G-I$ for arbitrary independent set $I$. The conclusion is deduced by making sure that $G'$ meets Table \[table2\] or Corollary \[corollary4\].
If every two vertices has an edge in $G'$, we obtain $$|G'|\ge\frac{n(b+a)}{b+2a+\Delta}>\frac{(b+a-1+2m)(b+a)}{a+\Delta}
>\frac{(b+2m+a-2)(b+a)}{\Delta+a}.$$ The conclusion holds in light of Corollary \[corollary4\].
If $I$ only contain one vertex, we yield $|V(G')|>\frac{(b+2m+a-1)(b+2a+\Delta)-\Delta-a}{\Delta+a}>\frac{(b+a-1+2m)(b+a)}{\Delta+a}$. Hence, $\delta(G')\ge\frac{(b-\Delta)b}{a+\Delta}+m$ and $$\max\{d_{G'}(u),d_{G'}(v)\}\ge\frac{|V(G')|(b-\Delta)}{b+a}=\frac{(n-1)(b-\Delta)}{b+a}$$ for any $uv\notin E(G')$. Hence, the result obtained in view of Table \[table2\].
If $|I|\ge2$ and $G'$ isn’t complete. Applying degree condition, we infer $|V(G')|\ge\frac{(b+a)n}{2a+b+\Delta}>\frac{(b+a)(b+2m+a-1)}{\Delta+a}$. If $\max\{d_{G'}(u),d_{G'}(v)\}<\frac{|V(G')|(b-\Delta)}{b+a}$ for some $uv\notin E(G')$, we arrive $$\frac{(|V(G')|+|I|)(b+a)}{b+2a+\Delta}\le
\max\{d_{G}(v),d_{G}(u)\}<\frac{|V(G')|(b-\Delta)}{b+a}+|I|,$$ which implies $$|V(G')|<\frac{(a+\Delta)(b+a)}{a^{2}+\Delta(2a+\Delta)}|I|\le\frac{(b+a)(a+\Delta)}{a^{2}+\Delta(2a+\Delta)}\frac{n(\Delta+a)}{2a+\Delta+b}=\frac{(b+a)n}{2a+\Delta+b}.$$ It contradicts $|I|\ge2$ and $\max\{d_{G}(v),d_{G}(u)\}\ge\frac{(b+a)n}{b+2a+\Delta}$. Thus, $$\max\{d_{G'}(v),d_{G'}(u)\}\ge\frac{|V(G')|(b-\Delta)}{b+a}$$ for any $uv\notin E(G')$. Further, we get $m+\frac{b(b-\Delta)}{\Delta+a}\le \delta(G')$ in view of $|I|\le\frac{(a+\Delta)n}{2a+b+\Delta}$ and $\frac{n(\Delta+a)}{b+2a+\Delta}+m+\frac{b(b-\Delta)}{a+\Delta}\le \delta(G)$. Therefore, the result is obtained from Table \[table2\].
Hence, we finish the proof of Theorem \[theorem5\]. By means of Table \[table2\] and Corollary \[corollary4\], Theorem \[theorem6\] can be checked in the similar techniques. We omit the detailed procedure. [$\Box$ ]{}
Tight of results {#tight}
----------------
To show the tight of Theorem \[theorem4\], Theorem \[theorem5\] and Theorem \[theorem6\], we need the following lemma follows from the corollary of Lemma \[lemma1\].
\[kkk\] Assume $G$ is a graph, functions $g,f$ are integer-valued on its vertex set meeting $f(x)\ge g(x)$ for every vertex $x$ in $G$. Set $m\in\Bbb N^{+}\cup\{0\}$. Then $G$ is fractional $(g,f,m)$-deleted iff for all disjoint subsets $T,S\subseteq V(G)$, we have $$f(S)+d_{G-S}(T)-g(T)\ge\max_{
|H|=m}\{\sum_{x\in T}d_{H}(x)-e_{H}(T,S)\}.$$
Let $b=a+\Delta$. Take $G=(bt+1)K_{1}\vee K_{at}\vee(bt+1)K_{1}$, where $t\in\Bbb N$ is a large number. Apparently, $n=2+(2b+a)t$. Set $f(x)=b$ and $g(x)=a$ for any vertex $x$ in $G$. We have $$\frac{(b+a)n}{b+\Delta+2a}>\delta(G)=(b+a)t+1>\frac{(b+a)n}{b+\Delta+2a}-1,$$ $$\frac{(b+a)n}{b+\Delta+2a}>\max\{d_{G}(u),d_{G}(v)\}=(b+a)t+1>\frac{(b+a)n}{b+\Delta+2a}-1,$$ $$\frac{2(b+a)n}{b+\Delta+2a}>\sigma_{2}(G)=2+2(b+a)t>\frac{2(b+a)n}{b+\Delta+2a}-1.$$ Let $I=(bt+1)K_{1}$. For $G'=K_{at}\vee(bt+1)K_{1}$, let $S=K_{at}$ and $T=(bt+1)K_{1}$. We confirm that $e_{H}(T,S)=\sum_{x\in T}d_{H}(x)$ for arbitrary $H\subseteq E(G')$ having $m$ edges. As a result, $$f(S)+d_{G-S}(T)-g(T)-(\sum_{x\in T}d_{H}(x)-e_{H}(T,S))=b(at)-a(bt+1)=-a.$$ To sum up, $G$ isn’t fractional ID-$(g,f,m)$-deleted due to Lemma \[kkk\] and $G'$ isn’t fractional $(g,f,m)$-deleted.
Specific case in setting $(g,f)=(a,b)$ {#lastsubsection}
--------------------------------------
The below degree conditions in Table \[table6\] in setting $g(x)=a$ and $f(x)=b$ are derived in terms of Corollary \[corollary5\], Table \[table5\], and the approaches in Subsection \[subsection2.4\] and Subsection \[subsection3.1\].
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
order of graph degree condition additional condition
------------------------------------------- ---------------------------------------------------------- ------------------------------------------------------------------------------
$n> $\delta(G)\ge\frac{n(b+a)}{2a+\Delta+b}$
\frac{(b+2a+\Delta)(b+a-2+2m)}{\Delta+a}$
$n> $\max\{d_{G}(x),d_{G}(y)\}\ge\frac{(b+a)n}{b+2a+\Delta}$ $\delta(G)\ge\frac{(\Delta+a)n}{b+2a+\Delta}+\frac{b(b-\Delta)}{\Delta+a}+m$
\frac{(b+2a+\Delta)(b+a+2m-1)}{\Delta+a}$
$n> $\sigma_{2}(G)\ge\frac{2(b+a)n}{b+2a+\Delta}$ $\delta(G)\ge\frac{(\Delta+a)n}{b+2a+\Delta}+\frac{b(b-\Delta)}{a+\Delta}+m$
\frac{(b+a-2+2m)(b+2a+\Delta)}{\Delta+a}$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Degree conditions in fractional ID-$(a,b,m)$-deleted setting[]{data-label="table6"}
One important thing we emphasize here is that the results presented in Table \[table6\] are also the extensions of Theorem 10-12 in Gao et. al. [@Gaowei4]. Moreover, in terms of the example presented in Subsection \[tight\], we ensure that these degree conditions in Table \[table6\] are also tight.
Conclusion
==========
In our work, we mainly discuss the degree conditions for the existence of fractional factor in the setting that $b-\Delta\ge f(x)-\Delta\ge g(x)\ge a$ for each vertex $x$ in $G$, and some elements of graph are forbidden. Our results reveal that $\Delta$ is a key factor in this setting, and it specifically points out how $\Delta$ plays a role in the conclusion.
Acknowledgments
===============
This research is partially supported by NSFC (Nos. 11761083, 11771402, 11671053).
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[^1]: Corresponding author: [email protected]
|
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---
abstract: 'From the more than two hundred partial orders for fuzzy numbers proposes in the literature, only a few are totals. In this paper, we introduce the notion of admissible orders for fuzzy numbers equipped with a partial order, i.e. a total order which refines the partial order. In particular, is given special attention when thr partial order is the proposed by Klir and Yuan in 1995. Moreover, we propose a method to construct admissible orders on fuzzy numbers in terms of linear orders defined for intervals considering a strictly increasing upper dense sequence, proving that this order is admissible for a given partial order.'
address:
- 'Departamento de Matemática y Física, Universidad de Magallanes, Punta Arenas, Chile'
- 'Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal, Brasil'
- 'Vicerrectoría Académica, Universidad de Magallanes, Punta Arenas, Chile'
- 'Departamento de Automática y Computación, Universidad Pública de Navarra, Pamplona, España '
- 'Departamento de Ciencias Exactas, Universidad de Los Lagos, Osorno, Chile'
author:
- 'Nicolás Zumelzu$^{a}$ and Benjamin Bedregal$^b$ and Edmundo Mansilla$^{c}$ and Humberto Bustince$^d$ and Roberto Díaz$^e$'
title: Admissible orders on fuzzy numbers
---
Fuzzy numbers ,orders on fuzzy numbers ,admissible orders
Introduction
============
Fuzzy numbers have been introduced by Zadeh [@zadeh1965fuzzy] to deal with imprecise numerical quantities in a practical way. The concept of a fuzzy number plays a fundamental role in formulating quantitative fuzzy variables. These are variables whose states are fuzzy numbers. When also the fuzzy numbers represent linguistic concepts, such as very small, small, medium, and so on, as interpreted in a particular context, the resulting constructs are usually called linguistic variables.
The study of admissible orders over the set of closed subintervals of $[0,1]$, i.e. orders which refines the natural order for intervals, starts with the work of Bustince *et al.* [@bustince2013generation]. Latelly, this notion was adapted for other domains in [@Ivanosca2016; @Annaxsuel2019; @Laura16a; @Laura17].
From the more than two hundred partial orders for fuzzy numbers proposes in the literature, only a few are totals, for example [@TAsmus2017; @valvis2009new; @wang2014total]. Moreover, no study on admissible orders for fuzzy numbers or for a subclass of them had been made so far. In order to overcome this lack and motivated mainly by the application potential of this subject, in this work we introduce and analyze the notion of admissible orders for fuzzy numbers with respect to a partial order and in particular we explore the case where this partial order is the given in [@Klir1995].
A well know method of generate admissible orders is one based on aggregations functions and in this work, we proposes a method which use two aggregation functions to construct an admissible order on fuzzy numbers. In addition, our tool can be amplified to use aggregate functions or functions that are not aggregation functions, as long as some basic properties are fulfilled.
This paper is organized as follow: In section 2, in addition to establishing the notation used, we recall some essential notions for the remains sections. In section 3 we see the more basic partial order on fuzzy numbers and a total order proposal in [@wang2014total].The notion of admissible orders for fuzzy numbers is study in section 4. Finally, section 5 provide some final remarks.
Preliminaries and basic results
===============================
In this section, we introduce notations, definitions and preliminaries facts which are used throughout this work. Given a poset ${\langle P,\leq\rangle}$ and $a,b\in P$, we denote by $a\parallel b$ when $a$ and $b$ are incomparable, i.e. when neither $a\leq b$ nor $b\leq a$. We will denote the set of real numbers by ${\mathds{R}}$.
Let’s consider the following basic facts [@yeh2006real].
\[uppercontinuidad\] Let $X\subseteq {\mathds{R}}$. The function ${f:X\longrightarrow {\mathds{R}}}$ is upper semi-continuous, if for every $a\in X$ and $\varepsilon>0$, there exists $\delta>0$ such that when, $|x-a|<\delta$ for a $x\in X$ then $f(x)<f(a)+\varepsilon$.
This definition of *upper semi-continuous* is not the same, but is equivalent, to the definition given in [@yeh2006real], considering the usual topology for real numbers.
[@yeh2006real] Let $X\subseteq {\mathds{R}}$. The function ${f:X\longrightarrow {\mathds{R}}}$ is right-continuous, if for every $\varepsilon>0$, there exists $\delta>0$, such that for each $a,x\in X$ if $x-a<\delta$ then $|f(x)-f(a)|< \varepsilon$. Analogously, ${f:X\longrightarrow {\mathds{R}}}$ is left-continuous, if for every $\varepsilon>0$, there exists $\delta>0$, such that when $\delta < x-a$ for some $a,x\in X$ then $|f(x)-f(a)|< \varepsilon$.
Admissible Orders on the Real Closed Interval Set
--------------------------------------------------
Let ${\mathds{I}}{\mathds{R}}$ be the set of all the closed intervals of reais numbers, i.e. $${\mathds{IR}}=\{[a,b] {\; : \;}a,b\in{\mathds{R}},~a\leq b\}.$$ Closed intervals of real numbers will be called just of intervals. Degenerate intervals, that is, intervals $[a,a]$, will be written in simplified form $[a]$. Given an interval $A$, their inferior extreme is denoted by $\underline{A}$ and their superior extreme is denoted by $\overline{A}$, i.e. $\underline{[a,b]}=a$ and $\overline{[a,b]}=b$ for every $[a,b]\in {\mathds{IR}}$.
Since intervals are set, then inclusion also is an important order. Observe that the inclusion order for intervals can be determined exclusively on their extremes as follows
$$[a,b]\subseteq [c,d] \Leftrightarrow c\leq a\wedge b\leq d$$
Auxiliarly, we also define the following strict order on ${\mathds{IR}}$:
$$[a,b]\Subset [c,d] \Leftrightarrow c< a\wedge b< d$$
Observe that $\subseteq\neq \Subset$. For example, $[3,4]\subseteq [3,5]$ but $[3,4]\not\Subset [3,5]$.
In [@miranker1981computer] consider the following order for ${\mathds{IR}}$:
$$[a,b]\leq_{KM} [c,d]\Leftrightarrow a\leq c\wedge b\leq d.$$
This order is not linear and in some situations a linear order is fundamental (see for example [@deng2012fuzzy]). Of course, there are infinitely many linear orders on ${\mathds{IR}}$. This motived [@bustince2013generation], in the context of interval-valued fuzzy sets, i.e. in $L([0,1])=\{[a,b]\in {\mathds{IR}}: 0\leq a\leq b\leq 1\}$, to introduce the notion of admissible linear orders. For Bustince an order only is admissible if it refines the usual order on $L([0,1])$. But, is clear that this notion can be addapted in a straight way for ${\mathds{IR}}$:
[@bustince2013generation] \[def-ordenadmisible1\] A relation $\preceq$ on ${\mathds{IR}}$ is called an admisible order, if
1. $\preceq$ is a linear order on ${\mathds{I}}{\mathds{R}}$,
2. for all $A$, $B$ on ${\mathds{IR}}$, $A\preceq B$ whenever $A\leq_{KM} B$.
\[Exemple2.1\] The following relations are admissible orders on ${\mathds{IR}}$:
1. The Lexical 1: $[a,b]\preceq_{Lex1} [c,d]\Leftrightarrow a< c\vee (a=c\wedge b\leq d)$;
2. The Lexical 2: $[a,b]\preceq_{Lex2} [c,d]\Leftrightarrow b< d\vee (b=d\wedge a\leq c)$;
3. Xu-Yager (adapted from [@xu2006some]): $[a,b]\preceq_{XY} [c,d]\Leftrightarrow a+b< c+d\vee (a+b=c+d\wedge b-a\leq d-c)$;
Fuzzy sets
----------
The following definitions can be found in [@Bector2005; @Klir1995] and in most of the introductory books on fuzzy sets theory. In all this section $X$ will be a non-empty reference set with generic elements denoted by $x$.
\[definicionsoporte\] A fuzzy set $A$ on $X$ is a function ${A:X\longrightarrow [0, 1]}$. In addition,
1. The support of $A$, is the set $S(A)=\{x\in X{\; : \;}A (x) > 0\}$;
2. The kernel of $A$, is the set $\ker(A)=\{x\in X{\; : \;}A (x) =1\}$;
3. Given $\alpha\in (0,1]$, the $\alpha$-cuts set of $A$ is the set ${\nicefrac{\Large{A}}{\alpha}} =\{x\in X {\; : \;}A(x)\geq \alpha\}$;
4. The height of $A$ is $h(A) = {\underset{x\in X}{\sup~}} A (x)$.
If $h(A) = 1$, then the fuzzy set $A$ is called of normal fuzzy set, otherwise, i.e. if $0<h(A)<1$, it is called subnormal. Clearly, $A$ is normal if and only if $\ker(A)\neq \emptyset$ and if $A$ is subnormal it can be normalized, for a new fuzzy set $A^*$ on $X$ where $A^\ast(x)=\frac{A(x)}{h(A)}$, for each $x\in X$.
A fuzzy set $A$ on ${\mathds{R}}$ is said to be a convex fuzzy set if its $\alpha$-cuts are (crisp) convex sets, i.e. for all $\alpha,t\in (0, 1]$ and $x,y\in {\nicefrac{\Large{A}}{\alpha}}$, $tx+(1-t)y\in{\nicefrac{\Large{A}}{\alpha}}$.
[@Klir1995 Theorem 1.1]\[teoconvexidade\] A fuzzy set $A$ on ${\mathds{R}}$ is convex iff $$A(\lambda x_1+(1-\lambda)x_2)\geq \min[A(x_1),A(x_2)]$$ for all $x_1$, $x_2\in{\mathds{R}}$ and $\lambda\in[0,1]$, where $\min$ denotes the minimum operator.
Fuzzy numbers
-------------
There are several different definitions of fuzzy numbers in the literature, as for example [@hanss2005applied; @Bector2005; @Klir1995; @lee2004first; @buckley2002introduction; @gomes2015fuzzy; @valvis2009new]. The most of them vary in the kind of continuity required for the membership function. For example, in [@gomes2015fuzzy; @valvis2009new] is considered upper semi-continuity whereas in [@hanss2005applied; @lee2004first] is required piecewise continuity and in [@Bector2005; @Klir1995; @buckley2002introduction] no continuity constraint is required. Another difference can be in that some one requires that the kernel of the fuzzy number be a singleton another only that it be non empty. Here we adopted the given in [@Klir1995].
\[def-FN\] A fuzzy set $A$ on ${\mathds{R}}$ is called a fuzzy number if it satisfies the following conditions
1. $A$ is normal,
2. ${\nicefrac{\Large{A}}{\alpha}}$ is a closed interval for every $\alpha\in (0, 1]$,
3. the support of $A$ is bounded.
Finally, $\mathcal{F}(\mathds{R})$ will denote the set of all fuzzy numbers.
\[sobreelsoporteconteoremadecaracterizaciondefuzzy\] Since the support of a fuzzy number of $A$ is bounded, there exists $\omega_1$, $\omega_2$ in ${\mathds{R}}$, s.t. $S(A)=\{x\in{\mathds{R}} {\; : \;}\omega_1<x<\omega_2\}$. Analogously, since the kernel of $A$ is convex then there exists a closed interval $[a,b]$, s.t. $\ker(A)=[a,b]$.
The next theorem gives a full characterization of fuzzy numbers.
[@Klir1995 Theorem 4.1] \[teoKlir19954-1\] Let $A$ be a fuzzy set on $\mathds{R}$. Then, $A\in \mathcal{F}({\mathds{R}})$ if and only if there exist a closed interval $[a, b]\neq \emptyset$, a function $l$ from $(-\infty, a)$ to $[0, 1]$ which is right-continuous, increasing in $[\omega_1,a)$ and $l(x)=0$ for each $x\in (-\infty,\omega_1)$, and a function $r$ from $(b,+\infty)$ to $[0, 1]$ which is left-continuous, decreasing in $(b,\omega_2]$ and $r(x)=0$ for each $x\in (\omega_2,+\infty,)$, such that $$\label{eqteoKlir19954-1}
A(x)=\left\{\begin{array}{ll}
1 & \textrm{ if }x \in [a, b]\\
l(x) & \textrm{ if } x \in (-\infty, a)\\
r(x) & \textrm{ if }x \in (b,+\infty).\\
\end{array}\right.$$
In the following theorem we rewrite the Theorem 1.1 of [@goetschel1986elementary] considering the parameterization defined by $\{[l^\ast(\alpha),r^\ast(\alpha)]:0\leq \alpha \leq 1\}$, where $l^\ast: [0,1]\rightarrow (-\infty,a]$ and $r^\ast:[0,1]\rightarrow [b,\infty)$ are defined by $$\label{eq-last-rast}
l^\ast(\alpha)=\inf\{x:l(x)\geq \alpha\}\mbox{ and } r^\ast(\alpha)=\sup\{x:r(x){\textcolor[rgb]{1.00,0.00,0.00}{\geq}} \alpha\}$$
\[teolevel\] Let real numbers $a\leq b$ and functions $l: (-\infty,a]\rightarrow [0,1]$ and $r:[b,\infty)\rightarrow [0,1]$. Then ${A:{\mathds{R}}\longrightarrow [0,1]}$ defined by $$A(x)=\sup\{\alpha\in [0,1] :l^\ast (\alpha)\leq x\leq r^\ast(\alpha)\}.$$ is a fuzzy number such that their $\alpha$-cuts are given by $$\label{parametrizacionalphacorte}
{\nicefrac{\Large{A}}{\alpha}}= \begin{cases}\begin{array}{ll}
[l^{\ast}(\alpha),r^{\ast}(\alpha)]& \textrm{ if }\alpha\in [0,1);\\
[a,b]&\textrm{if }\alpha=1.\end{array}\end{cases}$$ if, and only if, $l^\ast$ and $r^\ast$ satisfies the following conditions:
1. $l^{\ast}$ is a bounded increasing function;
2. $r^{\ast}$ is a bounded decreasing function,
3. $ l^{\ast}(1) \leq r^{\ast}(1)$,
4. for $0<\alpha \leq 1$, ${\underset{x\rightarrow \alpha^-}{\lim}}l^\ast(x)=l^\ast(\alpha)$ and ${\underset{x\rightarrow \alpha^-}{\lim}}r^\ast(x)=r^\ast(\alpha)$,
5. ${\underset{x\rightarrow 0^+}{\lim}}l^\ast(x)=l^\ast(0)$ and ${\underset{x\rightarrow 0^+}{\lim}}r^\ast(x)=r^\ast(0)$.
Consider the fuzzy number $A$ (see Figure \[ejemplo-alfa-cortes\]) given by: $$A(x)=\begin{cases} \begin{array}{ll}
1& \mbox{ if }x\in[3,4]\\ l(x)& \mbox{ if }x\in (-\infty,3)\\ r(x) & \mbox{ if }x\in(4,+\infty)
\end{array}\end{cases}$$ where $l$ and $r$ are: $$l(x)=\begin{cases}\begin{array}{ll}\frac{x+1}{4} &\mbox{ if }2\leq x<3\\
\frac{1}{2} & \mbox{ if }1\leq x< 2\\ 0 &\mbox{ if } x<1
\end{array}\end{cases}~~~~\textrm{ and }~~~~r(x)=\begin{cases} \begin{array}{ll}
\frac{20-3x}{8} & \mbox{ if }4<x\leq 5\\ \frac{6-x}{3} & \mbox{ if }5<x\leq 6\\ 0 & \mbox{ if }x>6\end{array}
\end{cases}$$
![Fuzzy Number $A$.[]{data-label="ejemplo-alfa-cortes"}](ejemplo-alfa-cortes.png){width="60.00000%"}
verify the conditions of the Theorem \[teoKlir19954-1\]. Let’s calculate the $\alpha$-cuts of $A$, starting with $l$, i.e.:
$$\label{lasteristo}
l^*(\alpha)=\left\{\begin{array}{ll}
3 &\textrm{ if } \alpha=1;\\
1-4\alpha &\textrm{ if } \alpha\in\left[\frac{3}{4},1\right);\\
2 &\textrm{ if } \alpha\in\left(\frac{1}{2},\frac{3}{4}\right);\\
1 &\textrm{ if } \alpha\in \left[0,\frac{1}{2}\right].
\end{array}\right.$$
To continue, with $r$ it’s analog, we have $$\label{rasteristo}
r^*(\alpha)=\left\{\begin{array}{ll}
4&\textrm{ if } \alpha=1;\\
\frac{20-8\alpha}{3}&\textrm{ if } \alpha\in\left[\frac{5}{8},1\right);\\
5&\textrm{ if } \alpha\in\left[\frac{1}{3},\frac{5}{8}\right);\\
6-3\alpha &\textrm{ if } \alpha\in\left[0,\frac{1}{3}\right),
\end{array}\right.$$ Therefore, from theorem \[teolevel\], we express the $\alpha$-cuts of $A$ given by: $${\nicefrac{\Large{A}}{\alpha}}=\left\{\begin{array}{ll}
[l^{-1}(\alpha),r^{-1}(\alpha)] & \textrm{ if }\alpha\in [0,1)\\
\left[a,b\right] & \textrm{ if }\alpha=1
\end{array}\right.
=\left\{\begin{array}{ll}
\left[4\alpha-1,\frac{20-8\alpha}{3}\right] & \textrm{ if }\alpha\in\left[\frac{3}{4},1\right)\\
\left[2,\frac{20-8\alpha}{3}\right] & \textrm{ if }\alpha\in\left[\frac{5}{8},\frac{3}{4}\right)\\
\left[2,5\right] & \textrm{ if }\alpha\in\left(\frac{1}{2},\frac{5}{8}\right)\\
\left[1,5\right] & \textrm{ if }\alpha\in\left[\frac{1}{3},\frac{1}{2}\right]\\
\left[1,6-3\alpha\right] & \textrm{ if }\alpha\in\left[0,\frac{1}{3}\right)\\
\left[3,4\right] & \textrm{ if }\alpha=1.\\
\end{array}\right.$$
For each interval $[a,b]\in {\mathds{IR}}$ their characteristic function $\widetilde{[a,b]}:{\mathds{R}}\rightarrow [0,1]$ defined by
$$\widetilde{[a,b]}(x)=\left\{\begin{array}{ll}
1 & \mbox{ if }x\in [a,b] \\
0 & \mbox{ if }x\not\in [a,b]
\end{array}
\right.$$ is a fuzzy number. So, in some sense, we can think that fuzzy number generalizes the interval set, i.e. that $U([0,1])\subseteq \mathcal{F}({\mathds{R}})$ and therefore ${\mathds{R}}\subseteq \mathcal{F}({\mathds{R}})$ too, once degenerated intervals can be seen as real numbers.
[@Klir1995 p. 109-110] Let $A,B\in \mathcal{F}({\mathds{R}})$ then the fuzzy sets $A\wedge B$ and $A\vee B$ defined by $$A\wedge B(x)={\underset{x=\min(y,z)}{\sup~}} \min(A(y),B(z)) \mbox{ and }
A\vee B(x)={\underset{x=\max(y,z)}{\sup~}} \min(A(y),B(z))$$ for each $x\in {\mathds{R}}$ is a fuzzy number. In addition, $\langle\mathcal{F}({\mathds{R}}),\wedge,\vee\rangle$ is a distributive lattice.
Order on fuzzy numbers
======================
Next the partial order for $\mathcal{F}({\mathds{R}})$ was propoced by Zadeh in [@zadeh1965fuzzy]
\[orderzadeh\] Let $A$ and $B$ be two fuzzy numbers. $$A\leq_Z B\Longleftrightarrow A (x ) \leq B(x) \textrm{ for all } x\in{\mathds{R}}.$$
The Figure-\[f:incomparables1\] present a case where $A\leq_Z B$.
The Zadeh’s order can be characterized in term of the inclusion order on their $\alpha$-cuts.
[@Klir1995 Theorem 2.3-(viii)] \[equivalencia-zadeh-moore\] Let $A$ and $B$ in $\mathcal{F}(\mathds{R})$. Then $A\leq_Z B$ if and only if $\forall\alpha\in(0,1]~~ {\nicefrac{\Large{A}}{\alpha}}\subseteq {\nicefrac{\Large{B}}{\alpha}}$.
The problem with this order is that it nor generalize the usual order on the real numbers. In fact, given $x,y\in {\mathds{R}}$ such that $x< y$, we have that $\widetilde{[x]}\not\leq_Z \widetilde{[y]}$.
Klir and Yuan in [@Klir1995] proposed the following partial order on $\mathcal{F}({\mathds{R}})$:
Let $A$ and $B$ in $\mathcal{F}(\mathds{R})$. Then $$A\leq_{KY} B \Longleftrightarrow A\wedge B=A.$$
[@Klir1995 p. 114]\[teoordenalphacorte\] Given fuzzy number $A$ and $B$, the following assertions are equivalents
1. $A\leq_{KY} B$;
2. $A\vee B=B$;
3. ${\nicefrac{\Large{A}}{\alpha}} \leq_{KM} {\nicefrac{\Large{B}}{\alpha}}$ for each $\alpha\in (0,1]$.
Observe that the Klir-Yuan partial order when restricted to intervals correspond to the Kulisch-Miranker order and when restrict to real numbers it agree with the usual order. The problem that there are several pair of fuzzy numbers which are non-comparable under this order. The Figure \[f:incomparables1\] and \[f:incomparables2\] present the two generic cases of pairs of fuzzy numbers which are non-comparable by the partial order $\leq_{KY}$ and therefore, $\leq_{KY}$ is not a linear order.
\[f:numeros\]
From the above observation, result clear the following characterization of the non-comparable fuzzy numbers for this order.
Let $A$ and $B$ be fuzzy numbers. $A$ and $B$ are non-comparable in the order $\leq_{KY}$, denoted by $A\parallel_{KY} B$ if, and only if,
1. there exist $\alpha\in (0,1]$ such that ${\nicefrac{\Large{A}}{\alpha}} \subseteq{\nicefrac{\Large{B}}{\alpha}}$ or ${\nicefrac{\Large{B}}{\alpha}} \subseteq {\nicefrac{\Large{A}}{\alpha}}$; or
2. there exists $\alpha,\beta\in (0,1]$ such that ${\nicefrac{\Large{A}}{\alpha}} <_{KM} {\nicefrac{\Large{B}}{\alpha}}$ and $\nicefrac{B}{\beta} <_{KM} \nicefrac{A}{\beta}$.
### Wang-Wang order
([@wang2014total])\[defdensidadS\] A set $S\subseteq (0,1]$ is an upper dense sequence in $(0, 1]$ if, for every point $x\in (0, 1]$ and any $\varepsilon > 0$, there exists $\delta\in S$ such that $\delta\in [x, x + \varepsilon)$.
Observe that in the previous definition, $x + \varepsilon$ not need belong to $[0,1]$ but $\delta\in S$ and therefore $\delta\in (0,1]$, i.e. $\delta\in [x, x + \varepsilon)\cap [x,1]$.
\[def3.3ww\] Let $A$ be fuzzy numbers. For a given upper dense sequence $S=(\alpha_i)_{i \in {\mathds{Z}}^+}$ in $(0,1]$, where ${\mathds{Z}}^+$ is the set of positive integers, we define ${c_i:\mathcal{F}(\mathds{R})\longrightarrow {\mathds{R}}}$ given by $$c_i(A)=\begin{cases} r^{\ast}(\alpha_{\frac{i}{2}})-l^{\ast}(\alpha_{\frac{i}{2}})\textrm{, if $i$ is even};\\
l^{\ast}(\alpha_{\frac{i+1}{2}})+r^{\ast}(\alpha_{\frac{i+1}{2}})\textrm{, if $i$ is odd}.\end{cases}$$
\[wang2014total-def\] Let $A$ and $B$ be two fuzzy numbers and an upper dense sequence $S=(\alpha_i)_{i \in {\mathds{Z}}^+}$ in $(0,1]$. We say that $A<_{WW}^SB$ when there exists a positive integer $n_0$ such that $c_{n_0}(A)<c_{n_0}(B)$ and $c_i(A)=c_i(B)$ for all positive integers $i<n_0$. We say that $A\leq_{WW}^S B$ if, and only if, $A<_{WW}^S B$ or $A=B$.
As is well know, any fuzzy set $A$ can be fully identified with their $\alpha$-cuts in the following sense:
$$A(x)=\sup_{\alpha\in (0,1]} \alpha\cdot \chi_{{\nicefrac{\Large{A}}{\alpha}}}(x)$$ where $\chi_{{\nicefrac{\Large{A}}{\alpha}}}$ is the characteristic function of the interval ${\nicefrac{\Large{A}}{\alpha}}$ which is called of decomposition theorem [@Klir1995 Theorems 2.5 – 2.7]. There is some variants of the decomposition theorem such as in [@Klir1995; @wang2014total]. In particular, Wang and Wang variant any fuzzy number is recovered from just a countably subset of their $\alpha$-cuts.
[@wang2014total Theorem 3] \[teo-decIV\] Let $A$ be a fuzzy number and $S$ be an upper dense sequence in $(0,1]$. Then $$A(x)=\sup_{\alpha\in S} \alpha\cdot \chi_{{\nicefrac{\Large{A}}{\alpha}}}(x)$$
\[coro-teo-decIV\] Let $A$ and $B$ be two fuzzy numbers and $S$ be an upper dense sequence in $(0,1]$. $A=B$ if, and only if, ${\nicefrac{\Large{A}}{\alpha}}={\nicefrac{\Large{B}}{\alpha}}$, for all $\alpha\in S$.
Straightforward from Theorem \[teo-decIV\].
[@wang2014total]\[ordenwang\] Let $S$ be an upper dense sequence in $(0,1]$. Then $\leq_{WW}^S$ is a linear order on ${\mathcal{F}(\mathds{R})}$.
Two different upper dense sequences $S_1$ and $S_2$ in $[0,1]$ can determines distint linear orders, i.e. $\leq_{WW}^{S_1}\neq \leq_{WW}^{S_2}$. Therefore, we are dealing with a family of linear orders (see Example 3 [@wang2014total]).
Admissible orders on fuzzy numbers
==================================
\[def-ordenadmisible\] Let $(\mathcal{F}(\mathds{R}),\leqq)$ be a poset. The order $\preceq$ is called an admisible order with respect to $(\mathcal{F}(\mathds{R}),\leqq)$, if
1. $\preceq$ is a linear order on $\mathcal{F}(\mathds{R})$,
2. for all $A$, $B$ in $\mathcal{F}(\mathds{R})$, $A\preceq B$ whenever $A\leqq B$.
Thus, an order $\preceq$ on $\mathcal{F}(\mathds{R})$ is admissible with respect to $(\mathcal{F}(\mathds{R}),\leqq)$, if it is linear and refine the order $\leqq$. In particular, when the order $\leqq$ is $\leq_{KY}$ we will call $\preceq$ just of admissible order on ${\mathcal{F}(\mathds{R})}$. Furthermore, if $\leqq$ is a linear order then $\preceq$ and $\leqq$ are the same.
Let $\preceq$ be an admissible order on ${\mathcal{F}(\mathds{R})}$. Then, neither there exists greatest nor smallest elements of $\mathcal{F}(\mathds{R})$.
Straightforward from Definition \[def-ordenadmisible\].
Let $A$ and $B$ in $\mathcal{F}(\mathds{R})$ and $S=(\alpha_i)_{i\in {\mathds{Z}}^+}$ be an strictly increasing upper dense sequence in $(0,1]$. We define $\underset{\alpha_i}{\min}\{A,B\}$ given by $$\min_{\alpha_i}\{A,B\}=\left\{\begin{array}{ll}
\min\{\alpha_i\in S: {\nicefrac{\large{A}}{\alpha_{i}}}\neq {\nicefrac{\large{B}}{\alpha_{i}}}\} & \mbox{ if $A\neq B$}; \\
1 & \mbox{ if $A=B$}.
\end{array}
\right.$$
Observe that, by Corollary \[coro-teo-decIV\], $\underset{\alpha_i}{\min}$ is well defined and that $\underset{\alpha_i}{\min}\{A,B\}=\underset{\alpha_i}{\min}\{B,A\}$.
\[lemanew2\] Let $A<_{KY}B$ and $S$ be an strictly increasing upper dense sequence in $(0,1]$. Then exists $\alpha\in(0,1]$ such that ${\nicefrac{\Large{A}}{\alpha}}\neq {\nicefrac{\Large{B}}{\alpha}}$, and for all $\varepsilon\in (0,\alpha)$ exists $\delta\in S$ that satisfies $\alpha-\varepsilon<\delta<\alpha$ and ${\nicefrac{\Large{A}}{\delta}}<_{KM}{\nicefrac{\Large{B}}{\delta}}$.
Let $A<_{KY}B$. Then $A\neq B$, so exists $\alpha\in(0,1]$ such ${\nicefrac{\Large{A}}{\alpha}}\neq {\nicefrac{\Large{B}}{\alpha}}$, i.e., ${\nicefrac{\Large{A}}{\alpha}}<_{KM} {\nicefrac{\Large{B}}{\alpha}}$. For all $\varepsilon \in(0,\alpha)$ we get $\alpha-\varepsilon<\alpha$. Since $S$ is an strictly increasing upper dense sequence in $(0,1]$ exists $\delta\in S$ such that $\alpha-\varepsilon<\delta<\alpha$. So by upper semi-continuity (Definition \[uppercontinuidad\]) and convexity of $A$ and $B$
1. $l^\ast_A(\delta)< l^\ast_B(\delta)$ and $r^\ast_A(\delta)\leq r^\ast_B(\delta)$ or
2. $l^\ast_A(\delta)\leq l^\ast_B(\delta)$ and $r^\ast_A(\delta)< r^\ast_B(\delta)$
Therefore, the proposition hold.
\[lemanew\] Let $S=(\alpha_i)_{i\in{\mathds{Z}}}$ be an strictly increasing upper dense sequence in $(0,1]$. If $A\neq B$ then $\{\alpha_i\in S:{\nicefrac{\large{A}}{\alpha_{i}}}\neq {\nicefrac{\large{B}}{\alpha_{i}}}\}\neq\{1\}$.
Straightforward from Proposition \[lemanew2\].
Let $S$ be an upper dense sequence in $(0,1]$. $S$ is an strictly increasing sequence in $(0,1]$ if, and only if, the order $\leq_{WW}^S$ is an admissible order on ${\mathcal{F}(\mathds{R})}$.
($\Rightarrow$) Suppose that $S$ is not strictly increascing, i.e. there exist $i<j$ such that $\alpha_j \leq \alpha_i$. Let $\alpha_m=\max \{ \alpha_k\in S : \alpha_k< \alpha_j\}$, the triangle fuzzy number $B=(0,1,2)$ and $A$ the fuzzy number
$$A(x)=\left\{\begin{array}{ll}
B(x) & \mbox{ if $x< \alpha_k$ or $x\geq 1$} \\
\frac{(1-\alpha_k)x+\alpha_i-\alpha_k}{1-\alpha_k} & \mbox{ otherwise} \\
\end{array} \right.$$ Clearly, $A<_{KY} B$. Since, $\underset{\alpha_m}{\min}\{A,B\}=\alpha_j$ and ${\nicefrac{\large{A}}{\alpha_{j}}}=[\alpha_k,2-\alpha_j] <_{KM}[\alpha_j,2-\alpha_j]= {\nicefrac{\large{B}}{\alpha_{j}}}$ then $l_A^{\ast}(\alpha_j)+r_A^{\ast}(\alpha_j) < l_B^{\ast}(\alpha_j)+r_B^{\ast}(\alpha_j)$. So, $c_{2j}(A) < c_{2j}(B)$. However, since $\alpha_i >\alpha_j$, $c_{2i}(A)= l_A^{\ast}(\alpha_i)+r_A^{\ast}(\alpha_i)\neq l_B^{\ast}(\alpha_i)+r_B^{\ast}(\alpha_i)=c_n(B)$ and therefore, once $i<j$, $A\not\leq_{WW}^S B$. Hence, by contraposition, $S$ is strictly increascing.
($\Leftarrow$) Definition \[def-ordenadmisible\] condition $i)$ is verified by Theorem \[ordenwang\]. Let $A$ and $B$ be fuzzy numbers such that $A\leq_{KY} B$. If $A=B$ then trivially $A\leq_{WW}^S B$. If $A<_{KY} B$, then for all $\alpha_i\in S$ we have ${\nicefrac{\large{A}}{\alpha_{i}}}\leq_{KM}{\nicefrac{\large{B}}{\alpha_{i}}}$, i.e. $[l_A^{\ast}(\alpha_i),r_A^{\ast}(\alpha_i)]\leq_{KM} [l_B^{\ast}(\alpha_i),r_B^{\ast}(\alpha_i)]$ for all $i\in{\mathds{Z}}^+$ and $\{\alpha_i\in S:{\nicefrac{\large{A}}{\alpha_{i}}}\neq {\nicefrac{\large{B}}{\alpha_{i}}}\}\neq \phi$. So, let $\alpha_j=\underset{\alpha_i}{\min}\{A,B\}$ then ${\nicefrac{\large{A}}{\alpha_{j}}}\neq {\nicefrac{\large{B}}{\alpha_{j}}}$ and because $A<_{KY} B$, then ${\nicefrac{\large{A}}{\alpha_{j}}}<_{KM}{\nicefrac{\large{B}}{\alpha_{j}}}$. So, $[l_A^{\ast}(\alpha_j),r_A^{\ast}(\alpha_j)]<_{KM} [l_B^{\ast}(\alpha_j),r_B^{\ast}(\alpha_j)]$ and therefore, $l_A^{\ast}(\alpha_j)+r_A^{\ast}(\alpha_j)< l_B^{\ast}(\alpha_j)+r_B^{\ast}(\alpha_j)$. So, taking $j_0=2j-1$ we have that $c_{j_0}(A)=l_A^{\ast}(\alpha_j)+r_A^{\ast}(\alpha_j)< l_B^{\ast}(\alpha_j)+r_B^{\ast}(\alpha_j)=c_{j_0}(B)$ and for each $n<j'$ we have that $m=\frac{n+1}{2} < j$. Since, $S$ is strictly increascing then $\alpha_m < \alpha_j$ which implies that ${\nicefrac{\large{A}}{\alpha_{m}}}={\nicefrac{\large{B}}{\alpha_{m}}}$. So, if $n$ is even then $c_n(A)=l_A^{\ast}(\alpha_m)+r_A^{\ast}(\alpha_m)=l_B^{\ast}(\alpha_m)+r_B^{\ast}(\alpha_m)=c_n(B)$ and if $n$ is odd then $c_n(A)=r_A^{\ast}(\alpha_m)-l_A^{\ast}(\alpha_m)=r_B^{\ast}(\alpha_m)+l_B^{\ast}(\alpha_m)=c_n(B)$. Therefore, $A\leq_{WW}^S B$.
\[adinter\] Let $\preceq$ be an order on ${\mathds{IR}}$, $A$, $B$ in $\mathcal{F}(\mathds{R})$ and $\alpha_m=\underset{\alpha_i}{\min}\{A,B\}$. Then, $$A\unlhd B\Longleftrightarrow {\nicefrac{\large{A}}{\alpha_{m}}} \preceq {\nicefrac{\large{B}}{\alpha_{m}}}.$$
\[ZBM\] Let $\preceq$ be an order on ${\mathds{IR}}$. The relation $\unlhd$ is an admissible order on ${\mathcal{F}(\mathds{R})}$.
Let $S$ be an strictly increasing upper dense sequence. We prove this conclusion considering five properties.
[:]{} Trivially the relation $\unlhd$ is reflexive.
[:]{} Let $A$, $B$ two number fuzzy such that $A\unlhd B$ and $B\unlhd A$. From the former ${\nicefrac{\large{A}}{\alpha_{m}}}\preceq{\nicefrac{\large{B}}{\alpha_{m}}}$ and ${\nicefrac{\large{B}}{\alpha_{m}}}\preceq {\nicefrac{\large{A}}{\alpha_{m}}}$ where $\alpha_m=\underset{\alpha_i}{\min}\{A,B\}$. So, because $\preceq$ is order the ${\nicefrac{\large{A}}{\alpha_{m}}}= {\nicefrac{\large{B}}{\alpha_{m}}}$, thus $\alpha_m\neq\min\{\alpha_i\in S:{\nicefrac{\large{A}}{\alpha_{i}}}\neq {\nicefrac{\large{B}}{\alpha_{i}}}\} $ and therefore $\alpha_m=1$, consequently by Corollary \[lemanew\] we get $A=B$.
[:]{} Let $A$, $B$, and $C$ be three fuzzy numbers such that $A\unlhd B$ and $B\unlhd C$. If $A=B$ or $B=C$ then trivially $A\unlhd C$. If $A\neq B$ and $B\neq C$ then $A\lhd B$ and $B\lhd C$. Hence ${\nicefrac{\large{A}}{\alpha_{j}}}\prec{\nicefrac{\large{B}}{\alpha_{j}}}$ and ${\nicefrac{\large{B}}{\alpha_{k}}}\prec{\nicefrac{\large{C}}{\alpha_{k}}}$, where $\alpha_j=\underset{\alpha_i}{\min}\{A,B\}$ and $\alpha_k=\underset{\alpha_i}{\min}\{B,C\}$. So, as $\preceq$ is order then ${\nicefrac{\large{A}}{\alpha_{r}}}\preceq {\nicefrac{\large{C}}{\alpha_{r}}}$ since cleary $\alpha_r=\min\{\alpha_j,\alpha_k\}=\underset{\alpha_i}{\min}\{A,C\}$. This means that relation is transitive.
[:]{} Let $A$ and $B$ be two fuzzy numbers such that $A\neq B$. Then by Corollary \[lemanew\] $\alpha_j=\underset{\alpha_i}{\min}\{A,B\}\neq 1$ and because $\preceq$ is total ${\nicefrac{\large{A}}{\alpha_{j}}}\prec{\nicefrac{\large{B}}{\alpha_{j}}}$ or ${\nicefrac{\large{B}}{\alpha_{j}}}\prec {\nicefrac{\large{A}}{\alpha_{j}}}$ and ${\nicefrac{\large{A}}{\alpha_{n}}}={\nicefrac{\large{B}}{\alpha_{n}}}$ for all $n<j$. Therefore $A\unlhd B$ or $B\unlhd A$. [:]{} Let $A$ and $B$ be two fuzzy numbers such that $A\leq_{KY} B$. Then, by Proposition \[teoordenalphacorte\], ${\nicefrac{\Large{A}}{\alpha}}\leq_{KM} {\nicefrac{\Large{B}}{\alpha}}$ for each $\alpha\in (0,1]$. So, in particular for $\alpha_m=\underset{\alpha_i}{\min}\{A,B\}$ we have that ${\nicefrac{\large{A}}{\alpha_{m}}}\leq_{KM} {\nicefrac{\large{B}}{\alpha_{m}}}$ and therefore, $A\unlhd B$.
Therefore, $\unlhd$ is an admissible orders.
Final Remarks
=============
In this paper, we generalize the notion of admissible order on the set of closed subinterval of $[0,1]$ with respect to the natural order on this set to admissible orders for fuzzy numbers equipped with an arbitrary order. Although of the Klir and Yuan order not be consensual as the natural order for the set of fuzzy numbers, most of the orders proposed for fuzzy numbers refines this order. So we deal the Klir-Yuan order as the “natural” one for ${\mathcal{F}(\mathds{R})}$ and explore the admissible order with respect to this order.
Applications of admissible orders on several domains has been well succed in several areas as can be seen in [@Laura17; @Bentkowska15; @BustinceGBKM13; @Laura16b] and the same happen with application of fuzzy numbers. So, result naturally that in a future enforce be made to develop interesting applications of admissible orders on $NFR$.
In [@bustince2013generation] a construction method of admissible orders over the set of closed subintervals of $[0,1]$ based on aggregation functions is provided and latellly generalized in [@Santana2020]. As a future work, we will intend to introduce a generation method of admissible order on ${\mathcal{F}(\mathds{R})}$.
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|
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---
abstract: 'We prove Bogolyubov-Ruzsa-type results for finite subsets of groups with small tripling, $|A^3|\leq O(|A|)$, or small alternation, $|AA\inv A|\leq O(|A|)$. As applications, we obtain a qualitative analog of Bogolyubov’s Lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox, and Zhao, and gives a quantitative version of previous work of the author, Pillay, and Terry.'
address: |
Department of Pure Mathematics and Mathematical Sciences\
University of Cambridge\
Cambridge, CB3 0WB, UK
author:
- Gabriel Conant
date: 'May 21, 2020'
title: On finite sets of small tripling or small alternation in arbitrary groups
---
Introduction {#sec:intro}
============
Freiman’s Theorem (see [@FreiKK], [@FreiFST]) is a combinatorial result in additive number theory which states that if $A$ is a finite subset of a torsion-free abelian group $G$, and $|A+A|\leq k|A|$ (i.e. $A$ has *small doubling*), then $A$ is contained in an $n$-dimensional arithmetic progression of length $c|A|$, where $c$ and $n$ depend only on $k$. In [@Ruz94], Ruzsa gave a new proof of this result, and a similar strategy was later used by Green and Ruzsa [@GrRuz] to prove a generalization of Freiman’s Theorem, involving coset progressions, for arbitrary abelian groups. A key part of this work is that a set of small doubling in an abelian group can be “Freiman-isomorphically" mapped to a dense set in a finite abelian group (see [@GrRuz Proposition 1.2]). This allows one to apply the following result, which Ruzsa [@Ruz94] adapted from Bogolyubov [@Bog39]. For comparison to our work, we state this result in two cases.
\[thm:Bogo\] Fix $r\in \Z^+$ and $\alpha\in\R^+$.
1. [(Bounded exponent case)]{.nodecor} Suppose $G$ is a finite abelian group of exponent at most $r$, and $A\seq G$ is such that $|A|\geq\alpha|G|$. Then there is a subgroup $H\leq G$ such that $[G:H]\leq r^{\alpha^{\nv 2}}$ and $H\seq 2A-2A$.
2. [(General case)]{.nodecor} Suppose $G$ is a finite abelian group, and $A\seq G$ is such that $|A|\geq\alpha|G|$, then there is a $(1/4,n)$-Bohr neighborhood $B$ in $G$ such that $n<\alpha^{\nv 2}$ and $B\seq 2A-2A$.
Bohr neighborhoods (see Definition \[def:Bohr\]) are certain kinds of well-structured sets which, in the abelian case, contain large coset progressions (preserved by Freiman isomorphism). This yields the “*Bogolyubov-Ruzsa Lemma*[^1] for finite abelian groups": if $G$ is abelian and $A\seq G$ is finite, with $|A+A|\leq k|A|$, then $2A-2A$ contains an $n$-dimensional coset progression of size $c|A|$, where $c$ and $n$ depend only on $k$. The conclusion of Freiman’s theorem (exchanging the arithmetic progression for one containing $A$) then follows after a little more work (see [@GrRuz Proposition 5.1]).
For an abelian group $G$, Freiman’s Theorem also yields a classification of **$k$-approximate subgroups** of $G$, i.e., finite symmetric subsets $A\seq G$ such that $A+A$ can be covered by $k$ translates of $A$. Approximate subgroups of arbitrary groups have been studied by many authors, culminating in the work of Breuillard, Green, and Tao [@BGT].
The goal of the present paper is to give generalizations of Bogolyubov’s Lemma to arbitrary finite groups, as well as similar statements about finite subsets of arbitrary groups whose product set growth can be controlled. For this, we focus on sets of small *tripling*, which satisfy Plunnecke-Ruzsa inequalities for product sets (as observed by Helfgott [@HelfGG], see Proposition \[prop:Ruz\]$(a)$). Motivated by the work of Hrushovski [@HruAG] on approximate groups, we also consider finite sets $A$ of *small alternation*, i.e. $|AA\inv A|\leq k|A|$ for some fixed $k$ (see Remark \[rem:alttrip\]).
Our main results, Theorems \[thm:mainbdd\] and \[thm:maingen\], are versions of the Bogolyubov-Ruzsa Lemma for finite sets of small tripling or alternation in arbitrary groups (with some further constraints). Our first application of these results is the following qualitative analogue of Bogolyubov’s Lemma (Theorem \[thm:Bogo\]) for arbitrary groups.
\[thm:Bogogen\] Fix a positive integer $r$ and a positive real number $\alpha$.
1. [(Bounded exponent case)]{.nodecor} Suppose $G$ is a finite group of exponent at most $r$, and $A\seq G$ is such that $|A|\geq\alpha|G|$. Then there is a normal subgroup $H\leq G$ such that $[G:H]\leq O_{r,\alpha}(1)$ and $H\seq (AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2$.
2. [(General case)]{.nodecor} Suppose $G$ is a finite group, and $A\seq G$ is such that $|A|\geq\alpha|G|$. Then there is a normal subgroup $H\leq G$ and a $(\delta,n)$-Bohr neighborhood $B$ in $H$, such that $[G:H],\delta\inv,n\leq O_{\alpha}(1)$ and $B\seq (AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2$.
This is proved in Section \[sec:results\]. For later applications, and also to illustrate the use of Bohr neighborhoods in the nonabelian setting, we prove (in Section \[sec:results\]) the following easy corollary of Theorem \[thm:Bogogen\]$(b)$. Call a (nontrivial) group $G$ **purely nonabelian** if no normal subgroup $H\leq G$ has a nontrivial abelian quotient (i.e. $[H,H]=H$ for all normal $H\leq G$). The class of purely nonabelian groups contains all nonabelian simple groups, and is closed under direct product by Goursat’s Lemma.
\[cor:pure\] Fix a positive real number $\alpha$. Suppose $G$ is a purely nonabelian finite group and $A\seq G$ is such that $|A|\geq\alpha|G|$. Then there is a normal subgroup $H\leq G$ such that $[G:H]\leq O_\alpha(1)$ and $H\seq (AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2$.
Before continuing to the next application, we state the following consequences for symmetric subsets of groups of bounded exponent (part $(a)$ follows immediately from Theorem \[thm:mainbdd\] and part $(b)$ is a special case of Theorem \[thm:Bogogen\]$(a)$, see Section \[sec:results\]).
\[cor:sym\]$~$
1. Fix positive integers $k$ and $r$. Suppose $G$ is a group of exponent $r$ and $A\seq G$ is finite and symmetric, with $|A^3|\leq k|A|$. Then there is a subgroup $H\leq G$ such that $A$ is covered by $O_{k,r}(1)$ translates of $H$ and $H\seq A^4$.
2. Fix a positive integer $r$ and a positive real number $\alpha$. Suppose $G$ is a finite group of exponent $r$ and $A\seq G$ is symmetric, with $|A|\geq\alpha|G|$. Then there is a normal subgroup $H\leq G$, of index $O_{\alpha,r}(1)$, such that $H\seq A^4$.
\[rem:feit\] We do not know if $H$ can also be made normal in part $(a)$ of the previous corollary. In addition to improving Theorem \[thm:NIPregexp\] below (see Remark \[rem:NIPpre\]), such a result could be quite interesting, depending on the methods used. For example, together with the Feit-Thompson Theorem and the Brauer-Fowler Theorem, this strengthening of Corollary \[cor:sym\]$(a)$ would imply that for any positive integer $r$, there are only finitely many finite simple groups of exponent $r$. This is a known fact, but its proof requires the classification of finite simple groups (e.g., [@BaGoPy Theorem 5.4]).
Our final applications are in the subject of arithmetic regularity (developed by Green [@GreenSLAG] for abelian groups). There has been a recent interest in strengthened arithmetic regularity lemmas for subsets of groups satisfying special tameness assumptions. This was initiated by the work of Terry and Wolf [@TeWo] on “stable arithmetic regularity" in $\F_p^n$, which is continued in [@CPT] and [@TeWo2]. Arithmetic regularity in the setting of bounded VC-dimension is considered in [@AFZ], [@CPTNIP], and [@SisNIP]. Given a group $G$ and $A\seq G$, define the **VC-dimension of $A$** to be the VC-dimension of the collection of left translates of $A$, i.e., the supremum of all integers $d$ such that, for some $d$-element set $X\seq G$, one has $\cP(X)=\{X\cap gA:g\in G\}$. In [@AFZ], Alon, Fox, and Zhao show that if $G$ is a finite *abelian* group of exponent at most $r$, and $A\seq G$ has VC-dimension at most $d$, then there is a subgroup $H\leq G$ of index $(1/\epsilon)^{d+o_{r,d}(1)}$, and a subset $D\seq G$ which is a (possibly empty) union of cosets of $H$, such that $|A\smd D|\leq \epsilon|G|$. A main tool in their proof is Theorem \[thm:Bogo\]$(a)$, and we will use Corollary \[cor:sym\]$(a)$ to give the following generalization to arbitrary groups.
\[thm:NIPregexp\] Fix positive integers $r$ and $d$. Suppose $G$ is a finite group of exponent at most $r$, and $A\seq G$ has VC-dimension at most $d$. Then, for any $\epsilon,\nu>0$, there is a subgroup $H$ of $G$, of index $O_{r,d,\nu}((1/\epsilon)^{d+\nu})$, which satisfies the following properties.
1. [(structure)]{.nodecor} There is a set $D\seq G$, which is a union of right cosets of $H$, such that $|A\smd D|\leq \epsilon|G|$.
2. [(regularity)]{.nodecor} There is a set $Z\seq G$, with $|Z|<\frac{1}{2}\epsilon^{1/2}|G|$, such that for any $x\in G\backslash Z$, either $|Hx\cap A|\leq\epsilon^{1/4}|H|$ or $|Hx\backslash A|\leq\epsilon^{1/4}|H|$.
\[rem:NIPpre\] There are several comments to be made about Theorem \[thm:NIPregexp\].
1. In [@AFZ], Alon, Fox, and Zhao conjecture that condition $(i)$ of Theorem \[thm:NIPregexp\] holds for a *normal* subgroup $H$ of index $\epsilon^{\nv O_{r,d}(1)}$. This would follow from the proof if one could show that Corollary \[cor:sym\]$(a)$ holds with $H$ being normal (see Remark \[rem:feit\]). However, it does follow from the proof that one can replace the subgroup $H$ with the intersection of its conjugates to obtain a normal subgroup of index $2^{\epsilon^{\nv O_{r,d}(1)}}$ satisfying conditions $(i)$ and $(ii)$ (see Remark \[rem:NIPreg\]).
2. In [@CPTNIP] (joint with Pillay and Terry), we gave a version of Theorem \[thm:NIPregexp\] in which $H$ is also normal, but without effective bounds on its index. One could instead use Corollary \[cor:sym\]$(b)$ to deduce this, yielding a very different proof compared to what is done in [@CPTNIP] (see Remark \[rem:NIPreg\]).
3. The “regularity" statement in condition $(ii)$ is not made explicit in [@AFZ], but follows implicitly from their methods (see Lemma \[lem:separate\]).[^2]
4. The $O_{r,d, \nu}$ constant in the statement of the theorem comes from Corollary \[cor:sym\]$(a)$ and so, unlike the abelian case, is not explicit (see Section \[sec:explicit\]).
In Section \[sec:NIP\] we also show that a qualitative version of Theorem \[thm:NIPregexp\] holds for the class of purely nonabelian finite groups (see Theorem \[thm:NIPregpna\]), which yields an interesting divergence between sets of bounded VC-dimension in nonabelian finite simple groups, compared to the abelian setting (see Corollary \[cor:NIPregpna\]).
We end this introduction with some discussion of our proofs. The results above involving groups of bounded exponent will be derived from Theorem \[thm:mainbdd\]. By the work of Hrushovski [@HruAG] and Breuillard, Green, and Tao [@BGT], approximate subgroups of groups with bounded exponent are close to genuine subgroups (see Theorem \[thm:BGTbdd\]). Morever, in any group, finite sets of small tripling are close to approximate subgroups by a result of Tao [@TaoPSE]. Together, these two facts imply a weaker version of Theorem \[thm:mainbdd\] (see Section \[sec:final\]). To prove our result, we will sharpen what is essentially the first step of the work in [@BGT], which is a theorem about sets of small doubling (proved independently by Croot and Sisask [@CrSi] and Sanders [@SanBS]). Namely, if $G$ is a group, $A\seq G$ is finite, and $|A^2|\leq k|A|$, then $A^2A^{\nv 2}$ contains $S^n$, for some symmetric $S\seq G$ of size $\Omega_{k,n}(|A|)$. In Section \[sec:CSS\], we reprove this result using the same techniques, but for sets of small tripling or small alternation, which leads to stronger conclusions. We also work in the setting of measures (similar to Massicot and Wagner [@MassWa]), so that this analysis can be applied later to pseudofinite subsets of ultraproducts of groups. We then prove Theorem \[thm:mainbdd\] in Section \[sec:proofbdd\].
For Theorem \[thm:maingen\], we will need to delve a bit deeper into the underlying methods of [@BGT] and [@HruAG], in particular, the ultraproduct construction. To prove the theorem, we will first prove a pseudofinite analogue, and then deduce the finitary version using an “ultraproduct of counterexamples". To simplify this discussion, and illustrate the leverage obtained by working with pseudofinite sets, we focus on the case of symmetric sets of small tripling. In this case, the pseudofinite analog of Theorem \[thm:maingen\] deals with a group $G$ and a *pseudofinite* (symmetric) subset $A\seq G$. In other words, $G$ is an ultraproduct of groups, and $A$ is an ultraproduct of finite (symmetric) subsets of those groups. We also assume $\langle A\rangle=A^m$ for some fixed $m$ (which holds, for example, if $A$ is an ultraproduct of uniformly dense subsets of finite groups). If $A$ has small tripling (formulated using a pseudofinite counting measure), then the Sanders-Croot-Sisask analysis from Section \[sec:CSS\] yields a symmetric set $S$ such that $S^8\seq A^4$. Moreover, $S$ itself has small tripling (in fact it is an approximate subgroup), allowing us to iterate the process. After infinitely many iterations, we obtain a decreasing sequence of symmetric subsets of $A^4$, whose intersection is a demonstrably “large" subgroup of $\langle A\rangle$ contained in $A^4$.
We now reach an obstacle, in that although $G$ is an ultraproduct of groups, the subgroup constructed above need not be an ultraproduct of subgroups of those groups. In order to salvage this, we move to a saturated elementary extension $G_*$ of $G$ (in a suitable first-order language). We then find a normal subgroup $\Gamma$ of $\langle A_*\rangle$ of small index (where $A_*$ is the interpretation of $A$ in $G_*$), which is contained in $A_*^4$ and is an intersection of countably many definable sets. By standard facts, $\langle A_*\rangle/\Gamma$ is a compact Hausdorff group under a certain topology controlled by definable sets in $G_*$. By a result of Pillay [@PiRCP], the connected component of $\langle A_*\rangle/\Gamma$ is a compact connected *abelian* group, and thus an inverse limit of tori, supplying us with Bohr neighborhoods in $\langle A_*\rangle$. Finally, in order to transfer these Bohr neighborhoods through the ultraproduct, we will use an approximation method from [@CPTNIP], and a result about approximate homomorphisms from [@AlGlGo]. This will yield Bohr neighborhoods in the original groups, and allow us to prove Theorem \[thm:maingen\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Tim Burness, Daniel Palacín, Anand Pillay, Caroline Terry, and Julia Wolf for helpful conversations. Thanks also to the University of Bristol School of Mathematics for their hospitality during the time this work was completed, and to the anonymous referees for valuable comments.
Definitions, main theorems, and corollaries {#sec:results}
===========================================
Before stating the main theorems, we set some notation and definitions (used throughout the paper). Let $G$ be a group. Given $n\geq 1$, let $G^{\times n}=G\times\stackrel{n}{\ldots}\times G$. Given $X,Y\seq G$, let $XY=\{xy:x\in X,~y\in Y\}$. Set $X^0=\{1\}$ and inductively define $X^{n+1}=X^nX$. Let $X\inv=\{x\inv:x\in X\}$. We say that $X\seq G$ is **symmetric** if $1\in X$ and $X=X\inv$. A **$Y$-translate of $X$** is a set of the form $aX$ where $a\in Y$. Given a set $X\seq G$, we let $\langle X\rangle$ denote the subgroup of $G$ generated by $X$, and we use the notation $\bar{X}$ for the set $X\cup X\inv\cup\{1\}$.
Let $\T^n$ denote the $n$-dimensional torus $\R/\Z\times \stackrel{n}{\ldots}\times\R/\Z$, considered as an additive group with identity $0$. Let $d_n$ denote the invariant metric on $\T^n$ induced by the product of the arclength metric on $\R/\Z$ (identified with $S^1$).
\[def:Bohr\] Given a group $G$, a positive integer $n$, and a positive real number $\delta$, we say that $B\seq G$ is a **$(\delta,n)$-Bohr neighborhood in $G$** if there is a homomorphism $\tau\colon G\to \T^n$ such that $B=B^n_{\tau,\delta}:=\{x\in G:d_n(0,\tau(x))<\delta\}$.
In the setting of abelian groups, Bohr neighborhoods are often used as replacements for subgroups in cases where few subgroups are available (e.g., in $\Z/p\Z$). In general, if $G$ is a group and $B=B^n_{\tau,\delta}$ is a $(\delta,n)$-Bohr neighborhood in $G$, then $B$ is symmetric, closed under conjugation, and contains the kernel $N$ of a homomorphism from $G$ to some $\T^n$ (so $G/N$ is abelian). While $B$ may not be closed under the group operation, one can obtain control in pairs by allowing the radius $\delta$ to vary. For instance, $B^2\seq B^n_{\tau,2\delta}$ by the triangle inequality. A more sophisticated manifestation of this idea can be found in the work of Bourgain [@BourgTAP]. Finally, if $G$ is finite then Bohr neighborhoods are “large", for instance $|B^n_{\tau,\delta}|\geq \delta^n|G|$ (see [@TaoVu Lemma 4.20] or [@CPTNIP Proposition 4.5]).
Recall from the introduction that we are interested in finite subsets $A$ of some group $G$, which either have *small tripling*, i.e., $|A^3|\leq k|A|$ for some fixed constant $k$, or have *small alternation*, i.e., $|AA\inv A|\leq k|A|$ for some fixed $k$.
\[rem:alttrip\] The notion of small alternation is motivated by Hrushovski’s definition of a *near-subgroup* from [@HruAG]. Our terminology is explained by Proposition \[prop:Ruz\]$(b)$, which shows that small alternation for a finite set $A$ in a group $G$ implies “very small" tripling for $AA\inv$ (see Section \[sec:Tao\], and especially Remark \[rem:HruTao\], for discussion on the relationship to approximate subgroups). Note that small tripling implies small doubling, and also small alternation due to the general Plunnecke-Ruzsa inequalties observed by Helfgott (see Proposition \[prop:Ruz\]$(a)$). For abelian groups, small alternation clearly implies small doubling, and it is well-known that small doubling implies small tripling (see [@Plunn]), making the three notions equivalent. However, in nonabelian groups, there are no general implications between small doubling and small alternation. For example let $G$ be the free product $H\ast F_2$ where $H$ is some finite group and $F_2$ is the free group on two generators, say $a$ and $b$. Set $A=H\cup\{a\}$ and $B=aHb$. Then $A$ satisfies small doubling but not small alternation, and $B$ satisfies small alternation but not small doubling.
We now state our two main theorems, which are Bogolyubov-Ruzsa-type statements for finite sets of small alternation or small tripling. Each statement involves two crucial assumptions, the first being either small tripling or small alternation for some finite set, and the second being one of the following options: $(1)$ bounded exponent of a certain subgroup, $(2)$ bounded generation of a certain subgroup, or $(3)$ both. Altogether, this yields six statements, which we have divided into two theorems, one for the bounded exponent case and the other for the general case. The two results are proved in Sections \[sec:proofbdd\] and \[sec:genproof\], respectively.
\[thm:mainbdd\] Fix positive integers $k$, $m$, and $r$. Let $G$ be a group, and fix a nonempty finite subset $A\seq G$.
1. [(small alternation)]{.nodecor} Suppose $|AA\inv A|\leq k|A|$ and $\langle AA\inv\rangle$ has exponent $r$.
1. There is a subgroup $H\leq\langle AA\inv\rangle$ such that:
1. $(AA\inv)^m$ is covered by $O_{k,m,r}(1)$ left cosets of $H$, and
2. $H\seq (AA\inv)^2$.
2. Assume $\langle AA\inv\rangle=(AA\inv)^m$. Then there is a normal subgroup $H\leq \langle AA\inv\rangle$, of index $O_{k,m,r}(1)$, such that $H\seq (AA\inv)^2$.
2. [(small tripling)]{.nodecor} Suppose $|A^3|\leq k|A|$ and $\langle A\rangle$ has exponent $r$.
1. There is a subgroup $H\leq\langle A\rangle$ such that:
1. $\bar{A}^m$ is covered by $O_{k,m,r}(1)$ left cosets of $H$, and
2. $H\seq (AA\inv)^2\cap A^2A^{\nv2}\cap (A\inv A)^2\cap A^{\nv 2} A^2$.
2. Assume $\langle A\rangle=\bar{A}^m$. Then there is a normal subgroup $H\leq \langle AA\inv\rangle$, of index $O_{k,m,r}(1)$, such that $H\seq (AA\inv)^2$.
\[thm:maingen\] Fix positive integers $k$ and $m$. Let $G$ be a group, and a fix a nonempty finite subset $A\seq G$.
1. [(small alternation)]{.nodecor} Suppose $|AA\inv A|\leq k|A|$ and $\langle AA\inv \rangle =(AA\inv)^m$. Then there are:
1. a normal subgroup $H$ of $\langle AA\inv\rangle$, of index $O_{k,m}(1)$, and
2. a $(\delta,n)$-Bohr neighborhood $B$ in $H$, with $\delta\inv,n\leq O_{k,m}(1)$,
such that $B\seq (AA\inv)^2$. Moreover, if $\langle AA\inv\rangle$ is abelian, then we may assume $H=\langle AA\inv\rangle$.
2. [(small tripling)]{.nodecor} Suppose $|A^3|\leq k|A|$ and $\langle A\rangle=\bar{A}^m$. Then there are:
1. a normal subgroup $H$ of $\langle A\rangle$, of index $O_{k,m}(1)$, and
2. a $(\delta,n)$-Bohr neighborhood $B$ in $H$, with $\delta\inv,n\leq O_{k,m}(1)$,
such that $B\seq (AA\inv)^2\cap A^2A^{\nv2}\cap (A\inv A)^2\cap A^{\nv 2} A^2$. Moreover, if $\langle A\rangle$ is abelian then we may assume $H=\langle A\rangle$.
Since the work of Breuillard, Green, and Tao [@BGT] on approximate groups makes several appearances in this paper, we take a moment to reconcile their work with the theorems above. First, Theorem \[thm:mainbdd\] strengthens the main result from [@BGT] on approximate subgroups of groups of bounded exponent (see Theorem \[thm:BGTbdd\]), in that we have replaced approximate subgroups with sets of small alternation or small tripling. As discussed in the introduction, this improvement is obtained by modifying the first step of the work in [@BGT], and then applying their structure theorem. (See also Section \[sec:Tao\], where discuss further consequences of our work for the structural results on approximate subgroups from [@BGT].)
To compare Theorem \[thm:maingen\] to [@BGT], we first quote one of their main results.
[[@BGT Theorem 1.6]]{.nodecor}\[thm:BGTmain\] Fix a positive integer $k$. Suppose $G$ is a group and $A\seq G$ is a finite $k$-approximate subgroup of $G$. Then there is a subgroup $H$ of $G$ and a finite normal subgroup $N$ of $H$ with the following properties:
1. $A$ is covered by $O_k(1)$ left translates of $H$;
2. $H/N$ is nilpotent and finitely generated of rank and step $O_k(1)$;
3. $A^4$ contains $N$ and a generating set for $H$.
For comparison, in both parts of Theorem \[thm:maingen\], the Bohr neighborhood $B$ contains the kernel $N$ of a homomorphism from $H$ to $\T^n$. Thus $H/N$ is a finite abelian group, which can be generated by $n\leq O_{k,m}(1)$ elements. Moreover, since $|B|\geq \Omega_{k,m}(|H|)$, we could replace $H$ by the the subgroup generated by $B$, and have that $B$ contains a generating set of $H$ (although possibly losing normality of $H$). Altogether, Theorem \[thm:maingen\] can be seen as an analog of Theorem \[thm:BGTmain\], where we obtain stronger conclusions for sets of small alternation or small tripling, under the extra “bounded generation" assumption coming from the parameter $m$. As with Theorem \[thm:mainbdd\], our proof of Theorem \[thm:maingen\] relies on [@BGT], although this time implicitly via a result of Pillay [@PiRCP] used in Proposition \[prop:Bohr\]. However, this dependence on [@BGT] could be avoided by using a generalization of Pillay’s result due to Nikolov, Schneider, and Thom [@NST].
Recall that the “Bogolyubov-Ruzsa Lemma" for *abelian* groups, discussed after Theorem \[thm:Bogo\], does not involve a “bounded generation" parameter $m$ like in Theorem \[thm:maingen\]. However, similar to Freiman’s Theorem, this result for abelian groups reduces to Theorem \[thm:Bogo\]$(b)$, using the fact that Bohr neighborhoods in abelian groups can be approximated by coset progressions (see [@TaoVu Lemma 4.22]), and that finite sets of small doubling in abelian groups have “good models" as dense sets in finite abelian groups (see [@GrRuz Proposition 2.1]). Thus, for the sake of completeness, we will explain in Remark \[rem:abelianBohr\] how to obtain $G=H$ in Theorem \[thm:Bogogen\]$(b)$ when $G$ is abelian.
The rest of this section is devoted to proving the theorems and corollaries in Section \[sec:intro\] (except for Theorem \[thm:NIPregexp\], which is proved in Section \[sec:NIP\]). We first consider Corollary \[cor:sym\] since it is immediate from the theorems above.
Part $(a)$ is immediate from Theorem \[thm:mainbdd\]. Part $(b)$ is immediate from Theorem \[thm:Bogogen\]$(a)$.
Part $(a)$. Fix a positive integer $r$ and a positive real number $\alpha$. Suppose $G$ is a finite group of exponent $r$ and $A\seq G$ is such that $|A|\geq\alpha|G|$. It is straightforward to show that $\langle A\rangle=\bar{A}^m$ for some $m\leq\lceil 3\alpha+1\rceil$ (see, e.g., [@MassWa Remark 4]). So we can apply Theorem \[thm:mainbdd\]$(2)(b)$, with $k=\lceil\alpha\inv\rceil$ and $m=\lceil 3\alpha+1\rceil$, to obtain a subgroup $K\leq \langle A\rangle$, of index $n=n(\alpha,r)$, such that $K\seq (AA\inv)^2\cap A^2A^{\nv2}\cap (A\inv A)^2\cap A^{\nv 2} A^2$. Note also that $[G:\langle A\rangle]\leq\lceil\alpha\inv\rceil$, and so $[G:K]\leq n\lceil\alpha\inv\rceil$. Now, if $H=\bigcap_{g\in G}gKg\inv$, then $[G:H]\leq [G:K]!\leq O_{\alpha,r}(1)$, $H$ is normal in $G$, and $H\seq (AA\inv)^2\cap A^2A^{\nv2}\cap (A\inv A)^2\cap A^{\nv 2} A^2$.
Part $(b)$. Fix a positive real number $\alpha$. Suppose $G$ is a finite group and $A\seq G$ is such that $|A|\geq\alpha|G|$. In analogy to part $(a)$, Theorem \[thm:maingen\]$(2)$ provides a subgroup $K\leq\langle A\rangle$ and a Bohr neighborhood $B^n_{\tau,\delta}\seq K$ such that $[G:K]$, $\delta\inv$, and $n$ are bounded above in terms of $\alpha$, and $B^n_{\tau,\delta}\seq (AA\inv)^2\cap A^2A^{\nv2}\cap (A\inv A)^2\cap A^{\nv 2} A^2$. If $H=\bigcap_{g\in G}gKg\inv$ and $B=B^n_{\tau,\delta}\cap H$, then $H$ is normal in $G$, $[G:H]\leq O_{\alpha}(1)$, and $B=B^n_{\tau{\upharpoonright}H,\delta}$. So $B$ and $H$ are as desired.
It is worth pointing out that part $(a)$ of Theorem \[thm:Bogogen\] also follows easily from part $(b)$, due to the fact that a $(\delta,n)$-Bohr neighborhood in a group of exponent $r>\delta\inv$ is a subgroup. We leave details to the reader. In a similar way, parts $(1b)$ and $(2b)$ of Theorem \[thm:mainbdd\] follows from parts $(1)$ and $(2)$ of Theorem \[thm:maingen\], respectively. On the other hand, the proof given below of Theorem \[thm:mainbdd\] is more direct, and does not require the model theoretic methods employed here, nor the work from [@CPTNIP] on approximate Bohr neighborhoods.
Fix $\alpha>0$. Suppose $G$ is a purely nonabelian finite group and $A\seq G$ is such that $|A|\geq\alpha|G|$. By Theorem \[thm:Bogogen\]$(b)$, there is a normal subgroup $H\leq G$ and a $(\delta,n)$-Bohr neighborhood $B\seq H$, such that $[G:H]\leq O_\alpha(1)$ and $B\seq (AA\inv)^2\cap (A\inv A)^2\cap A^2A^{\nv 2}\cap A^{\nv 2}A^2$. Since $B$ contains $\ker(\tau)$ for some homomorphism $\tau\colon H\to\T^n$, and $G$ is purely nonabelian, we must have $B=H$.
\[rem:simple\] Corollary \[cor:pure\] implies that for any $\alpha>0$, if $G$ is a nonabelian finite simple group of size at least $\Omega_\alpha(1)$, and $A\seq G$ is such that $|A|\geq\alpha |G|$, then $G=(AA\inv)^2= (A\inv A)^2= A^2A^{\nv 2}= A^{\nv 2}A^2$. Applied to the case of alternating groups $A_n$, this implies that the least upper bound on the index of $H$ in Theorem \[thm:Bogogen\]$(b)$ must be greater than $\frac{1}{2}\lfloor \alpha\inv\rfloor!$ (at least for $\alpha\leq \frac{1}{5}$). However, it should be noted that stronger results about dense sets in nonabelian finite simple groups are already known. In particular, if $G$ is a nonabelian finite simple group with $\log|G|\geq\Omega(\alpha^{\nv 6})$, and $A,B,C\seq G$ are such that $|A|,|B|,|C|\geq\alpha|G|$, then $G=ABC$.[^3] This follows from work of Gowers [@GowQRG] on quasirandom groups (as observed by Nikolov and Pyber [@NikPyDJ]; see see [@NikPyDJ Corollary 1], [@GowQRG Theorem 3.3], and [@GowQRG Theorem 4.7]). Similar results are shown by Hrushovski in [@HruAG] (e.g. [@HruAG Corollary 1.4]).
Ultraproducts of groups {#sec:G}
=======================
In this section we review the ultraproduct construction in the case of groups. The reader only interested in Theorem \[thm:maingen\] (the bounded exponent case) can skip this section. Throughout this section, let $(G_s)_{s\in\N}$ be a fixed sequence of groups, and fix a nonprincipal ultrafilter $\cU$ on $\N$. Let $G=\prod_{\cU}G_s$ be the ultraproduct of the sequence $(G_s)_{s\in\N}$ with respect to $\cU$. Explicitly, $G=(\prod_s G_{s\in\N})/\!\!\sim$, where $(a_s)\sim(b_s)$ if and only if $\{s:a_s=b_s\}\in\cU$. Recall that $G$ is a group under the (well-defined) operation $[(a_s)]\cdot [(b_s)]=[(a_s\cdot b_s)]$. A subset $X\seq G$ is **internal** if there is a sequence $(X_s)_{s\in\N}$, with $X_s\seq G_s$, such that $X=\prod_{\cU}X_s:=(\prod_{s\in \N}X_s)/\!\!\sim$. The collection of internal subsets of $G$ forms a Boolean algebra.
We also assume that $G$ is infinite, i.e., $\{s\in \N:|G_s|>n\}\in\cU$ for all $n\in\N$. As a result, we obtain the following saturation property of $G$.
\[fact:Keisler\] Suppose $(X_i)_{i=0}^\infty$ is a sequence of internal subsets of $G^{\times n}$ such that $\bigcap_{i=0}^k X_i\neq\emptyset$ for all $k\in\N$. Then $\bigcap_{i=0}^\infty X_i\neq\emptyset$.
Finally, we fix a distinguished internal set $A\seq G$ (so $A=\prod_{\cU}A_s$ for some $A_s\seq G_s$), and we assume that $A$ is nonempty and *pseudofinite* (i.e., $A_s$ is nonempty and finite for all $s\in\N$). With $A$ fixed, we define the **$|A|$-normalized pseudofinite counting measure** $\mu$ on the Boolean algebra of internal subsets of $G$. Specifically, given an internal set $X=\prod_{\cU}X_s$, define $$\mu(X)=\lim_{\cU}\frac{|X_s|}{|A_s|}\in\R_{\geq 0}\cup\{\infty\},$$ (where $\lim_{\cU} x_s=y$ if and only if, for all $\epsilon>0$, $\{s:|x_s-y|<\epsilon\}\in\cU$). Note that $\mu$ is a left and right invariant finitely additive measure on the internal subsets of $G$.
Properties of finite subsets of groups such as small alternation or small tripling can be formulated using $\mu$. For example, $\mu(A^3)<\infty$ if and only if for some fixed $k>0$, $\{s:|A_s^3|\leq k|A_s|\}\in\cU$. The fact that $\mu$ is controlled by discrete counting measures allows us to transfer the following Plunnecke-Ruzsa inequalities to $G$.
\[prop:Ruz\] Fix an internal set $X\seq G$, with $0<\mu(X)<\infty$.
1. Suppose $\mu(X^3)\leq k\mu(X)$ for some $k>0$. Then, for any $n\geq 1$ and $\epsilon_1,\ldots,\epsilon_n\in\{\nv1,1\}$, $\mu(X^{\epsilon_1}\cdot\ldots\cdot X^{\epsilon_n})\leq k^{O_n(1)}\mu(X)$.
2. Suppose $\mu(XX\inv X)\leq k\mu(X)$ for some $k>0$. Then $\mu((XX\inv)^n)\leq k^{O_n(1)}\mu(X)$ for any $n\geq 1$.
It suffices to fix $s\in\N$ and prove the claims for $G_s$ with $\mu$ replaced by the usual counting measure. In this setting, part $(a)$ is precisely the “discrete case" of [@TaoPSE Lemma 3.4] (first observed by Helfgott [@HelfGG Lemma 2.2]). So we only need to show part $(b)$. The proof is similar to that of [@TaoPSE Lemma 3.4] and relies on the triangle inequality for Ruzsa distance. In particular, given nonempty finite $X,Y\seq G_s$, the Ruzsa distance is between $X$ and $Y$ is defined as $$d(X,Y)=\log\left(\frac{|XY\inv|}{|X|^{1/2}|Y|^{1/2}}\right).$$ Then, for nonempty finite $X,Y,Z\seq G_s$, we have $d(X,Z)\leq d(X,Y)+d(Y,Z)$ (this is due to Ruzsa [@Ruz] in the commutative setting; see also [@TaoPSE Lemma 3.2]).
Now fix a nonempty finite set $X\seq G_s$, and assume $|XX\inv X|\leq k|X|$. By part $(a)$ (in $G_s$) it is enough to show $|(XX\inv)^3|\leq k^{O(1)}|X|$. For this, first note that $d(XX\inv,X\inv)\leq\log k$. So $d(XX\inv,XX\inv)\leq \log k^2$ by the triangle inequality, and thus $|(XX\inv)^2|\leq k^2|XX\inv|$. Then $d(X X\inv X,X)\leq \log k^2$. By the triangle inequality, $d(X X\inv X,X X\inv X)\leq \log k^4$, and thus $|(XX\inv)^3|\leq k^4|XX\inv X|\leq k^5|X|$.
Sanders-Croot-Sisask Analysis {#sec:CSS}
=============================
In this section, we prove Lemma \[lem:MWKP\], which is the main technical lemma of the paper. It is essentially a modification of a result of Croot-Sisask [@CrSi] and Sanders [@SanBS], which was later adapted by Breuillard, Green, and Tao [@BGT Section 5] for their results on the structure of approximate groups. In the model-theoretic setting, these same techniques were used by Massicot and Wagner [@MassWa] in their work on “definably amenable" approximate groups, and also by Krupi[ń]{}ski and Pillay [@KrPiAG]. Part $(a)$ of Lemma \[lem:MWKP\], which deals with sets of small alternation, is similar to some of Hrushovski’s work with near-subgroups, especially [@HruAG Corollary 3.11]. Our proof follows Sanders [@SanBS] (as do [@BGT], [@KrPiAG], and [@MassWa]), and makes the modifications necessary to work with sets of small tripling or small alternation, and also to account for working with internal sets in the case of ultraproducts.
In this section, we work with a fixed group $G$, a fixed subset $A\seq G$, and a finitely additive measure $\mu$, defined on a certain Boolean algebra of subsets of $G$ and taking values in $\R_{\geq 0}\cup\{\infty\}$. While one could formulate a precise axiomatic framework to allow for a more general setting, it will suffice for our purposes to further assume that one of the following two cases holds.
**Discrete case:** $A$ is a nonempty finite subset of $G$ and $\mu$ is the $|A|$-normalized counting measure: $\mu(X)=|X|/|A|$, defined for any $X\seq G$.
**Pseudofinite case:** $G$, $A$, and $\mu$ are as in Section \[sec:G\].
The reader only interested in Theorem \[thm:maingen\] can assume the discrete case and ignore the pseudofinite case. We call a set $X\seq G$ **measurable** if $\mu(X)$ is defined. Note that Proposition \[prop:Ruz\] makes sense in the discrete case if we remove the word “internal", and the statement remains true (this is what was shown in the proof).
\[lem:MWKP\] Fix $m,n\geq 1$ and a measurable set $X\seq G$, with $0<\mu(X)<\infty$.
1. Suppose $\mu(XX\inv X)\leq k\mu(X)$ for some $k\geq 1$. Then there is a measurable symmetric set $Y\seq G$ such that $Y^n\seq (X X\inv)^2$ and $(XX\inv)^m$ is covered by $O_{k,m,n}(1)$ $(XX\inv)^m$-translates of $Y$.
2. Suppose $\mu(X^3)\leq k\mu(X)$ for some $k\geq 1$. Then there is a measurable symmetric set $Y\seq G$ such that $Y^n\seq (XX\inv)^2\cap X^2X^{\nv 2}\cap (X\inv X)^2\cap X^{\nv 2}X^2$ and $\bar{X}^{m}$ is covered by $O_{k,m,n}(1)$ $\bar{X}^{m}$-translates of $Y$.
We will prove the two statements in parallel, as the arguments are similar. Let $X\seq G$ be a fixed internal set, with $0<\mu(X)<\infty$. Fix an integer $k\geq 1$. By “case $(a)$", we mean the assumption that $\mu(XX\inv X)\leq k\mu(X)$; and by “case $(b)$”, we mean the assumption that $\mu(X^3)\leq k\mu(X)$.
Before starting the argument, we first simplify case $(b)$. Define $$\Pi_1(X)=(XX\inv)^2,~\Pi_2(X)=X^2X^{\nv 2},~\Pi_3(X)=(X\inv X)^2,\text{ and }\Pi_4(X)=X^{\nv 2}X^2.$$ We claim that it suffices to find, for each individual $c\in\{1,2,3,4\}$, a set $Y_c$ as described but only with $Y_c^n\seq \Pi_c(X)$. To see this, we apply some elementary tools from [@BGT] (which transfer to the pseudofinite setting). So fix $m,n\geq 1$ and suppose that, for $c\in\{1,2,3,4\}$, we have measurable symmetric $Y_c\seq G$ such that $Y_c^{4n}\seq\Pi_c(X)$ and $\bar{X}^m$ is covered by $O_{k,m,n}(1)$ $\bar{X}^m$-translates of $Y_c$. Let $S=\bar{X}^{\max\{m,2\}}$. Then $S$ is an $O_{k,m}(1)$-approximate subgroup by Proposition \[prop:Ruz\]$(a)$ and [@BGT Corollary 5.2]. Setting $Y_*=\bigcap_{c=1}^4 Y^2_c$, we have $\mu(\bar{X}^m)\leq\mu(S)\leq O_{k,m,n}(\mu(Y_*))$ by [@BGT Corollary 5.9]. By [@BGT Lemma 5.1], there is a finite set $F\seq \bar{X}^m$ such that $|F|\leq\mu(\bar{X}^mY_*)/\mu(Y_*)$ and $\bar{X}^m\seq FY_*^2$. Since $Y_*\seq \bar{X}^2$, we have $\mu(\bar{X}^mY_*)\leq O_{k,m}(\mu(\bar{X}^m))$ by Proposition \[prop:Ruz\]$(a)$, which implies $|F|\leq O_{k,m,n}(1)$. So if we set $Y=Y_*^2$, then $Y$ is a measurable symmetric set, $\bar{X}^m$ is covered by $O_{k,m,n}(1)$ $\bar{X}^m$-translates of $Y$, and $Y^n= Y_*^{2n}\seq \bigcap_{c=1}^4 Y^{4n}_c\seq \bigcap_{c=1}^4\Pi_c(X)$, as desired.
So now in case $(b)$, we fix $c\in\{1,2,3,4\}$ and find $Y=Y_c$ with $Y^n\seq\Pi_c(X)$. Since $\Pi_1(X\inv)=\Pi_3(X)$ and $\Pi_2(X\inv)=\Pi_4(X)$, it suffices to assume $c\in\{1,2\}$. Set $$V=\begin{cases}
(XX\inv)^m & \text{in case $(a)$}\\
\bar{X}^{m} & \text{in case $(b)$}
\end{cases}
\makebox[.4in]{and}
Z=\begin{cases}
X\inv & \text{in case $(a)$, or case $(b)$ with $c=1$}\\
X & \text{in case $(b)$ with $c=2$.}
\end{cases}$$ Note that $V$ is symmetric. We now closely follow Sanders [@SanBS]. For $t\in (0,1]$, define $$\cB_t=\{B\seq X:\text{$B$ is internal and $\mu(B)\geq t\mu(X)$}\}.$$ Then $X\in \cB_t$ for all $t\in(0,1]$, and so we may define a function $f\colon (0,1]\to[1,\infty)$ such that $f(t) =\inf\{\mu(BZ)/\mu(X):B\in\cB_t\}$.
By Proposition \[prop:Ruz\], we may fix $\ell\geq 1$ such that $\ell\leq k^{O_m(1)}$ and $\mu(VX)\leq \ell\mu(X)$ (in case $(a)$, use $\mu((XX\inv)^mX)\leq \mu((XX\inv)^{m+1})$). By [@MassWa Lemma 11] (taken from [@SanBS]), we may choose $t\in (0,1]$ such that $t\inv\leq O_{k,m,n}(1)$ and $f(t^2/2\ell)\geq ((2n-1)/2n)f(t)$. Choose $B\in\cB_t$ such that $\mu(BZ)/\mu(X)\leq ((2n+1)/2n)f(t)$.
Define $Y_*=\{g\in V^2:\mu(gB\cap B)\geq (t^2/2\ell)\mu(X)\}$, and note that $1\in Y_*$, since $B\in\cB_t$ and $t>t^2/2\ell$.
*Claim 1*: $V$ is covered by $O_{k,m,n}(1)$ $V$-translates of $Y_*$.
*Proof*: Let $w=\lfloor 2\ell/t\rfloor$ and note that $w\leq O_{k,m,n}(1)$. Suppose, for a contradiction, that $V$ is not covered by $w$ $V$-translates of $Y_*$. Then we may construct a sequence $(g_i)_{i=0}^w$ from $V$ such that, for all $i\leq w$, $g_i\not\in\bigcup_{j<i}g_j Y_*$. For any $0\leq i<j\leq w$, $g_i\inv g_j\in V^2\backslash Y_*$, and so we have $\mu(g_iB\cap g_jB)<(t^2/2\ell)\mu(X)$. We also have $g_iB\seq VX$ for any $0\leq i\leq w$. Now we obtain a contradiction: $$\begin{aligned}
\ell\mu(X) &\geq \mu(VX) \geq \mu\left(\bigcup_{i\leq w}g_i B\right) \geq (w+1)\mu(B)-\sum_{i<j\leq w}\mu(g_iB\cap g_jB)\\
&> (w+1)t\mu(X)-\frac{w(w+1)t^2\mu(X)}{4\ell}= (w+1)\left(1-\frac{wt}{4\ell}\right)t\mu(X)\\
&> \ell\mu(X),\end{aligned}$$ where the last inequality uses $w\leq 2\ell/t< w+1$.[$\dashv_{\text{\scriptsize{Claim {1}}}}$]{}
Set $$W=\begin{cases}
(XX\inv)^2 & \text{in case $(a)$, or case $(b)$ with $c=1$}\\
X^2X^{\nv 2} & \text{in case $(b)$ with $c=2$.}
\end{cases}$$
*Claim 2*: $Y^n_*\seq W$.
*Proof*: We first show that $\mu(gBZ\smd BZ)<2\mu(BZ)/n$ for any $g\in Y_*$. To see this, note that if $g\in Y_*$ then $gB\cap B\in\cB_{t^2/2\ell}$, and so $$\mu(gBZ\cap BZ)\geq\mu((gB\cap B)Z)\geq f(t^2/2\ell)\mu(X)\geq \frac{2n-1}{2n}f(t)\mu(X)\geq\frac{2n-1}{2n+1}\mu(BZ).$$ So, for any $g\in Y_*$, $$\mu(gBZ\smd BZ)=2\mu(BZ)-2\mu(gBZ\cap BZ)\leq \frac{4}{2n+1}\mu(BZ)<\frac{2}{n}\mu(BZ).$$ Now fix $g_1,\ldots,g_n\in Y_*$ and, for $0\leq i\leq n$, let $h_i=\prod_{j\leq i} g_j$ (so $h_0=1$). Then $$\begin{aligned}
\mu(h_nBZ\smd BZ) \leq \mu\left(\bigcup_{i=0}^{n-1} h_i(g_{i+1}BZ\smd BZ)\right)\leq \sum_{i=0}^{n-1}\mu(g_{i+1}BZ\smd BZ)<2\mu(BZ).\end{aligned}$$ It follows that $h_nBZ\cap BZ\neq\emptyset$, which implies $h_n\in BZZ\inv B\inv\seq W$.[$\dashv_{\text{\scriptsize{Claim {2}}}}$]{}
Now, in the discrete case, we may take $Y=Y_*$ and the proof is finished. In the pseudofinite case, we must address the fact that $Y_*$ may not be internal. So suppose we are in the pseudofinite case.
*Claim 3*: $Y_*=\bigcap_{i=0}^\infty Y_i$ where, for each $i\in\N$, $Y_i$ is symmetric and internal, and contains $Y_{i+1}$.
*Proof*: Let $\beta=(t^2/2\ell)\mu(X)$, and so $Y_*=\{g\in V^2:\mu(gB\cap B)\geq \beta\}$. By assumption, $X$ and $B$ are internal and so we may choose sets $X_s,B_s\seq G_s$, for $s\in\N$, such that $X=\prod_\cU X_s$ and $B=\prod_\cU B_s$. Given $s\in\N$, set $$V_s=\begin{cases}
(X_sX_s\inv)^n & \text{in case $(a)$,}\\
\bar{X}_s^{n} & \text{in case $(b)$.}
\end{cases}$$ Note that each $V_s$ is symmetric, and $V=\prod_{\cU}V_s$. Given $i\in\N$ and $s\in\N$, define $$Y_{i,s}=\left\{g\in V_s^2:\frac{|gB_s\cap B_s|}{|G_s|}>\textstyle\beta-\frac{1}{i+1}\right\}.$$ Note that $Y_{i,s}\inv = Y_{i,s}$ for all $i,s\in\N$. Given $i\in\N$, let $Y_i=\prod_\cU Y_{i,s}$. Then, for any $i\in\N$, $Y\inv_i=Y_i$ and $Y_{i+1}\seq Y_i$. Moreover, $Y_*=\bigcap_{i=0}^\infty Y_i$. [$\dashv_{\text{\scriptsize{Claim {3}}}}$]{}
Fix $(Y_i)_{i=0}^\infty$ as in Claim 3. To finish the proof of the lemma in the pseudofinite case, it suffices to show that $Y^n_i\seq W$ for some $i\in\N$. Toward this end, we first show $Y_*^n=\bigcap_{i=0}^\infty Y^n_i$. We clearly have $Y_*^n\seq \bigcap_{i=0}^\infty Y^n_i$. For the other direction, fix $a\in \bigcap_{i=0}^\infty Y_i^n$. For $i\in\N$, define $$D_i=\{(g_1,\ldots,g_n)\in G^{\times n}:g_j\in Y_i\text{ for $1\leq j\leq i$, and $a=g_1\cdot\ldots \cdot g_n$}\}.$$ Then, for all $i\in\N$, $D_i$ is nonempty, internal, and $D_{i+1}\seq D_i$. By Fact \[fact:Keisler\], there is $(g_1,\ldots,g_n)\in \bigcap_{i=0}^\infty D_i$, and so $a=g_1\cdot \ldots \cdot g_n\in Y^n_*$.
Finally, since $\bigcap_{i=0}^{\infty} Y^n_i=Y_*^n\seq W$, it follows from Fact \[fact:Keisler\] that $Y^n_i\seq W$ for some $i\in\N$.
At this point, we have all necessary tools to proceed with the proof of Theorem \[thm:mainbdd\] (see Section \[sec:proofbdd\]). For Theorem \[thm:maingen\], we will need following corollary of the previous lemma, which is only meaningful in the pseudofinite case.
\[cor:MWKP\]$~$
1. Suppose $\mu(AA\inv A)\leq k<\infty$. Then there is a sequence $(X_n)_{n=0}^\infty$ of symmetric, internal subsets of $G$ such that $X_0\seq (AA\inv)^2$ and, for any $n\in\N$, $X^2_{n+1}\seq X_n$ and $(AA\inv)^4$ is covered by $O_{k,n}(1)$ $\langle AA\inv\rangle$-translates of $X_n$.
2. Suppose $\mu(A^3)\leq k<\infty$. Then there is a sequence $(X_n)_{n=0}^\infty$ of symmetric, internal subsets of $G$ such that $X_0\seq (AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2$ and, for any $n\in\N$, $X^2_{n+1}\seq X_n$ and $\bar{A}^8$ is covered by $O_{k,n}(1)$ $\langle A\rangle$-translates of $X_n$.
As in Lemma \[lem:MWKP\], we prove the two statements in parallel. Set $$(V,W,\Sigma)=\begin{cases}
((AA\inv)^4,(AA\inv)^2,\langle AA\inv\rangle) & \text{in case $(a)$,}\\
(\bar{A}^8,(AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2,\langle A\rangle) & \text{in case $(b)$.}
\end{cases}$$
We construct a sequence $(Y_n)_{n=0}^\infty$ of symmetric, internal subsets of $G$, such that $Y_0^8\seq W$ and, for all $n\in\N$, $Y_{n+1}^8\seq Y_n^4$ and $V$ is covered by $O_{K,n}(1)$ $\Sigma$-translates of $Y_n$. Given this, the result follows by setting $X_n=Y^4_n$.
By Lemma \[lem:MWKP\], there is a symmetric, internal set $Y_0\seq G$ such that $Y_0^8\seq W$ and $V$ is covered by $O_k(1)$ $V$-translates of $Y_0$. Suppose we have constructed $Y_0,\ldots,Y_n$ satisfying the desired properties. Note that $\mu(W)<\infty$ by Proposition \[prop:Ruz\], and so $\mu(Y_n)<\infty$ since $Y_n\seq Y^8_n\seq W$. Since $V$ is covered by $O_{k,n}(1)$ translates of $Y_n$, we have $0<\mu(V)\leq O_{k,n}(1)\mu(Y_n)$, and so $\mu(Y_n)>0$. Since $Y^3_n\seq W$, we also have $\mu(Y^3_n)\leq O_{k,n}(1)\mu(Y_n)$. By Lemma \[lem:MWKP\]$(b)$, there is a symmetric, internal $Y_{n+1}\seq G$ such that $Y_{n+1}^8\seq Y_n^4$ and $Y_n^8$ is covered by $O_{k,n+1}(1)$ $Y_n^8$-translates of $Y_{n+1}$. Since $Y_n\seq Y_n^8\seq W$, it follows that $Y_n$ is covered by $O_{k,n+1}(1)$ $W$-translates of $Y_{n+1}$. Since $V$ is covered by $O_{k,n}(1)$ $\Sigma$-translates of $Y_n$, it follows that $V$ is covered by $O_{k,n+1}(1)$ $\Sigma$-translates of $Y_{n+1}$.
Proof of Theorem \[thm:mainbdd\] {#sec:proofbdd}
================================
The following theorem is [@BGT Theorem 6.15]. It can also be deduced from [@HruAG Corollary 4.18].
\[thm:BGTbdd\] Let $G$ be a group of exponent $r$, and suppose $X\seq G$ is a $k$-approximate subgroup. Then $X^4$ contains a subgroup $H\leq G$ such that $X$ is covered by $O_{k,r}(1)$ left cosets of $H$.
We now give the proof of Theorem \[thm:mainbdd\].
We prove parts $(1)$ and $(2)$ of the theorem in two parallel cases. Fix positive integers $k$, $m$, and $r$. Let $G$ be a group and fix $A\seq G$ nonempty and finite, with $|AA\inv A|\leq k|A|$ in case $(1)$ and $|A^3|\leq k|A|$ in case $(2)$. Set $$(V,W,\Sigma)=\begin{cases}
(AA\inv,(AA\inv)^2,\langle AA\inv\rangle) & \text{in case $(1)$,}\\
(\bar{A},(AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2,\langle A\rangle) & \text{in case $(2)$.}
\end{cases}$$ By increasing $m$ if necessary, we may assume $W\seq V^m$ without loss of generality.
Assume $\Sigma$ has exponent $r$. By Lemma \[lem:MWKP\], there is a nonempty finite symmetric set $Y\seq G$ such that $Y^4\seq W$ and $V^m$ is covered by $O_{k,m}(1)$ left translates of $Y$. Since $Y^2\seq Y^4\seq W\seq V^m$, it follows that $Y$ is an $O_{k,m}(1)$-approximate group. By Theorem \[thm:BGTbdd\], $Y^4$ contains a subgroup $H\leq G$ such that $Y$ is covered by $O_{k,m,r}(1)$ left cosets of $H$. Then $H\seq W$ and $V^m$ is covered by $O_{k,m,r}(1)$ left cosets of $H$. This proves part $(a)$ in both cases $(1)$ and $(2)$.
For part $(b)$, suppose $\Sigma=V^m$. By part $(a)$ there is a subgroup $K\leq \Sigma$, of index $O_{k,m,r}(1)$, such that $K\seq W$. If $H=\bigcap_{g\in \Sigma}gKg\inv$, then $H$ is normal in $\Sigma$, $H\seq W$, and $[G:H]\leq [G:K]!\leq O_{k,m,r}(1)$.
Saturated extensions and approximate Bohr neighborhoods {#sec:G*}
=======================================================
Throughout this section, let $G$ be an ultraproduct constructed as in Section \[sec:G\]. We will now endow $G$ with a first-order structure, and then take a sufficiently saturated elementary extension $G_*$. Specifically, we define the **internal language of $G$**, denoted $\cL$, to be the group language together with a unary relation $R_X$ for any internal $X\seq G$. We view $G$ as an $\cL$-structure by interpreting each $R_X$ as $X$. We also view each $G_s$ as an $\cL$-structure by interpreting $R_X$ as some set $X(G_S)\seq G_s$, so that $X=\prod_{\cU}X(G_s)$. In particular, $G$ is also the ultraproduct of the sequence of *$\cL$-structures* $(G_s)_{s\in\N}$.
Now let $G_*$ be a sufficiently saturated elementary extension of $G$ with respect to the language $\cL$.[^4] When we say $X\seq G_*$ (resp. $X\seq G$) is *definable*, we mean definable in the language $\cL$ using parameters from $G_*$ (resp. from $G$). If we want to specify that $X$ is definable using parameters from some set $C$, we will say *$C$-definable*. Let $A_*$ be the interpretation in $G_*$ of the predicate in $\cL$ naming $A$.
Note that the measure $\mu$ naturally extends to $G$-definable subsets of $G_*$. In particular, given a $G$-definable set $X\seq G_*$, the interpretation $X(G)$ of $X$ in $G$ is internal, and so we let $\mu(X)=\mu(X(G))$. We say that a $G$-definable set $X\seq G_*$ is **pseudofinite** if $X(G)$ is an ultraproduct of finite sets.
Although it will not be necessary for our work, we recall that $\mu$ can be extended (not necessarily uniquely) to all definable subsets of $G_*$. For example, one can add a sort for $[0,1]$ and a function $f_\phi$ for each formula $\phi(x;{\bar{y}})$, from the home sort to $[0,1]$, which is interpreted as $f_\phi({\bar{b}})=\mu(\phi(G;{\bar{b}}))$. Then take $G_*$ to be a saturated extension in this larger language. See [@HPP Section 2].
A cardinal is **bounded** if it is strictly smaller than the saturation of $G_*$. A set $X\seq G_*$ is **type-definable** (resp. **countably type-definable**) if it is an intersection of a bounded (resp. countable) number of definable subsets of $G_*$.
Now suppose $\Sigma$ is a definable subgroup of $G_*$, and $\Gamma$ is a type-definable normal subgroup of $\Sigma$ such that $[\Sigma:\Gamma]$ is bounded. Call a set $X\seq \Sigma/\Gamma$ **closed** if $\pi\inv(X)$ is type-definable, where $\pi$ is the canonical projection from $\Sigma$. It is a standard fact that this defines a topology on $\Sigma/\Gamma$, called the **logic topology**, under which $\Sigma/\Gamma$ is a compact (Hausdorff) topological group. If $\Gamma$ is countably type-definable, then $\Sigma/\Gamma$ is second countable. See [@PilCLG Section 2] for details.
The rest of this section summarizes some tools from [@CPTNIP] concerning Bohr neighborhoods in $G_*$ and issues regarding their transfer to $G$ and the groups $G_s$.
Given a compact space $\cX$, we say that a map $f\colon G\to \cX$ is **definable** if $f\inv(C)$ is type-definable for any closed $C\seq \cX$. The next proposition is a special case of [@CPTNIP Proposition 5.1], and crucially relies on the result of Pillay [@PiRCP] that the connected component of a definable compactification of a pseudofinite group is abelian.
\[prop:Bohr\] Suppose $\Sigma$ is a $G$-definable pseudofinite subgroup of $G_*$, and $\Gamma\leq \Sigma$ is a countably type-definable bounded-index normal subgroup of $\Sigma$. Then there is a decreasing sequence $(X_i)_{i=0}^\infty$ of definable subsets of $\Sigma$ such that $\Gamma=\bigcap_{i=0}^\infty X_i$ and, for all $i\in\N$, there are:
1. a definable finite-index normal subgroup $H_i\leq \Sigma$, and
2. a definable homomorphism $\pi_i\colon H_i\to \T^{n_i}$, for some $n_i\in\N$,
such that $\Gamma\seq \ker\pi_i\seq X_i\seq H_i$. If, moreover, $\Sigma/\Gamma$ is abelian, then we may assume $H_i=\Sigma$ for all $i\in\N$.
In the setting of the previous proposition, the fact that $X_i$ is definable and contains $\ker\pi_i$ implies that it contains a Bohr neighborhood $B^{n_i}_{\pi_i,\epsilon_i}$ for sufficiently small $\epsilon_i>0$. However, these Bohr neighborhoods are not necessarily definable, and so we will need to approximate them by definable objects.
\[def:approx\] Fix a group $H$ and an integer $n\in\N$.
1. Given $\delta>0$, we say that a function $f\colon H\to\T^n$ is a **$\delta$-homomorphism** if $f(1)=0$ and, for all $x,y\in H$, $d_n(f(xy),f(x)+f(y))<\delta$.
2. Given $\delta,\epsilon>0$, we say that $Y\seq H$ is a **$\delta$-approximate $(\epsilon,n)$-Bohr neighborhood in $H$** if there is a $\delta$-homomorphism $f\colon H\to \T^n$ such that $Y=\{x\in H:d_n(f(x),0)<\epsilon\}$.
3. Assume $H$ is a definable subgroup of $G_*$, and $\pi\colon H\to \T^n$ is a definable homomorphism. Given an integer $t\geq 1$, we say that a decreasing sequence $(Y_i)_{i=0}^\infty$ of subsets of $H$ is a **definable $(t,\pi)$-approximate Bohr chain in $H$** if $\bigcap_{i=0}^\infty Y_i=\ker\pi$ and there is a decreasing sequence $(\delta_i)_{i=0}^\infty$ in $\R_{>0}$ converging to $0$ such that, for all $i\geq 0$, $Y_i=\{x\in H:d_n(f_i(x),0)<t\delta_i\}$ for some definable $\delta_i$-homomorphism $f_i\colon H\to \T^n$ with finite image.
Note that if $(Y_i)_{i=0}^\infty$ is a definable $(t,\pi)$-approximate Bohr chain in $H$, then each $Y_i$ is a $\delta_i$-approximate $(t\delta_i,n)$-Bohr set in $H$. It is also worth emphasizing that each $Y_i$ is indeed a *definable* subset of $H$ (see [@CPTNIP Proposition 5.3]). The next result is a special case of [@CPTNIP Lemma 5.4].
\[lem:Bapprox\] Suppose $H\leq G_*$ is definable and $\pi\colon H\to \T^n$ is a definable homomorphism for some $n\in\N$. Then, for any integer $t\geq 1$, there is a definable $(t,\pi)$-approximate Bohr chain $(Y_i)_{i=0}^\infty$ in $H$.
Finally, we state a special case of [@CPTNIP Corollary 4.4], which is an immediate consequence of [@AlGlGo Theorem 5.13].
\[prop:findBohr\] There is a real number $\theta>0$ such that if $H$ is a finite group, $n\in\N$, and $0<\delta<\theta$, then every $\delta$-approximate $(3\delta,n)$-Bohr neighborhood in $H$ contains a $(\delta,n)$-Bohr neighborhood in $H$.
Proof of Theorem \[thm:maingen\] {#sec:genproof}
================================
Transfer to $G_*$
-----------------
Throughout this subsection, let $G$ be an ultraproduct constructed as in Section \[sec:G\], and let $G_*$ be the saturated extension from Section \[sec:G\*\]. The goal of this subsection is to transfer the analysis in Section \[sec:CSS\] to the saturated group $G_*$. The main idea is that the decreasing sequence $(X_n)_{n=0}^\infty$ of internal sets constructed in Corollary \[cor:MWKP\] converges to a subgroup of $G$, which is “large" in a certain sense. By transferring the sequence first to $G_*$, we will have more precise control over exactly what this means, and it will be easier to find normal subgroups.
Recall that $A_*$ is the interpretation in $G_*$ of our distinguished internal set $A\seq G$.
\[lem:Gamma\]$~$
1. Suppose $\mu(A_*A\inv_* A_*)<\infty$. Then there is a countably type-definable subgroup $\Gamma\leq G_*$ such that $\Gamma\seq (A_*A_*\inv)^2$ and $\Gamma$ has index at most $2^{\aleph_0}$ in $\langle A_* A_*\inv\rangle$.
2. Suppose $\mu(A_*^3)<\infty$. Then there is a countably type-definable subgroup $\Gamma\leq G_*$ such that $\Gamma\seq (A_*A_*\inv)^2\cap A_*^2A_*^{\nv 2}\cap (A_*\inv A_*)^2\cap A_*^{\nv 2}A_*^2$ and $\Gamma$ has index at most $2^{\aleph_0}$ in $\langle A_*\rangle$.
Set $$(V,W,\Sigma)=\begin{cases}
(A A\inv)^2,(A A\inv)^2,\langle A A\inv\rangle) & \text{in case $(a)$}\\
(\bar{A}^4,(AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2,\langle A\rangle) & \text{in case $(b)$.}
\end{cases}$$ Let $(V_*,W_*,\Sigma_*)$ be similarly defined, but with $A_*$ in place of $A$.
Working first in $G$, we apply Corollary \[cor:MWKP\] to find a sequence $(Y_n)_{n=0}^\infty$ of symmetric, internal subsets of $G$ such that $Y_0\seq W$ and, for any $n\in\N$, $Y^2_{n+1}\seq Y_n$ and $V^2$ is covered by finitely many $\Sigma$-translates of $Y_n$. So, for any $n\in\N$, there is some $k_n\in\N$ such that $V^2$ is covered by finitely many $V^{k_n}$-translates of $Y_n$.
Now in $G_*$, let $X_n$ be the $\emptyset$-definable set given by the interpretation of the unary relation $R_{Y_n}$. By elementarity, $X_0\seq W_*$ and, for any $n\in\N$, $X_n$ is symmetric and internal, $X^2_{n+1}\seq X_n$, and $V_*^2$ is covered by finitely many $V_*^{k_n}$-translates of $X_n$.
Fix $n\in\N$, and let $F\seq \Sigma_*$ be finite such that $V_*^2\seq FX_n$. By induction on $k\geq 1$, we show that $V_*^k\seq F^kX_n$. The base case is given, so assume the result for $k\geq 1$. Then $V_*^{k+1}=V^k_*V_*\seq F^kX_nV_*\seq F^kW_*V_*\seq F^kV^2_*\seq F^{k+1}X_n$.
We have shown that, for any $n\in\N$, there is a countable set $F_n\seq \Sigma_*$ such that $\Sigma_*=FX_n$. Let $\Gamma=\bigcap_{n=0}^\infty X_n$, and note that $\Gamma$ is a countably type-definable subgroup of $G_*$, which is contained in $W_*$. Since $\Sigma_*$ is covered by countably many $\Sigma_*$-translates of $X_n$ for all $n\geq 1$, it follows that $\Gamma$ has index at most $2^{\aleph_0}$ in $\Sigma_*$.
\[cor:Gamma\]$~$
1. Suppose $\mu(A_*A\inv_* A_*)<\infty$. Then there is a countably type-definable subgroup $\Gamma\leq G_*$ such that:
1. $\Gamma\seq (A_* A_*\inv)^2$,
2. $\Gamma$ is normal in $\langle A_*A_*\inv\rangle$, and
3. $\Gamma$ has index at most $2^{\aleph_0}$ in $\langle A_* A_*\inv\rangle$.
2. Suppose $\mu(A_*^3)<\infty$. Then there is a countably type-definable subgroup $\Gamma\leq G_*$ such that:
1. $\Gamma\seq (A_*A_*\inv)^2\cap A_*^2 A_*^{\nv 2}\cap (A_*\inv A_*)^2\cap A_*^{\nv 2}A_*^2$,
2. $\Gamma$ is normal in $\langle A_*\rangle$, and
3. $\Gamma$ has index at most $2^{\aleph_0}$ in $\langle A_*\rangle$.
Set $$(V,W,\Sigma)=\begin{cases}
A_* A_*\inv,(A_* A_*\inv)^2,\langle A_* A_*\inv\rangle) & \text{in case $(a)$}\\
(\bar{A}_*,(A_*A_*\inv)^2\cap A_*^2A_*^{\nv 2}\cap (A_*\inv A_*)^2\cap A_*^{\nv 2}A_*^2,\langle A_*\rangle) & \text{in case $(b)$.}
\end{cases}$$ By Lemma \[lem:Gamma\], we have a countably type-definable subgroup $\Gamma_0\leq G_*$ such that $\Gamma_0\seq W$ and $\Gamma_0$ has index at most $2^{\aleph_0}$ in $\Sigma$. Let $\Gamma=\bigcap_{g\in\Sigma}g\Gamma_0g\inv$. Then $\Gamma$ is an intersection of at most $2^{\aleph_0}$ conjugates $g\Gamma_0g\inv$ with $g\in\Sigma$. So $\Gamma$ is a type-definable subgroup of $G_*$, which is normal in $\Sigma$ and has index at most $2^{2^{\aleph_0}}$ in $\Sigma$.
We now show that $\Gamma$ is countably type-definable of index at most $2^{\aleph_0}$ in $\Sigma$. For the first part, let $\cL_0\seq\cL$ be a countable language containing the language of groups, and unary predicates defining $A$ and $Y_n$ for $n\in\N$, where $Y_n$ are the predicates used to obtain $\Gamma_0$ (via the proof of Lemma \[lem:Gamma\]). Then $\Gamma$ is $\cL_0$-type-definable. Moreover, $\Gamma_0$ is $\cL_0$-type-definable over $\emptyset$ and, so $\sigma(\Gamma_0)=\Gamma_0$ for any $\sigma\in \Aut_{\cL_0}(G_*)$. Since $\Sigma$ is $\Aut_{\cL_0}(G_*)$-invariant, $\sigma(\Gamma)=\Gamma$ for any $\sigma\in\Aut_{\cL_0}(G_*)$, and so $\Gamma$ is $\cL_0$-type-definable over $\emptyset$. Since $\cL_0$ is countable, $\Gamma$ is countably type-definable.
Finally, let $\Gamma=\bigcap_{n=0}^\infty D_n$, where each $D_n$ is definable and (without loss of generality) contained in $\Sigma$. Since $\Gamma$ has bounded index in $\Sigma$, we may fix some bounded set $C\subset\Sigma$ such that $\Sigma =C\Gamma$. Fix $m,n\in\N$. Then $V^m\seq\Sigma= C\Gamma=CD_n$. By saturation of $G_*$, it follows that there is some finite $C_{n,m}\seq C$ such that $V^m\seq C_{n,m}D_n$. So, if $C_n=\bigcup_{m\in\N}C_{n,m}$, then $C_n$ is countable and $\Sigma = C_nD_n$. Once again, this implies that $\Gamma$ has index at most $2^{\aleph_0}$ in $\Sigma$.
Corollary \[cor:Gamma\] is a nonstandard Bogolyubov-Ruzsa-type statement about pseudofinite sets of small alternation or small tripling. However, since the subgroup $\Gamma$ is not necessarily definable, it cannot be directly transferred to statements about internal subsets of $G$ (which are needed in order to transfer to the finite groups $G_s$). For this, we need the material in Section \[sec:G\*\] on approximate Bohr neighborhoods.
Ultraproduct argument
---------------------
We now prove parts $(1)$ and $(2)$ of Theorem \[thm:maingen\] simultaneously. Given a group $G$ and a set $A\seq G$, let $$\Sigma(A) =\begin{cases}
\langle AA\inv\rangle \\
\langle A\rangle
\end{cases}
U(A) =\begin{cases}
AA\inv A \\
A^3
\end{cases}
V(A) =\begin{cases}
AA\inv & \text{\hspace{10pt}in part $(1)$}\\
\bar{A} & \text{\hspace{10pt}in part $(2)$,}
\end{cases}$$ $$\text{and }W(A) =\begin{cases}
(AA\inv)^2 & \text{in part $(1)$}\\
(AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2 & \text{in part $(2)$.}
\end{cases}$$ The ambient group $G$ is supressed from the notation, but this should cause no confusion in the following proof.
The next result is a restatement of Theorem \[thm:maingen\], which we will prove by taking an ultraproduct of counterexamples, and using the material in Section \[sec:G\*\] in order to transfer Bohr neighborhoods through ultraproducts and saturated extensions.
\[thm:gen2\] For any positive integers $k$ and $m$, there is an integer $s=s(k,m)$ such that the following holds. Suppose $G$ is a group and $A\seq G$ is finite such that $|U(A)|\leq k|A|$ and $\Sigma(A)=V(A)^m$. Then there are:
1. a normal subgroup $H\leq \Sigma(A)$, of index at most $s$, and
2. a $(\delta,n)$-Bohr neighborhood $B$ in $H$, with $\delta\inv,n\leq s$,
such that $B\seq W(A)$. Moreover, if $\Sigma(A)$ is abelian then we may assume $H=\Sigma(A)$.
Suppose not. Then for any $s\in\N$, we may fix a group $G_s$ and a finite set $A_s\seq G_s$ such that $|U(A_s)|\leq k|A_s|$, $\Sigma(A_s)=V(A_s)^m$, and there does not exist a normal subgroup $H\leq \Sigma(A_s)$ and a $(\delta,n)$-Bohr neighborhood $B$ in $H$ such that $[\Sigma(A_s):H],\delta\inv,n\leq s$ and $H\seq W(A_s)$. Note that $|\Sigma(A_s)|>\max\{s,m\}$, since otherwise we could take $H=\{1\}$.
Let $\cU$ be a nonprincipal ultrafilter on $\N$ and set $G=\prod_{\cU}G_s$. Let $A=\prod_{\cU}A_s$, and note that $A$ is an internal subset of $G$. We also have $U(A)=\prod_{\cU}U(A_s)$, $V(A)=\prod_{\cU}V(A_s)$, $W(A)=\prod_{\cU}W(A_s)$, and $$\Sigma(A)=\bigcup_{n\in\N}V(A)^n=\bigcup_{n\in\N}\prod_{\cU}V(A_s)^n=\prod_{\cU}V(A_s)^m= V(A)^m$$ Note, in particular, that $\Sigma(A)=\prod_{\cU}\Sigma(A_s)$ is infinite, and so $G$ is infinite. Let $\mu$ be the $|A|$-normalized pseudofinite counting measure on internal subsets of $G$. By [Ł]{}o[ś]{}’s Theorem, $\mu(U(A))\leq k<\infty$.
Let $G_*$ be a saturated elementary extension of $G$ in the internal language of $G$ (see Section \[sec:G\*\]), and let $A_*$ be the interpretation in $G_*$ of the predicate defining $A$ in $G$. By Lemma \[lem:Gamma\], there is a countably type-definable subgroup $\Gamma\leq G_*$ such that $\Gamma\seq W(A_*)$ and $\Gamma$ has index at most $2^{\aleph_0}$ in $\Sigma(A_*)$. Note also that $\Sigma(A_*)=V(A_*)^m$. In particular, $\Sigma(A_*)$ is $G$-definable and pseudofinite.
By Proposition \[prop:Bohr\] and saturation of $G_*$, there is a definable finite-index normal subgroup $H\leq\Sigma(A_*)$ and a definable homomorphism $\pi\colon H\to\T^n$, for some $n\in\N$, such that $\Gamma\seq\ker\pi\seq H\cap W(A_*)$. By Lemma \[lem:Bapprox\], there is a definable $(3,\pi)$-approximate Bohr chain $(Y_i)_{i=0}^\infty$ in $H$. By saturation, $Y_i\seq W(A_*)$ for sufficiently large $i\in\N$. So we may fix $\delta<\theta$, where $\theta$ is as in Proposition \[prop:findBohr\], and a definable $\delta$-homomorphism $f\colon H\to \T^n$ such that $Y:=\{x\in H:d(f(x),0)<3\delta\}\seq W(A^*)$.
Let $\Lambda=f(H)$, and note that $\Lambda$ is finite (see Definition \[def:approx\]$(3)$). Given $\lambda\in\Lambda$, let $F(\lambda)=f\inv(\lambda)\seq H$. Then each $F(\lambda)$ is definable. Set $r=[\Sigma(A_*):H]<\infty$.
Fix $\cL$-formulas $\phi(x;{\bar{y}})$, $\psi(x;{\bar{z}})$, and $\zeta_\lambda(x;{\bar{u}}_\lambda)$ for $\lambda\in \Lambda$, such that $H$ is defined by an instance of $\phi(x;{\bar{y}})$, $Y$ is defined by an instance of $\psi(x;{\bar{z}})$, and, for $\lambda\in\Lambda$, $F(\lambda)$ is defined by an instance of $\zeta_\lambda(x;{\bar{u}}_\lambda)$. Let $I\seq\N$ be the set of $s\in\N$ such that, for some tuples ${\bar{a}}_s$, ${\bar{b}}_s$, and ${\bar{c}}_{\lambda,s}$ (for $\lambda\in\Lambda$) from $G_s$, we have:
1. $\phi(x;{\bar{a}}_s)$ defines a normal subgroup $H_s$ of $\Sigma(A_s)=V(A_s)^m$ of index $r$,
2. for all $\lambda\in\Lambda$, $\zeta_\lambda(x;{\bar{c}}_{\lambda,s})$ defines a subset $F_s(\lambda)$ of $H_s$,
3. if $f_s\colon H_s\to \Lambda$ is defined so that $f_s(x)=\lambda$ if and only if $x\in F_s(\lambda)$, then $f_s$ is a well-defined $\delta$-homomorphism (from $H_s$ to $\T^n$),
4. $\psi(x;{\bar{b}}_s)$ defines a subset $Y_s$ of $H_s$, and $Y_s=\{x\in H_s:d(f_s(x),0)<3\delta\}$,
5. $Y_s\seq W(A_s)$.
Then $I\in\cU$ by [Ł]{}oś’s Theorem and elementarity (checking that $(i)$ through $(v)$ are first-order expressible is somewhat cumbersome, but fairly routine; see the proof of [@CPTNIP Lemma 5.6]). So we may fix some $s\in I$ such that $r,n,\delta\inv\leq s$. For this $s$, $Y_s$ is a $\delta$-approximate $(3\delta,n)$-Bohr set in $H_s$. By Proposition \[prop:findBohr\], there is a $(\delta,n)$-Bohr set $B\seq Y_s$. So $B\seq W(A_s)$, which contradicts the choice of $G_s$ and $A_s$.
Finally, if we assume $\Sigma(A)$ is abelian then, in the above proof, we may take $H=\Sigma(A_*)$ by Proposition \[prop:Bohr\], and thus assume $H_s=\Sigma(A_s)$ for all $s\in\N$.
\[rem:abelianBohr\] Suppose that in Theorem \[thm:gen2\] we further assume $G$ is abelian and $|A|\geq c|G|$ for some fixed $c>0$. Then we have $\langle A\rangle=\bar{A}^m$, where $m\leq \lceil3c\inv+1\rceil$), and $[G:\langle A\rangle]\leq \lceil c\inv\rceil$. Therefore, in the proof of the theorem, $\Sigma(A_*)$ has finite index in $G_*$, and so $\Gamma$ has index at most $2^{\aleph_0}$ in $G_*$. So we can carry out the rest of the proof with $G_*$ in place of $\Sigma(A_*)$, obtaining $H_s=G_s$ in the conclusion. Consequently, in Theorem \[thm:Bogogen\]$(b)$, if $G$ is abelian then we may take $H=G$.
Arithmetic regularity and VC-dimension {#sec:NIP}
======================================
The goal of this section is to prove Theorem \[thm:NIPregexp\]. As indicated in the introduction, the only ingredient in the work of Alon, Fox, and Zhao [@AFZ] requiring abelian groups is Theorem \[thm:Bogo\]$(a)$. The (qualitative) nonabelian version of this result for sets of small *tripling*, provided by Corollary \[cor:sym\]$(a)$, will be sufficient to essentially carry out the same proof as in [@AFZ] (see also Remark \[rem:Tao\]). The only extra work is in specifying the numerics and clarifying the “regularity" aspect the result (i.e. condition $(ii)$ of Theorem \[thm:NIPregexp\]). We also make some similar (but mostly qualitative) statements for purely nonabelian finite groups, and finite simple groups.
Let $G$ be a finite group. Given a subset $A\seq G$ and some $\epsilon>0$, define the **$\epsilon$-stabilizer of $A$** to be the set $\Stab_\epsilon(A):=\{x\in G:|xA\smd A|\leq\epsilon|G|\}$.
The following lemma, which we have extracted from the counting techniques done in [@AFZ], makes explicit the connection between $\epsilon$-stabilizers and strong arithmetic regularity involving subgroups.
\[lem:separate\] Let $G$ be a finite group and fix a subset $A\seq G$ and some $\epsilon>0$. Suppose $H$ is a subgroup of $G$ contained in $\Stab_\epsilon(A)$.
1. There is $D\seq G$, which is a union of right cosets of $H$, such that $|A\smd D|\leq \epsilon|G|$.
2. There is $Z\seq G$, with $|Z|<\frac{1}{2}\epsilon^{1/2}|G|$, such that for any $x\in G\backslash Z$, either $|Hx\cap A|\leq\epsilon^{1/4}|H|$ or $|Hx\backslash A|\leq\epsilon^{1/4}|H|$.
Let $\cC$ be the set of right cosets of $H$ in $G$. Given $C\in \cC$, define $P_C=(C\cap A)\times (C\backslash A)$. Let $P=\bigcup_{C\in \cC}P_C=\{(a,g)\in A\times G\backslash A:ga\inv\in H\}$, and note that $P_C\cap P_{C'}=\emptyset$ for distinct $C,C'\in\cC$. From the proof of [@AFZ Lemma 2.4], one obtains $$2\sum_{C\in\cC}|P_C|=2|P|=\sum_{x\in H}|xA\smd A|\leq\epsilon|G||H|.\tag{$\dagger$}$$ For part $(a)$, we continue to follow [@AFZ]. Let $D=\bigcup\{C\in\cC:|C\cap A|\geq|H|/2\}$. Then, by $(\dagger)$, $$|A\smd D| =\sum_{C\in\cC}\min\{|C\cap A|,|C|-|C\cap A|\}\leq \sum_{C\in\cC}\frac{2}{|H|}|P_C|\leq \epsilon|G|.$$ For part $(b)$, let $\cZ=\{C\in\cC:|P_C|>\epsilon^{1/2}|H|^2\}$. By $(\dagger)$, $$\textstyle\frac{1}{2}\epsilon|G||H|\geq \displaystyle \sum_{C\in\cC}|P_C|>\epsilon^{1/2}|H|^2|\cZ|.$$ So $|\cZ|<\frac{1}{2}\epsilon^{1/2}\frac{|G|}{|H|}$. Now set $Z=\bigcup_{C\in\cZ}C$. Then $|Z|<\frac{1}{2}\epsilon^{1/2}|G|$. Moreover, if $x\in G\backslash Z$ then $Hx\not\in\cZ$, and so $|P_{Hx}|\leq \epsilon^{1/2}|H|^2$, which implies $|Hx\cap A|\leq\epsilon^{1/4}|H|$ or $|Hx\backslash A|\leq\epsilon^{1/4}|H|$.
We can now prove Theorem \[thm:NIPregexp\], following the same steps as in [@AFZ].
Fix positive integers $r$ and $d$, and real numbers $\epsilon, \nu>0$. Suppose $G$ is a finite group of exponent at most $r$, and $A\seq G$ has VC-dimension at most $d$. Let $S=\Stab_\delta(A)$, where $\delta=(\epsilon/4)^{(d+ \nu)/d}/30^{ \nu/d}$. Note that $S$ is symmetric. Set $k=(30/\delta)^d$ and $p=d(d+ \nu)/ \nu$. It is an immediate consequence of *Haussler’s Packing Lemma* [@HaussPL], for sets systems of finite VC-dimension, that $|S|\geq|G|/k$ (see Lemmas 2.1 and 2.2 of [@AFZ]). Therefore, we cannot have $|S^{3^{i+1}}|>3^{p}|S^{3^i}|$ for all $i\leq \log_{3^{p}}(k)$. So we may fix some $t\leq \log_{3^{p}}(k)$ such that, setting $B=S^{3^t}$, we have $|B^3|\leq 3^{p}|B|$. By Corollary \[cor:sym\]$(a)$, there is a subgroup $H\leq G$ such that $H\seq B^4$ and $B$ is covered by $O_{r,d, \nu}(1)$ left translates of $H$. Since $|G|\leq k|B|$, we see that $H$ has index at most $O_{r,d, \nu}(k)=O_{r,d, \nu}((1/\epsilon)^{d+ \nu})$.
To finish the proof, it suffices by Lemma \[lem:separate\] to show that $H\seq\Stab_\epsilon(A)$. We have $|xA\smd A|\leq \delta |G|$ for all $x\in S$, and $H\seq B^4=S^{4\cdot 3^{t}}$. So, for any $x\in H$, $$|xA\smd A|\leq 4\cdot 3^t\delta|G|\leq 4 k^{1/p}\delta|G|=\epsilon|G|.\qedhere$$
\[rem:NIPreg\] We make some comments to follow up on Remark \[rem:NIPpre\].
1. Note that, in the proof of Theorem \[thm:NIPregexp\], if $K=\bigcap_{g\in G}gHg\inv$ then $K$ is normal of index at most $[G:H]!$ and $K\seq \Stab_\epsilon(A)$. So, if $[G:H]\leq O_{r,d, \nu}((1/\epsilon)^{d+ \nu})$, for some chosen $\epsilon, \nu>0$, then $\log [G:K]\leq O_{r,d, \nu}(\epsilon^{\nv(d+ \nu)}\log(\epsilon^{\nv 1}))$. Altogether, we have a statement identical to Theorem \[thm:NIPregexp\], but with a *normal* subgroup of index $2^{O_{r,d, \nu}((1/\epsilon)^{d+ \nu})}$. One reason a normal subgroup is desirable in this situation is that it implies a very strong graph regularity conclusion for the bipartite graph $xy\in A$ on $G$, in which the pieces of the regular partition are the cosets of $H$ (see [@CPTNIP Corollary 3.3]).
2. A non-effective version of Theorem \[thm:NIPregexp\], with a normal subgroup, can also be proved by applying Corollary \[cor:sym\]$(b)$ directly to $\Stab_\epsilon(A)$. Together with Haussler’s Packing Lemma, this would directly yield a normal subgroup of index $O_{r,d,\epsilon}(1)$ contained in $\Stab_\epsilon(A)$. It is interesting to note that a qualitative version of Theorem \[thm:NIPregexp\], with a normal subgroup, was already shown in [@CPTNIP] using fairly different techniques (although there are some aspects of the work in [@CPTNIP] which are not recovered here, including definability of the subgroup $H$ and stronger regularity statement).
Finally, we prove similar results about purely nonabelian finite groups and finite simple groups. To motivate our interest in this setting, we recall some of the previous work on arithmetic regularity for subsets of finite groups satisfying extra tameness properties. One example of such a property is bounded VC-dimension, which we have already discussed. Another important example is that of a **$d$-stable** subset $A$ of a group $G$, for some integer $d\geq 1$, which means there do not exist $a_1,\ldots,a_d,b_1,\ldots,b_d\in G$ such that $a_ib_j\in A$ if and only if $i\leq j$. Note that a $d$-stable set has VC-dimension at most $d-1$. Both of these properties were previously studied in the setting of Szemer[é]{}di regularity for graphs (see [@AFN], [@MaShStab]).
In [@CPT] (joint with Pillay and Terry), we showed that, given $d\geq 1$ and $\epsilon>0$, if $G$ is a finite group and $A\seq G$ is $d$-stable then there is a normal subgroup $H\leq G$, of index $O_{d,\epsilon}(1)$, and a union $D$ of cosets of $G$ such that $|A\smd D|\leq \epsilon|H|$. Informally, stable subsets of finite groups are structurally approximated by cosets of a bounded-index normal subgroup. In the setting of finite groups, this phenomenon was first investigated by Terry and Wolf [@TeWo], who proved a similar result for $G=\F_p^n$ with strong quantitative bounds, but with the approximation $|A\smd D|\leq \epsilon|G|$. (This was recently generalized to arbitrary finite abelian groups in [@TeWo2].)
In contrast, easy examples show that subgroups are not sufficient to control sets of bounded VC-dimension. For example, as noted in [@AFZ], if $p\geq 3$ is prime and $G=\Z/p\Z$ and $A=\{1,\ldots,\frac{p-1}{2}\}$, then $A$ has VC-dimension $2$, but $A$ cannot be approximated by cosets of a nontrivial subgroup of $G$. This is one reason to use Bohr neighborhoods in the formulation of arithmetic regularity for sets of bounded VC-dimension, which was done by Sisask in the abelian setting [@SisNIP], and independently in [@CPTNIP] for general finite groups. As we have seen above, if one introduces a uniform bound on the exponent of the groups, then subgroups can be used to approximate sets of bounded VC-dimension. So this motivates the following result that purely nonabelian groups (see Corollary \[cor:pure\]) also exhibit this behavior.
\[thm:NIPregpna\] Fix a positive integer $d$. Suppose $G$ is a purely nonabelian finite group, and $A\seq G$ has VC-dimension at most $d$. Then, for any $\epsilon>0$, there is a normal subgroup $H$ of $G$, of index $O_{d,\epsilon}(1)$, which satisfies the following properties.
1. [(structure)]{.nodecor} There is a set $D\seq G$, which is a union of cosets of $H$, such that $|A\smd D|\leq \epsilon|G|$.
2. [(regularity)]{.nodecor} There is a set $Z\seq G$, with $|Z|<\frac{1}{2}\epsilon^{1/2}|G|$, such that for any $x\in G\backslash Z$, either $|xH\cap A|\leq\epsilon^{1/4}|H|$ or $|xH\backslash A|\leq\epsilon^{1/4}|H|$.
Fix a purely nonabelian finite group $G$, a subset $A\seq G$ of VC-dimension at most $d$, and $\epsilon>0$. As in Theorem \[thm:NIPregexp\], if $S=\Stab_{\epsilon/4}(A)$ then $|S|\geq (\epsilon/120)^d|G|$. By Corollary \[cor:pure\] there is a normal subgroup $H\leq G$, of index $O_{d,\epsilon}(1)$, such that $H\seq S^4$. So $|xA\smd A|\leq \epsilon|G|$ for any $x\in H$. Now apply Lemma \[lem:separate\].
The previous theorem can also be deduced from [@CPTNIP Theorem 5.7], yielding further information as discussed in Remark \[rem:NIPreg\]$(2)$. On the other hand, the proof here seems more direct, and certainly uses a more acute application of VC-theory. (Both proofs involve identical uses of [@AlGlGo] and [@PiRCP]).
The work in [@CPT] on stable regularity implies that, for any $d\geq 1$ and $\epsilon>0$, if $G$ is a finite simple group of size $\Omega_{d,\epsilon}(1)$ and $A\seq G$ is $d$-stable, then $|A|\leq\epsilon|G|$ or $|A|\geq (1-\epsilon)|G|$.[^5] For the abelian case (i.e. $G=\Z/p\Z$), a quantitative lower bound on $p=|G|$, in terms of $d$ and $\epsilon$, could be deduced from [@TeWo2]. On the other hand, the example above, which shows that subgroups are not sufficient to approximate sets of bounded VC-dimension, takes place in *abelian* finite simple groups. This motivates the following corollary of Theorem \[thm:NIPregpna\].
\[cor:NIPregpna\] For any integer $d$ and any $\epsilon>0$, there is an integer $n=n(d,\epsilon)$ such that, if $G$ is a nonabelian finite simple group of size greater than $n$, and $A\seq G$ has VC-dimension at most $d$, then $|A|\leq\epsilon|G|$ or $|A|\geq (1-\epsilon)|G|$.
Using a similar strategy, we can give a direct proof of the previous corollary, which yields $\log (n(d,\epsilon))\leq O((\epsilon/90)^{\nv 6d})$ as an explicit bound. Namely, by the work of Gowers [@GowQRG] discussed in Remark \[rem:simple\], there is some $c>0$ such that if $G$ is a nonabelian finite simple group with $\log|G|\geq c(\epsilon/90)^{\nv 6d}$, and $S\seq G$ is such that $|S|\geq (\epsilon/90)^{\nv d}|G|$, then $G=S^3$. So fix such a $G$, and suppose $A\seq G$ is of VC-dimension at most $d$. By Haussler’s Packing Lemma, and choice of $c$, we have $G=(\Stab_{\epsilon/3}(A))^3=\Stab_\epsilon(A)$. Now apply Lemma \[lem:separate\].[^6]
Final remarks {#sec:final}
=============
Quantitative bounds {#sec:explicit}
-------------------
An obvious question at this point is on effective bounds for Theorems \[thm:Bogogen\], \[thm:mainbdd\], and \[thm:maingen\]. Our proof of Theorem \[thm:maingen\] used an ultraproduct construction, and did not give explicit bounds of any kind. While ultraproducts do not appear explicitly in our proof of Theorem \[thm:mainbdd\], they are similarly used in previous work on approximate groups (both in [@BGT] and [@HruAG]).
It is sometimes possible, with enough work, to reverse engineer effective bounds from arguments with ultraproducts, but these bounds are usually very bad (see, e.g., [@TaoH5P Chapter 7] for some discussion on this topic). Altogether, it seems that in order to obtain efficient bounds for the above results, one would need efficient bounds for results on approximate groups, or different proof strategy altogether.
Small tripling vs. approximate groups {#sec:Tao}
-------------------------------------
For the sake of completeness, we note that weaker versions of our main results can be obtained without the revised Sanders-Croot-Sisask analysis in Section \[sec:CSS\]. This is because of the following result of Tao, which follows from the proof of [@TaoPSE Theorem 3.9] (or see [@BGT Corollary 5.2]).
\[thm:Tao\] Suppose $A$ is a nonempty finite subset of a group $G$. If $|A^3|\leq k|A|$ then $\bar{A}^2$ is an $O(k^{O(1)})$-approximate group containing $A$.
Together with Theorem \[thm:BGTbdd\], one obtains a weaker version of Theorem \[thm:mainbdd\].
\[cor:Tao\] Fix positive integers $k$ and $r$. Let $G$ be a group of exponent $r$, and fix a finite subset $A\seq G$. Suppose $|A^3|\leq k|A|$. Then there is $H\leq\langle A\rangle$ such that $\bar{A}^2$ is covered by $O_{k,r}(1)$ left cosets of $H$ and $H\leq \bar{A}^8$.
\[rem:Tao\] Corollary \[cor:Tao\] could be used instead of Corollary \[cor:sym\]$(a)$ in the proof Theorem \[thm:NIPregexp\].
\[rem:HruTao\] Recall that if $A\seq G$ is finite and nonempty, with $|AA\inv A|\leq k|A|$, then $|(A\inv A)^3|\leq k^{O(1)}|A\inv A|$ by Proposition \[prop:Ruz\]$(b)$, and so $(AA\inv)^2$ is an $O(k^{O(1)})$-approximate group by Theorem \[thm:Tao\]. Altogether, this is essentially the “discrete case" of [@HruAG Corollary 3.11].
A weaker version of Theorem \[thm:maingen\] can also be formulated using Theorem \[thm:Tao\], but the proof would still require our work with saturated extensions and approximate Bohr neighborhoods, and so we will not go into it any further. On the other hand, the following statement about sets of small tripling in arbitrary groups follows by combining Theorem \[thm:Tao\] with the main structure theorems for approximate groups from Breuillard, Green, and Tao [@BGT].
\[thm:BGTtrip\] Fix a positive integer $k$. Suppose $G$ is a group and $A\seq G$ is finite and nonempty, with $|A^3|\leq k|A|$. Then there is a subgroup $H$ of $G$ and a finite normal subgroup $N$ of $H$ with the following properties:
1. $A$ is covered by $O_k(1)$ left cosets of $H$;
2. $H/N$ is nilpotent and finitely generated of rank and step $O_k(1)$;
3. $\bar{A}^8$ contains $N$ and a generating set for $H$.
Moreover, there is a coset nilprogression $P\seq \bar{A}^8$ of rank and step $O_k(1)$ such that $A$ is covered by $O_k(1)$ left translates of $P$.
The final clause of the previous theorem is a non-abelian analogue of the *Bogolyubov-Ruzsa Lemma* for finite abelian groups, which was stated after Theorem \[thm:Bogo\]. However, we have the qualitative discrepancy between $\bar{A}^8$ in condition $(iii)$ and $2A-2A$ in the abelian case. Given our earlier results, one naturally wonders if $\bar{A}^8$ can be replaced by $(AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2$. This also raises a similar question about small alternation. So we observe that these issues can be addressed simply by combining the results in [@BGT] with Lemma \[lem:MWKP\].
\[thm:BGT+\] Fix a positive integer $k$. Suppose $G$ is a group and $A\seq G$ is finite and nonempty. Furthermore,
1. assume $|AA\inv A|\leq k|A|$ and set $V=AA\inv$ and $W=(AA\inv)^2$, or
2. assume $|A^3|\leq k|A|$ and set $V=\bar{A}$ and $W=(AA\inv)^2\cap A^2A^{\nv 2}\cap (A\inv A)^2\cap A^{\nv 2}A^2$.
Then there is a subgroup $H$ of $G$ and a finite normal subgroup $N$ of $H$ with the following properties:
1. for all $m\geq 1$, $V^m$ is covered by $O_{k,m}(1)$ left cosets of $H$;
2. $H/N$ is nilpotent and finitely generated of rank and step $O_{k}(1)$;
3. $W$ contains $N$ and a generating set for $H$.
Moreover, there is a coset nilprogression $P\seq W$ of rank and step $O_{k}(1)$ such that for all $m\geq 1$, $V^m$ is covered by $O_{k,m}(1)$ left translates of $P$.
By Lemma \[lem:MWKP\], there is a symmetric set $Y\seq G$ such that $Y^8\seq W$ and $V^3$ is covered by $O_k(1)$ left translates of $Y$. Since $Y\seq V^2$, it follows that for all $m\geq 1$, $V^m$ is covered by $O_{k,m}(1)$ left translates of $Y$. Note also that $Y$ is an $O_k(1)$-approximate group. So by [@BGT Theorem 1.6], there are $N\trianglelefteq H\leq G$ such that $Y$ is covered by $O_k(1)$ left cosets of $H$, $H/N$ is nilpotent and finitely generated of rank and step $O_k(1)$, and $Y^4$ contains $N$ and a generating set for $H$. Moreover, by [@BGT Theorem 2.10], there is a coset nilprogression $P\seq Y^8$ of rank and step $O_k(1)$ such that $|Y|\leq O_k(|P|)$ and $Y$ is covered by $O_k(1)$ left translates of $P$. Altogether, the result follows by choice of $Y$.
[10]{}
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[^1]: This name is from Sanders [@SanBR], who gives a different proof of the result yielding better bounds.
[^2]: This was first observed by C. Terry.
[^3]: By [@GowQRG], the implied constant in $\Omega(\alpha^{\nv 6})$ is no more than $25^{\log(25)}$. Using the classification of finite simple groups, the overall bound can be improved to $|G|>(\lceil\alpha^{\nv 3}\rceil+1)!$ (see [@CollJCLG]).
[^4]: “Sufficiently saturated" typically means $\kappa$-saturated and strongly $\kappa$-homogeneous for some very large (e.g. strongly inaccessible) cardinal $\kappa$.
[^5]: This is a finitary analogue of the older fact that any definable subset of an (infinite) definably-connected stable group has measure $0$ or $1$ with respect to the unique Keisler measure.
[^6]: As in Remark \[rem:simple\], the work in [@CollJCLG] implies $n(d,\epsilon)\leq (\lceil (\epsilon/90)^{\nv 3d}\rceil+1)!$ in Corollary \[cor:NIPregpna\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Test beam measurements at the test beam facilities of DESY have been conducted to characterise the performance of the EUDET-type beam telescopes originally developed within the ${\ensuremath{\textrm{EUDET}}}$ project. The beam telescopes are equipped with six sensor planes using ${\ensuremath{\textrm{MIMOSA\,26}}}$ monolithic active pixel devices. A programmable Trigger Logic Unit provides trigger logic and time stamp information on particle passage. Both data acquisition framework and offline reconstruction software packages are available. User devices are easily integrable into the data acquisition framework via predefined interfaces.
The biased residual distribution is studied as a function of the beam energy, plane spacing and sensor threshold. Its standard deviation at the two centre pixel planes using all six planes for tracking in a 6GeV electron/positron-beam is measured to be $(2.88\,\pm\,0.08)\,\upmu\meter$. Iterative track fits using the formalism of General Broken Lines are performed to estimate the intrinsic resolution of the individual pixel planes. The mean intrinsic resolution over the six sensors used is found to be $(3.24\,\pm\,0.09)\,\upmu\meter$. With a 5GeV electron/positron beam, the track resolution halfway between the two inner pixel planes using an equidistant plane spacing of 20mm is estimated to $(1.83\,\pm\,0.03)\,\upmu\meter$ assuming the measured intrinsic resolution. Towards lower beam energies the track resolution deteriorates due to increasing multiple scattering. Threshold studies show an optimal working point of the ${\ensuremath{\textrm{MIMOSA\,26}}}$ sensors at a sensor threshold of between five and six times their RMS noise. Measurements at different plane spacings are used to calibrate the amount of multiple scattering in the material traversed and allow for corrections to the predicted angular scattering for electron beams.
author:
- |
H. Jansen${}^{\textrm{a,}}$, S. Spannagel${}^{\textrm{a}}$, J. Behr${}^{\textrm{a,}}$[^1], A. Bulgheroni${}^{\textrm{b,}}$[^2], G. Claus${}^{\textrm{c}}$, E. Corrin${}^{\textrm{d,}}$[^3], D. G. Cussans${}^{\textrm{e}}$, J. Dreyling-Eschweiler${}^{\textrm{a}}$, D. Eckstein${}^{\textrm{a}}$, T. Eichhorn${}^{\textrm{a}}$, M. Goffe${}^{\textrm{c}}$, I. M. Gregor${}^{\textrm{a}}$, D. Haas${}^{\textrm{d,}}$[^4], C. Muhl${}^{\textrm{a}}$, H. Perrey${}^{\textrm{a,}}$[^5], R. Peschke${}^{\textrm{a}}$, P. Roloff${}^{\textrm{a,}}$[^6], I. Rubinskiy${}^{\textrm{a,}}$[^7], M. Winter${}^{\textrm{c}}$\
${}^{\textrm{a}}$ Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany\
${}^{\textrm{b}}$ INFN Como, Italy\
${}^{\textrm{c}}$ IPHC, Strasbourg, France\
${}^{\textrm{d}}$ DPNC, University of Geneva, Switzerland\
${}^{\textrm{e}}$ University of Bristol, UK
bibliography:
- 'bibtex/refs.bib'
- 'refs.bib'
title: |
Performance of the EUDET-type\
beam telescopes
---
=1
Introduction {#sec:intro}
============
Beamlines {#sec:beamlines}
=========
Components of the EUDET-type beam telescopes {#sec:tscope}
============================================
The EUDAQ data acquisition framework {#sec:eudaq}
====================================
Offline analysis and reconstruction using EUTelescope {#sec:offline}
=====================================================
Track resolution studies {#sec:trackres}
========================
Considerations for DUT integrations {#sec:dutintegration}
===================================
Conclusion {#sec:conclusion}
==========
Data and materials {#data-and-materials .unnumbered}
==================
The datasets supporting the conclusions of this article are available from reference [@jansen_data]. The software used is available from the github repositories: 1) <https://github.com/eutelescope/eutelescope>, 2) <https://github.com/simonspa/eutelescope/>, branch *scattering* and 3) <https://github.com/simonspa/resolution-simulator>. For the presented analysis, these specific tags have been used: [@jansen_2016_49065] and [@spannagel_2016_48795].
Competing interests {#competing-interests .unnumbered}
===================
The authors declare that they have no competing interests.
Acknowledgements {#acknowledgements .unnumbered}
================
We are indebted to Claus Kleinwort for his counsel and numerous discussions. Also, we would like to thank Ulrich Kötz. The test beam support at DESY is highly appreciated. This work was supported by the Commission of the European Communities under the FP7 Structuring the European Research Area, contract number RII3-026126 (EUDET). Furthermore, strong support from the Helmholtz Association and the BMBF is acknowledged.
[^1]: Now at Institut für Unfallanalysen, Hamburg, Germany
[^2]: Now at KIT, Karlsruhe, Germany
[^3]: Now at SwiftKey, London, UK
[^4]: Now at SRON, Utrecht, Netherlands
[^5]: Now at Lund University, Sweden
[^6]: Now at CERN, Geneva, Switzerland
[^7]: Now at CFEL, Hamburg, Germany
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Multiparameter estimation theory offers a general framework to explore imaging techniques beyond the Rayleigh limit. While optimal measurements of single parameters characterizing a composite light source are now well understood, simultaneous determination of multiple parameters poses a much greater challenge that in general requires implementation of collective measurements. Here we show, theoretically and experimentally, that Hong-Ou-Mandel interference followed by spatially resolved detection of individual photons provides precise information on both the separation and the centroid for a pair of point emitters, avoiding trade-offs inherent to single-photon measurements.'
author:
- Michał Parniak
- Sebastian Borówka
- Kajetan Boroszko
- Wojciech Wasilewski
- Konrad Banaszek
- 'Rafał Demkowicz-Dobrzański'
bibliography:
- 'bibliografia.bib'
nocite:
- '[@Chrapkiewicz2016]'
- '[@Lipka2018]'
title: 'Beating the Rayleigh limit using two-photon interference'
---
Multiparameter quantum estimation emerges as a general framework to optimize information retrieval in a variety of experimental scenarios. The problem of imaging can be viewed as an important example of such a scenario, where the properties of an image, for example locations and intensities of point emitters or the moments of the image intensity distribution are the parameters to be estimated [@Rehacek2017; @Paur2016; @Yang2016; @Chrostowski2017; @Ragy2016; @Napoli2018; @Zhou2018]. A recently introduced family of superresolution imaging schemes [@Tsang2009; @Tsang2015; @Tsang2016; @Tsang2017; @Kerviche2017; @Dutton2018] based on spatial demultiplexing enable one to determine the separation of two nearby point sources below the Rayleigh limit, but require in principle perfect knowledge of the centroid [@Chrostowski2017]. Moreover, at the single photon level they are fundamentally incompatible with the measurement needed to estimate the centroid itself. Nonetheless, the effort to extract optimally information carried in light emitted naturally by a source [@Puschmann2005; @Tang2016; @Nair2016; @Nair2015; @Lupo2016; @Dutton2018; @Tham2017] may open up new applications compared to established approaches that require manipulations of the sample to be imaged [@Moerner2015].
A deeper insight rooted in the multiparameter estimation theory reveals a possible solution of the above incompatibility problem. Interestingly, in the strong subdiffraction regime where images of the sources overlap significantly, the problem can modelled as simultaneous estimation of the length and the rotation angle of a qubit Bloch vector [@Chrostowski2017]. From the theory of multiparameter estimation it then follows that, provided collective measurement on the photons (or qubits) are allowed, the incompatibility between the optimal individual measurements to estimate the centroid and the sources separation ceases to be an issue [@Bagan2006; @Vidrighin2014]. The question is how to realize such a collective measurement in practice.
In this Letter we exploit the advantages offered by the multiphoton interference approach, demonstrating a two-photon protocol for imaging of two point sources, where the centroid estimation is performed in the optimal way, and at the same time the sources separation parameter is estimated with a superresolution precision. The idea relies on the effect of two-photon interference and does not require pre-estimation of the centroid or fine-tuning of the measurement basis inherent to spatial mode demultiplexing schemes [@Tsang2009; @Tsang2015; @Tsang2016; @Tsang2017; @Kerviche2017; @Dutton2018], where any systematic error in centroid estimation propagates to separation estimation and significantly degrades the imaging protocol.
In Fig. 1(a) we depict a scenario where two photons emitted by a composite source arrive simultaneously at the input ports of the beamsplitter. The proposed protocol exploits both cross-coincidences between the output ports and double events in each port, detected with spatial resolution [@Jachura2016]. The number of cross-coincidences grows with the distinguishability of the two photons and therefore carries information about the separation between point sources. Most importantly, the proposed interferometric scheme does not require prior selection of the measurement basis or the axis of symmetry, as the two photons serve as a reference for each other. Furthermore, thanks to spatially-resolved detection this strategy will be shown to be robust against residual spectral distinguishability. Let us note that previous approaches to collective measurements relied on the fundamental advantage of using photonic entanglement [@Rozema2014], also for superresolution photolithography [@Liao2010; @Boyd2012; @DAngelo2001; @Boto2000], which is essentially different from our technique of simply utilizing the bosonic nature of photons.
The somewhat non-trivial demand of interfering two photons from a realistic classical (thermal) composite source on a beamsplitter could be realized by a photon number quantum nondemolition (QND) measuring device that preserve spatial properties of light, and upon registering single photons delays and redirects them so that they arrive together at the two beamsplitter input ports. Recent advances in storing and controlling single photons in quantum nonlinear media such as Rydberg atoms [@Firstenberg2013] as well as spatially-multimode quantum memories [@Parniak2017] with processing capabilities [@Parniak2018] could provide a viable way to realize the scheme. In particular, a $\pi$ phase shift induced by a single photon has already been achieved [@Tiarks2016] and current experiments already explore the Rydberg interactions in the trasverse spatial domain [@Busche2017]. The combination of a multimode quantum memory with the spatially-resolving QND measurement could follow the steps of experiments demonstrating optical storage in Rydberg media [@Distante2017; @Li2016], use alternative proposals such as nonlinearities induced by ac-Stark shifts [@Everett2016] or utilize novel solid-state systems with similar capabilities yet broader spectral bandwidths [@Yang2018a; @Walther2018].
{width="\columnwidth"}
To support the intuitions behind the discussed scheme let us compare the two-photon imaging scheme with direct imaging (DI) by modeling a problem of resolving a 1D image formed by two point sources. Let $\psi(x-x_0)$ be a 1D wave function representing the amplitude transfer function of a single source in the image plane centered at point $x_0$. We assume that this function is determined by well characterized properties of the imaging setup. In what follows we denote the corresponding single photon state characterized by $\psi(x-x_0)$ as ${\left| {x_0} \right\rangle}$.
Consider a situation where the image is produced as a result of an incoherent overlap of images of two point sources separated by a distance $\varepsilon$, located at $x_{+}=x_0+\varepsilon/2$ and $x_{-}=x_0-\varepsilon/2$. We may then write the spatial density matrix of a photon emitted from the system as $\rho=1/2({\left| {x_{+}} \right\rangle}{\left\langle {x_{+}}\right|}+{\left| {x_{-}} \right\rangle}{\left\langle {x_{-}}\right|})$.
In the DI scheme the probability distribution for the position of the detected photon is given by $p_{\boldsymbol{\theta}}(x)=\tfrac{1}{2}|\psi(x-x_{+})|^{2}+\frac{{1}}{2}|\psi(x-x_{-})|^{2}$, where $\boldsymbol{\theta}=((x_++x_-)/2,x_+-x_-)=(x_0,\varepsilon)$ represents the dependence on the estimated parameters. For any locally unbiased estimator, the covariance matrix for the estimated parameters can be lower bounded using the Cram[é]{}r-Rao inequality [@Kay1993]: $$\mathrm{Cov}\boldsymbol{\theta}\geq \frac{F^{-1}}{N},
\ F_{ij}=\int_{-\infty}^{\infty}{\textrm{d}}x\frac{\partial_{\theta_{i}}p_{\boldsymbol{\theta}}(x)\partial_{\theta_{j}}p_{\boldsymbol{\theta}}(x)}{p_{\boldsymbol{\theta}}(x)} \label{eq:fim},$$ where $F_{ij}$ is the Fisher information (FI) matrix per single photon, while $N$ represents the total number of photons registered. The bound is asymptotically saturable using e.g. max-likelihood estimator, hence $\lim_{N \rightarrow \infty} N \mathrm{Cov}\boldsymbol{\theta} = F^{-1}$. As the FI matrix is diagonal for the given problem, we can easily calculate the variances $\Delta^2x_{0} = (F^{-1})_{11}$, $\Delta^2\varepsilon = (F^{-1})_{22}$ the respective variances of the estimated parameters per single photon used. In case of DI the FI matrix yields the following precision for estimation in the leading order in $\varepsilon$: $$\begin{aligned}
(\Delta^2x_0)^{-1}_\mathrm{DI} = 1-\frac{\varepsilon^2}{4}, \\
(\Delta^2\varepsilon)^{-1}_\mathrm{DI} = \frac{\varepsilon^2}{8},\end{aligned}$$ where for concreteness we have assumed a Gaussian-shaped transfer function $\psi(x)=(2\pi)^{-1/4}\exp(-x^{2}/4)$, yielding intensity profile with standard deviation $1$ which can be regarded as a natural unit of distance in the problem. The above expansion is valid for small $\varepsilon$ when source point images are separated by a distance smaller than the transfer function spread, and clearly shows impossibility of precise estimation of $\varepsilon$ in the $\varepsilon \rightarrow 0$ limit.
Crucially, as observed in [@Tsang2016], a more fundamental bound based on the quantum FI matrix $F^Q$ [@Helstrom1976], which does not assume any particular measurement strategy and is based solely on the properties of the quantum state $\rho$ to be measured reads: $$\begin{aligned}
\label{eq:qfi}
(\Delta^2x_0)^{-1}_\mathrm{Q} = 1-\frac{\varepsilon^2}{4}, \\
(\Delta^2\varepsilon)^{-1}_\mathrm{Q} = \frac{1}{4},\end{aligned}$$ indicating a potential spectacular robustness of $\varepsilon$ estimation as the $\Delta^2 \varepsilon$ is constant irrespectively of how small $\varepsilon$ is. While the bound with $F$ being replaced by $F^Q$ is saturable for the problem considered, it requires collective measurements on many copies of $\rho$ [@Vidrighin2014; @Rehacek2017; @Chrostowski2017; @Ragy2016].
{width="\textwidth"}
We are now ready to quantify the precision of estimating $x_0$ and $\varepsilon$ in the two-photon (2P) interferometric scheme and contrast it with the above-mentioned strategies. Given $\rho^{\otimes 2}$ at the input ports of the beam-splitter, we calculate spatially-resolved propabilities for coincidences $p_{\mathrm{c}}(x_{1},x_{2})$ as well as double events $p_{\mathrm{{d}}}(x_{1},x_{2})$, from which the information about $x_0$ and $\varepsilon$ is drawn. Furthermore, we assume a known two-photon visibility $\mathcal{V}$ resulting from the operation of the non-demolition photon routing device before the beamsplitter. The resulting precision of estimation per single photon used, see Supplementary Material, expanded up to the second order in $\varepsilon$ reads: $$\begin{aligned}
\label{eq:2p-1}
(\Delta^2x_0)^{-1}_\mathrm{2P} &= 1-\frac{\varepsilon^2}{4}, \\
(\Delta^2\varepsilon)^{-1}_\mathrm{2P} &= \begin{cases} \frac{1}{8} + \frac{5}{128} \varepsilon^2 &
\label{eq:2p-2}
\mathcal{V} = 1 \\ \frac{4-\mathcal{V}^2}{32(1-\mathcal{V}^2)} \varepsilon^2 & \mathcal{V} < 1 \end{cases},\end{aligned}$$ while the expansion in case of imperfect visibility is valid in the regime where $\varepsilon^{2}\lesssim1-\mathcal{{V}}$. In case of perfect interference, we see that while keeping the optimality of $x_0$ estimation, we additionally obtain $\varepsilon$ estimation with precision reduced by approximately a factor of $2$ compared to the fundamental bound given in (5). This shows superiority of 2P over DI, with the added advantage that the measurement setting is fixed and does not require adjusting the measurement for $\varepsilon$ depending on preestimation of $x_0$. Here we would like to stress the importance of spatial information that is available in the experiment: if only the ratio of coincidence and double events was available, there would be no information on $x_0$ parameter at all, while the precision of $\varepsilon$ estimation shows a small reduction for finite $\varepsilon$ compared with Eq. \[eq:2p-2\] and reads: $\frac{1}{8} - \frac{5}{128} \varepsilon^2 + O(\varepsilon^4)$ when the visibility is equal to one.
The role of spatial information becomes more pronounced for finite visibilities $\mathcal{V}$, for which the spatial information always provides an advantage for all values of $\varepsilon$ compared to the case when we consider only the ratio of cross-coincidences and double-event where the precision reads $\mathcal{V}^2 [32(1-\mathcal{V}^2)]^{-1} \varepsilon^2+ O(\varepsilon^4)$. This is achieved as coincidences that arise due to finite visibility are characterized by a different spatial distribution than coincidences that are due to spatial separation. In both cases we recover the $\varepsilon^{2}$-scaling and thus for small $\varepsilon$ the advantage of the collective schemes over DI takes the form of a constant factor rather than favorable scaling. Nonetheless, as this factor scales as $(1-\mathcal{V})^{-1}$ the enhancement can be significant.
For a proof-of-principle experimental demonstration, we generated families of states $\rho^{\otimes 2}$ for a set of separations $\varepsilon$ (see Fig. 2(a) and Supplementary Material for details of the interferometric setup). In Fig. 2(c) and 2(d) we plot the final precision of estimation divided by the total number of photons used as a function of $\varepsilon$ (see Supplementary Material for details of data analysis). The proposed theory (for $\mathcal{{V}}=0.92$) accurately predicts the estimation precision for the given experimental parameters demonstrating a significant, over twofold enhancement over the DI scheme. The spatial resolution provides an advantage over the whole range of parameters, as it allows us to distinguish effects of finite visibility versus the reduced mode overlap due to source separation.
In Figure 2(c),(d) we additionally plot the theoretical predictions for $\mathcal{{V}}=0.99$ and perfect interference i.e. $\mathcal{{V}}=1$. The precision approaches a constant value for $\varepsilon\rightarrow0$ only for $\mathcal{V}=1$, but offers significant enhancement for realistic visibilities. Note that if information is drawn only from the number of coincidences to double events with no spatial resolution, we can still beat the DI scheme over a broad range of parameters, especially for small $\varepsilon$. This highlights the possibility to perform precise imaging with only single-pixel detectors.
Let us now provide a simple argument for the observed degree of precision enhancement. The approximately two-fold reduction of precision for $\varepsilon$ estimation for $\mathcal{V}=1$ in the 2P protocol compared to the fundamental bound is due to the fact that the protocol performs collective measurement on two photons only. The essence of the collective measurement is effective projection of $\rho^{\otimes 2}$ on symmetric and antisymmetric subspaces thanks to the properties of the Hong-Ou-Mandel interference. Such a measurement commutes with joint unitary rotation of the state $U^{\otimes 2} \rho^{\otimes 2} U^{\dagger \otimes 2}$ which represents the shift of the centroid $x_0$ in our model, and hence does not collide with the measurement optimal for extracting information on $x_0$. Theoretically, if collective measurements on arbitrary number of copies were possible, one could project $\rho^{\otimes N}$ state on subspaces corresponding to different irreducible representation of the permutation group which provides optimal information about the $\varepsilon$ parameter in the $N \rightarrow \infty$ limit and does not interfere with the optimal measurement of $x_0$ [@Bagan2006]. Thus, through harnessing more than two photons one would be able to approach and even saturate the quantum Cram[é]{}r-Rao bound .
Interestingly, in a slightly modified imaging scenario, the two-photon measurements may actually saturate the limit discussed above. Consider a different variant of the two-photon state impinging on the beamsplitter $$\rho_{11}=\frac{{1}}{2}({\left| {x_{+}} \right\rangle}{\left\langle {x_{+}}\right|}\otimes{\left| {x_{-}} \right\rangle}{\left\langle {x_{-}}\right|}+{\left| {x_{-}} \right\rangle}{\left\langle {x_{-}}\right|}\otimes{\left| {x_{+}} \right\rangle}{\left\langle {x_{+}}\right|}),$$ which represents a situation where the photons from the two sources always enter at different input ports of the beamsplitter. Such a two-photon state could be obtained from a pair of single-photon emitters excited simultaneously, where we would never observe two photons emitted from the same source. In this case, analogous calculations to the ones presented in Ref. [@Rehacek2017] for the $\rho$ state, lead to the quantum FI matrix which in the leading order in $\varepsilon$ remains unchanged, whereas the two photon experiment described above saturates the bound exactly: $$\begin{aligned}
\label{eq:2p-3}
(\Delta^2x_0)^{-1}_{\mathrm{2P},\rho_{11}} = (\Delta^2x_0)^{-1}_{\mathrm{Q},\rho_{11}} = 1, \\
\label{eq:2p-4}
(\Delta^2\varepsilon)^{-1}_{\mathrm{2P},\rho_{11}}= (\Delta^2\varepsilon)^{-1}_{\mathrm{Q},\rho_{11}} =\frac{1}{4}.\end{aligned}$$
Finally, it is insightful to juxtapose the presented scheme with the celebrated Hanbury Brown–Twiss (HBT) interferometry [@Hanbury1954; @Hanbury1956; @Brown1957; @Brown1958; @Fano1961]. The essential difference is that in our approach photon positions are measured in the image plane, while in the HBT scenario spatially resolved detection is implemented in the Fourier plane conjugate to the source. For photons arriving from point sources located at angular positions specified by wave vectors ${\bf k}_1$ and ${\bf k}_2$ and detectors placed at ${\bf r}_1$ and ${\bf r}_2$, HBT interference produces fringes whose spatial variation is proportional to the expression $\cos^2[({\bf k}_1 - {\bf k}_2)({\bf r}_1-{\bf r}_2)/2]$ [@Scully1997]. If the maximum distance $|{\bf r}_1-{\bf r}_2|$, which can be viewed as the aperture of the measuring system, is fixed, an attempt to retrieve the angular separation between the sources from HBT fringes will suffer from the Rayleigh curse in the limit $|{\bf k}_1
- {\bf k}_2| \rightarrow 0$. This is because for vanishing $|{\bf k}_1 -
{\bf k}_2|$ one will observe only a small fraction of the HBT fringe in the vicinity of its maximum.
In the case of the two-photon scheme presented here, we should emphasize the role of the prior QND measurement if superresolution is to be achieved with classical thermal light sources. While HBT interferometry works also with classical light sources, albeit with reduced visibility, the enhancement offered by our scheme stems from realizing two-photon interferometry sufficiently close to the dark fringe, i.e. with high visibility ${\cal V}$. In fact, since classical light sources can attain at most $50\%$ visibility of Hong-Ou-Mandel interference, formulas indicate that no significant improvement is possible over the DI scheme: for $\mathcal{V}=0.5$ we get $(\Delta^2\varepsilon)^{-1}_{2\textrm{P}}= \frac{5\varepsilon^2}{32}$ vs. $(\Delta^2\varepsilon)^{-1}_{\textrm{DI}}= \frac{\varepsilon^2}{8}$ in case of direct imaging.
In conclusion, we have demonstrated both theoretically and experimentally an imaging protocol that circumvents the difficulties in a multi-parameter estimation problem by use of a collective measurement. The presented experimental results conclusively confirm the possibility to exploit the inherent indistinguishability of photons to perform quantum-enhanced simultaneous estimation of source separation and centroid. With this proof-of-principle experiment we have also proposed a set of realistic schemes in which our protocol could be readily applied, even to gain additional information along the traditional single-photon DI scenario or other superresolution techniques. The general theory of super-resolved imaging [@Rehacek2017; @Chrostowski2017; @Zhou2018] implies that the same protocol might be directly applied in case of a more general light source distribution provided one would be interested in estimating its first and second moments.
This work has been funded by the National Science Centre (Poland) Projects No. 2017/25/N/ST2/01163, 2016/22/E/ST2/00559 and by the project “Quantum Optical Communication Systems” carried out within the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund. M.P. thanks M. Jachura for know-how transfer on the photon-pair source and M. Mazelanik for insightful discussions.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- '*List of authors in appendix*'
bibliography:
- 'references.bib'
title: |
A next-generation\
LHC heavy-ion experiment
---
Introduction {#sec:introduction}
============
With this document, we express our interest for a new LHC experiment, dedicated to the high-statistics study of the production of heavy flavour hadrons and of the soft electromagnetic and hadronic radiation produced in high-energy proton-proton and nuclear collisions. The apparatus would be centered on an ultra-low-mass silicon tracker, made with Complementary Metal-Oxide-Silicon (CMOS) Monolithic Active Pixel Sensors (MAPS) technology. Such an experiment could be installed during the LHC Long Shutdown 4 (LS4), at the Interaction Point 2 (IP2), where the ALICE experiment is currently installed.
The ALICE Collaboration is completing the construction of an upgraded Inner Tracking System (ITS2) with seven layers of CMOS MAPS, to be installed during LS2 [@ITS_upgrade]. Recent advances in this technology have made possible the fabrication of wafer-scale ultra-thin silicon detectors, and the ALICE Collaboration is currently considering a further upgrade (ITS3) in which the three innermost layers of ITS2 would be replaced with three truly cylindrical layers of such detectors, with a material budget of only 0.05% X$_0$ per layer [@LS3_upgrade].
This novel technology opens up the possibility of constructing a new, all-silicon tracker, with unprecedented low mass, that would allow reaching down to an ultra-soft region of phase space, to measure the production of very-low transverse momentum lepton pairs, photons and hadrons at the LHC. The detector would consist of a barrel and two end-caps made of layers of ultra-thin Si-sensors and cover the rapidity region $|\eta| < 4$. The ultra-low material thickness, combined with the placement of the first detector layers either inside the beam pipe or at a very close distance from its outer wall, would allow charged particle detection at transverse momenta of the order of a few tens of MeV/$c$.
The excellent timing resolution ($\simeq 20$ ps) achievable with CMOS detectors will provide particle identification information. Electrons at low momentum ($< 500$ MeV/$c$) will be separated from hadrons using time-of-flight information while, at higher momenta, electrons and photons will be identified in a dedicated shower detector. Removable converter structures will also allow the measurement of photons with the conversion method.
Such an experiment would, for instance, allow to measure the primordial electromagnetic radiation emitted by the Quark-Gluon Plasma (QGP) produced in nuclear collisions, providing key information for the understanding of the emergent properties of QCD matter. More generally, it would open a new window for the study of soft phenomena in hadronic collisions, allowing to address fundamental physics questions that could not be tackled so far.
The high-rate capabilities of MAPS will allow the experiment to run at significantly (a factor 20 to 50) higher luminosities than the upgraded ALICE experiment. It could therefore exploit all pA and AA luminosities that could be reached by accelerating ions from light to very heavy in the LHC [@Citron:2018lsq].
The detector concept discussed in this document provides unprecedented physics performance for heavy flavour studies, enabling the measurement of the production of exotic quarkonia and Multiply Heavy Flavoured (MHF) baryons and mesons in pp, p-A and nuclear collisions, for which theoretical uncertainties on the cross-section span across orders of magnitude, providing a significant new window on the properties of the Quark-Gluon Plasma.
The current ideas for a possible layout are presented in section \[sec:detector\_concept\] and some of the areas where such an experiment would have a significant impact are discussed in section \[sec:physics\_potential\].
Detector concept {#sec:detector_concept}
================
In the following we describe the key technologies, the conceptual layout and the main features of the proposed experimental apparatus, which is entirely based on CMOS technology.
Detector technology
-------------------
CMOS technology, which fueled the rapid growth of the information technology industry in the past 50 years, has also played and continues to play a crucial role in the remarkable development of detectors for High-Energy Physics (HEP) experiments. The amazing evolution of CMOS transistors in terms of speed, integration and cost decrease, allowed a continuous increase of density, complexity and performance of the front-end and readout circuits for HEP detectors. With the advent of CMOS MAPS, where the sensing layer and its readout circuitry are combined in a single integrated circuit, CMOS became also the technology for a new generation of vertex and tracking detectors. An important example of such an approach is represented by the new ALICE Inner Tracking System (ITS2) [@ITS_upgrade], which is based on a MAPS device, named ALPIDE, covering about 10m$^2$ of area with about 12.5 billion pixels.
The development of ALPIDE represents a quantum leap in the field of CMOS MAPS for single-particle detection, reaching unprecedented performance in terms of signal/noise ratio, spatial resolution, material budget and readout speed. Still, further significant improvements are possible by fully exploiting the rapid progress that this technology is making in the fields of imaging and time-of-flight measurements for consumer applications.
One of the features offered recently by CMOS imaging sensor technologies, called stitching, will allow developing a new generation of large-size MAPS with an area of up to , for wafer processes, and up to , for wafer processes. Moreover, the reduction of the sensor thickness to values of about 20-40$\mu$m will open the possibility of exploiting the flexible nature of silicon to implement large-area curved sensors. In this way, it will become possible to build cylindrical layers of silicon-only sensors, which will enable a dramatic reduction of the detector material thickness [@LS3_upgrade].
CMOS technology is also currently revolutionizing the field of 3D imaging, which has become a key sensing technology in a wide range of LiDAR (Light Detection And Ranging) applications in the field of robotic, automotive, medical and spacecraft systems. Time-of-flight (TOF) imagers based on CMOS Single Photon Avalanche Diodes (SPADs) feature very small pixel pitches, O(), and very high time resolution. Such a technology can be further optimized and tailored towards the measurement of minimum ionizing particles with a time resolution of the order of a few tens of picoseconds.
Detector layout and main parameters
-----------------------------------
The view of the experimental apparatus is shown in Fig. \[fig:detector\_layout\]. The detector, which covers the pseudorapidity region of $|\eta| < 4$ over the full azimuth, has a very compact layout with radial and longitudinal dimensions of 1.2m and 4m, respectively. It consists of a central barrel and two end-caps, which are embedded in a solenoid magnet (not shown in Fig. \[fig:detector\_layout\]). The moderately weak solenoidal field (0.5 T) of the ALICE magnet seems adequate to meet the requirements of high tracking efficiency at very low transverse momentum (a few tens of MeV$/c$) while preserving very good relative momentum resolution ($\approx 2\%$) at high transverse momentum (${\mbox{$p_{\rm T}$}}\approx 30~$GeV$/c$). However, the option of a magnet with a larger magnetic field (1 T or larger) will also be considered.
The central barrel, which covers the pseudorapidity region $|\eta| < 1.4$, consists of (from the inside out) an Inner Tracker (IT), with 3 layers located inside the beam pipe, the Outer Tracker (OT) with 7 layers, a Time-Of-Flight (TOF) detector for the identification of hadrons, as well as electrons at very low transverse momentum (${\mbox{$p_{\rm T}$}}< 500~$MeV$/c$), and an electromagnetic Shower Pixel Detector (SPD) for the identification of electrons and photons (${\mbox{$p_{\rm T}$}}> 500~$MeV$/c$). The two endcaps, which extend the acceptance to the pseudorapidity region $1.4 < |\eta| < 4$, contain each 4 disks in the IT, 6 disks in the OT and one disk in the SPD.
![Longitudinal view of the experimental apparatus. The detector is embedded in a solenoid magnet (not shown in the figure). The central barrel, which covers the pseudorapidity region $|\eta| < 1.4$, is composed of the IT, inside the beampipe (blue layers in the figure), the OT (yellow and green layers), the TOF (orange layers) and the SPD (outermost blue layers). Two endcaps, each consisting of a set of tracking disks and an SPD disk, extend the rapidity coverage to the region $|\eta| < 4$.[]{data-label="fig:detector_layout"}](detector_layout.pdf){width="\textwidth"}
The IT barrel is based on curved wafer-scale ultra-thin CMOS MAPS arranged in truly cylindrical layers, featuring an unprecedented low material budget of X$_0$ per layer, with the innermost layer positioned at only radial distance from the beam line. The pixel sensor contains approximately $10^6$ pixels per square centimeter, each measuring about , featuring a position resolution of better than . The IT layers will be located either in the secondary vacuum of a larger vacuum chamber, as shown in Fig. \[fig:detector\_layout\], or outside the beam pipe, with the innermost layers at a radial distance of from its outer wall. Four end-cap disks on each side complement the central barrel layers. A system of seven barrel layers and six endcap disks form the OT. They are based on the same technology as the sensors for the vertex detector, but their pixel size is increased to about in order to reduce the power density. These layers will provide a spatial resolution of about . The material budget of the OT layers will be about X$_0$ per layer.
A TOF detector consisting of three layers of CMOS MAPS with time resolution of the order of will surround the central tracker. Low-Gain Avalanche Detectors (LGAD) featuring such a time precision, which are being developed for the phase-II upgrade of the ATLAS and CMS experiments, represent a viable option. Given the moderately low radiation levels at the location of the time-of-flight system ($10^{12}$ 1MeV $n_{\rm eq}$/cm$^{2}$), CMOS SPADs also represent a very promising technology. These detectors are based on arrays of avalanche photodiodes reverse-biased above their breakdown voltage. SPAD detectors of recent generation feature a time jitter of tens of picoseconds. The number of layers needed for the realization of the TOF detector will depend on the time resolution and spatial fill factor achieved in the single layer.
The SPD is based on a stack-up of a few layers made of a dense passive material, e.g. lead, interleaved with layers of high-granularity pixel sensors, which allow counting individual particles generated in a shower. Electrons and positrons will be distinguished from photons by the presence of a track in the tracking layers. The contamination from pions will be very small due to the large ratio between the nuclear interaction length and the radiation length of lead ($\lambda_{n}/X_{0} \simeq 30$).
Tracking performance
--------------------
The tracking performance of this detector has been studied using a fast Monte Carlo tool. The code, which accounts for multiple scattering, detector occupancy and deterministic energy loss, provides accurate determination of the tracking resolution as a function of the detector configuration for both the spatial and the momentum components and a reliable estimate of the tracking efficiency.
An important measure of the achieved tracking precision is the track impact-parameter resolution, defined as the dispersion of the distribution of the Distance of Closest Approach (DCA) of the reconstructed tracks to their production vertex. It is the parameter that defines the capability of a vertex detector to separate secondary vertices of heavy-flavour decays from the interaction vertex. Two alternative configurations have been studied. In the first configuration, the three IT layers are contained inside the beam vacuum chamber, while the innermost OT layer is located outside of it at a radial distance of about 2mm from the outer wall. In the second configuration the IT layers are located outside the beam pipe with the innermost layers at about 2mm from the outer wall. The radial arrangement of the IT layers is identical for the two configurations. A comparison of the impact-parameter resolution for the two configurations is shown in Fig. \[fig:impact\_parameter\] for at typical pseudorapidity $|\eta| = 0.5$. Owing to its nearly zero mass, the detector exhibits a spectacular vertexing performance, with an impact parameter resolution that is below for ${\mbox{$p_{\rm T}$}}\simeq 1~$GeV$/c$ and remains below down to ${\mbox{$p_{\rm T}$}}\simeq 0.1~$GeV$/c$.
![Impact parameter resolution for primary electrons, charged pions and protons as a function of the transverse momentum for two configurations of the IT: inside the beam pipe (empty circles) and outside the beam pipe (filled circles).[]{data-label="fig:impact_parameter"}](impact_parameter){width="80.00000%"}
Physics potential {#sec:physics_potential}
=================
The ability to measure the production of leptons, photons and identified hadrons down to ${\mbox{$p_{\rm T}$}}$ scales of the order of a few tens of MeV/$c$, would provide significant advances in several areas. In the following, we list only some of the physics topics that would dramatically benefit from such an apparatus.
Heavy flavour and quarkonia
---------------------------
The unique tracking and vertexing capabilities of the proposed apparatus, combined with its high-luminosity capabilities, will provide an ideal tool for a comprehensive campaign of high-precision measurements of the production of open and hidden heavy flavour particles in nuclear collisions. Below, we list some examples of measurements where the proposed experiment will have a crucial impact.
The suppression of the production of quarkonia due to Debye screening of the strong interaction was one of the first predicted signatures of colour deconfinement [@Matsui:1986dk]. Anomalous suppression of the production of J/$\psi$ mesons (beyond the level that could be explained by conventional models, such as the interaction of the J/$\psi$ with comoving hadrons) was experimentally observed in nuclear collisions at the CERN Super Proton Synchrotron [@Alessandro:2004ap]. Measurements on Au-Au collisions at RHIC revealed similar levels of suppression as at the SPS [@Adare:2006ns].
Data from Pb-Pb collisions at the LHC have revealed a new regime for J/$\psi$ production [@Abelev:2013ila]: at the LHC the suppression is reduced compared to lower energies and, unlike at lower energies, it is very weakly dependent on the centrality of the collision and instead of decreasing, it increases with increasing ${\mbox{$p_{\rm T}$}}$. Such observations are consistent with the presence of a significant contribution at the LHC from a novel mechanism, which had been proposed in [@BraunMunzinger:2000px], for a recent review see [@Andronic:2017pug], for quarkonium production via statistical hadronization of initially uncorrelated heavy quark pairs at the phase boundary. Alternatively, a mechanism of continuous creation and destruction of charmonia in the QGP was proposed in [@Thews:2000rj]. In this new regime for J/$\psi$ production the above mentioned ${\mbox{$p_{\rm T}$}}$ dependence arises naturally, as predicted in [@Zhao:2011cv]. If the above picture is correct, measurements of exotic hadrons would provide a direct window on hadron formation from a deconfined Quark-Gluon Plasma, and spectacular effects would be expected for Multiply Heavy Flavoured (MHF) baryons and mesons. In such a scenario, the yields of MHF baryons relative to the number of produced charm quarks are predicted to be enhanced in AA relative to pp collisions. First predictions in this area were made in [@Becattini:2005hb], recent calculations are discussed in [@Cho:2017dcy; @Zhao:2016ccp; @He:2014tga; @Yao; @Andronic_exotica]. Enhancements are expected by as much as a factor 10$^2$ for the recently discovered $\Xi_{cc}$ baryon and even by as much as a factor 10$^3$ for the as yet undiscovered $\Omega_{ccc}$ baryon. The observation and precise quantification of such effects would represent a quantum jump for the study of the properties of deconfined matter. The observation of the effect in MHF hadrons would provide a key confirmation of the current interpretation of the LHC quarkonium results and open a crucial new window to study the mechanisms of hadron formation from a deconfined Quark-Gluon Plasma. Such studies are currently way beyond reach. The detection of MHF baryons at the LHC requires nucleon-nucleon luminosities of the order of 1 fb$^{-1}$ [@LHCb], with excellent tracking and secondary vertex and particle identification capabilities down to low transverse momenta to enable the reconstruction of complex cascades of weak decays of heavy flavour hadrons. The apparatus discussed in this document would provide the ideal detector for such measurements, combining ultra-low-mass tracking over a wide momentum range, particle identification and high-speed capabilities. Nucleon-nucleon luminosities in the fb$^{-1}$ range could be accumulated at the LHC in a few weeks with the use of lighter nuclei such as Xe or Kr [@Citron:2018lsq], bringing such measurements well within reach.
The proposed detector would also significantly enhance the capability for quarkonium physics. The systematics of charmonium production cannot be fully understood without a precise separation of prompt charmonia from those originating from B decays as well as a quantitative determination of the production yields of low transverse momentum $\chi_c$ states, that can decay strongly into prompt J$/\psi$, but are expected to be much more fragile, due to the significantly smaller binding energy. The separation of prompt charmonia from secondary charmonia from B decays in the proposed apparatus will be excellent, and the high luminosity should make even the rare $\psi$(2S) abundant. The detection of $\chi_c$ states involves the identification of a low-energy (300-400 MeV) photon in addition to a J$/\psi$ meson. Low-energy photons can be measured with high efficiency in the SPD. A resolution of about 5% around 400MeV is required, corresponding to $\frac{\delta E_{\gamma}}{E{\gamma}} \approx 3\%/\sqrt{E_{\gamma}(\mathrm{GeV})}$, which may be achievable in the SPD.
The proposed detector can also be used to shed new light on the nature and structure of the X, Y, Z charmonium-like states recently discovered, see, e.g., [@Esposito:2014rxa] and references therein. As a case in point we note that the X(3872) state which is well studied in pp collisions, but the whose nature is not yet understood, could be measured with precision also in nuclear collisions. Since this is a charmonium state, its yield is expected to be particularly enhanced at low transverse momenta (${\mbox{$p_{\rm T}$}}< 4$ GeV/$c$) [@Andronic:2019wva]. It would also be very interesting to perform a precision comparison of the transverse momentum spectra of particles of similar mass, but very different binding energies, such as $\psi$(2S) and X(3872).
Low-mass dileptons
------------------
### The low mass continuum, $0 < m < 3$ GeV
The strongly-interacting medium is expected to emit electromagnetic radiation during a significant part of its lifetime. At colliders, signals of an excess of dileptons with respect to the expectations from known hadronic sources have been observed both at RHIC [@Adam:2018tdm; @Adam:2018qev; @Adare:2015ila] and with still low significance at the LHC [@Acharya:2018nxm]. For LHC Run3 and Run4 the ALICE collaboration will upgrade its detector [@Abelevetal:2014cna] resulting in significant improvements for low-mass dilepton measurements compared to the results from the present RHIC and LHC experiments.
The nearly massless detector with high-precision tracking and vertexing and very high rate capability, discussed in this document, would allow pushing down in transverse momentum very close to the natural scale determined by the inverse radius of the system (about 100 MeV/$c$ in pp collisions, about 10 MeV/$c$ in Pb-Pb collisions), which would represent the “ultimate” dilepton experiment.
The very low mass and ${\mbox{$p_{\rm T}$}}$ cutoff would allow testing theoretical predictions in the region of phase space, currently beyond reach, where most of the radiation is emitted.
### Chiral symmetry restoration and the temperature of the hot QGP fireball
Up to now, no direct experimental evidence exists for the restoration of chiral symmetry in the hot and dense phase formed during a relativistic nucleus-nucleus collision. Indeed, the enhanced low-mass dilepton continuum found at the SPS, RHIC, and LHC, can be described well by assuming that the $\rho$ meson broadens in the medium without mass change [@Arnaldi:2008fw; @Adamova:2006nu; @Rapp:2009yu; @Adare:2015ila; @Acharya:2018nxm]. With an essentially mass-less detector as described above one can take an entirely new approach, based on a precision measurement of the thermal dilepton continuum starting from the $\rho$ meson and reaching up to masses of about 1.6 GeV. Guidance comes from inspecting the light flavor section of Fig. 51.3 of the latest PDG Review of Particle Physics [@Tanabashi:2018oca]. Here one can see that the R factor obtained from $e^+e^-$ collisions and representing the ‘vacuum’ in this context has a clear minimum near masses of 1.26 GeV, the mass of the a$_1$ meson. This meson is generally considered the chiral partner of the $\rho$ meson and as such does not couple to $e^+e^-$ pairs, hence the minimum. The idea for the proposed measurement is then to study this mass region in pp, p-Pb and Pb-Pb collisions to see whether the minimum near 1.26 GeV fills in as one goes from minimum bias pp to central Pb-Pb collisions.
With the dramatic improvements in vertexing from the detector described above, it will be possible to quantitatively measure and tag/reject non-prompt dileptons, revealing the true thermal continuum. We note that the only other known background for this measurement is from Drell-Yan production which at LHC energies is estimated to be negligible. With this approach one could not only provide a crucial test for chiral symmetry restoration in the $\rho$-a$_1$ sector but also make a precision determination of the temperature of the QGP from the analysis of the mass spectrum in the 1.8–3 GeV region.
Soft and ultra-soft photons
---------------------------
Measurements of real soft photons are notoriously difficult due to the huge background from $\pi^0$ decays and electron bremsstrahlung. Available measurements in ultra-relativistic nuclear collisions typically extend down to around 1 GeV, see [@Adam:2015lda] and references therein. In the proposed ultra-low mass tracker soft real photons could be measured using the conversion method pioneered by ALICE [@Adam:2015lda].
This should extend the photon transverse momentum range into the region of 50-100 MeV/$c$, thereby allowing tests of current predictions for radiation from the QGP in completely uncharted regions of phase space.
Very low ${\mbox{$p_{\rm T}$}}$ photons (1 MeV/$c$ $< p_{\rm{T}}^{\gamma}<$ 100 MeV/$c$) could be measured with a special, small spectrometer at forward rapidity in the range $3.5<|\eta|<5$.
The measurement of very soft electromagnetic radiation in the p$_T$ region below 100 MeV/$c$ and approaching 1 MeV/$c$ is of fundamental interest. The production of photons at such low transverse momenta arises as a consequence of the structure of all gauge theories, see, in particular, [@bloch_nordsieck; @low; @cahn; @strominger1; @strominger2]. According to resulting soft theorems the number of soft (real) photons actually diverges towards low ${\mbox{$p_{\rm T}$}}$, but, as discussed in [@strominger2], [*“in a highly controlled manner that is central to the consistency of the underlying quantum field theory”*]{}. It would be of prime importance to reach the experimental sensitivity to test this prediction. This would require measurements at very low ${\mbox{$p_{\rm T}$}}$, below 10 MeV/$c$, which could be achieved with a high-rapidity photon spectrometer. It is intuitively clear that the $1/p_{\rm T}$ divergence characteristic of photon bremsstrahlung and predicted by [@bloch_nordsieck; @low] will eventually be reached, but at what level and at which value of ${\mbox{$p_{\rm T}$}}$ will depend on the size and structure of the system, possibly on its quark content [@nachtmann_reiter].
To make progress in this area one would measure photon production for pp and pA and ultimately Pb-Pb collisions. Measurements performed at fixed target energies in bubble chambers [@Chliapnikov:1984ed; @Botterweck] and at the OMEGA Spectrometer [@Banerjee; @Belogianni_1; @Belogianni_2] reported an excess of photons at low transverse momenta (below 40 MeV/$c$) over the rate expected from hadronic bremsstrahlung, centered a positive centre-of-mass rapidity, while measurements performed with the HELIOS apparatus at central and backward rapidities [@antos] reported yields consistent with the expectation from hadronic bremsstrahlung. These results were analyzed by [@botz], in the framework of quark synchrotron radiation. It would be very important to follow up on this with a state-of-the-art measurement at the LHC.
Other topics
------------
An ultra-low mass, high-resolution and high-rate experiment as discussed in this document, would also allow to make major contributions to other areas, not covered here, such as precision studies of spectral distortions at low transverse momenta, coherent pion production, hadronisation at very-low transverse momenta, disoriented chiral condensates, femtoscopy, fluctuations and diffusion of conserved charges as well as the search for dark photons.
**Author List**
D. Adamová$^{\rm 72}$, G. Aglieri Rinella$^{\rm 23}$, M. Agnello$^{\rm 21}$, Z. Ahammed$^{\rm 102}$, D. Aleksandrov$^{\rm 67}$, A. Alici$^{\rm 5,39}$, A. Alkin$^{\rm 2}$, T. Alt$^{\rm 47}$, I. Altsybeev$^{\rm 82}$, D. Andreou$^{\rm 23}$, A. Andronic$^{\rm 79,104}$, F. Antinori$^{\rm 41}$, P. Antonioli$^{\rm 39}$, H. Appelshäuser$^{\rm 47}$, R. Arnaldi$^{\rm 43}$, I.C. Arsene$^{\rm 12}$, M. Arslandok$^{\rm 78}$, R. Averbeck$^{\rm 79}$, M.D. Azmi$^{\rm 10}$, X. Bai$^{\rm 79}$, R. Bailhache$^{\rm 47}$, R. Bala$^{\rm 75}$, L. Barioglio$^{\rm 16}$, G.G. Barnaföldi$^{\rm 105}$, L.S. Barnby$^{\rm 71}$, P. Bartalini$^{\rm 4}$, K. Barth$^{\rm 23}$, S. Basu$^{\rm 103}$, F. Becattini$^{\rm 18}$, C. Bedda$^{\rm 51}$, I. Belikov$^{\rm 92}$, F. Bellini$^{\rm 23}$, R. Bellwied$^{\rm 95}$, S. Beole$^{\rm 16}$, L. Bergmann$^{\rm 78}$, R.A. Bertens$^{\rm 99}$, M. Besoiu$^{\rm 54}$, L. Betev$^{\rm 23}$, A. Bhatti$^{\rm 79}$, A. Bianchi$^{\rm 16}$, L. Bianchi$^{\rm 16,95}$, J. Bielčík$^{\rm 25}$, J. Bielčíková$^{\rm 72}$, A. Bilandzic$^{\rm 77,87}$, S. Biswas$^{\rm 3}$, R. Biswas$^{\rm 3}$, D. Blau$^{\rm 67}$, F. Bock$^{\rm 23}$, M. Bombara$^{\rm 26}$, M. Borri$^{\rm 71}$, P. Braun-Munzinger$^{\rm 79}$, M. Bregant$^{\rm 89}$, G.E. Bruno$^{\rm 22}$, M.D. Buckland$^{\rm 97}$, H. Buesching$^{\rm 47}$, S. Bufalino$^{\rm 21}$, P. Buncic$^{\rm 23}$, J.B. Butt$^{\rm 7}$, A. Caliva$^{\rm 79}$, P. Camerini$^{\rm 15}$, F. Carnesecchi$^{\rm 5}$, J. Castillo Castellanos$^{\rm 93}$, F. Catalano$^{\rm 20}$, S. Chapeland$^{\rm 23}$, M. Chartier$^{\rm 97}$, C. Cheshkov$^{\rm 91}$, B. Cheynis$^{\rm 91}$, V. Chibante Barroso$^{\rm 23}$, D.D. Chinellato$^{\rm 90}$, P. Chochula$^{\rm 23}$, T. Chujo$^{\rm 101}$, C. Cicalo$^{\rm 40}$, F. Colamaria$^{\rm 38}$, D. Colella$^{\rm 38}$, M. Concas$^{\rm 43,I}$, Z. Conesa del Valle$^{\rm 46}$, G. Contin$^{\rm 97}$, J.G. Contreras$^{\rm 25}$, F. Costa$^{\rm 23}$, B. Dönigus$^{\rm 47}$, T. Dahms$^{\rm 77,87}$, A. Dainese$^{\rm 41}$, J. Dainton$^{\rm 97}$, A. Danu$^{\rm 54}$, S. Das$^{\rm 3}$, D. Das$^{\rm 80}$, S. Dash$^{\rm 34}$, A. Dash$^{\rm 65}$, G. David$^{\rm 83}$, A. De Caro$^{\rm 13}$, G. de Cataldo$^{\rm 38}$, A. De Falco$^{\rm 14}$, N. De Marco$^{\rm 43}$, S. De Pasquale$^{\rm 13}$, S. Deb$^{\rm 35}$, D. Di Bari$^{\rm 22}$, A. Di Mauro$^{\rm 23}$, T. Dietel$^{\rm 94}$, R. Divià$^{\rm 23}$, U. Dmitrieva$^{\rm 50}$, A. Dobrin$^{\rm 23}$, A.K. Dubey$^{\rm 102}$, A. Dubla$^{\rm 79}$, D. Elia$^{\rm 38}$, B. Erazmus$^{\rm 85}$, A. Erokhin$^{\rm 82}$, G. Eulisse$^{\rm 23}$, D. Evans$^{\rm 81}$, L. Fabbietti$^{\rm 77,87}$, M. Faggin$^{\rm 19}$, P. Fecchio$^{\rm 21}$, A. Feliciello$^{\rm 43}$, G. Feofilov$^{\rm 82}$, A. Fernández Téllez$^{\rm 30}$, A. Festanti$^{\rm 23}$, S. Floerchinger$^{\rm 49}$, P. Foka$^{\rm 79}$, S. Fokin$^{\rm 67}$, A. Franco$^{\rm 38}$, C. Furget$^{\rm 60}$, A. Furs$^{\rm 50}$, J. Gaardh[ø]{}je$^{\rm 68}$, M. Gagliardi$^{\rm 16}$, P. Ganoti$^{\rm 63}$, C. Garabatos$^{\rm 79}$, E. Garcia-Solis$^{\rm 6}$, C. Gargiulo$^{\rm 23}$, P. Gasik$^{\rm 77}$, M.B. Gay Ducati$^{\rm 56}$, M. Germain$^{\rm 85}$, P. Ghosh$^{\rm 102}$, P. Giubellino$^{\rm 43}$, P. Giubilato$^{\rm 19}$, P. Glässel$^{\rm 78}$, V. Gonzalez$^{\rm 79}$, O. Grachov$^{\rm 103}$, A. Grigoryan$^{\rm 1}$, S. Grigoryan$^{\rm 58}$, F. Grosa$^{\rm 21}$, J.F. Grosse-Oetringhaus$^{\rm 23}$, R. Guernane$^{\rm 60}$, T. Gunji$^{\rm 100}$, R. Gupta$^{\rm 75}$, A. Gupta$^{\rm 75}$, M.K. Habib$^{\rm 79}$, H. Hamagaki$^{\rm 62}$, J.W. Harris$^{\rm 106}$, D. Hatzifotiadou$^{\rm 5,39}$, S.T. Heckel$^{\rm 47}$, E. Hellbär$^{\rm 47}$, H. Helstrup$^{\rm 24}$, T. Herman$^{\rm 25}$, H. Hillemanns$^{\rm 23}$, C. Hills$^{\rm 97}$, B. Hippolyte$^{\rm 92}$, S. Hornung$^{\rm 79}$, P. Hristov$^{\rm 23}$, J.P. Iddon$^{\rm 97}$, S. Igolkin$^{\rm 82}$, G. Innocenti$^{\rm 23}$, M. Ippolitov$^{\rm 67}$, M. Ivanov$^{\rm 79}$, A. Jacholkowski$^{\rm 17}$, M. Jung$^{\rm 47}$, A. Jusko$^{\rm 81}$, M.K. Köhler$^{\rm 78}$, S. Kabana$^{\rm 85}$, A. Kalweit$^{\rm 23}$, A. Karasu Uysal$^{\rm 59}$, T. Karavicheva$^{\rm 50}$, U. Kebschull$^{\rm 57}$, R. Keidel$^{\rm 32}$, M. Keil$^{\rm 23}$, B. Ketzer$^{\rm 29}$, S.A. Khan$^{\rm 102}$, A. Khanzadeev$^{\rm 73}$, Y. Kharlov$^{\rm 69}$, A. Khuntia$^{\rm 35}$, B. Kim$^{\rm 45}$, J. Kim$^{\rm 78}$, M. Kim$^{\rm 45,78}$, J. Klein$^{\rm 43}$, C. Klein$^{\rm 47}$, C. Klein-Bösing$^{\rm 104}$, S. Klewin$^{\rm 78}$, A. Kluge$^{\rm 23}$, M.L. Knichel$^{\rm 23}$, C. Kobdaj$^{\rm 86}$, M. Kofarago$^{\rm 105}$, P.J. Konopka$^{\rm 23}$, V. Kovalenko$^{\rm 82}$, I. Králik$^{\rm 52}$, M. Krüger$^{\rm 47}$, L. Kreis$^{\rm 79}$, M. Krivda$^{\rm 81}$, F. Krizek$^{\rm 72}$, M. Kroesen$^{\rm 78}$, E. Kryshen$^{\rm 73}$, V. Kučera$^{\rm 45,72}$, C. Kuhn$^{\rm 92}$, L. Kumar$^{\rm 74}$, S. Kundu$^{\rm 65}$, S. Kushpil$^{\rm 72}$, M.J. Kweon$^{\rm 45}$, M. Kwon$^{\rm 11}$, Y. Kwon$^{\rm 107}$, P. Lévai$^{\rm 105}$, S.L. La Pointe$^{\rm 27}$, E. Laudi$^{\rm 23}$, T. Lazareva$^{\rm 82}$, R. Lea$^{\rm 15}$, L. Leardini$^{\rm 78}$, S. Lee$^{\rm 11}$, R.C. Lemmon$^{\rm 71}$, R. Lietava$^{\rm 81}$, B. Lim$^{\rm 11}$, V. Lindenstruth$^{\rm 27}$, A. Lindner$^{\rm 33}$, C. Lippmann$^{\rm 79}$, J. Liu$^{\rm 97}$, J. Lopez Lopez$^{\rm 78}$, C. Lourenco$^{\rm 23}$, G. Luparello$^{\rm 44}$, S.M. Mahmood$^{\rm 12}$, A. Maire$^{\rm 92}$, V. Manzari$^{\rm 38}$, Y. Mao$^{\rm 4}$, A. Marín$^{\rm 79}$, M. Marchisone$^{\rm 91}$, G.V. Margagliotti$^{\rm 15}$, M. Marquard$^{\rm 47}$, P. Martinengo$^{\rm 23}$, S. Masciocchi$^{\rm 79}$, M. Masera$^{\rm 16}$, E. Masson$^{\rm 85}$, A. Mastroserio$^{\rm 38}$, A.M. Mathis$^{\rm 77,87}$, A. Matyja$^{\rm 88,99}$, M. Mazzilli$^{\rm 22}$, M.A. Mazzoni$^{\rm 42}$, L. Micheletti$^{\rm 16}$, A.N. Mishra$^{\rm 55}$, D. Miskowiec$^{\rm 79}$, B. Mohanty$^{\rm 65}$, M. Mohisin Khan$^{\rm 10,II}$, A. Morsch$^{\rm 23}$, T. Mrnjavac$^{\rm 23}$, V. Muccifora$^{\rm 37}$, D. M[ü]{}hlheim$^{\rm 104}$, S. Muhuri$^{\rm 102}$, J.D. Mulligan$^{\rm 61,106}$, M.G. Munhoz$^{\rm 89}$, R.H. Munzer$^{\rm 47}$, H. Murakami$^{\rm 100}$, L. Musa$^{\rm 23}$, B. Naik$^{\rm 34}$, B.K. Nandi$^{\rm 34}$, R. Nania$^{\rm 5,39}$, T.K. Nayak$^{\rm 65,102}$, D. Nesterov$^{\rm 82}$, G. Nicosia$^{\rm 98}$, S. Nikolaev$^{\rm 67}$, V. Nikulin$^{\rm 73}$, F. Noferini$^{\rm 5,39}$, J. Norman$^{\rm 60}$, A. Nyanin$^{\rm 67}$, V. Okorokov$^{\rm 70}$, C. Oppedisano$^{\rm 43}$, J. Otwinowski$^{\rm 88}$, K. Oyama$^{\rm 62}$, M. Płoskoń$^{\rm 61}$, Y. Pachmayer$^{\rm 78}$, A.K. Pandey$^{\rm 34}$, C. Pastore$^{\rm 38}$, J. Pawlowski$^{\rm 49}$, H. Pei$^{\rm 4}$, T. Peitzmann$^{\rm 51}$, D. Peresunko$^{\rm 67}$, M. Petrovici$^{\rm 33}$, R.P. Pezzi$^{\rm 56}$, S. Piano$^{\rm 44}$, E. Prakasa$^{\rm 36}$, S.K. Prasad$^{\rm 3}$, R. Preghenella$^{\rm 39}$, F. Prino$^{\rm 43}$, C.A. Pruneau$^{\rm 103}$, I. Pshenichnov$^{\rm 50}$, M. Puccio$^{\rm 16}$, J. Pucek$^{\rm 25}$, E. Quercigh$^{\rm 23}$, D. Röhrich$^{\rm 9}$, L. Ramello$^{\rm 20}$, F. Rami$^{\rm 92}$, S. Raniwala$^{\rm 76}$, R. Raniwala$^{\rm 76}$, R. Rath$^{\rm 35}$, I. Ravasenga$^{\rm 21}$, A. Redelbach$^{\rm 27}$, K. Redlich$^{\rm 64,III}$, F. Reidt$^{\rm 23}$, K. Reygers$^{\rm 78}$, V. Riabov$^{\rm 73}$, P. Riedler$^{\rm 23}$, W. Riegler$^{\rm 23}$, D. Rischke$^{\rm 48}$, C. Ristea$^{\rm 54}$, S.P. Rode$^{\rm 35}$, M. Rodríguez Cahuantzi$^{\rm 30}$, D. Rohr$^{\rm 23}$, A. Rossi$^{\rm 19,41}$, R. Rui$^{\rm 15}$, A. Rustamov$^{\rm 66}$, A. Rybicki$^{\rm 88}$, K. Šafařík$^{\rm 23,25}$, R. Sadikin$^{\rm 36}$, S. Sadovsky$^{\rm 69}$, R. Sahoo$^{\rm 35}$, P. Sahoo$^{\rm 35}$, P.K. Sahu$^{\rm 53}$, J. Saini$^{\rm 102}$, V. Samsonov$^{\rm 73}$, P. Sarma$^{\rm 28}$, H.S. Scheid$^{\rm 47}$, R. Schicker$^{\rm 78}$, A. Schmah$^{\rm 78}$, M.O. Schmidt$^{\rm 78}$, C. Schmidt$^{\rm 79}$, Y. Schutz$^{\rm 23,92}$, K. Schweda$^{\rm 79}$, E. Scomparin$^{\rm 43}$, J.E. Seger$^{\rm 8}$, S. Senyukov$^{\rm 92}$, A. Seryakov$^{\rm 82}$, R. Shahoyan$^{\rm 23}$, N. Sharma$^{\rm 74}$, S. Siddhanta$^{\rm 40}$, T. Siemiarczuk$^{\rm 64}$, R. Singh$^{\rm 65}$, M. Sitta$^{\rm 20}$, H. Soltveit$^{\rm 78}$, M. Spyropoulou-Stassinaki$^{\rm 63}$, J. Stachel$^{\rm 78}$, T. Sugitate$^{\rm 31}$, S. Sumowidagdo$^{\rm 36}$, X. Sun$^{\rm 4}$, J. Takahashi$^{\rm 90}$, C. Terrevoli$^{\rm 19,95}$, A. Toia$^{\rm 47}$, N. Topilskaya$^{\rm 50}$, S. Tripathy$^{\rm 35}$, S. Trogolo$^{\rm 16}$, V. Trubnikov$^{\rm 2}$, W.H. Trzaska$^{\rm 96}$, B.A. Trzeciak$^{\rm 51}$, T.S. Tveter$^{\rm 12}$, A. Uras$^{\rm 91}$, G.L. Usai$^{\rm 14}$, G. Valentino$^{\rm 98}$, L.V.R. van Doremalen$^{\rm 51}$, M. van Leeuwen$^{\rm 51}$, P. Vande Vyvre$^{\rm 23}$, M. Vasileiou$^{\rm 63}$, V. Vechernin$^{\rm 82}$, L. Vermunt$^{\rm 51}$, O. Villalobos Baillie$^{\rm 81}$, T. Virgili$^{\rm 13}$, A. Vodopyanov$^{\rm 58}$, S.A. Voloshin$^{\rm 103}$, G. Volpe$^{\rm 22}$, B. von Haller$^{\rm 23}$, I. Vorobyev$^{\rm 77}$, Y. Wang$^{\rm 4}$, M. Weber$^{\rm 84}$, A. Wegrzynek$^{\rm 23}$, D.F. Weiser$^{\rm 78}$, S.C. Wenzel$^{\rm 23}$, J.P. Wessels$^{\rm 104}$, J. Wiechula$^{\rm 47}$, U. Wiedemann$^{\rm 23}$, J. Wilkinson$^{\rm 39}$, B. Windelband$^{\rm 78}$, M. Winn$^{\rm 46}$, N. Xu$^{\rm 4}$, K. Yamakawa$^{\rm 31}$, Z. Yin$^{\rm 4}$, I.-K. Yoo$^{\rm 11}$, J.H. Yoon$^{\rm 45}$, A. Yuncu$^{\rm 78}$, V. Zaccolo$^{\rm 15,43}$, C. Zampolli$^{\rm 23}$, A. Zarochentsev$^{\rm 82}$, B. Zhang$^{\rm 4}$, X. Zhang$^{\rm 4}$, C. Zhao$^{\rm 12}$, V. Zherebchevskii$^{\rm 82}$, D. Zhou$^{\rm 4}$, Y. Zhou$^{\rm 4}$, Y. Zhou$^{\rm 68}$, G. Zinovjev$^{\rm 2}$.
$^{1}$ A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia\
$^{2}$ Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine\
$^{3}$ Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India\
$^{4}$ Central China Normal University, Wuhan, China\
$^{5}$ Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche ‘Enrico Fermi’, Rome, Italy\
$^{6}$ Chicago State University, Chicago, Illinois, United States\
$^{7}$ COMSATS University Islamabad, Islamabad, Pakistan\
$^{8}$ Creighton University, Omaha, Nebraska, United States\
$^{9}$ Department of Physics and Technology, University of Bergen, Bergen, Norway\
$^{10}$ Department of Physics, Aligarh Muslim University, Aligarh, India\
$^{11}$ Department of Physics, Pusan National University, Pusan, Republic of Korea\
$^{12}$ Department of Physics, University of Oslo, Oslo, Norway\
$^{13}$ Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo Collegato INFN, Salerno, Italy\
$^{14}$ Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy\
$^{15}$ Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy\
$^{16}$ Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy\
$^{17}$ Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Catania, Italy\
$^{18}$ Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Florence, Italy\
$^{19}$ Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Padova, Italy\
$^{20}$ Dipartimento di Scienze e Innovazione Tecnologica dell’Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy\
$^{21}$ Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy\
$^{22}$ Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy\
$^{23}$ European Organization for Nuclear Research (CERN), Geneva, Switzerland\
$^{24}$ Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway\
$^{25}$ Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic\
$^{26}$ Faculty of Science, P.J. Šafárik University, Košice, Slovakia\
$^{27}$ Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany\
$^{28}$ Gauhati University, Department of Physics, Guwahati, India\
$^{29}$ Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany\
$^{30}$ High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico\
$^{31}$ Hiroshima University, Hiroshima, Japan\
$^{32}$ Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation (ZTT), Worms, Germany\
$^{33}$ Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania\
$^{34}$ Indian Institute of Technology Bombay (IIT), Mumbai, India\
$^{35}$ Indian Institute of Technology Indore, Indore, India\
$^{36}$ Indonesian Institute of Sciences, Jakarta, Indonesia\
$^{37}$ INFN, Laboratori Nazionali di Frascati, Frascati, Italy\
$^{38}$ INFN, Sezione di Bari, Bari, Italy\
$^{39}$ INFN, Sezione di Bologna, Bologna, Italy\
$^{40}$ INFN, Sezione di Cagliari, Cagliari, Italy\
$^{41}$ INFN, Sezione di Padova, Padova, Italy\
$^{42}$ INFN, Sezione di Roma, Rome, Italy\
$^{43}$ INFN, Sezione di Torino, Turin, Italy\
$^{44}$ INFN, Sezione di Trieste, Trieste, Italy\
$^{45}$ Inha University, Incheon, Republic of Korea\
$^{46}$ Institut de Physique Nucléaire d’Orsay (IPNO), Institut National de Physique Nucléaire et de Physique des Particules (IN2P3/CNRS), Université de Paris-Sud, Université Paris-Saclay, Orsay, France\
$^{47}$ Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany\
$^{48}$ Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany\
$^{49}$ Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany\
$^{50}$ Institute for Nuclear Research, Academy of Sciences, Moscow, Russia\
$^{51}$ Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht, Netherlands\
$^{52}$ Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia\
$^{53}$ Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India\
$^{54}$ Institute of Space Science (ISS), Bucharest, Romania\
$^{55}$ Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico\
$^{56}$ Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil\
$^{57}$ Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany\
$^{58}$ Joint Institute for Nuclear Research (JINR), Dubna, Russia\
$^{59}$ KTO Karatay University, Konya, Turkey\
$^{60}$ Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France\
$^{61}$ Lawrence Berkeley National Laboratory, Berkeley, California, United States\
$^{62}$ Nagasaki Institute of Applied Science, Nagasaki, Japan\
$^{63}$ National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece\
$^{64}$ National Centre for Nuclear Research, Warsaw, Poland\
$^{65}$ National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India\
$^{66}$ National Nuclear Research Center, Baku, Azerbaijan\
$^{67}$ National Research Centre Kurchatov Institute, Moscow, Russia\
$^{68}$ Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark\
$^{69}$ NRC Kurchatov Institute IHEP, Protvino, Russia\
$^{70}$ NRNU Moscow Engineering Physics Institute, Moscow, Russia\
$^{71}$ Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom\
$^{72}$ Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy, Czech Republic\
$^{73}$ Petersburg Nuclear Physics Institute, Gatchina, Russia\
$^{74}$ Physics Department, Panjab University, Chandigarh, India\
$^{75}$ Physics Department, University of Jammu, Jammu, India\
$^{76}$ Physics Department, University of Rajasthan, Jaipur, India\
$^{77}$ Physik Department, Technische Universität München, Munich, Germany\
$^{78}$ Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany\
$^{79}$ Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany\
$^{80}$ Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India\
$^{81}$ School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom\
$^{82}$ St. Petersburg State University, St. Petersburg, Russia\
$^{83}$ State University of New York, Stony Brook, New York, United States\
$^{84}$ Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria\
$^{85}$ SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France\
$^{86}$ Suranaree University of Technology, Nakhon Ratchasima, Thailand\
$^{87}$ Technische Universität München, Excellence Cluster ’Universe’, Munich, Germany\
$^{88}$ The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland\
$^{89}$ Universidade de São Paulo (USP), São Paulo, Brazil\
$^{90}$ Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil\
$^{91}$ Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, Lyon, France\
$^{92}$ Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France\
$^{93}$ Université Paris-Saclay Centre d’Études de Saclay (CEA), IRFU, Department de Physique Nucléaire (DPhN), Saclay, France\
$^{94}$ University of Cape Town, Cape Town, South Africa\
$^{95}$ University of Houston, Houston, Texas, United States\
$^{96}$ University of Jyväskylä, Jyväskylä, Finland\
$^{97}$ University of Liverpool, Liverpool, United Kingdom\
$^{98}$ University of Malta, Msida, Malta\
$^{99}$ University of Tennessee, Knoxville, Tennessee, United States\
$^{100}$ University of Tokyo, Tokyo, Japan\
$^{101}$ University of Tsukuba, Tsukuba, Japan\
$^{102}$ Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India\
$^{103}$ Wayne State University, Detroit, Michigan, United States\
$^{104}$ Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany\
$^{105}$ Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary\
$^{106}$ Yale University, New Haven, Connecticut, United States\
$^{107}$ Yonsei University, Seoul, Republic of Korea\
$^{\rm I}$ Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy\
$^{\rm II}$ Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India\
$^{\rm III}$ Also at: Institute of Theoretical Physics, University of Wroclaw, Poland\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We demonstrate the eclipsing binary detection performance of the Gaia variability analysis and processing pipeline using Hipparcos data. The automated pipeline classifies 1067 (0.9%) of the 118204 Hipparcos sources as eclipsing binary candidates. The detection rate amounts to 89% (732 sources) in a subset of 819 visually confirmed eclipsing binaries, with the period correctly identified for 80% of them, and double or half periods obtained in 6% of the cases.'
address: 'Department of Astronomy, University of Geneva, CH 1290 Versoix ()'
author:
- Berry Holl
- Nami Mowlavi
- 'Isabelle Lecoeur-Taïbi'
- Fabio Barblan
- Lorenzo Rimoldini
- Laurent Eyer
- Maria Süveges
- Leanne Guy
- 'Diego Ordoñez-Blanco'
- Idoia Ruiz
- Krzysztof Nienartowicz
title: |
Automated eclipsing binary detection:\
applying the Gaia CU7 pipeline to Hipparcos
---
Introduction
============
The Gaia mission is expected to observe $\sim$1 billion sources, among which 0.4 to 7 million are expected to be eclipsing binaries [as summarized in @2013EAS....64..399H]. The detection and characterisation of those eclipsing binaries are distributed over two Coordination Units (CUs) of the Gaia Data Processing and Analysis Consortium. CU7 has the task to identify eclipsing binaries, find their orbital periods, and sub-classify them. This information is then passed to CU4, who will model the eclipsing binaries and derive stellar and orbital parameters [@2012ocpd.conf...59S]. The processing loop is schematized in Fig. 1 of [@2013EAS....64..399H].
In this short contribution, we demonstrate the eclipsing binary detection performance of the current Gaia CU7 pipeline on the Hipparcos data [@1997ESASP1200.....E]. Because the Hipparcos time sampling and mean number of observations are similar to Gaia, it a good dataset to test the performance of the Gaia processing pipeline.
The Hipparcos data set is described in Sect. \[sect:referenceSet\] and the results are presented in Sect. \[sect:results\]. Conclusions are drawn in Sect. \[sect:conclusions\].
Hipparcos eclipsing binaries {#sect:referenceSet}
============================
We define a reference set of 819 Hipparcos eclipsing binaries after visual inspection of the light curves, chosen among the list of eclipsing binaries published in Vol. 11 of the “Hipparcos catalogue of periodic variable stars” of [@1997ESASP1200.....E], to which additional Hipparcos sources are added that are flagged as eclipsing binaries in the January 2014 revision of the AAVSO catalogue [@Watson13]. We must note that the periods published in the Hipparcos catalog were derived from Hipparcos light curves for only 682 sources. The periods published for the remaining 137 sources were taken from the literature. We thus do not expect our automated pipeline to recover the periods of all those later sources.
Gaia CU7 pipeline applied to Hipparcos {#sect:results}
======================================
The Gaia CU7 pipeline, outlined in [@Eyer15], can be divided in four per-source analysis steps. They are briefly described in the following sections, with our application to the 115423 Hipparcos sources that have at least one good observation (flag 0 or 1) in their time series. Note that the applied pipeline configuration is simplified with respect to what is planned for official (Gaia) data processing.
----------------------------- ----------- --------- ----------- --------
Module
\#sources % \#sources %
Input time series 115423 100.00% 819 100.0%
Variability detection 15568 13.49% 819 100.0%
Supervised clas.: eclipsing 1598 1.38% 752 91.8%
SOS Eclipsing binaries 1067 0.92% 732 89.4%
----------------------------- ----------- --------- ----------- --------
: Pipeline processing result from top to bottom for the whole Hipparcos catalog (left), and for the eclipsing binaries reference set (right).[]{data-label="tab:resultSelectivity"}
Variability detection
---------------------
Variable sources are detected using a $\chi^2$ test with a p-value threshold of $p<10^{-4}$. This reduces the list of time series to be processed to 15568, see Table \[tab:resultSelectivity\].
Characterisation
----------------
Periodic sources are searched using the unweighted Lomb-Scargle method , and multi-harmonic Fourier series are fitted to their light curves.
Classification
--------------
The 15,568 variable sources are classified into 23 different variable types using a supervised Random Forest classifier [@Breiman01]. Input to the classifier are various Hipparcos specific attributes [detailed in @2011MNRAS.414.2602D] which are derived from the light curve model parameters determined in the previous step, together with the parallax, and V-I colour. Eclipsing binaries are represented by one class containing types ‘EA’, ‘EB’, and ‘EW’. We base our training set on [@2011MNRAS.414.2602D], which however includes 72% of the eclipsing binaries in our reference data set (Sect. \[sect:referenceSet\]). To make the reference set more independent of the training set, we train our classifier with only half of the Dubath training set, containing only 37% of the reference eclipsing binaries. Although this reduces the precision of the classifier, it strengthens the power of the reference set to evaluate the pipeline performance. The (ten-fold cross-validation) confusion matrix of the training set has a high completeness[^1] of 94.7% and a contamination[^2] of only 7.5% for eclipsing binaries.
Table \[tab:resultSelectivity\] shows that, applied to all Hipparcos data, the trained Random Forest classifier predicts 1598 sources to be of type eclipsing binary (selecting those with probability $>0.5$), which includes 92% of the reference eclipsing binaries.
![Period recovery for the 732 eclipsing binaries from the reference set that were identified as eclipsing binaries. The results from the unweighted Lomb-Scargle period search (left), and the ‘corrected’ periods using our Specific Object Study module ‘Eclipsing Binaries’ (right) recovering the correct period in most cases. []{data-label="fig:periodRecovery"}](GreatBarcelona_BH_figurePeriodRecovery.pdf){width="\textwidth"}
Specific Object Studies (SOS): Eclipsing Binaries
-------------------------------------------------
This post-classification step processes all 1598 eclipsing binary candidates provided by supervised classification. It aims at finding the best period and at sub-classifying the eclipsing binaries based on a two-Gaussian model description of the eclipses in the folded light curves. The detection and period identification algorithms will be detailed in [@HollInPrep], and the two-Gaussian modeling and sub-classification algorithm in [@MowlaviInPrep]. Basically, the automated procedures test the goodness of the model fits for several fractions and multiples of the computed Lomb Scargle period. If not satisfactory, the procedure is repeated for additional periods found with phase-dispersion minimisation [@Jurkevich71; @Stellingwerf78; @1997ApJ...489..941S] and String Length [@Lafler65; @Burke70]. The best period is retained from those tests. The eclipsing binary is then sub-classified based on the two-Gaussian model parameters as described in [@MowlaviInPrep].
This last step of per-source pipeline processing confirms 1067 eclipsing binaries (Table \[tab:resultSelectivity\]), of which 89% (732) are in our reference set of eclipsing binaries. Figure \[fig:periodRecovery\] shows that of these 732 reference eclipsing binaries, SOS Eclipsing Binaries recovers[^3] the periods listed in the Hipparcos catalog for 80% of the sources, and double or half periods for 6%. As mentioned in Sect. \[sect:referenceSet\], the periods in the Hipparcos catalog were taken from literature for 137 sources. The SOS Eclipsing Binaries package identifies 113 of those and recovers the period for 48%, and double or half the period for 7%.
Conclusions and discussion {#sect:conclusions}
==========================
Applying the Gaia CU7 pipeline to the 118204 Hipparcos time series, 1067 (0.9%) are identified as eclipsing binary candidates. Validating the results against a reference set of 819 visually identified eclipsing binaries we find that 732 (89%) are included, for which the Hipparcos period is recovered in 80% of the cases. Assuming that the reference set contains all eclipsing binaries detectable in Hipparcos data translates into a completeness of 89% and contamination of 31% of our 1067 candidates. Further investigation of this 31% ‘contamination’ is planned to be included in [@HollInPrep].
Given the similarities between Hipparcos and Gaia observations, the good detection rate on Hipparcos data suggests that the current CU7 variability analysis and processing pipeline is in good shape to automatically detect eclipsing binaries in Gaia too.
[99]{} Burke , 1970, J. Roy. Astron. Soc. Canada 64, 353 Breiman L., 2001, Mach. Learn., 45, 5 Dubath, P., Rimoldini, L., S[ü]{}veges, M., 2011, MNRAS, 414, 2602 Dworetsky, M.M., 1983, MNRAS 203, 917 ESA, 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200 Eyer, L., Evans, D. W., Mowlavi, N. these proceedings Holl, B., , in preparation Holl, B., Mowlavi, N., Eyer, L., 2013, EAS Publications Series, 64, 399 Jurkevich, I., 1971, Astrophys. Space Sci. 13, 154 Lafler, J., Kinman, T.D., 1965, ApJ Suppl. 11, 216 Lomb, N. R. 1976, Astrophys. Space Sci, 39, 447 Mowlavi, N., , in preparation Scargle, J. D. 1982, Ap. J., 263, 835 Schwarzenberg-Czerny, A. 1997, ApJ, 489, 941 Siopis, C., & Sadowski, G. 2012, Orbital Couples: Pas de Deux in the Solar System and the Milky Way, 59 Stellingwerf, R.F., 1978, Ap. J. 224, 953 Watson, C., Henden, A. A., & Price, A. 2013, VizieR Online Data Catalog, 1, 2027
[^1]: Type completeness = number of correctly classified sources / number in training set.
[^2]: Type contamination = 1 - number of correctly classified sources / number classified.
[^3]: $\hbox{The period X*P}_{true}\hbox{ is recovered if } \left| \hbox{P}_{true} - \hbox{P}_{found}/X \right| \left(\Delta T /\hbox{P}_{true}\right)<0.1\hbox{P}_{true}$, with $\Delta T$ the time-series duration and $\hbox{P}_{true}$ the Hipparcos or literature period, see [@2011MNRAS.414.2602D].
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{
"pile_set_name": "ArXiv"
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abstract: 'An imaging survey of CO(1$-$0), HCN(1$-$0), and HCO$^+$(1$-$0) lines in the centers of nearby Seyfert galaxies has been conducted using the Nobeyama Millimeter Array and the RAINBOW interferometer. Preliminary results reveal that 3 Seyferts out of 7 show abnormally high HCN/CO and HCN/HCO$^+$ ratios, which cannot occur even in nuclear starburst galaxies. We suggest that the enhanced HCN emission originated from X-ray irradiated dense obscuring tori, and that these molecular line ratios can be a new diagnostic tool to search for “pure” AGNs. According to our HCN diagram, we suggest that NGC 1068, NGC 1097, and NGC 5194 host “pure” AGNs, whereas Seyfert nuclei of NGC 3079, NGC 6764, and NGC 7469 may be “composite” in nature.'
author:
- 'K. Kohno'
- 'S. Matsushita'
- 'B. Vila-Vilaró'
- 'S. K. Okumura, T. Shibatsuka, M. Okiura'
- 'S. Ishizuki, R. Kawabe'
title: Dense Molecular Gas and Star Formation in Nearby Seyfert Galaxies
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
Dense molecular matter is considered to play various roles in the vicinity of active galactic nuclei (AGNs). The presence of dense and dusty interstellar matter (ISM), which obscures the broad line regions in AGNs, is inevitable at a few pc - a few 10 pc scale according to the unified model of Seyfert galaxies. This circumnuclear dense ISM could be a reservoir of fuel for nuclear activity, and also be a site of massive star formation. In order to investigate dense molecular matter in the centers of Seyfert galaxies, we have conducted an imaging survey of CO(1$-$0), HCN(1$-$0), and HCO$^+$(1$-$0) lines in nearby Seyfert galaxies using the Nobeyama Millimeter Array (NMA). High resolution HCN observations of Seyfert galaxies are of interest because unusually strong HCN emission has been reported in the type-2 Seyfert galaxies NGC 1068 (Jackson et al. 1993; Tacconi et al. 1994; Helfer & Blitz 1995) and NGC 5194 (Kohno et al. 1996). The HCN/CO integrated intensity ratios in brightness temperature scale, $R_{\rm HCN/CO}$ hereafter, within the central $r \sim$ a few 10 pc region exceed 0.4, which [*is never observed in non-Seyfert galaxies*]{} including nuclear starburst galaxies ($R_{\rm HCN/CO}
< 0.3$; see Kohno et al. 1999 and references therein).
CO, HCN, and HCO$^+$ Images of Seyferts
=======================================
Figure 1 shows a part of the CO images of Seyfert galaxies obtained with the NMA so far. Some Seyfert galaxies have also been observed using the RAINBOW interferometer, which is a 7 elements combined array consisting of six 10 m dishes (NMA) and NRO 45 m telescope; see Sofue et al. (2001) for NGC 3079, and see Okiura et al. (2001, this volume) for NGC 7469.
These CO images show a wide variety of gas morphologies in the central kpc regions of Seyferts, just as in the case of [*normal*]{} spirals (Sakamoto et al. 1999). Single dish CO surveys aiming to determine the global amount of molecular gas in Seyferts have revealed that there is no significant difference between the [*total*]{} amounts of molecular gas in Seyfert and quiescent spirals (e.g., Vila-Vilaró et al. 1998). Our higher angular resolution CO images may already suggest that the accumulation of molecular gas in the central kpc region is still insufficient for Seyfert activity.
Figure 2 shows the HCN and HCO$^+$ images of Seyfert galaxies. Except for NGC 5194 and NGC 6951, HCN and HCO$^+$ lines were observed simultaneously, thanks to the wide (1024 MHz) band width of a new spectro-correlator (UWBC; Okumura et al. 2000). This enables us to measure HCN/HCO$^+$ ratios ($R_{\rm HCN/HCO^+}$) accurately (systematic error must be less than a few %).
We find significant enhancement of HCN toward the nucleus of the type-1 Seyfert galaxy, NGC 1097 (Storchi-Bergmann et al. 1997). As demonstrated in Figure 3 and 4, the $R_{\rm HCN/CO}$ in the center of NGC 1097 is enhanced up to 0.34. This is the 3rd detection of abnormally ($R_{\rm HCN/CO} > 0.3$) enhanced HCN after NGC 1068 and NGC 5194. We have already mapped another 4 Seyferts (NGC 3079, NGC 6764, NGC 6951, and NGC 7469). We therefore find that 3 Seyferts out of 7 show extreme enhancement of HCN based on our preliminary data.
Another intriguing point is the remarkable weakness of HCO$^+$ emission in NGC 1068 and NGC 1097; in Figure 2, we find that $R_{\rm HCN/HCO^+}$ is 2.3 and 2.1 in NGC 1068 and NGC 1097, respectively. The ratios decrease to about unity in the circumnuclear starburst region. Preliminary results of HCO$^+$ in M51 also show very weak HCO$^+$ ($R_{\rm HCN/HCO^+} > 2$; Shibatsuka et al., in preparation).
Discussions
===========
We have observed extremely strong HCN emission in 3 Seyferts out of 7. What is the nature of these “HCN-enhanced Seyferts”? Here we compare the observed line ratios in Seyferts with those in nuclear starburst galaxies, which were also measured with similar angular resolutions. In Figure 5, it is immediately evident that Seyferts without abnormal HCN enhancements, i.e. NGC 3079, NGC 6764, and NGC 7469, show $R_{\rm HCN/CO}$ and $R_{\rm HCN/HCO^+}$ values just comparable to those in nuclear starbursts; they have $R_{\rm HCN/CO}$ less than 0.3, and $R_{\rm HCN/HCO^+}$ ranging from 0.5 to 1.5. On the other hand, HCN-enhanced Seyferts, i.e. NGC 1068 and NGC 1097, also have very high $R_{\rm HCN/HCO^+}$ values ($> 2$). Note that Nguyen-Q-Rieu et al. (1992) reported a very high $R_{\rm HCN/HCO^+}$ in NGC 3079 and Maffei 2 ($>3$), yet our new simultaneous measurements gave moderate ($\sim 1$) ratios.
We propose that these two groups in our “HCN diagram” (Figure 5) can be understood in terms of “AGN - nuclear starburst connection” (note that this should not be confused with “AGN - starburst cohabitation”, which often refers to the association of AGN with star formation on galactic scales in AGN hosts). In the Seyferts with line ratios comparable to those in nuclear starburst galaxies, it seems likely that nuclear starburst (presumably in the dense molecular torus) is associated with the Seyfert nucleus (i.e., “composite”). In the nuclear regions of composite Seyferts, HCO$^+$ fractional abundance is expected to increase due to frequent supernova (SN) explosions. In fact, in evolved starbursts such as M82, where large scale outflows have occurred due to numerous SN explosions, HCO$^+$ is often stronger than HCN (e.g. Nguyen-Q-Rieu et al. 1992). On the other hand, the HCN-enhanced Seyferts, which shows $R_{\rm HCN/CO} > 0.3$ and $R_{\rm HCN/HCO^+} > 2$, would host “pure” AGNs, where there is no associated nuclear starburst activity. In such a condition, the HCN line can be very strong because it has been predicted that fractional abundance of HCN is enhanced by strong X-ray radiation from AGN (Leep & Dalgarno 1996), resulting in abnormally high $R_{\rm HCN/CO}$ and $R_{\rm HCN/HCO^+}$ values. Our interpretation is supported by other wavelength data; for instance, NGC 1068 has been claimed as a pure Seyfert (Cid Fernandes et al. 2001 and references therein), whereas NGC 6764 (Schinnerer et al. 2000) and NGC 7469 (Genzel et al. 1995) have a composite nature. We need further analysis to validate the proposed interpretation, but if it is the case, this will serve as a new way to investigate the nature of AGNs; although this technique requires high angular resolution observations in order to avoid contaminations from extended circumnuclear star-forming regions, it has some advantages (e.g., not being affected by dust extinction).
We are indebted to the NRO staff for their efforts in improving the performance of the array.
Cid Fernandes, R., Heckman, T. M., Schmitt, H., González Delgado, R. M., & Storchi-Bergmann, T. 2001, , in press (astro-ph/0104186)
Genzel, R., Weitzel, L., Tacconi-Garman, Blietz, M., Krabbe, A., Lutz, D., & Sternberg, A. 1995, , 444, 129
Helfer, T., Blitz, L. 1995, , 450, 90
Hummel, E., van der Hulst, J. M., & Keel, W. C. 1987, , 172, 32
Jackson, J. M., Paglione, T. D., Ishizuki, S., & Rieu, N. Q. 1993, , 418, L13
Kohno, K., Kawabe, R., Tosaki, T., & Okumura, K. S. 1996, , 461, L29
Kohno, K., Kawabe, R., & Vila-Vilaró, B. 1999, , 511, 157
Leep, S. & Dalgarno, A. 1996, , 306, L21
Nguyen-Q-Rieu, Jackson, J. M., Henkel, C., Truong-Bach, & Mauersberger, R. 1992, , 399, 521
Okumura, S. K., et al. 2000, , 52, 393
Quillen, A. C., Frogel, J. A., Kuchinski, L. E., & Terndrup, D. M. 1995, , 110, 156
Sakamoto, K., Okumura, K. S., Ishizuki, S., & Scoville, N. Z. 1999, , 124, 403
Schinnerer, E., Eckart, A., & Boller, T. 2000, , 545, 205
Storchi-Bergmann, T., Eracleous, M., Ruiz, M. T., Livio, M., Wilson, A. S., & Filippenko, A. V. 1997, , 489, 87
Sofue, Y., Koda, J., Kohno, K., Okumura, S. K., Honma, M., Kawamura, A., & Irwin, J. A. 2001, , 547, L115
Tacconi, L. J., Genzel, R., Blietz, M., Cameron, M., Harris, A. I., & Madden, S. 1994, , 426, L77
Vila-Vilaró, B., Taniguchi, Y., & Nakai, N. 1998, , 116, 1553
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{
"pile_set_name": "ArXiv"
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---
author:
- 'L. Freisem'
- 'G. S. M. Jansen'
- 'D. Rudolf'
- 'K. S. E. Eikema'
- 'S. Witte'
bibliography:
- 'REF/library.bib'
title: 'Spectrally resolved single-shot wavefront sensing of broadband high-harmonic sources'
---
Introduction
============
Accurate measurements in imaging and optical metrology often rely on a precise knowledge of the incident beam parameters. The Hartmann wavefront sensor, first introduced in the year 1900 to calibrate telescopes [@Hartmann], quickly became a standard tool to characterize optical wavefronts. The sensor consists of an opaque plate containing a structured array of apertures, and the wavefront information is retrieved by measuring the propagation of light transmitted through the holes. Detecting the positions of the aperture transmission with respect to a known reference provides amplitude and local phase tilts of a monochromatic wavefront simultaneously, from which the wavefront is then reconstructed. A widely used modification to increase the sensitivity of Hartmann sensors is the lenslet array introduced by Platt and Shack [@Shack] in 1971. In the regime of extreme ultraviolet (XUV) radiation, where lens arrays are challenging, the traditional hole array was shown to be a useful approach for the characterization of monochromatic synchrotron beams [@mercere_hartmann_2003] and high-harmonic generation (HHG) sources [@valentin_high-order_2008].
{width="\textwidth"}
High-harmonic generation sources [@popmintchev_attosecond_2010; @bartels_generation_2002; @rudawski_high-flux_2013] are becoming a mature table-top source of coherent XUV radiation, used in many applications such as nanoscale imaging [@gardner_subwavelength_2017; @zurch_real-time_2014; @baksh_wide-field_2016], soft-X-ray spectroscopy [@cousin_high-flux_2014] and attosecond physics [@krausz_attosecond_2009; @kraus_measurement_2015; @silva_spatiotemporal_2015]. However, a significant limitation of Hartmann sensors is their inability to provide spectral sensitivity. This is particularly important for the characterization of HHG beams, as it is known that the wavefronts of different harmonics can be substantially different [@zerne_phase-locked_1997]. Spectral wavefront variations in HHG may arise as a result of the phase matching geometry [@He_2009; @frumker_order-dependent_2012], and can convey information about quantum path interferences in the HHG process [@schapper_spatial_2010]. Introducing a controlled spectrally dependent wavefront tilt actually is the basis for the attosecond lighthouse effect, which can be used to produce isolated attosecond pulses [@quere_attosecond_2012; @kim_photonic_2013]. To extract spectrally resolved wavefront information, several methods have been demonstrated, such as slit scanning combined with grating-based spectrometry [@frumker_frequency-resolved_2009; @lloyd_complete_2013], and lateral shearing interferometry [@austin_lateral_2011; @mang_simultaneous_2014]. While these methods retrieve spectrally resolved wavefront information, they all depend on some form of mechanical scanning in obtaining a complete dataset, and therefore can only measure average wavefronts over many pulses. Due to the nonlinear nature of the HHG process, the influence of small driving pulse variations on the phase matching process can lead to wavefront fluctations on a shot-by-shot basis. Such fluctuations may in turn have consequences in experiments where knowledge of the input parameters typically limits the achievable accuracy, such as EUV scatterometry [@ku_euv_2016] and coherent diffractive imaging [@ge_impact_2013].
In this article we introduce a novel wavefront sensor concept called a Spectroscopic Hartmann Mask (SHM), that provides wavefront data for multiple spectral components directly in a single camera exposure. We replace the apertures of the Hartmann mask by transmission gratings which, in addition to the regular Hartmann spot pattern, produce replicas of this spot pattern at the $\pm 1^{\mathrm{st}}$ diffraction orders as shown schematically in Fig. \[fig:Scheme\](a). These diffracted beams have similar sensitivity to local wavefront tilts at each aperture, while the different harmonics can be clearly separated in the image. This approach enables the isolation and analysis of Hartmann spot patterns for the individual wavelengths of all harmonics in a HHG beam simultaneously from a single camera recording. The SHM wavefront sensor concept enables fast characterization of full HHG beams, even on a single-pulse basis for sufficiently bright beams, providing a unique tool for characterization of broadband short-wavelength sources and sensing of the underlying physics of the HHG process.
{width="65.00000%"}
Spectrally resolved wavefront characterization
==============================================
Spectroscopic Hartmann mask design
----------------------------------
A scanning electron microscope image of the SHM is displayed in Fig. \[fig:Scheme\]b, along with insets that show the individual gratings in more detail. To ensure clear separation of the spot patterns for the different harmonics, the mask design and the distance to the camera are important parameters. If the SHM-to-camera distance is too small, the first-order diffraction spots are not spread out sufficiently, leading to overlapping signal on the detector. Conversely, if the distance is too large, diffraction resulting from the apertures leads to larger spot sizes, resulting in overlapping spots of neighboring apertures. Using Fresnel diffraction theory to simulate the detected intensity patterns, it is readily possible to find designs that satisfy the requirements for different HHG spectra. The SHM design shown in Fig. \[fig:Scheme\]b enables wavefront measurements for all harmonics produced in argon using a fundamental wavelength of 810 nm. Moreover, there is some flexibility for measurements at both longer and shorter wavelengths by changing the distance between mask and camera.
Compared to conventional Hartmann masks, the SHM design does have a significantly lower spatial density of apertures. This requirement stems from the fact that each aperture now leads to a multitude of diffraction spots for the different harmonics, which all need to be spatially separated in order to retrieve accurate wavefront data. However, the sampling density in the presented design still allows wavefront characterization up to the fourth-order Zernike polynomials, which is sufficient for many practical purposes. Optimized designs with increased sampling density can be envisaged with more modelling efforts.
The SHM is fabricated by milling holes into a 200 nm thick, $1\times1$ mm sized silicon nitride membrane, covered with 5 nm chromium and 27 nm gold, using the focused ion-beam technique. The individual apertures are circular with a radius $r=10~\mu$m, and contain a transmission grating with a pitch $d=0.5~\mu$m. With this design, positioning the SHM at 3 cm from the camera ensures clear separation of high harmonics in the 20-60 nm wavelength range generated using 810 nm fundamental wavelength, while diffraction from the circular apertures themselves remains limited. To avoid overlap between the diffraction from adjacent apertures, the gratings are tilted at $17.5^{\circ}$ with respect to the row of apertures, which ensures that the rows of diffraction spots are separated by 90 $\mu$m on the camera. In addition, we orient the gratings in two different directions. Having multiple diffraction directions allows a higher spatial sampling density of the beam while avoiding overlapping spots on the camera. Furthermore, it enables detection of systematic effects (more detail given below), reducing the sensitivity to angular misalignment and making the sensor essentially self-referencing.
Measuring HHG wavefront data
----------------------------
To demonstrate spectrally resolved wavefront measurements of high-harmonic beams, we use the output of a Ti:sapphire-seeded non-collinear optical parametric chirped-pulse amplifier (NOPCPA). The 1 mJ, 25 fs, 810 nm laser pulses are focused with an $f=25$ cm lens into a pulsed gas jet, resulting in harmonics produced in argon or krypton covering the $25-56$ nm wavelength range. We use an adjustable iris to optimize HHG phase matching, leading to a Gaussian EUV beam. A $200$ nm thick aluminum foil is used to block the infrared radiation. An XUV-sensitive CCD-camera (Andor Ikon-L) is used to detect the radiation at approximately 63 cm away from the HHG source. The SHM is mounted on a translation stage and positioned in the beam at 3 cm before the camera.
A camera recording of the SHM diffraction pattern is shown in Fig. \[fig:Diff\]. Each aperture leads to an array of diffraction spots in the direction perpendicular to the grating lines, corresponding to the individual harmonics. As a guide to the eye, the diffraction from two apertures with different grating orientations are highlighted by the dashed boxes (note that the color coding corresponds to Fig. \[fig:Scheme\](b)). In addition to the normal Hartmann spot pattern from the direct transmission in the center of the image, the individual harmonics produce clearly separated diffraction spots in the $\pm 1^{\mathrm{st}}$ diffraction orders, of which the positions can be determined accurately by a standard two-dimensional Gaussian fitting procedure. As schematically depicted already in Fig. \[fig:Scheme\](a), these first-order diffraction spots are sensitive to the local wavefront tilts. Therefore they can be used to do wavefront reconstruction, for all harmonic orders simultaneously. In the present experiment up to nine harmonics are observed: the wavefronts of all the individual harmonics can be reconstructed from the single camera image of Fig. \[fig:Diff\]. To increase the accuracy of the spot determination further, we recorded images with exposure times up to 5 seconds in argon, and 10 seconds in krypton for the results described in section \[sec:results\].
![\[fig:angle\_scheme\] Schematic overview of the transmission of a single spectroscopic Hartmann aperture. The center of mass of the +1 and -1 diffraction orders approximates the position of the direct transmission. The quality of this approximation depends on the incident wavefront tilt as well as the angle between camera and wavefront sensor.](IMG/tilt_measurement_geometry_camera_angle){width="\linewidth"}
SHM diffraction pattern analysis
--------------------------------
Once the respective positions of all diffraction spots corresponding to each wavelength have been determined, all the required information for reconstructing the wavefronts is in principle available. Because of the diffraction geometry, there are some subtle differences compared to the traditional Hartmann mask analysis. A schematic overview of the diffraction geometry for a single aperture is drawn in Fig. \[fig:angle\_scheme\]. At a distance $L$ behind the SHM, the $\pm 1^{\mathrm{st}}$ diffraction orders appear at positions $y_{\pm 1}$. The main principle of the SHM is that a nonzero wavefront tilt angle $\theta_i$ leads to a shift of the $\pm 1^{\mathrm{st}}$ diffraction orders that is similar to the shift in the direct transmission position $y_0$. Therefore, the center of mass $(y_{+1} + y_{-1})/2$ of these diffraction positions can be used as input for the wavefront reconstruction algorithm. However, as indicated in Fig. \[fig:angle\_scheme\], there are two additional effects that need to be taken into account. First of all, for a non-zero $\theta_{i}$, the shift of the +/-1 diffraction orders is not perfectly symmetric, leading to an offset between their combined center of mass position and $y_0$. Secondly, there may be a finite angle $\beta$ of the camera with respect to the SHM. In this case, the measured positions of the diffraction spots are given by $h_{\pm 1}$, which are related to the expected positions as $h_{\pm 1} = y_{\pm 1} \cos{\theta_{\pm 1}} / \cos{\left(\theta_{\pm 1} \mp \beta\right)}$. Overall, the position of the measured spot position for the $m^\mathrm{th}$ diffraction order at a wavelength $\lambda$ can be written as: $$h_m = \frac{L(m \lambda - d \sin(\theta_i))}{d \sin(\beta+\arccos(\frac{m \lambda}{d}-\sin(\theta_i)))}
\label{Eq:hm}$$ where $d$ is the grating pitch and $L$ is the distance between SHM and camera.
Starting from Eq. \[Eq:hm\] and using a Taylor approximation to quantify the effect of small variations in $\theta_i$ and $\beta$, we find an expression for the center of mass position: $$\begin{gathered}
(h_{-1}+h_{1})/2 \approx \\ h_0 + \frac{L \lambda^2}{\lambda^2-d^2}\beta+(L-\frac{L d^3 \sqrt{d^2-\lambda^2}}{(d^2-\lambda^2)^2})\theta_i
+ O(\beta^2 \theta_i + \beta \theta_i^2).
\label{Eq:taylor_angles}\end{gathered}$$ From this expression, it is clear that the effects of the two angles are independent up to the third-order Taylor terms, so that these effects can readily be mitigated. As $\beta$ is a constant number for all apertures, the effect of the $\beta$-dependent term is to shift the center of mass position of all diffraction spots of a given wavelength by an equal amount. Therefore only the linear Zernike terms (tip and tilt) would be affected. By having multiple grating orientations in the SHM, the effect of this constant offset can be fully eliminated, as only the subset of apertures diffracting in the plane of rotation of $\beta$ is affected by this offset. The presence of an angle $\beta$ in either direction will therefore show up as a different tip/tilt measurement for both subsets of apertures, from which the offsets can be deduced and corrected for. As a result, spectrally-dependent variations in tip and tilt can be quantified with the SHM.
The $\theta_i$-term in Eq. \[Eq:taylor\_angles\] means that the center of mass position displaces more than the direct transmission in case of a local wavefront tilt along the grating diffraction direction. This effect needs to be corrected for to avoid a slight overestimation of the wavefront tilt. The size of this error depends only on known parameters and can be calculated to be $\sim 0.5\%$ per degree wavefront tilt at a wavelength of 30 nm. It it therefore straightforward to correct for this effect by first performing a wavefront reconstruction, finding an initial value for $\theta_i$, and subsequently correcting the center of mass position using the term from Eq. \[Eq:taylor\_angles\]. This correction could be performed in multiple iterations, but we find that one correction step suffices for an accurate wavefront reconstruction. Finally, the higher-order terms in Eq. \[Eq:taylor\_angles\] are found to be at least a factor $10^3$ smaller than the linear terms and can be safely ignored.
Wavefront reconstruction
------------------------
The analysis of the SHM diffraction patterns yields a set of Hartmann spot patterns for all the individual harmonics. These patterns can then be compared to the reference structure to find a displacement for every aperture on the sensor. This comparison yields the gradient of the wavefront $\nabla\phi\left(\vec{r}\right) = \Delta \vec{r} /L$, in which $\Delta r$ is the spot displacement with respect to the calibration. We characterized the reference pattern at 0.7 $\mu$m spatial resolution from an image of the SHM recorded by an optical microscope. This pattern is then rotated and centered with respect to the measured data.
A calibration of the distance $L$ is required for the analysis. To this end we recorded the SHM pattern from Krypton harmonics with the sensor placed at three different distances from the camera, with a precisely known displacement $\Delta L$ between them. These images allowed calibration of the harmonic wavelengths from the measurement geometry (Fig. \[fig:angle\_scheme\]), upon which a consistent value for $L$ could be retrieved. As an independent verification of the wavelengths of the different harmonics, spatially resolved Fourier-transform spectroscopy (FTS) [@Jansen2016] was used to measure the spectra of the individual spots on the camera. In our case where multiple harmonics are detected, $L$ can in principle also be obtained from the spot separation of consecutive harmonics on the camera image, if the assumption of an equal frequency spacing of twice the driving laser frequency between harmonics holds. We find that this approach leads to a slightly less accurate wavelength calibration, but still provides good wavefront reconstructions from a single SHM pattern.
From the measured local wavefront tilts the total wavefront can then be reconstructed using a two-dimensional integration of the measured gradient. In order to do so, a bi-cubic spline interpolation is used to resample the data from the non-rectangular arrangement of subpupils to convert it to a rectangular grid. The complete wavefront is obtained by numerical integration of the gradient.
![\[fig:WFs\] Reconstructed wavefronts for nine harmonics produced in argon, visualized by calculating their interference pattern with a plane wave. The center wavelength of the harmonic is indicated below each image.](IMG/WFsFLIPPED_onecolumn){width="0.9\linewidth"}
Wavefront characterization of a high-harmonic beam {#sec:results}
==================================================
Wavefront reconstructions of individual harmonics
-------------------------------------------------
Figure \[fig:WFs\] shows the reconstructed wavefronts for a range of nine harmonics generated in argon. To visualize these wavefronts and show their small features in addition to the curvature, they are displayed as how they would interfere with a plane wave. As the HHG beam undergoes free-space propagation from its generation point to the SHM, we measure a diverging beam with a significant spherical wavefront curvature. The brightest harmonics (33.5, 36.4 and 39.9 nm) appear to be nearly diffraction-limited, while the other harmonics show some deviations from an ideal spherical wavefront.
Because an uncertainty in the position of each aperture affects the final wavefront reconstruction in a different and complex way, we analyze the achieved wavefront accuracy by a Monte Carlo type analysis. From the center positions and uncertainties of the Gaussian fits of all the individual diffraction spots, we construct a distribution of Hartmann spot patterns that are all statistically consistent with the original measurement. We reconstruct wavefronts for all these spot patterns and compare the results. For the measurement shown in Fig. \[fig:WFs\], we find a wavefront reproducibility better than $\lambda/9$. The result is more accurate for the brightest harmonics, which show an uncertainty of $\lambda/32$.
![\[fig:ZP\] Wavelength-dependence of several beam parameters retrieved from a SHM measurement on HHG in Ar and Kr. Error bars constitute one standard deviation. **(a)** Linewidth of individual harmonics. **(b)** RMS deviation of the wavefronts from a spherical wave. The Marechal-criterion of $\lambda/14$ is indicated by a dashed line. **(c)** Radius of curvature (RoC) of the respective HHG wavelengths. **(d)** Astigmatism magnitude as calculated from the Zernike terms. **(e)** Reconstructed HHG spectra, normalized to the brightest argon line. The krypton spectrum is scaled by $2\times$ for visibility purposes.](IMG/wavefront_analysis){width="0.92\columnwidth"}
The information acquired from such a SHM measurement is not limited to wavefronts. Since each SHM aperture is a small transmission grating, a spectrum can be obtained for all apertures in the sensor. This enables a measurement of the linewidth of the harmonics as well as the spatial intensity distribution. Since a non-zero linewidth would result in elliptical spots, an accurate measurement of the linewidth can be made by fitting the spots with a 2D Gaussian that has different widths parallel and perpendicular to the diffraction direction. The width of each harmonic can then be extracted by deconvolution as $\mathrm{FWHM}=\sqrt{\left(\sigma_w^2-\sigma_h^2\right)}$, where $\sigma_w$ and $\sigma_h$ are the $1/e$-widths in the parallel and perpendicular directions respectively. This analysis assumes that the Gaussian shape is appropriate for both the spectrum and the diffraction spot shape.
Spectroscopic wavefront analysis
--------------------------------
To quantify the wavefront aberrations we expand the retrieved wavefronts in terms of Zernike polynomials, including the lowest 11 polynomial terms from $Z_0$ to $Z_3$ and $Z_4^0$ [@wyant_basic_1992]. The unit circle on which the Zernike polynomials are defined has a diameter of 740 $\mu$m and is chosen to fit within the region covered by the sensor apertures, ensuring accurate Zernike coefficients within this region.
Figure \[fig:ZP\] shows a set of results from such a wavefront analysis in terms of Zernike polynomials over a range of detected harmonic wavelengths, for HHG produced in both argon and krypton. As discussed above, the SHM provides some information about spectral parameters such as the linewidth of the harmonics, which is displayed in Fig. \[fig:ZP\](a). At shorter wavelengths the harmonics are more broadband, and we observe broader harmonics in krypton than in argon. The harmonics in krypton also appear slightly blue-shifted (Fig. \[fig:ZP\](e)), which together with the broader harmonics is indicative of the rather high intensity used for HHG in krypton. Figure \[fig:ZP\](b) shows the root-mean-square (RMS) deviation of the wavefronts from a perfect spherical shape, which can be viewed as a measure of the deviations from a diffraction-limited wavefront. Within the measurement accuracy, the wavefront deviations are found to be either at, or slightly above, the Marechal criterion of $\lambda/14$ (dashed line), indicating that the XUV beams are in general close to diffraction-limited.
While the strongest harmonics appear close to diffraction-limited, stronger wavefront aberrations are observed for the weaker harmonics. As an example, Fig. \[fig:ZP\](d) shows the magnitude of astigmatism, calculated as the root mean square of the $Z_2^{+2}$ and $Z_2^{-2}$ Zernike terms corresponding to straight and diagonal astigmatism. We found astigmatism to form the main contribution to the wavefront deviations observed in Fig. \[fig:WFs\], although there are small amounts of coma and trifoil present in the wavefronts as well.
One further parameter which is of interest is the radius of curvature (RoC) of the spherical wavefront component of the individual harmonics, as it has previously been reported that there can be a variation of the apparent focal distance with high-harmonic number [@frumker_order-dependent_2012]. Retrieving the RoC from the SHM data involves a combination of the defocus, spherical aberration and astigmatism Zernike terms [@wyant_basic_1992]. From our measurement data, the observed radii of curvature (Fig. \[fig:ZP\](c)) roughly match the source-to-mask distance of $60~\mathrm{cm}$, but no clear trend can be observed. This may be attributed to different phase-matching conditions in the high-harmonic generation. At the shortest wavelengths a deviation towards longer RoC is retrieved, although the error bars for these low-intensity harmonics also increase significantly.
![\[fig:Misalign\] Measured relative wavefront tilt in the horizontal (tilt, (a)) and vertical (tip, (b)) directions as a function of wavelength, retrieved from the SHM analysis, for HHG produced in argon and krypton with 810 nm fundamental wavelength. A significant wavelength-dependent wavefront tilt in the horizontal plane (tilt) is observed, indicating the presence of a spatial chirp of the HHG beam as schematically drawn in the inset to (a).](IMG/tilt_analysis_withcomaterm_inset){width="\columnwidth"}
Detection of HHG pulse front tilts
----------------------------------
A remarkable feature of the SHM is the ability to measure relative wavefront tilts between the individual harmonics. Especially for ultrafast pulsed sources, the ability to measure tilts allows a characterization of spatiotemporal coupling such as pulse front tilts, and associated effects such as the attosecond lighthouse effect [@quere_attosecond_2012; @kim_photonic_2013]. Since a wavefront tilt leads to a constant displacement of all spots on the camera, an accurate tilt measurement can only be performed with a Hartmann sensor if the exact position of the reference grid with respect to the camera is calibrated. For the SHM, an absolute wavefront tilt can in principle be determined if an accurate calibration of $\beta$ is performed. Such a calibration would require measurements of at least two collinear monochromatic beams with different wavelengths and is challenging to perform, while the physical significance of an absolute tilt measurement is limited.
Knowledge of a possible relative tilt between different harmonics is relevant for many ultrafast experiments, as it constitutes a spatial chirp, which is equivalent to a tilt of the pulse front. Such a spatial chirp can modify the outcome of experiments that are sensitive to space-time coupling of ultrashort pulses. In a SHM wavefront measurement, we can use any of the detected harmonics as reference. Because in a SHG measurement the wavefronts of all harmonics are measured simultaneously in a common geometry, the relative difference in tip and tilt for different harmonics can readily be measured.
In this analysis the effect of a possible camera tilt needs to be included. As already discussed before with Eq. \[Eq:taylor\_angles\], a camera tilt along the diffraction direction leads to an additional displacement of the center-of-mass positions that needs to be corrected to obtain correct spot positions for wavefront reconstruction. Since the displacement perpendicular to the grating direction is only sensitive to the wavefront tilt, an SHM containing gratings with two different directions can directly isolate this effect and measure the actual wavefront tilt relative to the reference wavelength.
In this way we extract relative wavefront tilts for the individual high-harmonics produced in argon and krypton. The resulting wavefront tilt and tip are shown in Figs. \[fig:Misalign\](a) and \[fig:Misalign\](b). We find wavefront-tilt variations up to 0.1 milliradians across our full HHG spectrum. In comparison, the divergence of the high harmonic beams is approximately 1 milliradians, which means that even though the individual harmonics remain largely overlapped, there is a spectral variation and associated pulse front tilt in the HHG beam. We observe similar pulse front tilts in argon and krypton, indicating that this tilt is most likely due to a pulse front tilt in the driving laser beam [@quere_attosecond_2012; @kim_photonic_2013]. This is to be expected, as the employed beam path contains some birefringent components with a slight intentional misalignment in the horizontal direction [@Jansen2016]. Similar geometries have been used to intentionally induce strong tilts in the HHG beam, leading to the attosecond lighthouse effect [@kim_photonic_2013]. The SHM provides a sensitive measurement device to quantify the presence of even small tilts, aiding the alignment and optimization of broadband HHG wavefronts and spatiotemporal couplings.
Conclusions
===========
In summary, we have demonstrated a spectrally-resolved single-shot wavefront sensor for broadband high-harmonic beams. To achieve this capability, we designed a specific Hartmann mask that incorporates transmission gratings to acquire wavefront measurements of the individual harmonics. A detailed analysis of a single measured pattern provides information on harmonic wavefronts, relative intensities, and spectral linewidths. Given sufficient sensitivity and incident flux, this information can in principle be retrieved for individual HHG pulses, rather than for an average over many shots. This feature opens up the prospect of detecting shot-to-shot variations of HHG beams, which has already been shown to be important information for sensitive experiments [@Kunzel:15].
The calibration procedure of the SHM is not more complicated than for a conventional Hartmann sensor, as only the SHM-camera distance and the orientation of the mask need to be determined, and we have shown that these parameters can even be achieved from a wavefront measurement in a self-consistent manner, if several harmonics are present in the beam. Although the current work demonstrates spectrally resolved wavefront measurements for wavelengths between 25 and 50 nm, this technique is not limited to these wavelengths. By changing the SHM design parameters such as the grating pitch and distance to the camera, the SHM method can be extended to wavelengths ranging from soft x-rays to infrared. By further refining the analysis procedure, it may be possible to extend the SHM technology to the characterization of more complex or partially continuous spectra.
Funding Information {#funding-information .unnumbered}
===================
The project has received funding from the European Research Council (ERC) (ERC-StG 637476) and the Netherlands Organisation for Scientific Research (NWO).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We show that deciding whether a sparse univariate polynomial has a $p$-adic rational root can be done in ${{\mathbf{NP}}}$ for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic $p$-adic Newton polygon. We thus improve the best previous complexity upper bound of ${\mathbf{EXPTIME}}$. We also prove an unconditional complexity lower bound of ${{\mathbf{NP}}}$-hardness with respect to randomized reductions for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss how our results complement analogous results over the real numbers.'
author:
- |
Martin Avendaño$^{^{^\text{{\normalsize $*$}}}}$\
\
\
Ashraf Ibrahim$^{^{^\text{{\normalsize $*$}}}}$\
\
\
J. Maurice Rojas\
\
\
Korben Rusek$^{^{^\text{{\normalsize $*$}}}}$\
\
\
title: 'Near $\text{\scalebox{2}[2]{${{\mathbf{NP}}}$}}$-Completeness for Detecting $\text{\scalebox{2}[2]{$p$}}$-adic Rational Roots in One Variable'
---
Introduction
============
The fields ${\mathbb{R}}$ and ${\mathbb{Q}}_p$ (the reals and the $p$-adic rationals) bear more in common than just completeness with respect to a metric: increasingly, complexity results for one field have inspired and motivated analogous results in the other (see, e.g., [@cohenqe; @vandenef] and the pair of works [@few] and [@amd]). We continue this theme by transposing recent algorithmic results for sparse polynomials over the real numbers [@brs] to the $p$-adic rationals, sharpening the underlying complexity bounds along the way (see Theorem \[thm:qp\] below).
More precisely, for any commutative ring $R$ with multiplicative identity, we let ${{\text{{\tt FEAS}}}}_R$ — the [**$R$-feasibility**]{} \[1\][[**problem**]{} (a.k.a. Hilbert’s Tenth Problem over $R$ [@h10])]{} — denote the problem of deciding whether an input polynomial system $F\!\in\!\bigcup_{k,n\in{\mathbb{N}}} ({\mathbb{Z}}[x_1,\ldots,x_n])^k$ has a root in $R^n$. (The underlying input size is clarified in Definition \[dfn:basic\] below.) Observe that ${{\text{{\tt FEAS}}}}_{\mathbb{R}}$, ${{\text{{\tt FEAS}}}}_{\mathbb{Q}}$, and $\{{{\text{{\tt FEAS}}}}_{{\mathbb{F}}_q}\}_{q \text{ a prime power}}$ are central problems respectively in algorithmic real algebraic geometry, algorithmic number theory, and cryptography.
In particular, for any prime $p$ and $x\!\in\!{\mathbb{Z}}$, recall that the [**$p$-adic valuation**]{}, ${{\mathrm{ord}}}_p x$, is the greatest $k$ such that $p^k|x$. We can extend ${{\mathrm{ord}}}_p(\cdot)$ to ${\mathbb{Q}}$ by ${{\mathrm{ord}}}_p\left(\frac{a}{b}\right)\!:=\!{{\mathrm{ord}}}_p(a)-{{\mathrm{ord}}}_p(b)$ for any $a,b\!\in\!{\mathbb{Z}}$; and we let $|x|_p\!:=\!p^{-{{\mathrm{ord}}}_p x}$ denote the [**$p$-adic norm**]{}. The norm $|\cdot|_p$ defines a natural metric satisfying the ultrametric inequality and ${\mathbb{Q}}_p$ is, to put it tersely, the completion of ${\mathbb{Q}}$ with respect to this metric. This metric, along with ${{\mathrm{ord}}}_p(\cdot)$, extends naturally to the [**$p$-adic complex numbers**]{} ${\mathbb{C}}_p$, which is the metric completion of the algebraic closure of ${\mathbb{Q}}_p$ [@robert Ch. 3].
We will also need to recall the following containments of complexity classes: ${\mathbf{P}}\!\subseteq\!{{\mathbf{ZPP}}}\!\subseteq\!{{\mathbf{NP}}}\!\subseteq\cdots\subseteq\!{\mathbf{EXPTIME}}$, and the fact that the properness of [**every**]{} inclusion above (save ${\mathbf{P}}\!\subsetneqq\!{\mathbf{EXPTIME}}$) is a major open problem [@lab; @papa]. The definitions of the aforementioned complexity classes are reviewed briefly in the Appendix (see also [@papa] for an excellent textbook treatment).
The Ultrametric Side: Relevance and\
Results {#sub:padic}
------------------------------------
Algorithmic results over the $p$-adics are central in many computational areas: polynomial time factoring algorithms over ${\mathbb{Q}}[x_1]$ [@lll], computational complexity [@antsv], studying prime ideals in number fields [@cohenant Ch. 4 & 6], elliptic curve cryptography [@lauder], and the computation of zeta functions [@denefver]. Also, much work has gone into using $p$-adic methods to algorithmically detect rational points on algebraic plane curves via variations of the [**Hasse Principle**]{}[^1] (see, e.g., [@colliot; @bjornhasse2; @bjornbm]). However, our knowledge of the complexity of deciding the existence of solutions for [**sparse**]{} polynomial equations over ${\mathbb{Q}}_p$ is surprisingly coarse: good bounds for the number of solutions over ${\mathbb{Q}}_p$ in one variable weren’t even known until the late 1990s [@lenstra2]. So we focus on precise complexity bounds for one variable.
\[dfn:basic\] Let $f(x)\!:=\!\sum^m_{i=1} c_i
x^{a_i}\!\in\!{\mathbb{Z}}[x_1,\ldots,x_n]$where $x^{a_i}\!:=\!x^{a_{1i}}_1\cdots x^{a_{ni}}_n$, $c_i\!\neq\!0$ for all $i$, and the $a_i$ are pair-wise distinct. We call such an $f$ an [**$\pmb{n}$-variate $\pmb{m}$-nomial**]{}. Let us also define\
${\mathrm{size}}(f)\!:=\!\sum^m_{i=1}
\log_2\left[(2+|c_i|)(2+|a_{1,i}|)\cdots
(2+|a_{n,i}|)\right]$\
and, for any $F\!:=\!(f_1,\ldots,f_k)\!\in\!({\mathbb{Z}}[x_1,\ldots,x_n])^k$, wedefine ${\mathrm{size}}(F)\!:=\!\sum^k_{i=1}{\mathrm{size}}(f_i)$. Finally, we let ${{\mathcal{F}}}_{n,m}$ denote the subset of ${\mathbb{Z}}[x_1,\ldots,x_n]$ consisting of polynomials withexactly $m$ monomial terms [$\diamond$]{}
For instance, ${\mathrm{size}}(1+cx^{99}_1+x^d_1)\!=\!\Theta(\log(c)+\log(d))$. So the degree, $\deg f$, of a polynomial $f$ can sometimes be exponential in its size. Note also that ${\mathbb{Z}}[x_1]$ is the disjoint union $\bigsqcup_{m\geq 0}{{\mathcal{F}}}_{1,m}$.
\[dfn:qp\] Let ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}$ denote the problem of deciding, for an input polynomial system $F$$\in\!\bigcup_{k,n\in{\mathbb{N}}}({\mathbb{Z}}[x_1,\ldots,x_n])^k$ [**and**]{} an input prime $p$, whether $F$ has a root in ${\mathbb{Q}}^n_p$. Also let ${{\mathbb{P}}}\!\subset\!{\mathbb{N}}$ denote the set of primes and, when ${{\mathcal{I}}}$ is a family of such pairs $(F,p)$, we let ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({{\mathcal{I}}})$ denote the restriction of ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}$ to inputs in ${{\mathcal{I}}}$. The underlying input sizes for ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}$ and ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({{\mathcal{I}}})$ shall be ${\mathrm{size}}_p(F)\!:=\!{\mathrm{size}}(F)+\log p$ (cf. Definition \[dfn:basic\]). Finally, let $({\mathbb{Z}}\times ({\mathbb{N}}\cup \{0\}))^\infty$ denote the set of all infinite sequences of pairs $((c_i,a_i))^\infty_{i=1}$ with $c_i\!=\!a_i\!=\!0$ for $i$ sufficiently large. [$\diamond$]{}
Note that ${\mathbb{Z}}[x_1]$ admits a natural embedding into $({\mathbb{Z}}\times ({\mathbb{N}}\cup \{0\}))^\infty$ by considering coefficient-exponent pairs in order of increasing exponents, e.g., $a+bx^{99}+x^{2001} \mapsto
((a,0),(b,99),(1,2001),(0,0),(0,0),\ldots)$. [$\diamond$]{}
While there are now randomized algorithms for factoring $f\!\in\!{\mathbb{Z}}[x_1]$ over ${\mathbb{Q}}_p[x_1]$ with expected complexity polynomial in ${\mathrm{size}}_p(f)+\deg(f)$ [@cantorqp] (see also [@chistov]), no such algorithms are known to have complexity polynomial in ${\mathrm{size}}_p(f)$ alone. Our main theorem below shows that such algorithms are hard to find because their existence is essentially equivalent to the ${\mathbf{P}}\!=\!{{\mathbf{NP}}}$ problem. Moreover, we obtain new sub-cases of ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times{{\mathbb{P}}})$ lying in ${\mathbf{P}}$.
\[thm:qp\]\
1. ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({{\mathcal{F}}}_{1,k}\times {{\mathbb{P}}})\!\in\!{\mathbf{P}}$ for $k\!\in\!\{0,1,2\}$.\
2. For any $f(x_1)\!=\!c_1+c_2x^{a_2}_1+c_3x^{a_3}_1\!\in\!{\mathbb{Z}}[x_1]$ with the points$\{(0,{{\mathrm{ord}}}_p(c_1)),(a_2,{{\mathrm{ord}}}_p(c_2)),(a_3,{{\mathrm{ord}}}_p(c_3))\}$ [**non**]{}-collinear, and $p$ [**not**]{} dividing $a_2$, $a_3$, or $a_3-a_2$, we can decide theexistence of a root in ${\mathbb{Q}}_p$ for $f$ in ${\mathbf{P}}$.\
3. There is a countable union of algebraic hypersurfaces $E\!\subsetneqq\!{\mathbb{Z}}[x_1]\times {{\mathbb{P}}}$, with natural density $0$, such that ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}(({\mathbb{Z}}[x_1]\times {{\mathbb{P}}})\setminus E)\!\in\!{{\mathbf{NP}}}$. Furthermore, we can decide in ${\mathbf{P}}$ whether an $f\!\in\!{{\mathcal{F}}}_{1,3}$ also lies in $E$.\
4. If ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times {{\mathbb{P}}})\!\in\!{{\mathbf{ZPP}}}$ then ${{\mathbf{NP}}}\!\subseteq\!{{\mathbf{ZPP}}}$.\
5. If the Wagstaff Conjecture is true, then ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1])$$\in\!{\mathbf{P}}\Longrightarrow {\mathbf{P}}\!=\!{{\mathbf{NP}}}$, i.e., we can strengthen Assertion (4) above.
The Wagstaff Conjecture, dating back to 1979 (see, e.g., [@bs Conj. 8.5.10, pg. 224]), is the assertion that the least prime congruent to $k$ mod $N$ is $O(\varphi(N)\log^2 N)$, where $\varphi(N)$ is the number of integers in $\{1,\ldots,N\}$ relatively prime to $N$. Such a bound is significantly stronger than the known implications of the [**Generalized Riemann Hypothesis (GRH)**]{}. [$\diamond$]{}
While the real analogue of Assertion (1) is known (and easy), the stronger real analogue ${{\text{{\tt FEAS}}}}_{\mathbb{R}}({{\mathcal{F}}}_{1,3})\!\in\!{\mathbf{P}}$ to Assertion (2) was unknown until [@brs Thm. 1.3]. We hope to strengthen Assertion (2) to ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({{\mathcal{F}}}_{1,3}\times{{\mathbb{P}}})\!\in\!{\mathbf{P}}$ in future work. In fact, we can attain polynomial complexity already for more inputs in ${{\mathcal{F}}}_{1,3}\times {{\mathbb{P}}}$ than stated above, and this is clarified in Section \[sec:qp\].
Note that ${\mathbb{Q}}_p$ is uncountable and thus, unlike ${{\text{{\tt FEAS}}}}_{{\mathbb{F}}_p}$, ${{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_p}$ does [**not**]{} admit an obvious succinct certificate. Indeed, while it has been known since the late 1990’s that ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}\!\in\!{\mathbf{EXPTIME}}$ relative to our notion of input size [@mw1; @mw2], we are unaware of any earlier algorithms yielding ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1,\ldots,x_n]\times{{\mathbb{P}}})\!\in\!{{\mathbf{NP}}}$ for any fixed $n$: even ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({{\mathcal{F}}}_{1,4}\times{{\mathbb{P}}})\text{\scalebox{1}[.85]{$\stackrel{?}{\in}$}}
{{\mathbf{NP}}}$ and ${{\text{{\tt FEAS}}}}_{\mathbb{R}}({{\mathcal{F}}}_{1,4})\text{\scalebox{1}[.85]{$\stackrel{?}{\in}$}}{{\mathbf{NP}}}$ are open questions.[^2] Practically speaking, zero density means that under most reasonable input restrictions, the algorithmic speed-up in Assertion (3) is valid over a significantly large fraction of inputs.
Let $T$ denote the family of pairs $(f,p)\!\in\!{\mathbb{Z}}[x_1]\times {{\mathbb{P}}}$ with $f(x_1)\!=\!a+bx^{11}_1+cx^{17}_1
+x^{31}_1$ and let $T^*\!:=\!T\setminus E$. Then there is a sparse $61\times 61$ structured matrix ${{\mathcal{S}}}$ (cf. Lemma \[lemma:syl\] in Section \[sub:transfer\] below), whose entries lie in $\{0,1,31,a,b,11b,c,17c\}$, such that $(f,p)\!\in\!T^* \Longleftrightarrow p\!\not|\!\det {{\mathcal{S}}}$. So by Theorem \[thm:qp\], ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}(T^*)\!\in\!{{\mathbf{NP}}}$, and Corollary \[cor:lots\] in Section \[sec:qp\] below tells us that for large coefficients, $T^*$ occupies almost all of $T$. In particular, letting $T(H)$ (resp. $T^*(H)$) denote those pairs $(f,p)$ in $T$ (resp. $T^*$) with \[1\][$|a|,|b|,|c|,p\!\leq\!H$, we have $\frac{\#T^*(H)}{\#T(H)}\!\geq\!\left(1-\frac{61}{H}\right)
\left(1-\frac{31\log_2(124H)}{H}\right)$.]{} For instance, one can check via [Maple]{} that\
$(-973+21x^{11}_1-2x^{17}_1 +x^{31}_1,p)\!\in\!T^*$\
for all but $352$ primes $p$. [$\diamond$]{}
The exceptions in Assertion (3) appear to be due to the presence of [**ill-conditioned**]{} polynomials: $f$ having a root $\zeta$ with the ($p$-adic) norm of $f'(\zeta)$ very small — a phenomenon of approximation present in complete fields like ${\mathbb{R}}$, ${\mathbb{C}}$, and ${\mathbb{Q}}_p$. Curiously, the real analogue of Assertion (3) remains unknown [@brs Sec. 1.2].
As for lower bounds, while it is not hard to show that the full problem ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}$ is ${{\mathbf{NP}}}$-hard from scratch, the least $n$ making ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1,\ldots,x_n]\times {{\mathbb{P}}})$ ${{\mathbf{NP}}}$-hard appears not to have been known unconditionally. In particular, a weaker version of Assertion (4) was found recently, but only under the truth of an unproved hypothesis on the distribution of primes in arithmetic progresion [@myqua Main Thm.]. Assertion (4) thus also provides an interesting contrast to earlier work of H. W. Lenstra, Jr. [@lenstra1], who showed that one can actually find all [**low**]{} degree factors of a sparse polynomial (over ${\mathbb{Q}}[x_1]$ as opposed to ${\mathbb{Q}}_p[x_1]$) in polynomial time.
Random Primes and Tropical Tricks {#sub:key}
---------------------------------
The key to proving our lower bound results (Assertions (4) and (5) of Theorem \[thm:qp\]) is an efficient reduction from a problem discovered to be ${{\mathbf{NP}}}$-hard by David Alan Plaisted: deciding whether a sparse univariate polynomial vanishes at a complex $D{^{\text{\underline{th}}}}$ root of unity [@plaisted; @mega]. Reducing from this problem to its analogue over ${\mathbb{Q}}_p$ is straightforward, provided ${\mathbb{Q}}^*_p$ contains a cyclic subgroup of order $D$ where $D$ has sufficiently many distinct prime divisors. We thus need to consider the factorization of $p-1$, which in turn leads us to primes congruent to $1$ modulo certain integers.
While efficiently constructing random primes in [**arbitrary**]{} arithmetic progressions remains a famous open problem, we can now at least efficiently build random primes $p$ such that $p$ is moderately sized but $p-1$ has many prime factors. We use the notation $[j]\!:=\!\{1,\ldots,j\}$ for any $j\!\in\!{\mathbb{N}}$.
\[thm:von\] For any $\delta\!>\!0$, a failure probability ${\varepsilon}\!\in\!(0,1/2)$, and $n\!\in\!{\mathbb{N}}$, we can find — within $O\!\left((n/{\varepsilon})^{\frac{3}{2}+\delta} +
\left(n\log(n)+\log\frac{1}{{\varepsilon}}\right)^{7+\delta}
\right)$ randomized bit operations — a sequence $P\!=\!(p_i)^n_{i=1}$ of consecutive primes and a positive integer $c$ such that\
$\log(c),\log\left(\prod\limits^n_{i=1}p_i\right) =
O(n\log(n)+\log(s/{\varepsilon}))$\
and, with probability $\geq\!1-{\varepsilon}$, the number $p\!:=\!1+c\prod\limits^n_{i=1} p_i$ is prime.
Theorem \[thm:von\] and its proof are inspired in large part by an algorithm of von zur Gathen, Karpinski, and Shparlinski [@von Algorithm following Fact 4.9]. In particular, they used an intricate random sampling technique [@von Thm. 4.10] to show, in our notation, that the enumerative analogue of ${{{{\text{{\tt FEAS}}}}_{{\mathbb{F}}_{\stackrel{\text{\scalebox{.8}[.8]{prime}}}
{\text{\scalebox{.8}[.8]{powers}}}}}}}({\mathbb{Z}}[x_1,x_2])$ is $\#{\mathbf{P}}$-hard [@von Thm. 4.11]. Note in particular that neither of Theorem 4.10 of [@von] or Theorem \[thm:von\] above implies the other.
Our harder upper bound results (Assertions (2) and (3) of Theorem \[thm:qp\]) will follow from an arithmetic analogue of toric deformations. Here, this simply means that we find ways to reduce problems involving general $f\!\in\!{\mathbb{Z}}[x_1]$ to similar problems involving binomials. As a warm-up, let us recall that the convex hull of any subset $S\!\subseteq\!{\mathbb{R}}^2$ is the smallest convex set containing $S$. Also, an edge of a polygon $P\!\subset\!{\mathbb{R}}^2$ is called [**lower**]{} iff it has an inner normal with positive last coordinate, and the [**lower hull**]{} of $P$ is simply the union of all its lower edges.
\[lemma:newt\] (See, e.g., [@robert Ch. 6, sec. 1.6].) Given any polynomial $f(x_1)\!:=\!\sum^m_{i=1}c_ix^{a_i}_1\!\in\!{\mathbb{Z}}[x_1]$, we define its [**$p$-adic Newton polygon**]{}, ${\mathrm{Newt}}_p(f)$, to be the convex hull of the points $\{(a_i,{{\mathrm{ord}}}_p c_i)\; | \; i\!\in\!\{1,\ldots,m\}\}$. Then the number of roots of $f$ in ${\mathbb{C}}_p$ with valuation $v$, counting multiplicities, is [**exactly**]{} the horizontal length of the lower face of ${\mathrm{Newt}}_p(f)$ with inner normal $(v,1)$. [$\blacksquare$]{}
For the polynomial\
$f(x_1)\!:=\!243x^6-3646x^5+18240x^4-35310x^3+29305x^2
-8868x+36$, the polygon ${\mathrm{Newt}}_3(f)$ can easily be verified to resemble the following illustration:\
\
Note in particular that there are exactly $3$ lower edges, and their respective horizontal lengths and inner normals are $2$, $3$, $1$, and $(1,1)$, $(0,1)$, and $(-5,1)$. Lemma \[lemma:newt\] then tells us that $f$ has exactly $6$ roots in ${\mathbb{C}}_3$: $2$ with $3$-adic valuation $1$, $3$ with $3$-adic valuation $0$, and $1$ with $3$-adic valuation $-5$. Indeed, one can check that the roots of $f$ are exactly $6$, $1$, and $\frac{1}{243}$, with respective multiplicities $2$, $3$, and $1$. [$\diamond$]{}
The binomial associated to summing the terms of $f$ corresponding to the vertices of a lower edge of ${\mathrm{Newt}}_p(f)$ containing no other point of the form $(a_i,{{\mathrm{ord}}}_p c_i)$ in its interior is called a [**lower binomial**]{}.
\[lemma:ai\] Suppose $f(x_1)\!=\!c_1+c_2x^{a_2}_1+c_3x^{a_3}_1\!\in\!{\mathbb{Z}}[x]$, the points $\{(0,{{\mathrm{ord}}}_p(c_1)),(a_2,{{\mathrm{ord}}}_p(c_2)),(a_3,{{\mathrm{ord}}}_p(c_3))\}$ are-collinear, and $p$ is a prime [**not**]{} dividing $a_2$, $a_3$, or $a_3-a_2$. Then the number of roots of $f$ in ${\mathbb{Q}}_p$ is exactly the number of roots of the $p$-adic lower binomials of $f$ in ${\mathbb{Q}}_p$. [$\blacksquare$]{}
Our last lemma follows easily (taking direct limits) from a more general result ([@ai Thm. 4.5]) relating the number of roots of $f$ with the number of roots of its lower binomials over ${\mathbb{Z}}/p^N{\mathbb{Z}}$ for $N$ sufficiently large.
Our main results are proved in Section \[sec:qp\], after the development of some additional theory below.
Background and Ancillary\
Results {#sec:back}
=========================
Our lower bounds will follow from a common chain of reductions, so we will begin by reviewing the fundamental problem from which we reduce. We then show how to efficiently construct random primes $p$ such that $p-1$ has many prime factors in Section \[sub:agp\], and conclude with some quantitative results for transferring complexity results over ${\mathbb{C}}$ to ${\mathbb{Q}}_p$ in Section \[sub:transfer\].
Roots of Unity and NP-Completeness {#sub:cyclo}
----------------------------------
Recall that any Boolean expression of one of the following forms:\
$(\heartsuit)$ $y_i\vee y_j \vee y_k$, $\neg y_i\vee y_j \vee y_k$, $\neg y_i\vee \neg y_j \vee y_k$, $\neg y_i\vee \neg y_j \vee \neg y_k$,\
with $i,j,k\!\in\![3n]$,\
is a ${\mathtt{3CNFSAT}}$ [**clause**]{}. Let us first refine slightly Plaisted’s elegant reduction from ${\mathtt{3CNFSAT}}$ to feasibility testing for univariate polynomial systems over the complex numbers [@plaisted Sec. 3, pp. 127–129].
\[dfn:plai\] Letting $P\!:=\!(p_{1},\ldots,p_{n})$ denote any strictly increasing sequence of primes, let us inductively define a semigroup homomorphism ${{\mathcal{P}}}_P$ — the [**Plaisted morphism with respect to $P$**]{} — from certain Boolean expressions in the variables $y_1,\ldots,y_n$ to ${\mathbb{Z}}[x_1]$, as follows:[^3] (0) $D_P\!:=\!\prod^n_{i=1}p_{i}$, (1) ${{\mathcal{P}}}_P(0)\!:=\!1$, (2) ${{\mathcal{P}}}_P(y_i)\!:=\!x^{D_P/p_{i}}_1-1$, (3) ${{\mathcal{P}}}_P(\neg B):=$ $(x^{D_P}_1-1)/{{\mathcal{P}}}_P(B)$, for any Boolean expression $B$ for which ${{\mathcal{P}}}_P(B)$ has already been defined, (4) ${{\mathcal{P}}}_P(B_1\vee B_2)\!:=\!\mathrm{lcm}({{\mathcal{P}}}_P(B_1),{{\mathcal{P}}}_P(B_2))$, for any Boolean expressions $B_1$ and $B_2$ for which ${{\mathcal{P}}}_P(B_1)$ and ${{\mathcal{P}}}_P(B_2)$ have already been defined. [$\diamond$]{}
\[lemma:plai\] [@plaisted Sec. 3, pp. 127–129] Suppose $P\!=\!(p_i)^n_{k=1}$ is an increasing sequence of primes with $\log(p_{k})\!=\!O(k^\gamma)$ for some constant $\gamma$. Then, for all $n\!\in\!{\mathbb{N}}$ and any clause $C$ of the form $(\heartsuit)$, we have ${\mathrm{size}}({{\mathcal{P}}}_P(C))$ polynomial in $n$. In particular, ${{\mathcal{P}}}_P$ can be evaluated at any such $C$ in time polynomial in $n$. Furthermore, if $K$ is any field possessing $D_P$ distinct ${D_P}{^{\text{\underline{th}}}}$ roots of unity, then a ${\mathtt{3CNFSAT}}$ instance $B(y)\!:=C_1(y)\wedge \cdots \wedge
C_k(y)$ has a satisfying assignment iff the univariate polynomial system $F_B\!:=\!({{\mathcal{P}}}_P(C_1), \ldots,{{\mathcal{P}}}_P(C_k))$ has a root $\zeta\!\in\!K$ satisfying $\zeta^{D_P}-1$. [$\blacksquare$]{}
Plaisted actually proved the special case $K\!=\!{\mathbb{C}}$ of the above lemma, in slightly different language, in [@plaisted]. However, his proof extends verbatim to the more general family of fields detailed above.
Randomization to Avoid Riemann\
Hypotheses {#sub:agp}
-------------------------------
The result below allows us to prove Theorem \[thm:von\] and further tailor Plaisted’s clever reduction to our purposes. We let $\pi(x)$ the number of primes $\leq\!x$, and let $\pi(x;M,1)$ denote the number of primes $\leq\!x$ that are congruent to $1 \; {\mathrm{mod}}\; M$.
(very special case of [@carmichael Thm. 2.1, pg. 712]) There exist $x_0\!>\!0$ and an $\ell\!\in\!{\mathbb{N}}$ such that for each $x\!\geq\!x_0$, there is a subset ${{\mathcal{E}}}(x)\!\subset\!{\mathbb{N}}$ of finite cardinality $\ell$ with the following property: If $M\!\in\!{\mathbb{N}}$ satisfies $M\!\leq\!x^{2/5}$ and $a\not|M$ for all $a\!\in\!{{\mathcal{E}}}(x)$ then $\pi(x;M,1)\!\geq\!\frac{\pi(x)}{2\varphi(M)}$. [$\blacksquare$]{}
For those familiar with [@carmichael Thm. 2.1, pg. 712], the result above follows immediately upon specializing the parameters there as follows:\
$(A,{\varepsilon},\delta,y,a)\!=\!(49/20,1/2,2/245, x,1)$\
(see also [@von Fact 4.9]).
The AGP Theorem enables us to construct random primes from certain arithmetic progressions with high probability. An additional ingredient that will prove useful is the famous recent [**AKS algorithm**]{} for deterministic polynomial-time primality checking [@aks]. Consider now the following algorithm.
\
\[algor:primes\][**Input:**]{} A constant $\delta\!>\!0$, a failure probability ${\varepsilon}\!\in\!(0,1/2)$, a positive integer $n$, and the constants $x_0$ and $\ell$ from the AGP Theorem.\
[**Output:**]{} An increasing sequence $P\!=\!(p_j)^n_{j=1}$ of primes such that $\log p\!=\!O(n\log(n)+\log(1/{\varepsilon}))$ and, with probability $1-{\varepsilon}$, $p\!:=\!1+c\prod^n_{i=1} p_i$ is prime. In particular, the output always gives a true declaration as to the primality of $p$.
[**Description:**]{}
1. [Let $L\!:=\!\lceil 2/{\varepsilon}\rceil\ell$ and compute the first $nL$ primes $p_1, \ldots,$ $p_{nL}$ in increasing order.]{}
2. [Define (but do not compute) $M_j\!:=\!\prod\limits^{jn}_{k=(j-1)n+1} p_k$ for any $j\!\in\!{\mathbb{N}}$. Then compute $M_L$, $M_i$ for a uniformly random $i\!\in\![L]$, and $x\!:=\!\max\left\{x_0,
17,
1+M^{5/2}_L
\right\}$. ]{}
3. [\[1\][Compute $K\!:=\!\lfloor (x-1)/M_i\rfloor$ and $J\!:=\!\lceil 2\log(2/{\varepsilon})\log x\rceil$.]{}]{}
4. [Pick uniformly random $c\!\in\![K]$ until one either has $p\!:=\!1+cM_i$ prime, or one has $J$ such numbers that are each composite (using primality checks via the AKS algorithm along the way). ]{}
5. [If a prime $p$ was found then output\
“[$1+c\prod^{in}_{j=(i-1)n+1}p_j$ is a prime that works!]{}”\
and stop. Otherwise, stop and output\
“[I have failed to find a suitable prime. Please forgive me.]{}” [$\diamond$]{}]{}
\[rem:tran\] In our algorithm above, it suffices to find integer approximations to the underlying logarithms and square-roots. In particular, we restrict to algorithms that can compute the $\log_2 {{\mathcal{L}}}$ most significant bits of $\log {{\mathcal{L}}}$, and the $\frac{1}{2}\log_2 {{\mathcal{L}}}$ most significant bits of $\sqrt{{{\mathcal{L}}}}$, using\
$O((\log {{\mathcal{L}}})(\log \log {{\mathcal{L}}})\log \log \log {{\mathcal{L}}})$\
bit operations. Arithmetic-Geometric Mean Iteration and (suitably tailored) Newton Iteration are algorithms that respectively satisfy our requirements (see, e.g., [@dan] for a detailed description). [$\diamond$]{}
[**Proof of Theorem \[thm:von\]:**]{} It clearly suffices to prove that Algorithm \[algor:primes\] is correct, has a success probability that is at least $1-{\varepsilon}$, and works within\
$O\!\left(\left(\frac{n}{{\varepsilon}}\right)^{\frac{3}{2}+\delta}+
(n\log(n)+\log(1/{\varepsilon}))^{7+\delta}\right)$\
randomized bit operations, for any $\delta\!>\!0$. These assertions are proved directly below. [$\blacksquare$]{}
[**Proving Correctness and the Success Probability Bound for Algorithm \[algor:primes\]:**]{} First observe that $M_1,\ldots,M_{L}$ are relatively prime. So at most $\ell$ of the $M_i$ will be divisible by elements of ${{\mathcal{E}}}(x)$. Note also that $K\!\geq\!1$ and $1+cM_i\!\leq\!1+KM_i\!\leq\!1+((x-1)/M_i)M_i\!=\!x$ for all $i\!\in\![L]$ and $c\!\in\![K]$.
Since $x\!\geq\!x_0$ and $x^{2/5}\!\geq\!(x-1)^{2/5}\!\geq\!\left(M^{5/2}_i\right)^{2/5}\!=\!M_i$ for all $i\!\in\![L]$, the AGP Theorem implies that with probability $\geq 1-\frac{{\varepsilon}}{2}$ (since $i\!\in\![\lceil 2/{\varepsilon}\rceil \ell]$ is uniformly random), the arithmetic progression $\{1+M_i,\ldots,1+KM_i\}$ contains at least $\frac{\pi(x)}{2\varphi(M_i)}\!\geq\!\frac{\pi(x)}{2M_i}$ primes. In which case, the proportion of numbers in $\{1+M_i,\ldots,1+KM_i\}$ that are prime is $\frac{\pi(x)}{2KM_i}\!>\!\frac{\pi(x)}{2+2KM_i}\!>\!
\frac{x/\log x}{2x}\!=\!\frac{1}{2\log x}$, since $\pi(x)\!>\!x/\log x$ for all $x\!\geq\!17$ [@bs Thm. 8.8.1, pg. 233]. So let us now assume that $i$ is fixed and $M_i$ is not divisible by any element of ${{\mathcal{E}}}(x)$.
Recalling the inequality $\left(1-\frac{1}{t}\right)^{ct}\!\leq\!e^{-c}$ (valid for all $c\!\geq\!0$ and $t\!\geq\!1$), we then see that the AGP Theorem implies that the probability of [**not**]{} finding a prime of the form $p\!=\!1+cM_i$ after picking $J$ uniformly random $c\!\in\![K]$ is $\left(1-\frac{1}{2\log x}\right)^J \!\leq\!\left(1-\frac{1}{2\log x}
\right)^{2\log(2/{\varepsilon})\log x}\!\leq\!e^{-\log(2/{\varepsilon}) }\!=\!
\frac{{\varepsilon}}{2}$.
In summary, with probability $\geq\!1-\frac{{\varepsilon}}{2}
-\frac{{\varepsilon}}{2}\!=\!1-{\varepsilon}$, Algorithm \[algor:primes\] picks an $i$ with $M_i$ not divisible by any element of ${{\mathcal{E}}}(x)$ and a $c$ such that $p\!:=\!1+cM_i$ is prime. In particular, we clearly have that $\log p \!=\!O(\log(1+KM_i))\!=\!O(n\log(n)+\log(s/{\varepsilon}))$. [$\blacksquare$]{}
[**(Complexity Analysis of Algorithm \[algor:primes\]):**]{} Let $L'\!:=\!nL$ and, for the remainder of our proof, let $p_{i}$ denote the $i{^{\text{\underline{th}}}}$ prime. Since $L'\!\geq\!6$, $p_{L'}\!\leq L'(\log(L') + \log \log L')$ by [@bs Thm. 8.8.4, pg. 233]. Recall that the primes in $[{{\mathcal{L}}}]$ can be listed simply by deleting all multiples of $2$ in $[{{\mathcal{L}}}]$, then deleting all multiples of $3$ in $[{{\mathcal{L}}}]$, and so on until one reaches multiples of $\lfloor \sqrt{{{\mathcal{L}}}}\rfloor$. (This is the classic sieve of Eratosthenes.) Recall also that one can multiply an integer in $[\mu]$ and an integer $[\nu]$ within $O((\log\mu)(\log \log\nu)(\log\log\log \nu)
+(\log\nu)(\log \log\mu) \log\log\log \mu)$ bit operations (see, e.g., [@bs Table 3.1, pg. 43]). So let us define the function $\lambda(a):=(\log\log a)\log\log\log a$.
[**Step 0:**]{} By our preceding observations, it is easily checked that Step 0 takes $O(L'^{3/2}\log^3 L')$ bit operations.
[**Step 1:**]{} This step consists of $n-1$ multiplications of primes with $O(\log L')$ bits (resulting in $M_L$, which has $O(n\log L')$ bits), multiplication of a small power of $M_L$ by a square root of $M_L$, division by an integer with $O(n\log L')$ bits, a constant number of additions of integers of comparable size, and the generation of $O(\log L)$ random bits. Employing Remark \[rem:tran\] along the way, we thus arrive routinely at an estimate of\
$O\left(n^2(\log L')\lambda(L')+\log(1/{\varepsilon})\lambda(1/{\varepsilon}))
\right)$\
for the total number of bit operations needed for Step 1.
[**Step 2:**]{} Similar to our analysis of Step 1, we see that Step 2 has bit complexity\
$O((n\log(L')+\log(1/{\varepsilon}))\lambda(n\log L'))$.
[**Step 3:**]{} This is our most costly step: Here, we require\
$O(\log K)\!=\!O(n\log(L')+\log(1/{\varepsilon}))$\
random bits and $J\!=\!O(\log x)\!=\!O(n\log(L')+\log(1/{\varepsilon}))$ primality tests on integers with $O(\log(1+cM_i))\!=\!O(n\log(L')+\log(1/{\varepsilon}))$ bits. By an improved version of the AKS primality testing algorithm [@aks; @lp] (which takes $O(N^{6+\delta})$ bit operations to test an $N$ bit integer for primality), Step 3 can then clearly be done within\
$O\!\left((n\log(L')+\log(1/{\varepsilon}))^{7+\delta}\right)$\
bit operations, and the generation of $O(n\log(L')+\log(1/{\varepsilon}))$ random bits.
[**Step 4:**]{} This step clearly takes time on the order of the number of output bits, which is just $O(n\log(n)+\log(1/{\varepsilon}))$ as already observed earlier.
[**Conclusion:**]{} We thus see that Step 0 and Step 3 dominate the complexity of our algorithm, and we are left with an overall randomized complexity bound of\
$O\!\left(L'^{3/2}\log^3(L')+ \left(n\log(L')+\log(1/{\varepsilon})\right)^{7+\delta}
\right)$\
$=O\!\left(\left(\frac{n}{{\varepsilon}}\right)^{3/2}\log^3(n/{\varepsilon})
+\left(n\log(n)
+\log(1/{\varepsilon}) \right)^{7+\delta}\right)$\
$=O\!\left(\left(\frac{n}{{\varepsilon}}\right)^{\frac{3}{2}+\delta}+
\left(n\log(n)+\log(1/{\varepsilon})\right)^{7+\delta}
\right)$\
randomized bit operations. [$\blacksquare$]{}
Transferring from Complex Numbers to p-adics {#sub:transfer}
--------------------------------------------
\[prop:sos\] Given any $f_1,\ldots,f_k\!\in\!{\mathbb{Z}}[x_1]$ with maximum coefficient absolute value $H$, let $d_i\!:=\!\deg f_i$ and\
$\tilde{f}(x_1):=
x^{d_1}_1f_1(x_1)f_1(1/x_1)+\cdots+x^{d_k}_1f_k(x_1)f_k(1/x_1)$.\
Then $f_1\!=\cdots=\!f_k\!=\!0$ has a root on the complex unit circle iff $\tilde{f}$ has a root on the complex unit circle. In particular, if $f_i\!\in\!{{\mathcal{F}}}_{1,\mu_i}$ and $\mu_i\!\leq\!m$ for all $i$, then $\tilde{f}\!\in\!{{\mathcal{F}}}_{1,\mu}$ for some $\mu$ with $\mu\!\leq\!((m-1)m+1)k$ and $\tilde{f}$ has maximum coefficient bit-size $O(\log(kmH))$. [$\blacksquare$]{}
Proposition \[prop:sos\] follows easily upon observing that $f_i(x_1)f_i(1/x_1)\!=\!|f_i(x_1)|^2$ for all $i\!\in\![k]$ and any $x_1
\!\in\!{\mathbb{C}}$ with $|x_1|\!=\!1$.
\[lemma:syl\] \[1\][(See, e.g., [@gkz94 Ch. 12, Sec. 1, pp. 397–402].)]{} Suppose $f(x_1)\!=\!a_0+\cdots+a_dx^{d}_1$ and $g(x_1)\!=\!b_0+\cdots+b_{d'}x^{d'}_1$ are polynomials with indeterminate coefficients. Define their [**Sylvester matrix**]{} to be the $(d+d')\times (d+d')$ matrix
\[.8\][${{\mathcal{S}}}_{(d,d')}(f,g)\!:=\!\begin{bmatrix}
a_0 & \cdots & a_d & 0 & \cdots & 0 \\
& \ddots & & & \ddots & \\
0 & \cdots & 0 & a_0 & \cdots & a_d \\
b_0 & \cdots & b_{d'} & 0 & \cdots & 0 \\
& \ddots & & & \ddots & \\
0 & \cdots & 0 & b_0 & \cdots & b_{d'}
\end{bmatrix}
\begin{matrix}
\\
\left. \rule{0cm}{.9cm}\right\}
d' \text{ rows}\\
\left. \rule{0cm}{.9cm}\right\}
d \text{ rows} \\
\\
\end{matrix}$]{}\
\[1\][and their [**Sylvester resultant**]{} to be ${{\mathcal{R}}}_{(d,d')}(f,g)\!:=\!\det {{\mathcal{S}}}_{(d,d')}(f,g)$.]{} Then, assuming $f,g\!\in\!K[x_1]$ for some field $K$ and $a_db_{d'}\!\neq\!0$, we have that $f\!=\!g\!=\!0$ has a root in the algebraic closure of $K$ iff ${{\mathcal{R}}}_{(d,d')}(f,g)\!=\!0$. Finally, if we assume further that $f$ and $g$ have complex coefficients of absolute value $\leq\!H$, and $f$ (resp. $g$) has exactly $m$ (resp. $m'$) monomial terms, then $|{{\mathcal{R}}}_{(d,d')}(f,g)|\!\leq\!
m^{d'/2}m'^{d/2}H^{d+d'}$. [$\blacksquare$]{}
The last part of Lemma \[lemma:syl\] follows easily from Hadamard’s Inequality (see, e.g., ).
\[lemma:red\] Suppose $D\!\in\!{\mathbb{N}}$ and $f\!\in\!{\mathbb{Z}}[x_1]\!\setminus\!\{0\}$ has degree $d$, exactly $m$ monomial terms, and maximum coefficient absolute value $H$. Also let $p$ be any prime congruent to $1$ mod $D$. Then $f$ vanishes at a complex $D{^{\text{\underline{th}}}}$ root of unity $\Longleftrightarrow
f$ vanishes at a $D{^{\text{\underline{th}}}}$ root of unity in ${\mathbb{Q}}_p$. [$\blacksquare$]{}
Note that $x^2_1+x_1+1$ vanishes at a $3{^{\text{\underline{rd}}}}$ root of unity in ${\mathbb{C}}$, but has [**no**]{} roots at all in ${\mathbb{F}}_5$ or ${\mathbb{Q}}_5$. Hence our congruence assumption on $p$ in Lemma \[lemma:red\]. [$\diamond$]{}
[**Proof of Lemma \[lemma:red\]:**]{} First note that by our assumption on $p$, ${\mathbb{Q}}_p$ has $D$ distinct $D{^{\text{\underline{th}}}}$ roots of unity: This follows easily from Hensel’s Lemma (cf. the Appendix) and ${\mathbb{F}}_p$ having $D$ distinct $D{^{\text{\underline{th}}}}$ roots of unity. Since ${\mathbb{Z}}\hookrightarrow{\mathbb{Q}}_p$ and ${\mathbb{Q}}_p$ contains all $D{^{\text{\underline{th}}}}$ roots of unity by construction, the equivalence then follows directly from Lemma \[lemma:syl\]. [$\blacksquare$]{}
A Remark on Natural Density
---------------------------
Let us now introduce the [**$\pmb{{{\mathcal{A}}}}$-discriminant**]{} and clarify how often our $p$-adic speed-ups hold for inputs with bounded coefficients.
Write any $f\!\in\!{\mathbb{C}}[x_1]$ as $f(x_1)\!=\!\sum^m_{i=1}c_ix^{a_i}_1$ with $0\!\leq\!a_1\!<\cdots<\!a_m$. Letting ${{\mathcal{A}}}\!=\!\{a_1,\ldots,a_m\}$, andfollowing the notation of Lemma \[lemma:red\], we then define ${{\mathcal{D}}}_{{\mathcal{A}}}(f)$ to be ${{\mathcal{R}}}_{(a_m-a_1,a_m-a_2)}\left.\left(\frac{f(x_1)}
{x^{a_1}_1},\left.\frac{\partial\left(\frac{f(x_1)}{x^{a_1}_1}\right)}
{\partial x_1}\right/x^{a_2-1}\right)\right/c_m$\
to be the [**$\pmb{{{\mathcal{A}}}}$-discriminant**]{} of $f$ (see also [@gkz94 Ch. 12, pp. 403–408]). Finally, if $c_i\!\neq\!0$ for all $i$, then we call ${\mathrm{Supp}}(f)\!:=\!\{a_1,\ldots,a_m\}$ the [**support**]{} of $f$. [$\diamond$]{}
\[cor:lots\] For any subset ${{\mathcal{A}}}\!\subset\!{\mathbb{N}}\cup\{0\}$ of cardinality $m$, let ${{\mathcal{I}}}_{{\mathcal{A}}}$ denote the family of pairs $(f,p)\!\in\!{\mathbb{Z}}[x_1]\times {{\mathbb{P}}}$ with $f(x)\!=\!\sum^m_{i=1}c_ix^{a_i}_1$ and let ${{\mathcal{I}}}^*_{{\mathcal{A}}}$ denote the subset of ${{\mathcal{I}}}_{{\mathcal{A}}}$ consisting of those pairs $(f,p)$ with $p\not\!|{{\mathcal{D}}}_{{\mathcal{A}}}(f)$. Also let ${{\mathcal{I}}}_{{\mathcal{A}}}(H)$ (resp. ${{\mathcal{I}}}^*_{{\mathcal{A}}}(H)$) denote those pairs $(f,p)$ in ${{\mathcal{I}}}_{{\mathcal{A}}}$ (resp. ${{\mathcal{I}}}^*_{{\mathcal{A}}}$) where $|c_i|\!\leq\!H$ for all $i\!\in\![m]$ and $p\!\leq\!H$. Then $\frac{\#{{\mathcal{I}}}^*_{{\mathcal{A}}}(H)}{\#{{\mathcal{I}}}_{{\mathcal{A}}}(H)}\!\geq\!\left(1-\frac{(2d-1)m}{H}\right)
\left(1-\frac{d\log_2(dmH)}{H}\right)$. [$\blacksquare$]{}
Our corollary above follows easily from our proof of Assertion (3) of Theorem \[thm:qp\] via an application of Lemma \[lemma:syl\] and the Schwartz-Zippel Lemma [@schwartz], and is [**not**]{} used in any of our proofs.
The Proof of Theorem 1.4 {#sec:qp}
========================
[**(Assertion (1): $\pmb{{{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({{\mathcal{F}}}_{1,m}\times{{\mathbb{P}}})\!\in\!{\mathbf{P}}}$ for $m\!\leq\!2$):**]{} First note that the case $m\!\leq\!1$ is trivial: such a univariate $m$-nomial has no roots in ${\mathbb{Q}}_p$ iff it is a nonzero constant. So let us now assume $m\!=\!2$.
Next, we can easily reduce to the special case $f(x)\!:=\!x^d-\alpha$ with $\alpha\!\in\!{\mathbb{Q}}$, since we can divide any input by a suitable monomial term, and arithmetic over ${\mathbb{Q}}$ is doable in polynomial time. The case $\alpha\!=\!0$ always results in the root $0$, so let us also assume $\alpha\!\neq\!0$. Clearly then, any $p$-adic root $\zeta$ of $x^d-\alpha$ satisfies $d{{\mathrm{ord}}}_p\zeta\!=\!{{\mathrm{ord}}}_p\alpha$. Since we can compute ${{\mathrm{ord}}}_p\alpha$ and reductions of integers mod $d$ in polynomial-time [@bs Ch. 5], we can then assume that $d|{{\mathrm{ord}}}_p\alpha$ (for otherwise, $f$ would have no roots over ${\mathbb{Q}}_p$). Replacing $f(x_1)$ by $p^{-{{\mathrm{ord}}}_p\alpha}f(p^{{{\mathrm{ord}}}_p\alpha/d}x_1)$, we can assume further that ${{\mathrm{ord}}}_p\alpha\!=\!{{\mathrm{ord}}}_p\zeta\!=\!0$. In particular, if ${{\mathrm{ord}}}_p\alpha$ was initially a nonzero multiple of $d$, then $\log \alpha\!\geq\!d\log_2 p$. So ${\mathrm{size}}(f)\!\geq\!d$ and our rescaling at worst doubles ${\mathrm{size}}(f)$.
Letting $k\!:=\!{{\mathrm{ord}}}_p d$, note that $f'(x)\!=\!dx^{d-1}$ and thus ${{\mathrm{ord}}}_p f'(\zeta)\!=\!{{\mathrm{ord}}}_p(d)+(d-1){{\mathrm{ord}}}_p\zeta\!=\!k$. So by Hensel’s Lemma (cf. the Appendix), it suffices to decide whether the ${\mathrm{mod}}\ p^\ell$ reduction of $f$ has a root in $({\mathbb{Z}}/p^\ell{\mathbb{Z}})^*$, for $\ell\!=\!1+2k$. Note in particular that ${\mathrm{size}}(p^\ell)\!=\!
O(\log(p){{\mathrm{ord}}}_p d)\!=\!O(\log(p)\log(d)/\log p)\!=\!O(\log d)$ which is linear in our notion of input size. By Lemma \[lemma:qp\] of the Appendix, we can then clearly decide whether $x^d-\alpha$ has a root in $({\mathbb{Z}}/p^\ell{\mathbb{Z}})^*$ within ${\mathbf{P}}$ (via a single fast exponentiation), provided $p^\ell\!\not\in\!\{8,16,32,\ldots\}$.
To dispose of the remaining cases $p^\ell\!\in\!\{8,16,32,\ldots\}$, first note that we can replace $d$ by its reduction mod $2^{\ell-2}$ since every element of $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*$ has order dividing $2^{\ell-2}$, and this reduction can certainly be computed in polynomial-time. Let us then write $d\!=\!2^h d'$ where $2\!\!\not\!|d'$ and $h\!\in\!\{0,
\ldots,\ell-3\}$, and compute $d''\!:=\!1/d'
\ {\mathrm{mod}}\ 2^{\ell-2}$. Clearly then, $x^d-\alpha$ has a root in $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*$ iff $x^{2^h}-\alpha'$ has a root in $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*$, where $\alpha'\!:=\alpha^{d''}$ (since exponentiation by any odd power is an automorphism of $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*$). Note also that $\alpha'$, $d'$, and $d''$ can clearly be computed in polynomial time.
Since $x^{2^h}-\alpha'$ always has a root in $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*$ when $h\!=\!0$, we can then restrict our root search to the cyclic subgroup $\{1,5^2,5^4,5^6,\ldots,5^{2^{\ell-2}-2}\}$ when $h\!\geq\!1$ and $\alpha'$ is a square (since there can be no roots when $h\!\geq\!1$ and $\alpha'$ is not a square). Furthermore, we see that $x^{2^h}-\alpha'$ can have no roots in $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*$ if ${{\mathrm{ord}}}_2\alpha'$ is odd. So, by rescaling $x$, we can assume further that ${{\mathrm{ord}}}_2\alpha'\!=\!0$, and thus that $\alpha'$ is odd. Now an odd $\alpha'$ is a square in $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*$ iff $\alpha'\!\equiv\!1 \; {\mathrm{mod}}\; 8$ [@bs Ex. 38, pg. 192], and this can clearly be checked in ${\mathbf{P}}$. So we can at last decide the existence of a root in ${\mathbb{Q}}_2$ for $x^d-\alpha$ in ${\mathbf{P}}$: Simply combine fast exponentiation with Assertion 3 of Lemma \[lemma:qp\] again, applied to $x^{2^h}-\alpha'$ over the cyclic group $\{1,5^2,5^4,5^6,\ldots,5^{2^{\ell-2}-2}\}$.
[**(Assertion (2): $\pmb{{{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({{\mathcal{F}}}_{1,3}\times{{\mathbb{P}}})\!\in\!{\mathbf{P}}}$ for non-flat $\pmb{{\mathrm{Newt}}_p(f)}$):**]{} First note that $x\!\in\!{\mathbb{Q}}_p\setminus{\mathbb{Z}}_p \Longleftrightarrow \frac{1}{x}\!\in\!p{\mathbb{Z}}_p$. Letting $f^*(x)\!:=\!x^{\deg f}f(1/x)$ denote the reciprocal polynomial of $f$, note that the set of $p$-adic rational roots of $f$ is simply the union of the $p$-adic integer roots of $f$ and the reciprocals of the $p$-adic integer roots of $f^*$. So we need only show we can detect roots in ${\mathbb{Z}}_p$ in ${\mathbf{P}}$.
As stated, Assertion (2) then follows directly from Lemma \[lemma:ai\].
So let us now concentrate on extending polynomiality to some of our exceptional inputs: Writing $f(x)\!=\!c_1+c_2x^{a_2}+c_3x^{a_3}$ as before, let us consider the special case where $f\!\in\!{{\mathcal{F}}}_{1,3}$ has a degenerate root in ${\mathbb{C}}_p$ and $\gcd(a_2,a_3)\!=\!1$. Note that we now allow $p$ to divide any number from$\{a_2,a_3,a_3-a_2\}$. (It is easily checked that the collinearity condition fails for such polynomials since their $p$-adic Newton polygons are line segments.) The $\{0,a_2,a_3\}$-discriminant of $f$ then turns out to be $\Delta:=(a_3-a_2)^{a_3-a_2}a^{a_2}_2c^{a_3}_2-
(-a_3)^{a_3}c^{a_3-a_2}_1c^{a_2}_3$ (see, e.g., [@gkz94 Prop. 1.8, pg. 274]). In particular, while one can certainly evaluate $\Delta$ with a small number of arithmetic operations, the bit-size of $\Delta$ can be quite large. However, we can nevertheless efficiently decide whether $\Delta$ vanishes for integer $c_i$ via [**gcd-free bases**]{} (see, e.g., [@brs Sec. 2.4]). Thus, we can at least check whether $f$ has a degenerate root in ${\mathbb{C}}_p$ in ${\mathbf{P}}$.
Given an $f$ as specified, it is then easily checked that if $\zeta\!\in\!{\mathbb{C}}_p$ is a degenerate root of $f$ then the vector $[c_1,c_2\zeta^{a_2},c_3\zeta^{a_3}]$ must be a right null vector for the matrix $\begin{bmatrix}1 & 1 & 1 \\ 0 & a_2 & a_3
\end{bmatrix}$. In other words, $[c_1,c_2\zeta^{a_2},c_3\zeta^{a_3}]$ is a mutiple of $[\alpha, \beta,\gamma]$ for some integers $\alpha,\beta,\gamma$ with size polynomial in ${\mathrm{size}}(f)$. Via the extended Euclidean algorithm [@bs Sec. 4.3], we can find $A$ and $B$ (also of size polynomial in ${\mathrm{size}}(f)$) with $Aa_2+Ba_3\!=\!1$. So then we obtain that\
$\left(\frac{c_2\zeta^{a_2}}{c_1}\right)^A
\left(\frac{c_3\zeta^{a_3}}
{c_1}\right)^B\!=\!\frac{c^A_2c^B_3}{c^{A+B}_1}\zeta\!=\!
\left(\frac{\beta}{\alpha}\right)^A\left(\frac{\gamma}
{\alpha}\right)^B$.\
In other words, $f$ has a rational root, and thus this particular class of $f$ always has $p$-adic rational roots. [$\blacksquare$]{}
[**(Assertion (3): $\pmb{{{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times {{\mathbb{P}}})\!\in\!{{\mathbf{NP}}}}$ for most inputs):**]{} Just as in our reduction from ${\mathbb{Q}}_p$ to ${\mathbb{Z}}_p$ in the beginning of our last proof, it is enough to show that, for most $f$, roots in ${\mathbb{Z}}_p$ admit succinct certificates. We can also clearly assume that $f$ is not divisible by $x_1$.
Observe now that the $p$-adic valuations of all the roots of $f$ in ${\mathbb{C}}_p$ can be computed in polynomial-time. This is easily seen via two facts: (1) convex hulls of subsets of ${\mathbb{Z}}^2$ can be computed in polynomial-time (see, e.g., [@edelsbrunner]), and (2) the valuation of any root of $f(x)\!=\!\sum^m_{i=1}c_ix^{a_i}$ is a ratio of the form $\frac{{{\mathrm{ord}}}_p(c_i)-{{\mathrm{ord}}}_p(c_j)}{a_j-a_i}$, where $(a_i,{{\mathrm{ord}}}_p(c_i))$ and $(a_j,{{\mathrm{ord}}}_p(c_j))$ are respectively the left and right vertices of a lower edge of ${\mathrm{Newt}}_p(f)$ (cf. Lemma \[lemma:newt\] of the Appendix). Since ${{\mathrm{ord}}}_p(c_i)\!\leq\!\log_p(c_i)\!\leq\!{\mathrm{size}}(c_i)$, note in particular that every root $\zeta\!\in\!{\mathbb{C}}_p$ of $f$ satisfies $|{{\mathrm{ord}}}_p\zeta|\!\leq\!2\max_i{\mathrm{size}}(c_i)\!\leq\!2{\mathrm{size}}(f)\!<\!
2{\mathrm{size}}_p(f)$.
Since ${{\mathrm{ord}}}_p({\mathbb{Z}}_p)\!=\!{\mathbb{N}}\cup\{0\}$, we can clearly assume that ${\mathrm{Newt}}_p(f)$ has an edge with non-positive integral slope, for otherwise $f$ would have no roots in ${\mathbb{Z}}_p$. Letting $a$ denote the smallest nonzero exponent in $f$, $g(x)\!:=\!f'(x)/x^{a-1}$, and $\zeta\!\in\!{\mathbb{Z}}_p$ any $p$-adic integer root of $f$, note then that ${{\mathrm{ord}}}_p f'(\zeta)\!=\!(a-1){{\mathrm{ord}}}_p(\zeta)+{{\mathrm{ord}}}_p g(\zeta)$. Note also that\
${{\mathcal{D}}}_{{\mathcal{A}}}(f)\!=\!{\mathrm{Res}}_{a_m,a_m-a_1}(f,g)$\
so if $p\not\!\!|{{\mathcal{D}}}_{{\mathcal{A}}}(f)$ then $f$ and $g$ have no common roots in the algebraic closure of ${\mathbb{F}}_p$ by Lemma \[lemma:syl\]. In particular, $p\!\!\not|{{\mathcal{D}}}_{{\mathcal{A}}}(f)\Longrightarrow g(\zeta)\!\not
\equiv\!0 \; {\mathrm{mod}}\; p$; and thus $p\!\!\not\!|{{\mathcal{D}}}_{{\mathcal{A}}}(f,g)\Longrightarrow
{{\mathrm{ord}}}_p f'(\zeta)\!=\!(a-1){{\mathrm{ord}}}_p(\zeta)$. Furthermore, by the convexity of the lower hull of ${\mathrm{Newt}}_p(f)$, it is clear that ${{\mathrm{ord}}}_p(\zeta)\!\leq\!
\frac{{{\mathrm{ord}}}_p c_i -{{\mathrm{ord}}}_p c_0}{a_1}\!\leq\!
\frac{2\max_i \log_p|c_i|}{a_1}$. So $p\not\!|{{\mathcal{D}}}_{{\mathcal{A}}}(f)\Longrightarrow
{{\mathrm{ord}}}_p f'(\zeta)\!<\!2{\mathrm{size}}(f)$.
Our fraction of inputs admitting a succinct certificate will then correspond precisely to those $(f,p)$ such that $p\!\!\not\!|{{\mathcal{D}}}_{{\mathcal{A}}}(f)$. In particular, let us define $E$ to be the union of all pairs $(f,p)$ such that $p|{{\mathcal{D}}}_{{\mathcal{A}}}(f)$, as ${{\mathcal{A}}}$ ranges over all finite subsets of ${\mathbb{N}}\cup\{0\}$. It is then easily checked that $E$ is a countable union of hypersurfaces.
Fix $\ell\!=\!4{\mathrm{size}}(f)$. Clearly then, by Hensel’s Lemma, for any $(f,p)\!\in\!({\mathbb{Z}}[x_1]\times {{\mathbb{P}}})\setminus E$, $f$ has a root $\zeta\!\in\!{\mathbb{Z}}_p \Longleftrightarrow f$ has a root $\zeta_0\!\in\!{\mathbb{Z}}/p^\ell{\mathbb{Z}}$. Since $\log(p^\ell)\!=\!O({\mathrm{size}}(f)\log p)\!=\!O({\mathrm{size}}_p(f)^2)$, and since arithmetic in ${\mathbb{Z}}/p^\ell{\mathbb{Z}}$ can be done in time polynomial in $\log(p^\ell)$ [@bs Ch. 5], we have thus at last found our desired certificate: a root $\zeta_0\!\in\!({\mathbb{Z}}/p^\ell{\mathbb{Z}})^*$ of $f$ with $\ell\!=\!4{\mathrm{size}}(f)$.
To conclude, the assertion on checking whether trinomial inputs lie in $E$ follows immediately from our earlier observations on deciding the vanishing of $\Delta$. In particular, instead of applying gcd-free bases, we can instead simply use recursive squaring and efficient ${\mathbb{F}}_p$-arithmetic. [$\blacksquare$]{}
[**(Assertion (4): $\pmb{{{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times {{\mathbb{P}}})}$ is ${{\mathbf{NP}}}$-hard under ${{\mathbf{ZPP}}}$-reductions):**]{} We will prove a (${{\mathbf{ZPP}}}$) randomized polynomial-time reduction from ${\mathtt{3CNFSAT}}$ to ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times{{\mathbb{P}}})$, making use of the intermediate input families $\{({\mathbb{Z}}[x_1])^k\; | \; k\!\in\!{\mathbb{N}}\}$ and ${\mathbb{Z}}[x_1]\times\{x^D_1-1\;
| \; D\!\in\!{\mathbb{N}}\}$ along the way.
Toward this end, suppose $B(y)\!:=\!C_1(y)\wedge\cdots\wedge C_k(y)$ is any ${\mathtt{3CNFSAT}}$ instance. The polynomial system $({{\mathcal{P}}}_P(C_1),\ldots,$${{\mathcal{P}}}_P(C_k))$, for $P$ the first $n$ primes (employing Lemma \[lemma:plai\]), then clearly yields the implication${{\text{{\tt FEAS}}}}_{\mathbb{C}}(\{({\mathbb{Z}}[x_1])^k\; | \;
k\!\in\!{\mathbb{N}}\})\!\in\!{\mathbf{P}}\Longrightarrow {\mathbf{P}}\!=\!{{\mathbf{NP}}}$. Composing this reduction with Proposition \[prop:sos\], we then immediately obtain the implication ${{\text{{\tt FEAS}}}}_{\mathbb{C}}({\mathbb{Z}}[x_1]\times\{x^D_1-1\; | \;
D\!\in\!{\mathbb{N}}\})\!\in\!{\mathbf{P}}\Longrightarrow {\mathbf{P}}\!=\!{{\mathbf{NP}}}$.
At this point, we need only find a means of transferring from ${\mathbb{C}}$ to ${\mathbb{Q}}_p$. This we do by preceding our reductions above by a judicious (possibly new) choice of $P$. In particular, by applying Theorem \[thm:von\] with ${\varepsilon}\!=\!1/3$ (cf. Lemma \[lemma:red\]) we immediately obtain the implication ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}(({\mathbb{Z}}[x_1]\times\{x^D_1-1\; | \;
D\!\in\!{\mathbb{N}}\})\times {{\mathbb{P}}})\!\in\!{{\mathbf{ZPP}}}\Longrightarrow {{\mathbf{NP}}}\!\subseteq\!{{\mathbf{ZPP}}}$.
To conclude, observe that any root $(x,y)\!\in\!{\mathbb{Q}}^2_p\setminus\{(0,0)\}$ of the quadratic form $x^2-py^2$ must satisfy $2{{\mathrm{ord}}}_p x\!=\!1+2{{\mathrm{ord}}}_p y$ — an impossibility. Thus the only $p$-adic rational root of $x^2-p y^2$ is $(0,0)$ and we easily obtain a polynomial-time reduction from ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}(({\mathbb{Z}}[x_1]\times\{x^D_1-1\; | \;
D\!\in\!{\mathbb{N}}\})\times{{\mathbb{P}}})$ to ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times{{\mathbb{P}}})$: simply map any instance $(f(x_1),x^D_1-1,p)$ of the former problem to $(f(x_1)^2-(x^D_1-1)^2p,p)$. So we are done. [$\blacksquare$]{}
[**(Assertion (5): $\pmb{{{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times {{\mathbb{P}}})}$ is ${{\mathbf{NP}}}$-hard, assuming Wagstaff’s Conjecture):**]{} If we also have the truth of the Wagstaff Conjecture then we simply repeat our last proof, replacing our AGP Theorem-based algorithm with a simple brute-force search. This maintains polynomial complexity, but with the added advantage of completely avoiding randomization. [$\blacksquare$]{}
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank David Alan Plaisted for his kind encouragement, and Eric Bach, Sidney W. Graham, and Igor Shparlinski for many helpful comments on primes in arithmetic progression. We also thank Matt Papanikolas for valuable $p$-adic discussions. Finally, we thank an anonymous referee for insightful comments that greatly helped clarify our presentation.
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Appendix: Additional Background
===============================
Let us first recall briefly the following complexity classes (see also [@papa] for an excellent textbook treatment):
- [ The family of decision problems which can be done within time polynomial in the input size.[^4]]{}
- [ The family of decision problems admitting a randomized polynomial-time algorithm giving a correct answer, or a report of failure, the latter occuring with probability $\leq\!\frac{1}{2}$. ]{}
- [ The family of decision problems where a “[Yes]{}” answer can be [**certified**]{} within time polynomial in the input size.]{}
- [ The family of decision problems solvable within time exponential in the input size.]{}
The classical Hensel’s Lemma can be phrased as follows.
\[lemma:hensel\] [@robert Pg. 48] Suppose $f\!\in\!{\mathbb{Z}}_p[x_1]$ and $\zeta_0\!\in\!{\mathbb{Z}}_p$ satisfies $f(\zeta_0)\!\equiv\!0 \ ({\mathrm{mod}}\ p^\ell)$ and ${{\mathrm{ord}}}_p f'(\zeta_0)\!<\!\frac{\ell}{2}$. Then there is a root $\zeta\!\in\!{\mathbb{Z}}_p$ of $f$ with $\zeta\!\equiv\!\zeta_0 \
({\mathrm{mod}}\ p^{\ell-{{\mathrm{ord}}}_p f'(\zeta_0)})$ and ${{\mathrm{ord}}}_p f'(\zeta)\!=\!{{\mathrm{ord}}}_p
f'(\zeta_0)$. [$\blacksquare$]{}
The final tool we will need is a standard lemma on binomial equations over certain finite groups. Recall that for any ring $R$, we denote its unit group by $R^*$.
\[lemma:qp\] (See, e.g., [@bs Thm. 5.7.2 & Thm. 5.6.2, pg. 109]) Given any cyclic group $G$, $a\!\in\!G$, and an integer $d$, the following 3 conditions are equivalent:\
1. the equation $x^d\!=\!a$ has a solution $a\!\in\!G$.\
2. the order of $a$ divides $\frac{\#G}{\gcd(d,\#G)}$.\
3. $a^{\#G/\gcd(d,\#G)}\!=\!1$.\
Also, ${\mathbb{F}}^*_q$ is cyclic for any prime power $q$, and $({\mathbb{Z}}/p^\ell{\mathbb{Z}})^*$ is cyclic for any $(p,\ell)$ with $p$ an odd prime or $\ell\!\leq\!2$. Finally, for $\ell\!\geq\!3$, $({\mathbb{Z}}/2^\ell{\mathbb{Z}})^*\!=\!
\{\pm 1,\pm 5,\pm 5^2,\pm 5^3,\ldots,\pm 5^{2^{\ell-2}-1} \ {\mathrm{mod}}\ 2^\ell\}$. [$\blacksquare$]{}
[^1]: If $F(x_1,\ldots,x_n)\!=\!0$ is any polynomial equation and $Z_K$ is its zero set in $K^n$, then the Hasse Principle is the assumption that \[$Z_{\mathbb{C}}$ smooth, $Z_{\mathbb{R}}\!\neq\!\emptyset$, and $Z_{{\mathbb{Q}}_p}\!\neq\!\emptyset$ for all primes $p$\] implies $Z_{\mathbb{Q}}\!\neq\!\emptyset$ as well. The Hasse Principle is a theorem when $Z_{\mathbb{C}}$ is a quadric hypersurface or a curve of genus zero, but fails in subtle ways already for curves of genus one (see, e.g., [@bjornhasse1]).
[^2]: An earlier result claiming ${{{{\text{{\tt FEAS}}}}_{{\mathbb{Q}}_\mathrm{primes}}}}({\mathbb{Z}}[x_1]\times {{\mathbb{P}}})\!\in\!{{\mathbf{NP}}}$ for “most” inputs [@myqua Main Thm.] appears to have fatal errors in its proof.
[^3]: Throughout this paper, for Boolean expressions, we will always identify $0$ with “[False]{}” and $1$ with “[True]{}”.
[^4]: Note that the underlying polynomial depends only on the problem in question (e.g., matrix inversion, shortest path finding, primality detection) and not the particular instance of the problem.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'E.Tremou , M. Garcia-Marin, J. Zuther, A. Eckart, M. Valencia-Schneider, M. Vitale , C. Shan'
bibliography:
- 'llqso.bib'
date: 'Received Month xx, 201x; accepted xxx x, 201x'
subtitle: 'III. Optical spectroscopic properties and activity classification'
title: 'A low-luminosity type-1 QSO sample'
---
Introduction {#sec:intro}
============
Accretion of matter onto a supermassive black hole (SMBH) at the center of a galaxy is the main energy source of galaxies hosting an active galactic nucleus (AGN). In a similar vein, the centers of starburst galaxies are not considered to be very active in terms of nonthermal emission arising from nuclear accretion and star formation processes originate the energy output instead. However, the association between the AGN activity and the star formation mechanism is still undetermined in galaxy evolution scenarios. Hence, reliable classification frames are vital to establish the activity of the galaxies.
Classification of AGN depends upon many parameters. Various studies focusing on selection criteria, morphology, and line widths, have produced a variety of classification schemes . The emission line spectra of extragalactic sources has proven to be a reliable approach to diagnose the origin of the ionizing emission in a galaxy. In particular, the information contained in the relative intensities of the emission lines in the visible domain have been used by @baldwin1981 [BPT diagrams], [@1987ApJS...63..295V], and more recently by [@2006MNRAS.372..961K]. The main idea is to discriminate between the different excitation mechanisms operating on the line emitting gas. Depending on the contribution of the AGN, the galaxies can be categorized as quasi-stellar objects (QSOs), from the high-power tail of the distribution in BPT diagrams down to Seyferts and low-ionization nuclear wmission line eegions [LINERs; @1980BAAS...12..809H].
The nature of LINERs has been a long debate, with several explanations being offered to account for it. Ionization by shocks was one of the first, [@Heckman1980], with young hot stars being responsible for it [@Terlevich1985; @Dopita1995]. Pre-main sequence stars ionization [@Cid2004] have also been proposed as ionization sources. Ionization by low-luminosity AGNs is a favored explanation [@Ferland1983; @Halpern1983; @Ho1997], in which case they would constitute the main fraction of the AGN population. More recently, using radial emission-line surface brightness profiles, @Singh2013 found that the class of LINER galaxies are not generally uniquely powered by a central AGN. They postulate that the excess LINER-like emission is ionization by evolved stars during the short, but very hot and energetic phase known as post-AGB.
Starburst galaxies are mainly ionized by hot stars [@1977ApJ...217..928H; @1992ApJ...397L..79F; @1992ApJ...399L..27S; @1998AJ....116...55M; @2000PASP..112..753B]. One main feature of the optical spectra in these sources is the presence of emission lines. Narrow permitted and forbidden emission lines (300-1000 km/s width) originating in the narrow line region (NLR) and especially broad lines (2000-6000 km/s width) originating in the broad line region (BLR) are considered to be an unambiguous indication of an AGN [@oster1987]. Some objects, such as QSOs and Seyfert 1 galaxies, show both types of lines. Seyfert 2s and LINERs show only narrow line emission.
Activity classification and diagnostic schemes relying on optical spectroscopic observations need to be handled carefully. The extended emission of the host galaxy may contaminate the point source nuclear spectral light. Early studies discussed the observational effects on early-type galaxies, which appear to be redder in their centers [@1961ApJS....5..233D; @1963AJ.....68..237H] . The color-aperture relation in early-type spiral galaxies has been shown to depend on the redshift and the size of the aperture . The H$\alpha$ emission line is widely used for estimations of star formation rates (SFR, e.g., @Kennicutt1983) and more recent analyses have dealt with the impact of aperture, which can lead to substantial fractions of the emission-line flux loss [@1995AJ....110.1602Z; @2003ApJ...591..827P]. Furthermore, [@2003ApJ...599..971H] concluded that the H$\alpha$ SFRs of SDSS samples are overestimated for nearby or most massive galaxies. The comparison with the radio SFRs led to a large deviation of the H$\alpha$ SFRs and the quantification of aperture effect was challenging. [@2005PASP..117..227K] investigated the effect of aperture on metalicity, extinction, and star formation rate by studying the integrated and nuclear spectra of a sample of galaxies as a function of both galaxy type and luminosity. They found that for flux covering fractions $<$ 20% of the galaxy light, the difference between the nuclear and global metalicity, extinction and star formation rate is substantial.
The influence of the atmospheric seeing effects on measurements of the \[O\]/H$\beta$ flux ratio and, in particular, for the case of the NLR of Seyfert galaxies is discussed in a study by [@1983ApJ...270...71P], using models of the surface brightness distribution. Their model resulted in a systematic effect of the \[O\]/H$\beta$ flux ratio at the 25% level for apertures close to the size of the NLR (usual case for the near Seyfert 1 galaxies). show the importance of the seeing effects in variability studies of low-luminosity Seyfert 1 galaxies by simulating the seeing variations. Since the BLR and the continuum source are not resolved, the seeing can drive the uncertainties on the measured ratio of the narrow line flux to BLR.
Recently, @2014MNRAS.441.2296M calculated the mean dispersion for the diagnostic line ratios used in the standard BPT diagram with respect to the central aperture of central extraction to obtain an estimate of the uncertainties resulting from aperture effects. They found that the starlight subtraction does not significantly change the effect of a different placement in the BPT diagrams, which results from a fiber and standard slit observational methods.
The main purpose of this paper is to characterize a sample of LLQSOs via the analysis of their optical spectra. We aim to classify their nuclear activity and to provide a more in-depth study of the aperture effect using a unique sample of low-luminosity AGN in the local universe. The sample members are among the closest AGN and have been studied at high angular resolution and with a variety of methods . In light of the importance and rarity of this sample, a complete and robust characterization of these galaxies is a crucial first step toward securing its full scientific return in terms of future studies with current and next generation observing facilities (e.g., the Atacama large millimeter array - ALMA, the square kilometer arry - SKA, the James Webb telescope - JWST).
This paper is structured as follows. In Section \[sample\], we introduce the low-luminosity QSO sample and its selection criteria. The data we used for this study are presented in Section \[obser\]. The spectroscopic analysis that we followed is described in Section \[anal\]. Section \[fitres\] presents the fitting results and draws some general sample characteristics based on them. The activity classification scheme of the sample is shown in Section \[class\], also comparing the results obtained using different observational techniques. Moreover, we compare the activity classification for both fiber and long-slit methods. Finally, we discuss the impact of the aperture effect in Section \[discussion\] and summarize in Section \[concl\].
The low-luminosity QSO sample {#sample}
=============================
Our low-luminosity QSO sample is drawn from the Hamburg/ESO QSO survey , a wide angle ($\approx$9500 deg$^2$) Southern Hemisphere survey for optically bright QSOs. The HES selects Type 1 AGNs up to a redshift of about 3.2, and has a brightness limit of $B_{J} \lesssim 17.3$. Because of variations on the observed field, this brightness limit has a dispersion of $\sim$ 0.5 mag.
Classic QSO detection techniques [e.g., @1983ApJ...269..352S] tend to miss luminous AGN, especially at low redshift. This introduces a distance-dependent incompleteness, which is not uncommon in QSOs surveys. Comparatively, the method used by the HES facilitates the inclusion of bright extended objects, which are located at the faint end of the distribution (17 $\lesssim B_{J} \lesssim$18). We selected a subsample of 99 objects from the HES (see Table \[tab:LLQSOSample\]). The main selection criteria was that they should all have a small cosmological distance, $z \leq 0.06$, thus ensuring the presence of the stellar CO(2-0) band head in the near-infrared (NIR) $K$-band (e.g., ).
![Absolute $B_J$ magnitude distribution of the LLQSOs sample.[]{data-label="fig:magn"}](Histogram_mag.ps){width="\columnwidth"}
Figure \[fig:magn\] presents the absolute $B_J$ magnitude distribution of the sample. Absolute magnitudes, $M(B_J)$, were estimated from the foreground extinction corrected apparent magnitudes and assuming an AGN power-law, $F_\nu\propto \nu^{-\alpha}$ with index $\alpha=0.5$ [@2001AJ....122..549V]. The absolute magnitude covers the range $-23\lesssim M(B_J)\lesssim -16$.
As shown in Fig\[fig:redshift\], following the commonly used demarcation between QSOs and Seyfert1 $M_{B} = −21.5 + 5log h_{0}$ our subsample falls into the Seyfert 1 region, slightly under luminous to be classified as QSOs. We then chose to name the objects presented here as low-luminosity QSOs (LLQSOs).
[cllccl]{}
\
------------------------------------------------------------------------
[ID]{} & [HES name]{} &[6dFGS name]{} & [RA \[deg\]]{} & [DEC \[deg\]]{} & [redshift]{}\
\
------------------------------------------------------------------------
[ID]{} & [HES name]{} &[6dFGS name]{} & [RA \[deg\]]{} & [DEC \[deg\]]{} & [redshift]{}\
1 & HE0003-5023 & & 1.42917 & -50.11530 &0.0334\
2 & HE0021-1810 & & 5.91417 & -17.89810 &0.0535\
3 & HE0021-1819 &g0023554-180251 & 5.98042 & -18.04720 &0.0532\
4 & HE0022-4546[$^\ddagger$]{} &g0025013-452955 & 6.25500 & -45.49830 &0.056\
5 & HE0036-5133[$^\ddagger$]{} &g0039159-511702 & 9.81583 & -51.28390 &0.0288\
6 & HE0038-0758 & & 10.21960 & -7.70278 &0.054\
7 & HE0040-1105 &g0042369-104922 & 10.65330 & -10.82250 &0.042\
8 & HE0045-2145 &g0047413-212927 & 11.92210 & -21.49080 &0.0214\
9 & HE0051-2420 &g0053544-240437 & 13.47670 & -24.07670 &0.056\
10 & HE0103-3447 & & 16.44420 & -34.52920 &0.057\
11 & HE0103-5842 & & 16.32080 & -58.43780 &0.0257\
12 & HE0108-1631 &g0111143-161555 & 17.80920 & -16.26500 &0.052\
13 & HE0108-4743[$^\ddagger$]{} &g0111097-472735 & 17.79040 & -47.46000 &0.0239\
14 & HE0111-1506[$^\ddagger$]{} &g0113499-145057 & 18.45750 & -14.84920 &0.0527\
15 & HE0114-0015 & & 19.26500 & 0.00750 &0.0456\
16 & HE0119-0118 & & 20.49920 & -1.04028 &0.0547\
17 & HE0122-5137 & & 21.25080 & -51.36580 &0.052\
18 & HE0125-1904 & & 22.02790 & -18.80860 &0.043\
19 & HE0126-0753 &g0129067-073830 & 22.27750 & -7.64167 &0.056\
20 & HE0149-3626 &g0151419-361116 & 27.92460 & -36.18780 &0.0335\
21 & HE0150-0344 & & 28.25580 & -3.49000 &0.0478\
22 & HE0203-0031 &g0206160-001729 & 31.56620 & -0.29139 &0.0424\
23 & HE0212-0059 &g0214336-004600 & 33.64000 & -0.76667 &0.0264\
24 & HE0224-2834 &g0226257-282059 & 36.60710 & -28.34970 &0.0605\
25 & HE0227-0913[$^\ddagger$]{} &g0230055-085953 & 37.52250 & -8.99806 &0.0164\
26 & HE0232-0900 &g0234378-084716 & 38.65710 & -8.78778 &0.043\
27 & HE0236-3101 & & 39.68790 & -30.80670 &0.062\
28 & HE0236-5224 & & 39.58210 & -52.19220 &0.045\
29 & HE0253-1641 &g0256027-162916 & 44.01080 & -16.48780 &0.032\
30 & HE0257-2434 &g0259305-242254 & 44.87710 & -24.38170 &0.035\
31 & HE0323-4204 & & 51.25920 & -41.90500 &0.058\
32 & HE0330-1404 &g0333078-135433 & 53.28210 & -13.90940 &0.04\
33 & HE0332-1523 &g0334245-151340 & 53.60210 & -15.22810 &0.035\
34 & HE0336-5545 & & 54.52620 & -55.60000 &0.059\
35 & HE0342-2657[$^\ddagger$]{} &g0345032-264820 & 56.26330 & -26.80530 &0.058\
36 & HE0343-3943[$^\ddagger$]{} &g0345125-393429 & 56.30210 & -39.57500 &0.0431\
37 & HE0345+0056 & & 56.91750 & 1.08722 &0.031\
38 & HE0349-4036 &g0351417-402759 & 57.92330 & -40.46640 &0.0582\
39 & HE0351+0240 & & 58.53920 & 2.82500 &0.034\
40 & HE0358-3713 &g0400407-370506 & 60.16960 & -37.08530 &0.051\
41 & HE0359-3841 &g0401462-383320 & 60.44250 & -38.55580 &0.059\
42 & HE0403-3719 &g0405017-371115 & 61.25670 & -37.18780 &0.0552\
43 & HE0412-0803 &g0414527-075540 & 63.71920 & -7.92806 &0.0379\
44 & HE0429-0247 &g0431371-024124 & 67.90420 & -2.69028 &0.041\
45 & HE0429-5343 & & 67.66670 & -53.61560 &0.04\
46 & HE0433-1028 &g0436223-102234 & 69.09250 & -10.37580 &0.0355\
47 & HE0433-1150[^1] & & 68.88540 & -11.74030 &0.058\
48 & HE0436-4717 & & 69.36710 & -47.19140 &0.053\
49 & HE0439-0832 & & 70.47500 & -8.44306 &0.045\
50 & HE0444-0513 & & 71.83580 & -5.13722 &0.0442\
51 & HE0447-0404 &g0450251-035903 & 72.60420 & -3.98389 &0.022\
52 & HE0521-3630[$^\ddagger$]{} &g0522580-362731 & 80.74170 & -36.45890 &0.0553\
53 & HE0535-4224 &g0537331-422230 & 84.38750 & -42.37500 &0.035\
54 & HE0608-5606 &g0609175-560658 & 92.32330 & -56.11610 &0.0318\
55 & HE0853-0126 &g0856178-013807 & 134.07401 & -1.63528 &0.0597\
56 & HE0853+0102 & & 133.97600 & 0.85278 &0.052\
57 & HE0934+0119[^2] & & 144.25400 & 1.09528 &0.0503\
58 & HE0949-0122 &g0952191-013644 & 148.07899 & -1.61222 &0.0197\
59 & HE1011-0403 &g1014207-041841 & 153.58600 & -4.31139 &0.0586\
60 & HE1013-1947 & & 153.98500 & -20.04080 &0.0547\
61 & HE1017-0305 & & 154.88699 & -3.33750 &0.0492\
62 & HE1029-1831[^3] &g1031573-184633 & 157.98900 & -18.77610 &0.0404\
63 & HE1107-0813 & & 167.45200 & -8.50417 &0.0583\
64 & HE1108-2813 &g1110480-283004 & 167.70000 & -28.50080 &0.024\
65 & HE1126-0407 & & 172.31900 & -4.40222 &0.0601\
66 & HE1136-2304 &g1138510-232135 & 174.71300 & -23.36000 &0.027\
67 & HE1143-1810 & & 176.41901 & -18.45470 &0.0329\
68 & HE1237-0504[$^\ddagger$]{} & & 189.91400 & -5.34444 &0.0084\
69 & HE1248-1356 &g1251324-141316 & 192.88499 & -14.22140 &0.0145\
70 & HE1256-1805[$^\ddagger$]{} & & 194.67900 & -18.36000 &0.014\
71 & HE1310-1051 &g1313058-110742 & 198.27400 & -11.12830 &0.034\
72 & HE1319-3048 &g1321582-310426 & 200.49200 & -31.07360 &0.0448\
73 & HE1328-2508 &g1331138-252410 & 202.80800 & -25.40280 &0.026\
74 & HE1330-1013 &g1332391-102853 & 203.16299 & -10.48140 &0.0225\
75 & HE1338-1423 & & 205.30400 & -14.64440 &0.0418\
76 & HE1346-3003 &g1349193-301834 & 207.33000 & -30.30970 &0.0161\
77 & HE1348-1758 &g1351295-181347 & 207.87300 & -18.22970 &0.012\
78 & HE1353-1917 &g1356367-193145 & 209.15300 & -19.52890 &0.0349\
79 & HE1417-0909 & & 215.02600 & -9.38694 &0.044\
80 & HE2112-5926 & & 318.96500 & -59.23170 &0.0317\
81 & HE2128-0221 &g2130499-020814 & 322.70801 & -2.13750 &0.0528\
82 & HE2129-3356 &g2132022-334254 & 323.00900 & -33.71500 &0.0293\
83 & HE2204-3249 &g2207450-323502 & 331.93799 & -32.58390 &0.0594\
84 & HE2211-3903[$^\ddagger$]{} &g2214420-384823 & 333.67499 & -38.80670 &0.0398\
85 & HE2221-0221[$^\ddagger$]{} & & 335.95700 & -2.10361 &0.057\
86 & HE2222-0026 &g2224353-001104 & 336.14700 & -0.18444 &0.0581\
87 & HE2231-3722 &g2234409-370644 & 338.67099 & -37.11220 &0.043\
88 & HE2233+0124 & & 338.92499 & 1.65917 &0.0564\
89 & HE2236-3621 & & 339.77200 & -36.09810 &0.06\
90 & HE2251-3316 &g2253587-330014 & 343.49399 & -33.00360 &0.056\
91 & HE2254-3712 &g2257390-365607 & 344.41199 & -36.93530 &0.038\
92 & HE2301-3517 & & 346.15500 & -35.02000 &0.04\
93 & HE2302-0857 & & 346.18100 & -8.68583 &0.0471\
94 & HE2306-3246 &g2309192-322958 & 347.32999 & -32.49940 &0.052\
95 & HE2322-3843 & & 351.35101 & -38.44690 &0.0359\
96 & HE2323-6122 &g2326376-610602 & 351.65701 & -61.10030 &0.0413\
97 & HE2337-2649 &g2340321-263319 & 355.13300 & -26.55530 &0.0496\
98 & HE2343-5235 & & 356.43500 & -52.31000 &0.035\
99 & HE2354-3044 & & 359.36700 & -30.46110 &0.0307\
The LLQSO sample has been studied intensively at multiple wavelengths with both photometry and spectroscopy. For all sample members accessible from the northern hemisphere, we obtained millimeter measurements of CO(1-0) and CO(2-1) [39 out of 99 objects @bertram2007]. For 27 of the CO detected objects, we also obtained 21 cm HI measurements [@konig2009]. Both observations show that LLQSO hosts are rich in cold molecular/atomic gas ($M_{\rm H_2}\approx 5\times 10^9
M_\odot$, $M_{\rm HI}\approx 10^{10}M_\odot$) and that the molecular gas is concentrated in the central kpc (; @Moser2012). In the near-infrared (NIR), so far we studied nine LLQSOs with the VLT. For eight of those, we obtained ISAAC long-slit $K$-band spectra and for the ninth, we obtained adaptive-optics-assisted $H+K$ imaging spectroscopy with SINFONI [@fischer_lrs_2008]. The current NIR data reveal predominantly late-type hosts with a high incidence of bars. The AGN-subtracted colors in the NIR are typical for nonactive, late-type galaxies.
More recently, @Busch2013B studied a subsample of 20 of our LLQSOs, performing aperture photometry and a decomposition into bulge, disk, bar, and bar components in the NIR. In good agreement with @fischer_lrs_2008, the analysis reveals that 50% of hosts are disk galaxies, 86% of them barred. The study also reveals stellar and black hole masses lower than those typical for brighter QSOs. Also, these LLQSOs do not follow the M$_{BH}$-L$_{bulge}$ relation for inactive galaxies.
Data used {#obser}
=========
The optical spectroscopic analysis of the LLQSO sample is based on two sets of observations. The first data set is from the HES [@reimers1996], in which follow-up observations were carried out to confirm the type-1 character of the survey candidates spectroscopically. In addition to this, we have used data from the 6 Degree Field Galaxy survey (6dFGS), an optical spectroscopy public database [@jones2004; @jones2009]. The 6df data generally offers a better spectral resolution, and combined with the HES allows us to discuss aspects like aperture effect and its impact in the results. The redshift distribution of both data sets is very comparable (see Fig.\[fig:hist\_redshift\]), with no bias of the 6dFGS subsample toward lower or higher redshifts. The HES provides 71 and the 6dFGS 58 spectra of our 99 LLQSO sample (see details in Table\[tab:LLQSOSample\]).
The HES data
------------
The HES used three ESO telescopes (3.6 m, 2.2 m, 1.52 m) to obtain long-slit, low-resolution spectra ($R\approx
165- 938$, i.e ${\rm FWHM}\approx 1800-320\,{\rm km}\,{\rm s}^{-1}$) during five observing campaigns between 1990 and 1994. Depending on the seeing conditions, the slit width varied between $1\farcs 5$ and $2\farcs 5$, while the slit position angle was always east-west. A detailed description of the instrument setup and data reduction is given in [@reimers1996].
We optimally extracted spectra with an algorithm similar to that of @1986PASP...98..609H. Leaving only the amplitudes as free parameters, the extraction procedure minimizes contamination from nearby sources, while at the same time the extraction windows (and subsequent photometry) are limited to apertures the size of the seeing disk. We applied corrections for atmospheric extinction using a standard La Silla extinction curve [@1995Msngr..80...34B]. The flux calibration accuracy, monitored by standard star spectra, is in many cases better than $\lesssim 20$%, but because of instrumental limitations, data should not be considered spectrophotometric [@reimers1996] However, since we are using flux ratios, the lack of spectrophotometrically calibrated data does not affect the results.
The archive 6dFGS data
----------------------
The 6dFGS is designed to measure the redshift and peculiar velocities of a NIR sample of galaxies selected from the Two Micron All Sky Survey (2MASS). In addition, a variety of targets from other surveys were included as filler programs. A detailed description of the survey can be found in [@jones2004]. The instrument used to carry out the 6dFGS was the Six-Degree Field multiobject fiber spectrograph facility (6dF, @Parker1998), located at the 1.2 m Anglo-Australian Observatory’s UK Schmidt Telescope (UKST). It is able to record 150 simultaneous spectra over a 5.7 deg field. Spectra are obtained with a $6\farcs7$ diameter fiber, using two separate V and R gratings, which together give $R\approx 1000$ (corresp. to about ${\rm FWHM}\approx 300\,{\rm km}\,{\rm s}^{-1}$) and cover at least 4000- 7500 Å[.]{} Typical signal-to-noise ratios are $\sim 10$ per pixel. The same fixed average spectral transfer function is assumed for each plate all the time. Differences in the transfer function between individual fibers are corrected for by the flat-fielding. All data are flux calibrated using 6dF observations of the standard stars Feige110 and EG274. See [@jones2004; @jones2009] for full details on the spectra processing. We retrieved the already reduced data for our cross-matched sample from the publicly accessible online database[^4]. We make use of 6dFGS data release 3 (DR3), which is the final redshift release.
Spectral line fitting {#anal}
=====================
The analysis of the optical spectra for both HES and 6dFGS focuses on seven emission lines: H$\alpha$ $\lambda$ 6562Å (hereafter H$\alpha$), H$\beta$ $\lambda$ 4861Å (hereafter H$\beta$), \[N[\]]{} $\lambda$$\lambda$ 6548, 6583Å, \[O[\]]{} $\lambda$$\lambda$ 4959, 5007Å, \[S\] $\lambda$$\lambda$ 6717,6731Å, and \[O\] $\lambda$ 6300Å. These lines were selected to classify the host galaxies using the standard BPT diagnostic diagrams [@baldwin1981].
The basic assumption here is that, since the hosts in the HES sources are usually relatively faint, the stellar contribution to the Balmer-line measurements is, in general, negligible.
There are, however, several cases in which Fe and MgIb 5180 features are clearly seen, so we have assessed their impact in the fitting results. It is not uncommon for Seyfert1 galaxies to show broad and Fe emission lines features in the blue region of the optical spectra [@Dong2011]. This emission surrounds and potentially contaminates the H$\beta$+\[O\] complex, although its presence and strength is not directly related to the lines under study. For the 6dFGS galaxies in our sample, 20% exhibit clear Fe features, whereas 24% show unclear or low-level traces of it. In our HES data, these values are 30% and 53%, respectively. We estimate that, in galaxies with a strong Fe contribution, this may lead to a maximum uncertainty of 15% in the emission lines flux measurements of the H$\beta$ and \[O\] lines. Depending on the case, this would add up an uncertainty of up to 0.13 dex on the O\[\]/H$\beta$ axis of the BPT diagnostic diagrams.
Old stellar populations are responsible for the MgIb absorption feature that can bias the narrow lines fitting results [@Bica1986]. In this case 25% of our 6dFGS and 27% of our HES source clearly exhibit it, whereas 22% and 17% of the studies 6dFGS and HES spectra, respectively, present it at a low level. In this case, we do not aim to study the galaxies’ stellar populations; we hence chose to fit the nebular emission lines using restricted spectral windows, and locally fit the continuum in each case with a first order polynomial fit. When broad absorption features, such as that of MgIb, are present, they would be accounted for as part of the continuum. This approach is not far from the standard SDSS estimate of the stellar continuum using a sliding median, which has been proven to be adequate for strong emission lines. This may be problematic if the aim is to recover weaker stellar or Balmer absorption lines [@Tremonti2004], but they are not the focus of the present paper.
For each spectrum, the total integrated flux, central wavelength, and width of the relevant emission lines have been measured fitting Gaussian functions using the MPFIT IDL software package [^5] [@Markwardt2009]. We fit all lines using narrow components, except for H$\alpha$ and H$\beta$ , which present both narrow and broad (coming from the narrow and broad line regions, respectively). It is well known that, under certain conditions such as line blending, the fitting algorithms may derive different solutions. Some of our HES spectra suffer from line blending, especially in the H$\alpha$+\[N\] complex and \[S\] spectral regions. Therefore we decided to apply some restrictions to increase the reliability of our results.
The H$\beta$ and \[O\] lines were analyzed simultaneously (see Fig.\[fig:fittinghb\]). The H$\beta$ broad component was fitted with no constraints. We assumed the three narrow components have the same width and kinematics. Additionally, we fit the \[O\] lines with two Gaussians with the intensity ratio fixed to the theoretical value of 3 [@2007MNRAS.374.1181D].
Similarly, the H$\alpha$ and \[N\] lines complex was simultaneously fitted (Fig. \[fig:fittingha\]), and the H$\alpha$ broad component was left free. The narrow components were assumed to have the same kinematics and width. For the \[N\] doublet, the intensity ratio was set to 3 as well [@1989agna.book.....O; @1997ApJS..112..391H]. The \[O\] lines (Fig.\[fig:fittingoi\]) were individually fitted with a single component, with no particular constrains. Finally, as the \[S\] line ratio can be used to estimate the electron density of the emitting gas [e.g., @1989agna.book.....O], we only assumed that both lines share the same kinematics (see Fig. \[fig:fittingsii\]).
With the intention of having a more robust result, we tried to impose the same FWHM to all narrow emission lines. This approach did not work, because in many cases, especially for the HES data, strong blending was affecting the H$\alpha$+\[N\] complex. This made it very difficult for the fit to converge, especially considering that in most cases an extra broad component was needed to fit H$\alpha$. Because of this and the inherent difficulties of the performed fits in the H$\beta$ region due to the presence of Fe lines and uncertainties in the result of the spectral slope, we chose not to impose conditions between different spectral regimes. Instead, we demonstrated that the results were consistent for both narrow and broad components after the fits were finished (see Section 5).
General fitting results {#fitres}
=======================
We have analyzed the optical spectra of 71 observed with the HES (long-slit spectroscopy) and 58 sources obtained with the 6dFGS (fiber spectroscopy). Tables \[tab:6df\] and \[tab:hes\] give information on the emission line flux ratios and line widths for 6dFGS and HES data, respectively. Because of low SNR in the HES spectra, in most cases the \[S\] and \[O\] spectral fits did not converge. As a result, for our analysis of the HES data set we only used the Balmer, \[N\], and \[O\] emission lines.
Width of emission lines {#emibroad}
------------------------
The 6dFGS line widths have been deconvolved according to the spectrometer resolution. For the HES sample we are giving observed widths; this is because a variety of different resolutions were used in the observations. The HES widths should hence be considered an upper limit. In this particular case this is not critical, as the integrated flux would not change and we are using the width information only to characterize and give a quality flag to the spectral fits.\
Figure\[fig:broad\_both\] compares the FWHM values of the fitted broad hydrogen recombination lines components, for both samples. All sources have FWHM larger than 1000 km s$^{-1}$, but the values are very scattered and do not follow any trend. Based on a study of more than 1000 local (z $\lesssim$ 0.4) NLSy1, @Zhou2006 found that the widths of the H$\alpha$ and H$\beta$ broad components do follow a linear correlation. They find the best fit to be: $FWHM(H\alpha) = 0.861 \times FWHM(H\beta)$. This relation has been positively tested for normal Seyfert 1 galaxies (M. Valencia-Schneider, priv. comm.), and reflects the fact that the lines are coming from different layers of the BLR. Forty-three% and 51% of the 6dFGS and HES galaxies follow the expected behavior within a 25% dispersion (see blue and green circles in Figure\[fig:broad\_both\]). Although the HES spectrometer resolution is different for the H$\alpha$ and H$\beta$ spectral regimes, we assume that the considered 25% uncertainty takes that into consideration. Both H$\beta$ and H$\alpha$ narrow components should have similar line widths, as their emission comes from the same region and trace identical mechanisms. In this case, 68% and 44% of the 6dfGS and HES, repectively, exhibit the same FWHM for both narrow lines within a 25% deviation (see blue and green circles in Fig.\[fig:narrow\_both\]). The mean FWHM and standard deviations of both full samples and selected galaxies are given in Table\[tab:averages\]. As the spectral resolution in this investigation is limited, the diagnostic emission line ratios may still contain hidden contributions of broad lines in H$\alpha$ and H$\beta$. However, as these lines are broad, and probably of low intensity, they only have a minor influence on the diagnostic ratios and on the conclusions we base them on.\
These departures in line widths measurements, between the lines of the same survey and average survey values, are not unexpected. The differences between survey results, especially in the H$\alpha$ narrow components, can be mostly accounted for by the better spectral resolution of the 6dfGS. Also, strong line blending, uncertainties in the result of the slope of the spectra, or contamination from iron lines in the case of H$\beta$, can be responsible for departures from the expected correlations. This has an impact on the measured fluxes, and may have an impact on the galaxies’ classification. For this reason, from now on we separate the two galaxy groups (following or not the expected correlations).
\(a) Results for the HES. (b) Results for the 6df survey.
-------------------- --------------- --------------- --------------- ---------------
Line 6df \[km/s\] HES \[km/s\] 6df \[km/s\] HES \[km/s\]
All Gxs. All Sel. Sel.
H$\beta_{Broad}$ 4120$\pm$2545 5179$\pm$2012 3754$\pm$1563 5610$\pm$1917
H$\beta_{Narrow}$ 501$\pm$147 1184$\pm$349 479$\pm$110 1156$\pm$331
H$\alpha_{Broad}$ 3783$\pm$4431 4301$\pm$1806 2926$\pm$1180 4623$\pm$1490
H$\alpha_{Narrow}$ 487$\pm$140 1093$\pm$527 469$\pm$114 1065$\pm$300
-------------------- --------------- --------------- --------------- ---------------
: Hydrogen recombination lines average FWHM and standard deviation for all studied galaxies, and for the galaxies selected following the critieria defined in Section \[emibroad\].[]{data-label="tab:averages"}
Narrow line Seyfert 1 subsample {#nls1}
-------------------------------
Classically, narrow line Seyfert galaxies (NLS1s) are defined by the width of their optical Balmer emission lines, such as H$\beta$, in combination with the relative weakness of the \[O\] $\lambda 5007\AA$ emission line, i.e. FWHM of the broad H$\beta$ component less than 2000 km/s and \[O\] / H$\beta_{\rm total}< 3$ [e.g., @1985ApJ...297..166O; @1989ApJ...342..224G; @2008RMxAC..32...86K]. There is some controversy as to whether NLS1 are a special class of AGN or an extension of classical Type 1 objects. @Valencia2012 found that the observed parameters in Type 1 AGN are continuous, with no particular difference between sources with FWHM(H$\beta_{broad}$) above or below 2000 km/s. This fact is inconsistent with the existence of two populations. We, however, chose to find out the relevance of the NLS1 galaxies populations, to compare with the results in the literature.
In our sample, and based on the 6dFGS data, we detect six galaxies that fulfill the classic NLS1 requirements (IDs 5, 8, 29, 54, 77, 91, see also Table\[tab:6df\]). At this point, we do not consider the often associated Fe emission . Two out of six NLS1 galaxies are consistent with previous references as NLS1 (IDs 5,77 Gruppe et al.2003 and Dietrich et al. 2005, respectvely) in the literature, while one of them is classified as Sy1.9 [ID$\sim$29 in Tab. \[tab:6df\]; @veron]. Three galaxies of the NLS1 group have not been classified as such before (IDs 8,54,91).
Population A vs. Population B
-----------------------------
AGNs are separated into radio-loud and radio-quiet objects, but to our knowledge there has not been any deep analysis or dedicated observations to verify the radio loudness of the sample under study. In the context of optical emission line properties, Seyfert1, NLS1, Seyfert2, QSOs, and broad absorption line QSO and LINERs are all radio-quiet galaxies. The radio-loud galaxies can be broadly divided into low-excitation and high-excitation classes [@Hine1979; @Laing1994]. Low-excitation objects do not show both broad and strong-narrow emission lines. They can exhibit weak narrow emission lines that may be excited by a different mechanism [@Baum1995]; their optical and X-ray nuclear emission are consistent with being generated in a jet [@Hardcastle2006]. Comparatively, the optical emission line spectra of high-excitation objects (narrow-line radio galaxies) is similar to those of Seyfert 2 galaxies. The small class of broad-line radio galaxies, which show relatively strong nuclear optical continuum emission (Grandi & Osterbrock 1978) and likely compress some low-luminosity, radio-loud quasars. Based on this, and with no radio information, the present sample should be comprised of radio-quiet galaxies, radio-loud high excitation and broad line radio galaxies. @Sulentic2000 [@Sulentic2000b] identified two radio-quiet AGN populations. Population A is an almost purely radio-quiet, with FWHM$\leq$4000 km s$^{-1}$, generally strong Fe emission and a soft X-ray excess. Comparatively, Population B has FWHM$\geq$4000 km s$^{-1}$ and optical properties largely indistinguishable from flat spectrum radio-loud sources, including usually weak Fe emission. A possible interpretation sees population A as lower BH mass/high accretion rate sources and population B/radio-loud sources as the opposite. Because of the better spectral resolution and accuracy on the spectral fits, we used the 6dFGS data of our sample to study the prevalence of Populations A and B. Twenty-four of the galaxies with H$\beta$ broad component have FWHM$\lesssim$4000 km s$^{-1}$, consistent with Population A, 33% of them have strong Fe emission. In addition 12% have weak Fe. As for Population B sources, 17 galaxies have FWHM$\gtrsim$4000 km s$^{-1}$, with 30% having weak and 18% strong Fe emission. Fifty% of our sources with H$\beta$ broad emission fit well into the categories designated by @Sulentic2000 [@Sulentic2000b]. We hypothezise that the remaining 50% are low-luminosity, radio-loud quasars.
Galaxies with double broad components
-------------------------------------
Two out of the 87 galaxies with HES data, HE0236-3101 (ID 27, see Fig.\[fig:hb\]) and HE0236-5224 (ID 28), give a better H$\beta$ fit result when using a double broad component. Already @1991ApJ...382L..63C found the double broad component in HE 0236-3101, whereas the case of HE0236-5224 has never been described before. No 6dFGS data was available for these galaxies, so further emission lines cross checking was not possible.\
Galaxies with no broad components
---------------------------------
A few galaxies do not present one or both of the expected broad emission lines associated with the hydrogen recombination lines. When both broad lines are not detected, we classify those sources as no broad line emitters (N.B.E in Tables\[tab:6df\] and \[tab:hes\]). We find two 6dFGS (sources ID 13 and 51) and one HES sources (ID 51) classified as N.B.E., which makes a total number of two sources in the entire sample. This means that the sources are not Type 1 Seyferts, or that the broad components are too weak and buried in the noise. Tables\[tab:6df\] and \[tab:hes\] give details on other sources in which one broad component, either H$\alpha$ or H$\beta$, is not detected. In those cases the nondetections are likely to be lines with low SNR, or lines subject to strong line blending (in particualr, for the H$\alpha$ lines in HES sources) that makes the fit more uncertain.
The NLR electron density
------------------------
We derived the electron density of 43 sources of the 6dFGS, using the \[S\]$\lambda$6716/\[S\]$\lambda$6731 lines ratio as a function of electron density at 10$^{4}$K (see @Osterbrock2006). The results are clearly skewed, with values clustered toward lower densities, between 100 and 1000 N$_{e}$/cm$^{3}$ (see Figure \[fig:electron\_density\_hist\]). Densities of the order of $\log \left(\frac{N_{e}}{{\rm cm}^{-3}}\right) \sim 2.2$ are typical for the narrow-line regions of broad line AGN [e.g., @2007ApJ...670...60X].
![Distribution of the electron density as estimated from the \[S\]$\lambda$6716/\[S\]$\lambda$6731 using the function given by @Osterbrock2006.[]{data-label="fig:electron_density_hist"}](Histogram_density.ps){width="\columnwidth"}
Optical emission lines diagnostic diagrams for the nearby LLQSOs sample {#class}
=======================================================================
The quality of the 6dFGS spectra is sufficiently good to clearly detect all optical emission lines involved in the three classic BPT [@baldwin1981] diagnostic diagrams. For the HES spectra, the detection of the \[O\] and \[S\] lines is quite challenging and the fitting results uncertain, hence we do not present them here. Figs. \[fig:6dfnii\] and \[fig:hesnii\] illustrate the \[O\]/H$\beta$ versus \[N\]/H$\alpha$ diagnostic diagram for our nearby sample based on the 6dFGS and HES. Figs.\[fig:6dfoi\] and \[fig:6dfsii\] show the \[S\]/H$\alpha$ and \[O\]/H$\alpha$ versus \[N\]/H$\alpha$ diagramas for the 6dfGS sources. Results are compared to the distribution of the SDSS galaxies used by [@kewley2006]. The latter sample selects nearby galaxies within the redshift range $0.04 < z < 0.1$, to ensure the results are not dominated by aperture effects from the SDSS fiber spectra (3 $\arcsec$ diameter).
Diagram H AGN/Sey Comp LINER
--------------------------- ----- --------- ------ -------
$[$N$]$/H$\alpha_{6dfGS}$ 18% 64% 18% —
$[$N$]$/H$\alpha_{HES}$ 29% 55% 16% —
$[$S$]$/H$\alpha_{6dfGS}$ 12% 68% — 20%
$[$O$]$/H$\alpha_{6dfGS}$ 19% 62% — 19%
: Percentages of LLQSOs classified using the BPT optical diagnostic diagrams. Only sources following the narrow hydrogen recombination lines criteria described in Sec.\[emibroad\] and represented in Figs.\[fig:6dfnii\] to \[fig:6dfsii\] are used here.[]{data-label="tab:perc_classif"}
Consistent with their classification, most sources of the LLQSOs sample populate the higher excitation region of the diagram. Very few galaxies lie in the densely populated star-forming branches of the SDSS sample with log(\[O\]/H$\beta$) $<$ 0.0
The HES and 6dfGS results from the \[N\]-based diagram compare well, especially considering the differences in spectral resolution and aperture used in the observations (slit widths of between $1\farcs 5$ and $2\farcs 5$ for the HES and a circular aperture of 6$\farcs$ in the 6dfGS). The other two emission-line diagnostic diagrams are not equally sensitive to the physical processes that are taking place. For instance, the \[N\]/H$\alpha$ and \[S\]/H$\alpha$ diagrams are more sensitive to star formation with respect to the \[O\]/H$\alpha$ diagram, which has proven to be more sensitive to shocks [cf., @2008ApJS..178...20A]. In spite of this, the percentage of galaxies classified as AGN/Seyfert remains quite constant: in all three classification schemes, between 50% and 68% of the sources are classified as AGN/Seyfert (see Table\[tab:perc\_classif\]). The HES results show the lower AGN activity, but there the quality of the spectra may be biasing the results. Depending on the diagram used, 10 to 35% of the LLQSOs are classified as H. This is consistent with AGNs showing star-forming activity potentially due to the emission from the extended AGN host galaxy that falls into the aperture used. As discussed by @2014MNRAS.441.2296M, in a study using simulations as well, the starlight coming from the host galaxy is responsible for the H classification. Between about 15 and 20% of the cases exhibit composite emission, and 20% of the sources are LINER. The presence of LINER-like emission is unexpected but not impossible. Nuclear LINER emission can be produced by an AGN, but extra nuclear LINER-like emission can be explained by fast shocks [@1995ApJ...455..468D] or photoionization by late-type stars.
One additional particular case, which we should stress is the example of a galaxy that can be clearly discriminated in the diagnostic diagram of 6dFGS due to its extreme location (see Figure \[fig:6dfnii\]); the galaxy with id 25 (HE 0227-0913) is located in the lower left part of 6dFGS classification diagram, but unfortunately the lack of the HES data do not allow us to compare the two classifcations. Previous studies by [@2001ApJS..136...61S] found that the galaxy has strong Balmer lines and the ratio \[O\]/H$\beta$ is uncharacteristically low for the narrow line region of a Seyfert galaxy, which agrees with 6dFGS results.
![\[N\]/H$\alpha$ versus \[O\]/H$\beta$ diagnostic diagram for the 6dFGS data. The solid line is the theoretical maximum starburst limit (Kewley et al. 2001). The sources that are located above the line are classified as AGN. Galaxies that lie below the star formation demarcation dashed line (Kauffmann et al. 2003a) are classified as H region-like galaxies. The galaxies that lie between the two demarcation lines are on the AGN-H mixing sequence and are called composite galaxies. The dark cloud represents SDSS galaxies from a previous study by [@kewley2006]. Green circles represent galaxies that follow the narrow lines width criteria (see Sect.\[emibroad\]), and red squares represent those that do not.[]{data-label="fig:6dfnii"}](BPT_NII_Backgr_6df.eps){width="7cm"}
![Same as Fig.\[fig:6dfnii\], except with the HES data. Blue circles represent galaxies that follow the narrow lines width criteria (see Sect.\[emibroad\]), and red squares represent those that do not.[]{data-label="fig:hesnii"}](BPT_NII_Backgr_hes.eps){width="7cm"}
![\[O\]/H$\alpha$ versus \[O\]/H$\beta$ diagnostic diagram for the 6dFGS data. The demarcation lines are from [@kewley2006]. The solid line represents the division between Seyferts and LINERs. The dark cloud represents SDSS galaxies from a previous study by [@kewley2006]. Green circles represent galaxies that follow the narrow lines width criteria (see Sect.\[emibroad\]), and red squares represent those that do not.[]{data-label="fig:6dfoi"}](BPT_OI_Backgr_6df.eps){width="7cm"}
![\[S\]/H$\alpha$ versus \[O\]/H$\beta$ diagnostic diagram for the 6dFGS data. The demarcation lines are from [@kewley2006]. The solid line represents the division between Seyferts and LINERs. The dark cloud represents SDSS galaxies from a previous study by [@kewley2006]. Green circles represent galaxies that follow the narrow lines width criteria (see Sect.\[emibroad\]), and red squares represent those that do not.[]{data-label="fig:6dfsii"}](BPT_SII_Backgr_6df.eps){width="7cm"}
Discussion
==========
The spectroscopic analysis of the LLQSOs sample helped us examine the properties of these systems. Notably, we probed the widths of the emission lines and the activity schemes of the samples. Moreover, we compared these features using two observational methods (fiber and typical long-slit spectroscopy).
Our targets comprise members of the Seyfert 1 category showing a typical range of emission line widths, FWHM $>$ 1000 km s$^{-1}$ and the widths of the H$\alpha$ and H$\beta$ broad components follow a linear correlation in agreement to [@Zhou2006].
The physical interpretation of these additional components is not straightforward. For example, HE 0236-3101 presents double broad components,while earlier studies have shown that this galaxy is an accretion disk candidate among luminous galaxies [@1991ApJ...382L..63C]. These broad Balmer emission line profiles are consistent with models of inclined small relativistic accretion disks around a massive black hole [@1988MNRAS.230..353P; @1989ASNYN...3...19C; @1990ApJ...365L..51H].
Previous X-ray studies of the double-peaked emitters (Balmer lines) such as HE 0203-0031 show that the illumination of the accretion disk requires an external power to produce lines of this strength due to insufficient local power [@2006ApJ...651..749S]. However, the fact that the X-ray emission of double-peaked emitters as a class does not differ from that of normal AGN with similar properties suggests that a peculiarity of the X-ray emission structure and mechanism is not responsible for the occurrence of double-peaked Balmer lines in AGN. On the other hand, the presence of double narrow components may be an indicator for superwinds. Galaxies in the local universe with large IR luminosties (L$_{IR}$ $>$ 10$_{44}$ erg/s), large IR excesses (L$_{IR}$ / L$_{OPT}$ $>$ 2), and/or warm far-IR colors (flux density at 60 micron greater than 50% of the flux density at 100 micron) drive superwinds [@ham]. However, the lack of relevant data and the low spectral resolution prevents us from further conclusions.
The fraction of NLS1s in the 6dfGS data presented is about 10$\%$, which is consistent with previous findings [e.g., @2008RMxAC..32...86K]. Nevertheless, there is some discussion, whether NLS1s really represent a class of their own [cf., @mv2013 and references therein].
[@Xu2007] introduced the “zone of avoidance", where AGNs with lines FWHM H$\beta$ $>$ 2000km s$^{-1}$ avoid low densities, whereas NLSy1 galaxies show a wider distribution in the NLR density, including a number of objects with low densities. Outflows may play a key role in driving differences in the NLR between NLS1 and BLS1 (Broad Line Seyfert 1), and consequently the zone of avoidance can be explained [@Xu2007]. Additionally, they discuss a number of explanations, such as supersolar metalicities, temperature effects, starburst contributions in NLS1, and the effect of NLR extent, which cannot support the idea of zone of avoidance in density. The NLR electron density values are consistent with those typically measured in our sample (see Fig\[fig:ne\_avoidance\]), although some sources are located in the zone of avoidance. Starburst contribution in a fraction of our sample can lead to lower measured density due to lower average density of H regions. Starburst activity is boosted in the young objects, since they are still in the process of growing their black holes [@mathur]. Moreover, the geometry of the NLR has been suggested to be responsible for the lower average densities, since the density declines outward in the extended NLR, .
![FWHM of the H$\beta$ broad component vs. \[S\]$\lambda$6716/$\lambda$6731 for our sample as measured with the 6df survey. Filled symbols represent galaxies for which the FWHM restriction considered between the broad components of H$\alpha$ and H$\beta$ is fulfilled. The shaded area marks the zone of avoidance for AGNs with broad lines (FWHM(H$\beta$) $>$ 2000 km s$^{-1}$)](Electron_density_Komossa.ps){width="\columnwidth"}
, as defined by @Xu2007. \[fig:ne\_avoidance\]
The fraction of LINERs (20%) was not expected in our sample but their presence can be explained. The \[S\]/H$\alpha$ and the \[O\]/H$\alpha$ diagram are considered to be more effective in separating Seyfert from LINER galaxies [@kewley2006]. The SDSS galaxy sample [@kewley2006] shows that the galaxies that are classified as composite using the \[N\]/H$\alpha$ diagram are mostly located within the star-forming sequence on the \[S\]/H$\alpha$ diagram. This tendency seems to be followed by our sample as well. The \[O\]/H$\alpha$ diagram is more sensitive to shocks and therefore is a more reliable tracer of changes in the ionization conditions due to fast shocks. The other diagrams are more sensitive to star formation. Star formation activity can be distributed all over the disk and can produce different ionizing continua. Additionally, the trend in late-type galaxies studied by [@2014MNRAS.441.2296M] agrees with our LINER fraction. Spiral galaxies show lower \[O\] values with increasing aperture. Hence, the unpredictable LINER activity in LLQSOs sample could also be justified as a result of aperture efffect.
{width="1.\linewidth"}
Aperture effect vs. sources variability
---------------------------------------
One unique aspect of the present study is the comparison between two data sets that have been taken with considerably different apertures. When comparing the data sets it is challenging to distinguish between differences that are due to variablility, either in the BLR, NLR or outer regions, or to the aperture effect. This would be better addressed using deconvolution techniques to separate the central point source from the background$/$host emission (see, e.g., @Lucy2003 for successful deconvolution of QSO long-slit data). However, these methods are not applicable in this study. In the case of 6dFGS, each galaxy is observed using a fiber, a single 1D spectra is obtained and hence there is not spatial information. As for the HES, the already extracted 1D spectra we have access to exhibit a variety of SNR in the studied emission lines. H$\alpha$ and \[O\]$\lambda$5007 are usually strong with a SNR of the order of 40 or more, depending on the case. As for H$\beta$ its strength varies from about 3 to more than 10. These observations used slits of between 1“.5 and 2”.5 width. Assuming the extracted galaxy spectra was distributed over a number of N pixels across the slit, this would reduce the observed SNR by a factor of $\sqrt{N}$. In many cases this would render the spatial signal of the H$\beta$ line too weak to be useful to perform a deconvolution. In any case, to apply spectral deconvoution techniques (at least in our long-slit data) we would need the 2D long-slit information and ideally (although not necessarily) a neighboring star to model the PSF. The present discussion is meant to clearly point out the issues faced when comparing different spectral data sets, and to clearly constrain the possible nature of the differences found and exploit the data we have in the best possible way.
When comparing the HES and 6sFGS spectra, several of our sources presented remarkable differences (see example in left panel of Fig.\[fig:spectral\_comparison\]) in the relative strength on the emission lines and in the detection as well (e.g., H$\beta$ broad component). After carefully cross-matching the sources and applying the narrow line widths quality criteria (as defined in Section\[anal\]), we ended up comparing 12 LLQSOs. For some of these sources the classification in the BPT diagram significantly changes depending on the data used (see Fig.\[fig:cross\_both\] Left), whereas for others the changes are not significant (see Fig.\[fig:cross\_both\] Right). We estimated the average travel distance for these sources to be $\sim$ 0.4 dex in both axes of the BPT diagrams (three times more than the maximum estimated uncertainty of 0.13 dex derived from not correcting the Fe contribution; see section \[anal\]).
From purely the instrumental point of view, there are two main reasons why a source observed with two different instruments can render different classifications. First, the differences in instrumentation setup; and second, differences in the spectral resolution (see discussion in Sect.\[emibroad\]). In the HES case, the slit width varied between $1\farcs 5$ and $2\farcs 5$ and was placed using an east-west slit position angle, while the 6dfGS uses a 67 circular fiber centered in the source. As illustrated in the right panel of Fig.\[fig:spectral\_comparison\], this means that different regions of the galaxy are being observed, which could easily lead to differences in the spectra and in classification. In the context of the aperture effect, two main effects can be described: dilution and inclusion [@storchi].
- Inclusion of new emitting regions: regions of star formation or ionization cones included in one of the apertures used and rejected by the other could easily be responsible for a change in the source classification. Inclusion effects could change a source classification from H to Composite or AGN, or vice-versa.
- Dilution of the emitting region: Dilution effects can be responsible for an AGN classification moving from higher to lower excitation regions, or simply to composite/H. In this context, large amounts of starlight could simply enhance the H contribution to the source.
These effects are difficult to categorize in an absolute way, as the results really depend on the galaxy geometry and internal structure. However, cases in which the classification of a source is almost the same (as in Fig.\[fig:cross\_both\] Right) point toward the source being dominated by its nuclear and close circumnuclear emission.
There is, however, the possibility of sources being variable. Here two different scenarios are plausible: variability of the emission coming from the narrow lines (e.g., NLR and stellar light) and variability in the BLR. The latter would not have a direct impact in the BPT classification, which uses only the narrow lines, but may affect the fits and dilute the stellar component. The lifetime of an H region is of the order of 10$^{6}$ years, in that period of time there are several events that, if observed, would change a galaxy classification. The bright Helium pulse flashes that occur periodically in AGB stars last for about hundreds years every 10$^{3}\sim$ 10$^5$ years, depending on the stellar mass. These events, even if they are coming from a stellar cluster, are not measurable in the timescales of our observations (ranging from 2 to 12 years). Supernovae explosions can be very bright events, of about M$_B$=-15 to -19 [@Richardson2002], or 10$^{10}$L$\sun$, which would make them detectable in some of our sample cases (see Fig.\[fig:magn\]). They, however, decay very rapidly in timescales of the order of 100-200 days. This makes them difficult to observe in extragalactic observations, but that possibility is open. Finally, the NLR and BLR could potentially undergo variability. The NLR densities are much smaller than in the BLR (n$_{e}$ of 10$^8$ cm$^{-3}$ vs. a maximum of about 10$^{4}$ cm$^-3$ for our LLQSOs, see Fig.\[fig:electron\_density\_hist\]). Variability in the BLR goes much faster than in the NLR, and in the NLR it lasts longer, as the lower density changes do not propagate as fast. So in the timescales of our observations (2-12 years) a change in the BLR could be measured, but not in the NLR.
One way of verifying if the BLR has not undergone variability is to represent the ratio between the narrow and broad line components of the H$\beta$ line (see Fig.\[fig:broad-vs\_narrow\]). We chose to use H$\beta$ for this because the spectral resolution, and hence the line fits, are better in that region. For three sources, the relation between the broad and narrow line components remains constant between the years of the HES and 6dfGS observations. This proves the consistency between the convolution and flux conservation of both data sets. Two of them (\#55 and \#46) essentially show no variation in classification based in the narrow lines (see left panel of Fig.\[fig:cross\_both\] ), meaning that the sources are nuclear dominated and have not undergone any variability. The third source (\#3) with a constant H$\beta$ line ratio in time shows a relative shift on the activity classification and interestingly located in the zone of avoidance with lower averaged NLR density. Source \#38 has a constant classification based on the BPT, but its H$\beta$ broad-to-narrow relation changes, making it a candidate for BLR variability. Finally, sources \#7, \#81, and \#43, as well as the rest of the studied LLQSOs, could be suffering from aperture effect as well as from variability. We do not have enough information from the data at hand to disentangle this effects.
We have also tested the ratio of the broad over narrow H$\beta$ components for the cross-matching sources. As mentioned in Section \[emibroad\], our HES sources have not been corrected for instrumental broadening. By using FWHM ratios, we are eliminating any bias that would show up by comparing just measured widths. Figure \[fig:broad-vs\_narrow\_fwhm\] shows that the importance of the broad over the narrow component is systematically higher for our 6dFGS cross-matched sources. This indicates that the better resolution of the data have a direct impact in the results, and that the HES results are clearly biased.
![Ratio of the H$\beta$ broad component over H$\beta$ narrow component for the cross-matched sources of both samples, which follow the broad lines width quality criteria defined in Section 5.1.[]{data-label="fig:broad-vs_narrow"}](Hbeta_broad_over_narrow.ps){width="\columnwidth"}
![Ratio of the FWHM of the H$\beta$ broad component over FWHM of the H$\beta$ narrow component for the cross-matched sources of both samples, which follow the broad lines width quality criteria defined in Section 5.1.[]{data-label="fig:broad-vs_narrow_fwhm"}](Hbeta_broad_over_narrow_FWHM.ps){width="\columnwidth"}
Summary {#concl}
=======
We presented an optical spectroscopy study of a sample of 99 LLQSOs, using two different sets of data. The HES data, observed with a slit and low spectral resolution, and the public 6dFGS data, whose spectra were obtained using a fiber and medium spectral resolution. The aims of the present investigation were to characterize the sample optical spectral properties, classify their activity, and study the effect of changes in aperture vs. variability of the BLR and$/$or NLR. We found the narrow and broad line widths to be within the expected values, on average several thousand km$/$s for the broad and $\lesssim$1000km$/$s for the narrow components. However, the results were sensitive to the spectral resolution and continuum characteristics of the spectra. For this reason, several fitting constrains and post-processing quality flags were used to ensure the consistency of the results. A small number of galaxies presented no broad component, but it is likely that they were buried in the data noise. We also found two sources with spectral characteristics consistent with the presence of double broad components, and six galaxies that comply with the classic NLS1 requirements. As for the NLR electron density value of our sources, it exhibits lower densities $\sim$ 2 to 3 cm$^{-3}$, consistent with NLR regions of broad line AGNs. The results are in good agreement with the zone of avoidance presented by [@Xu2007]. We also tested the relevance of Population A vs. Population B. Using only optical-based indicators, we find that 50% of our sources with H$\beta$ broad emission fit well into the @Sulentic2000 [@Sulentic2000b] radio-quiet sources definition. The remaining sources could be interpreted as low-luminosity, radio-loud quasars.
We presented the activity classification schemes (BPT diagrams) for both surveys, showing an AGN/Seyfert activity of between 50$-$60%. The starburst contribution throughout the galaxy might control the LINER and H classification of our sample. We justified the notable differences in classification from the comparison of the HES and 6dfGS spectra in the context of the aperture effect. It is challenging to quantify these effects since they depend on the galaxy structure and geometry. Alternatively, the differences measured could be due to variability in the BLR, although the nature of our data prevents us giving from a more precise quantification.
The HES data was kindly provided by Lutz Wisotzki, who also participated in discussions and gave extensive contributions to the paper. Special thanks to Marios Karouzos for very useful suggestions and discussions. This work is partially a result of the collaborative project between Korea Astronomy and Space Science Institute and Yonsei University through DRC program of Korea Research Council of Fundamental Science and Technology(DRC-12-2-KASI). This work has also been supported by the National Research Foundation of Korea grant 2012-8-0582. Part of this work was supported by the German [*Deutsche Forschungsgemeinschaft, DFG*]{} project numbers SFB956 and SFB494. Macarena Garcia-Marin is supported by the German federal department for education and research (BMBF) under the project number 50OS1101. Mariangela Vitale is member of the International Max-Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne, supported by the Max Planck Society. JZ acknowledges support by the German Academic Exchange Service (DAAD) under project number 50753527. This project has made use of the Final Release of 6dFGS data [@jones2004; @jones2009]. This research has also made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. While working on the current research, Ned’s Wright cosmology calculator was also used [@2006PASP..118.1711W].
=
[cccccccccc]{}
\
------------------------------------------------------------------------
Comments & ID & \[N\]/H$\alpha$ & \[O\]/H$\beta$ & \[S\]/H$\alpha$ & \[O\]/H$\alpha$ &&\
& & & & & &Narrow Component& Broad Component& Narrow Component& Broad Component\
\
------------------------------------------------------------------------
Comments & ID & \[N\]/H$\alpha$ & \[O\]/H$\beta$ & \[S\]/H$\alpha$ & \[O\]/H$\alpha$ &&\
& & & & & &Narrow Component& Broad Component& Narrow Component& Broad Component\
$\ddagger$ & 3 & 0.40 & 8.31 & 0.66 & 3.45 & 423.5 & 2844.6 & 465.8 & 4612.9\
No \[O\]$\ddagger$$\dagger$ & 4 & 0.23 & 6.01 & 0.37 & — & 585.2 & 2669.4 & 472.8 & 3385.4\
NLS1 $\ddagger$ & 5 & 0.26 & 1.46 & 0.15 & 0.02 & 675.3 & 2382.9 & 608.0 & 1537.8\
$\ddagger$$\dagger$& 7 & 0.41 & 11.85 & 1.49 & 0.50 & 345.5 & 2182.2 & 423.4 & 3116.7\
NLS1$\ddagger$ & 8 & 0.78 & 0.23 & 0.31 & 0.03 & 326.1 & 1246.9 & 335.7 & 526.2\
No H$\beta$ broad $\ddagger$ & 9 & 0.47 & 3.97 & 0.77 & 0.21 & 496.5 & 2813.1 & 408.7 & —\
$\ddagger$ & 12 & 0.45 & 1.78 & 0.44 & 0.04 & 324.9 & 1488.9 & 403.4 & 3248.1\
[**N.B.E.**]{} & 13 & 0.70 & 3.32 & 0.48 & 0.12 & 409.1 & 328.8 & — & —\
out & 14 & — & — & — & — & — & — & — & —\
No H$\beta$ broad $\ddagger$ & 19 & 0.85 & 7.80 & 0.43 & 0.22 & 381.7 & 33756.0 & 439.0 & —\
No H$\beta$ broad & 20 & 0.48 & 6.97 & 0.93 & 3.64 & 311.1 & 4241.7 & 438.9 & —\
No \[O\], Uncertain \[N\] & 22 & — & 13.04 & 0.02 & — & 377.8 & 6275.2 & 532.4 & 3512.8\
$\ddagger$ & 23 & 1.34 & 24.99 & 0.53 & 0.32 & 502.3 & 3072.6 & 561.1 & —\
$\ddagger$$\dagger$ & 24 & 0.47 & 8.88 & 0.84 & 0.24 & 481.9 & 3709.1 & 477.5 & 4371.6\
No \[O\] $\ddagger$ & 25 & 0.04 & 0.39 & 0.82 & — & 546.7 & 1937.2 & 718.3 & 3254.4\
No \[O\] $\ddagger$$\dagger$ & 26 & 0.48 & 11.32 & 0.72 & — & 558.8 & 4791.1 & 558.0 & 5502.8\
NLS1 $\dagger$ & 29 & 0.91 & 10.49 & 0.78 & 0.18 & 410.6 & 1803.9 & 622.2 & 1848.8\
$\dagger$ & 30 & 0.62 hhhh & 3.49 & 0.36 & 0.17 & 492.5 & 3230.4 & 392.3 & 3453.0\
No H$\beta$ broad & 32 & 1.52 & 2.87 & 0.26 & 0.10 & 652.2 & 3982.4 & 484.3 & —\
$\ddagger$ & 33 & 0.39 & 2.90 & 0.68 & 0.14 & 384.5 & 2183.9 & 407.4 & 4065.7\
$\ddagger$ & 35 & 0.20 & 5.57 & 1.27& 1.07 & 433.4 & 3214.3 & 466.8 & 5237.6\
& 36 & 5.26 & 3.15 & 1.53 & 7.30 & 597.7 & 4540.5 & 285.2 & 14403.6\
$\ddagger$ & 38 & 0.80 & 3.84 & 0.40 & 0.16 & 475.2 & 5248.5 & 537.3 & 4705.2\
$\ddagger$$\dagger$ & 40 & 0.72 & 6.59 & 0.25 & 0.20 & 393.7 & 2990.3 & 476.5 & 4172.0\
$\ddagger$$\dagger$ & 41 & 0.45 & 1.69 & 0.28 & 0.12 & 391.9 & 3162.2 & 481.9 & 4683.6\
& 42 & 0.57 & 10.24 & 0.88 & 0.79 & 402.9 & 9959.6 & 541.7 & 5226.5\
$\ddagger$$\dagger$ & 43 & 0.12 & 11.27 & 0.91 & 0.36 & 448.8 & 3489.2 & 421.6 & 4773.0\
$\ddagger$$\dagger$ & 44 & 0.25 & 6.49 & 0.37 & 0.06 & 494.2 & 2433.9 & 422.8 & 3320.1\
No \[O\], \[S\] $\ddagger$$\dagger$ & 46 &0.60 & 3.44 & 0.20 & — & 832.8 & 3159.7 & 899.0 & 4588.0\
[**N.B.E.**]{} $\ddagger$ & 51 & 0.15 & 2.75 & 1.29 & 0.20 & 332.5 & — & 392.2 & —\
out & 52 & — & — & 0.46 & 1.63 & — & 3187.0 & — & —\
out & 53 & — & — & 0.90 & — & — & 7369.8? & — & —\
NLS1 $\ddagger$$\dagger$ & 54 & 0.20 & 3.52 & 1.34 & 0.28 & 428.1 & 1734.1 & 434.4 & 1886.5\
No \[O\] $\ddagger$ & 55 & 0.28 & 0.98 & 0.57 & — & 451.7 & 1596.7 & 509.5 & 3286.1\
$\ddagger$ & 58 & 0.13 & 2.62 & 0.58 & 0.18 & 558.5 & 1982.9 & 503.3 & 3192.6\
No \[O\] & 59 & 0.39 & 3.12 & 0.08 & — & 823.3 & 2829.7 & 557.7 & 2268.6\
$\dagger$ & 62 & 0.65 & 2.00 & 0.33 & 0.07 & 408.8 & 2209.0 & 589.4 & 3248.9\
$\ddagger$$\dagger$ & 64 & 0.65 & 1.91 & 0.25 & 0.55 & 486.2 & 2352.0 & 479.1 & 3014.8\
No H$\beta$ broad & 66 & 0.39 & 8.50 & 0.90 & 0.40 & 596.9 & 3314.9 & 471.8 & —\
No H$\beta$ broad No \[O\] $\ddagger$ & 69 & 1.14 & — & 0.29 & — & 369.0 & 2915.6 & 431.6 & —\
$\ddagger$$\dagger$ & 71 & 0.13 & 9.42 & 0.95 & 0.73 & 417.5 & 2744.8 & 455.9 & 3688.3\
$\dagger$ & 72 & 1.96 & 2.22 & 0.17 & 3.42 & 285.4 & 1922.6 & 1146.3 & 2860.3\
& 73 & 1.53 & 11.99 & 0.64 & 0.25 & 554.8 & 3768.3 & 743.6 & 7556.9\
$\dagger$ & 74 & 0.36 & 2.83 & 0.22 & 1.18 & 626.4 & 2147.3 & 471.6 & 2510.8\
$\dagger$ & 76 & 1.02 & 13.39 & 0.89 & 0.37 & 524.1 & 4919.8 & 761.5 & 4811.7\
NLS1 & 77 & 0.01 & 2.78 & 1.53 & 3.59 & 816.4 & 2625.9 & 398.1 & 1501.5\
$\ddagger$$\dagger$ & 78 & 0.76 & 8.01 & 0.65 & 0.10 & 368.3 & 5198.1 & 373.4 & 7138.5\
$\ddagger$ & 81 & 0.13 & 4.31 & 1.34 & 0.65 & 405.4 & 1741.4 & 414.1 & 3406.6\
No H$\beta$ & 82 & — & 9.28 & — & — & 369.1 & 4515.9 & — & —\
No \[O\]& 83 & 1.00 & — & 0.63 & 0.24 & 509.8 & 4742.2 & 392.0 & —\
$\ddagger$$\dagger$ & 84 & 0.95 & 7.56 & 0.24 & 0.15 & 476.9 & 5253.2 & 382.9 & 7523.7\
out & 86 & — & — & — & 0.29 & — & 3174.9 & — & —\
$\ddagger$$\dagger$ & 87 & 0.13 & 8.69 & 1.10 & 1.00 & 446.4 & 2620.8 & 412.4 & 3118.6\
No H$\beta$ broad, No \[O\] $\ddagger$ & 90 & 0.48 & 1.54 & 1.30 & — & 385.4 & 2647.9 & 342.0 & —\
No \[O\], NLS1 $\dagger$& 91 & 0.28 & 0.72 & 0.16 & — & 856.7 & 1124.4 & — & 1126.7\
**[N.B.E.]{} $\ddagger$$\dagger$ & 94 & 0.59 & 2.96 & 0.50 & 0.16 & 623.1 & — & 549.3 & —\
$\ddagger$$\dagger$ & 96 & 1.22 & 13.07 & 0.10 & 0.17 & 700.0 & 1462.7 & 628.4 & 2204.3\
& 97 & 1.14 & 17.16 & 0.52 & 0.59 & 330.2 & 4815.9 & 457.2 & 11037.7\
**
=
[cccccccc]{}
\
------------------------------------------------------------------------
Comments & ID & \[N\]/H$\alpha$ & \[O\]/H$\beta$ & &\
& & & &Narrow Component& Broad Component& Narrow Component& Broad Component\
\
------------------------------------------------------------------------
Comments & ID & \[N\]/H$\alpha$ & \[O\]/H$\beta$ & &\
& & & &Narrow Component& Broad Component& Narrow Component& Broad Component\
& 1 & 0.23 & 3.23 & 1127 & 3396 & 672 & 2264\
No H$\beta$ broad & 2 & 1.08 & 8.24 & 479 & 11920 & 746 & —\
$\ddagger$$\dagger$ & 3 & 0.53 & 5.87 & 777 & 5460 & 667 & 6394\
$\ddagger$$\dagger$ & 6 & 0.52 & 3.42 & 842 & 4590 & 999 & 5293\
$\ddagger$ &7 & 0.05 & 4.34 & 921 & 1432 & 1151 & 3647\
No H$\beta$ broad & 8 & 0.89 & 1.89 & 932 & 1474 & 1561 & —\
$\ddagger$ & 9 & 2.45 & 1.73 & 1550 & 1545 & 1377 & 4162\
$\dagger$ & 11 & 0.84 & 5.86 & 931 & 3201 & 1379 & 3645\
$\ddagger$$\dagger$ & 12 & 0.45 & 6.10 & 1082 & 6440 & 1115 & 7589\
$\dagger$ & 15 & 0.43 & 1.09 & 785 & 3784 & 1114 & 4392\
$\ddagger$$\dagger$ & 16 & 0.76 & 8.53 & 1201 & 4559 & 1489 & 5299\
$\ddagger$$\dagger$ & 17 & 0.29 & 1.17 & 1391 & 5485 & 1674 & 7038\
$\ddagger$$\dagger$ & 18 & 0.17 & 4.95 & 1667 & 4445 & 1600 & 5186\
$\dagger$ & 19 & 0.70 & 7.80 & 866 & 4612 & 1234 & 7079\
$\ddagger$$\dagger$ & 20 & 1.72 & 18.21 & 666 & 4839 & 602 & 6178\
No H$\alpha$ broad & 21 & 0.26 & 1.46 & 802 & —- & 1540 & 1495??\
$\dagger$ & 22 & 0.15 & 12.9 & 2217 & 7737 & 789 & 7502\
& 23 & 0.20 & 3.30 & 2067 & 7451 & 967 & 5115\
No H$\alpha$ fitted & 24 & — & 8.51 & — & — & 600 & 5653\
& 27 & 0.65 & 1.49 & 831 & 4486 & 1633 & 3879(blue)/3400.94(blue)\
No H$\alpha$ fitted & 28 & — & 16.25 & — & — & 1113 & 6043(blue)/5220(red)\
& 29 & 0.88 & 3.81 & 671 & 2172 & 1329 & 4079\
& 30 & 0.73 & 2.38 & 1036 & 3346 & 1630 & 6834\
$\ddagger$$\dagger$ & 31 & 0.54 & 4.91 & 977 & 4362 & 848 & 5660\
No H$\beta$ broad & 32 & 1.39 & 2.62 & 855 & 3672 & 1148 & —-\
$\ddagger$$\dagger$ & 33 & 0.38 & 2.36 & 1226 & 3678 & 1186 & 4544\
$\ddagger$ & 34 & 0.61 & 4.58 & 777 & 4396 & 1015 & 7184\
$\ddagger$$\dagger$ & 37 & 0.10 & 0.54 & 928 & 4021 & 1223 & 6046\
$\ddagger$$\dagger$ & 38 & 0.50 & 3.87 & 669 & 3900 & 677 & 5224\
$\dagger$ & 39 & 0.11 & 5.98 & 1704 & 3616 & 1187 & 3462\
$\dagger$ & 40 & 0.69 & 13.98 & 698 & 4208 & 525 & 4059\
No H$\beta$ broad & 41 & 0.29 & 0.24 & 765 & 2414 & 1405 & —-\
$\dagger$ & 42 & 1.99 & 9.48 & 1576 & 8127 & 1066 & 9432\
$\ddagger$$\dagger$ & 43 & 0.23 & 10.79 & 1039 & 4870 & 1203 & 5948\
$\dagger$ & 44 & 0.99 & 1.16 & 1142 & 4541 & 1609 & 6897\
$\ddagger$ & 45 &0.19& 2.36 &584 &4367 & 568 & 1846\
$\ddagger$$\dagger$ & 46 & 0.70 & 3.34 & 999 & 3371 & 1209 & 3628\
& 47 & 0.39 & 3.96 & 1871 & 4589 & 1203 & 4087\
$\dagger$ & 48 & 1.85 & 1.90 & 602 & 4487 & 1317 & 6247\
$\ddagger$$\dagger$ & 49 & 0.13 & 1.82 & 1172 & 3906 & 1187 & 4531\
**[N.B.E.]{} $\dagger$ & 51 & 0.12 & 1.01 & 993 & — & 2107 & —\
$\ddagger$ & 53 & 0.55 & 0.62 & 1008 & 4725 & 1110 & 9256\
& 54 & 0.20 & 1.30 & 1047 & 3230 & 1401 & 5279\
$\ddagger$ & 55 & 0.21 & 0.95 & 883 & 2878 & 1093 & 4679\
$\ddagger$$\dagger$ & 57 & 0.24 & 1.98 & 1194 & 3012 & 1589 & 3486\
& 58& 0.09 & 1.77 & 1264 & 3787 & 1784 & 6782\
$\ddagger$$\dagger$ & 59 & 0.25 & 1.58 & 1516 & 3031 & 1279 & 2994\
No H$\alpha$ broad & 60 & 0.25 & 2.59 & 1143 & —- & 1606 & 4556\
$\dagger$ & 61 & 1.23 & 3.97 & 586 & 4403 & 1280 & 5129\
$\dagger$ & 62 & 0.78 & 2.09 & 1067 & 4321 & 1566 & 5043\
& 63&0.24&0.38&498&5125 & — & —\
& 64&0.63& 1.35 &1675 & 3878 & 1012 & 2356\
& 65 & 0.10 & 1.42 & 1969 & 6494 & 1466 & 3019\
No H$\beta$ broad & 66 & 0.36 & 9.73 & 400 & 3724 & 771 & —-\
$\ddagger$ & 69 &1.56& 23.3 & 890 & 4171 & 849 & 2387\
$\dagger$ & 71 & 0.11 & 6.22 & 433 & 3119 & 738 & 4491\
$\ddagger$$\dagger$ & 72 & 0.47 & 0.93 & 1285 & 3028 & 1704 & 3498\
out & 73 & — & — & — & — & — & —\
$\dagger$ & 74 & 0.07 & 0.79 & 1129 & 4841 & 591 & 6317\
& 75&0.51& 2.16 &3646 &5551 & 1020 & 4540\
$\ddagger$ & 78&0.02& 9.98 &1324 &1608 & 1618 & 7918\
$\ddagger$ & 79 & 0.15 & 4.34 & 681 & 3613 & 602 & 6201\
& 80 & 0.51 & 3.94 & 934 & 3271 & 1339 & 2769\
No H$\alpha$ broad $\ddagger$ & 81 & 0.24 & 2.88 & 1179 & —- & 1277 & 3507\
$\dagger$ & 82 & 1.00 & 16.65 & 2106 & 5046 & 1290 & 5822\
$\ddagger$$\dagger$ & 90 & 0.34 & 3.76 & 1699 & 6255 & 1544 & 7234\
No H$\beta$ broad & 91 & 0.57 & 0.48 & 527 & 1727 & 1299 & —-\
$\ddagger$ & 92 & 0.53 & 2.74 & 1024 & 3663 & 1325 & 7774\
$\ddagger$$\dagger$ & 93 & 1.11 & 10.70 & 670 & 8877 & 891 &11825\
$\dagger$ & 97 & 0.62 & 6.39 & 519 & 5877 & 1227 & 7343\
$\dagger$ & 98 & 0.37 & 6.89 & 500 & 1767 & 764 & 1892\
$\ddagger$ & 99 & 0.32 & 0.79 & 1184 & 3672 & 1153 & 2625\
**
[^1]: Spectral resolution about 770km/s.
[^2]: Spectral resolution about 880km/s.
[^3]: Spectral resolution about 880km/s.
[^4]: <http://www-wfau.roe.ac.uk/6dFGS/>
[^5]: <http://www.physics.wisc.edu/~craigm/idl/fitting.html>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[Using the fractional moment method it is shown that, within the Hartree-Fock approximation for the disordered Hubbard Hamiltonian, weakly interacting Fermions at positive temperature exhibit localization, suitably defined as exponential decay of eigenfunction correlators. Our result holds in any dimension in the regime of large disorder and at any disorder in the one dimensional case. As a consequence of our methods, we are able to show Hölder continuity of the integrated density of states with respect to energy, disorder and interaction.]{}'
address: 'Department of Mathematics, Michigan State University, East Lansing MI 48823, USA'
author:
- Rodrigo Matos and Jeffrey Schenker
date: 'November 25, 2016'
title: 'Localization and IDS Regularity in the Disordered Hubbard Model within Hartree-Fock Theory'
---
Introduction
============
Our goal in this note is to study Anderson localization in the context of infinitely many particles. We shall formulate our results for the disordered Hubbard model within Hartree-Fock theory. However, as the techniques involved are quite flexible, we expect that similar statements can be made in a more general framework, under appropriate modifications of the decorrelation estimates on section \[improvesec\]. The (deterministic) Hubbard model under Generalized Hartree-Fock Theory has been discussed (at zero and positive temperature) by Lieb, Bach and Solovej in [@B-Lieb-S] but, to the best of our knowledge, the localization properties of the disordered version of this model remained unexplored, even in the context of restricted Hartree-Fock theory, up to the present work. The main difficulty lies on the addition of a self-consistent effective field, which will be random and non-local by nature, to a random Schrödinger operator. The conclusion of this note can be summarized as follows: under technical assumptions, the results on (single-particle) Anderson localization obtained in the non-interacting setting in the regimes of large disorder (in dimension $d\geq 2$) and at any disorder (in dimension $d=1$), remain valid under the presence of sufficiently weak interactions. More specifically, in the regime of strong disorder this is accomplished in any dimension by theorem \[main\] below. Theorem \[1dloc\] contains the improvement in dimension one, where any disorder strength leads to localization, provided the interaction strength is taken sufficiently small. Our methods contain various bounds in the fluctuations of the effective interaction which are interesting on their own right and potentially useful on different contexts. To exemplify this, we prove Hölder regularity of the integrated density of states (IDS) with respect to various parameters by adapting arguments of [@H-K-S], which is the content of theorem \[thmids\].
Discussion of the results and main obstacles
--------------------------------------------
Mathematically, our setting can be understood as an Anderson-type model $H_{\omega}=H_0+\lambda U_{\omega}$ where the values of the random potential $U$ at different sites are correlated in a highly non-local and self-consistent fashion. The correlations are governed by a nonlinear function of $H_{\omega}$, as explained on section \[motivation\]. In comparison to the recent result on Hartree-Fock theory for lattice fermions in [@Duc], achieved via multiscale analysis, we use the fractional moment method to establish exponential decay of the eigenfunction correlators at large disorder in any dimension but also at any disorder in dimension one. In particular, in the above regimes we obtain for any $t>0$, exponential decay (on expectation) for the matrix elements of the Hamiltonian evolution, which means that, on average, $|\mel{m}{e^{-itH}}{n}|$ decays exponentially on $|m-n|$.
The result of complete localization in dimension one in such interacting context is new and deserves attention on its own. Its main technical difficulty lies on the non-local correlations of the potential, which means that standard tools such as Furstenberg’s theorem and Kotani theory are not available. Moreover, a large deviation theory for the Green’s function is a further obstacle to establishing dynamical localization even if one obtains uniform positivity of the Lyapunov exponent. We overcome these challenges using ideas of [@A-W-B Chapter 12], where arguments reminiscent of the proof of the main result in [@K-S] are presented. We then obtain positivity of the Lyapunov exponent at any disorder using uniform positivity for the Lyapunov exponent of the Anderson model, combined with an explicit bound on how this quantity depends on the interaction strength, see theorem \[uniformposh\]. When it comes to establishing a large deviation theorem, our modification of the argument in [@A-W-B Theorem 12.8] relies on quantifying the decorrelations on the effective potential, which is presented on lemma \[mixinglem\] in the form of a strong mixing statement. It is worth clarifying that, since our proof is based on fractional moments, we have not established localization in one dimension for rough potentials as in [@C-K-M]. Moreover, the gap assumption in [@Duc] is replaced by working at positive temperature thus our results do not apply to Hartree-Fock ground states.
Hartree-Fock theory
-------------------
Hartree-Fock theory has been widely applied in computational physics and chemistry. It also has a rich mathematical literature which goes well beyond the scope of random operators, see for instance [@H-Lewin-S],[@G-H-Lewin],[@B-Lieb-S],[@B-L-L-S],[@Lieb-Simon] and references therein.
Background on Localization for interacting systems
--------------------------------------------------
The main results of this note lie in between the vast literature on (non-interacting) single particle localization and the recent efforts to study many particle systems, as in the case of an arbitrary, but finite, number of particles in the series of works by Chulaevsky-Suhov [@C-S1],[@C-S2],[@C-S3] and Aizenman-Warzel [@A-W-P]. In comparison to the later, we only seek for a single-particle localization result but allow for infinitely many interactions, which occur in the form of a mean field. In comparison to the recent developments on spin chains, as the study of the XY spin chain in [@H-S-St] and the droplet spectrum of the XXZ quantum spin chain in [@E-K-St] and [@B-W], the notions of localization for a single-particle effective Hamiltonian are more clear and can be displayed from pure point spectrum to exponential decay of eigenfunctions and exponential decay of eigenfunction correlators. The later is agreed to be the strongest form of single particle localization and it is what we accomplish in this manuscript. If fact, dynamical localization in the form of theorems \[1dloc\] and \[main\] implies pure point spectrum via the RAGE theorem (see [@Stolz Proposition 5.3]) and exponential decay of eigenfunctions, see [@A-W-B Theorem 7.2 and Theorem 7.4].
Definitions and Statement of the Main Result
============================================
Notation
--------
In what follows, $\mathbb{Z}^d$ will be equipped with the norm $|n|=|n_1|+...+|n_d|$ for $n=(n_1,...,n_d)$. Given a subset $\Lambda\subset \mathbb{Z}^d$, we define $\ell^2(\Lambda):=\{\varphi:\Lambda \rightarrow \mathbb{C}\,|
\sum_{n\in \Lambda}|\varphi(n)|^2<\infty\}$ and, for $\varphi\in \ell^2(\Lambda)$, we let $\|\varphi\|_{\ell^2(\Lambda)}:=\left(\sum_{n\in \Lambda}|\varphi(n)|^2<\infty\}\right)^{1/2}$. Throughout this note, $\eta$ will be a positive constant and $F_{\beta,\kappa}$ will denote the Fermi-Dirac function at inverse temperature $\beta>0$ and chemical potential $\kappa$: $$\label{fermidirac}
F_{\beta,\kappa}(z)=\frac{1}{1+e^{\beta(z-\kappa)}}.$$ We shall omit the dependence on the above parameters whenever it is clear from the context. For many of our bounds, the specific form of (\[fermidirac\]) is not important and $F$ could denote a fixed function which is analytic on the strip $\mathcal{S}=\{z\in \mathbb{C}:\,\, |\mathrm{Im}z|<\eta\}$ and continuous up to the boundary of $\mathcal{S}$, in which case we define $\|F\|_{\infty}:=\sup_{z\in \mathcal{S}}|F(z)|$. For the function $F_{\beta,\kappa}$ in (\[fermidirac\]) one can take $\eta=\frac{\pi}{2\beta}$. However, to obtain robust results which incorporate delicate fluctuations, further properties of the Fermi-Dirac function are necessary. Namely, in section \[improvesec\] we use the the fact that $tF(t)$ is bounded as $t\to \infty$ and that $t(1-F(t))$ is bounded $t\to -\infty$. These properties will also play a role in the decoupling estimates needed in the proof of theorem \[1dloc\] but could be relaxed if one is only interested in the large disorder proof of theorem \[main\] for a specific distribution with heavy tails (for instance, the Cauchy distribution).
Our main goal is to study localization properties of non-local perturbations of the Anderson model $H_{\mathrm{And}}:=-\Delta+\lambda V_{\omega}$ which naturally arise in the context of Hartree-Fock theory for the Hubbard model. The random potential $V_{\omega}$ is the multiplication operator on $\ell^2(\mathbb{Z}^d)$ defined as $$\label{potentialdef}\left(V_{\omega}\varphi\right)(n)=\omega_n\varphi(n)$$ for all $n\in \mathbb{Z}^d$ and $\{\omega_n\}_{n\in \mathbb{Z}^d}$ are independent, identically distributed random variables on which we impose technical assumptions described in the next paragraph. The hopping operator $\Delta$ is the discrete Laplacian on $\mathbb{Z}^d$, defined via $$\left(\Delta\varphi\right)(n)=\sum_{|m-n|=1}\left(\varphi(m)-\varphi(n)\right).$$ The proofs of localization via fractional moments usually do not require the hopping to be dictated by $\Delta$; below we will replace $\Delta$ by a more general operator $H_0$ whose matrix elements decay sufficiently fast away from the diagonal. It is technically useful to formulate some of our results in finite volume, i.e, we will work with restrictions of the operators to $\ell^2(\Lambda)$ but the estimates obtained will be volume independent, meaning that all the constants involved are independent of $\Lambda\subset \mathbb{Z}^d$. We will use $\mathds{1}_{\Lambda}$ to denote the characteristic function of $\Lambda$ as well as the natural projection $P_{\Lambda}:\ell^2(\mathbb{Z}^d) \rightarrow \ell^2(\Lambda)$. With these preliminaries we are ready to define the Schrödinger operators studied in this work.
Definition of the operators
---------------------------
Let $H_{\mathrm{And}}=H_0+\lambda V_{\omega}$ be the Anderson model on $\ell^2\left(\mathbb{Z}^d\right)$ where:
1. [ $$\zeta(\nu):=\sup_{m}\sum_{n\in \mathbb{Z}^d}|H_0(m,n)|\left(e^{\nu|m-n|}-1\right)<\frac{\eta}{2}, \,\,\,\,\mathrm{for\, some}\,\,\nu>0 \,\,\mathrm{fixed}.$$]{}
2. [ $V_{\omega}$ is defined as in $(\ref{potentialdef})$ and the random variables $\{\omega(n)\}_{n\in \mathbb{Z}^d}$ are independent, identically distributed with a density $\rho$: $$\mathbb{P}\left(\omega(0)\in I\right)=\int_{I} \rho(x)\,dx,\,\,\,\,\,\mathrm{for}\,\,\, I\subset \mathbb{R}\,\,\,\mathrm{a\,\, Borel\,\, set\,}.$$]{}
3. [ We also assume that $\mathrm{supp}\,\rho=\mathbb{R}$ with $$\label{flucassump}\frac{\rho(x)}{\rho(x')}\geq e^{-c_1(\rho)|x-x'|(1+c_2(\rho)\max\{\,|x|,|x'|\,\})}$$ for some $c_1(\rho)>0$ and $c_2(\rho)\geq 0$ and any $x,x'\in \mathbb{R}$.]{}
Before stating the remaining assumptions on $\rho$, we need to introduce some notation. Assume that $\rho$ satisfies (\[flucassump\]). Let $$\label{upperdensity}
\overline{\rho}(x)=\frac{\rho(x)}{\int^{\infty}_{-\infty}\rho(\alpha)h(x-\alpha)\,d\alpha}$$ where $$\label{fluctintegral3}h(x)=
\left\{
\begin{array}{lll}
e^{-{\overline{c}}_{\rho}|x|} \;\,\,\,\,\, \mathrm{if} \;c_2(\rho)=0. \\
\\
e^{-{\overline{c}}_{\rho}|x|^2}\,\,\,\,\, \; \mathrm{if} \;c_2(\rho)>0. \;\;\;
\end{array}
\right.$$
1. \[fluctintegral1\][ The function $\overline{\rho}$ is bounded for some ${\overline{c}}_{\rho}>0$. ]{}
\[remark2\] The technical assumptions $(A_3)-(A_4)$ will be needed for the large disorder result of theorem \[main\] below. They include, for instance, the Cauchy distribution, the Gaussian, and the exponential distribution $\rho(v)=\frac{m}{2}e^{-m|v|}$.
The above assumptions will suffice to show localization at large disorder on theorem \[main\] below. To show complete localization in dimension one, theorem \[1dloc\] will also require a moment condition on $\overline{\rho}$, which is the following.
1. \[momentassumption\][ For some $\varepsilon>0$ and some ${\overline{c}}_{\rho}>0$, $\int^{\infty}_{-\infty}|x|^{\varepsilon}\overline{\rho}(x)\,dx<\infty$.]{}
\[remark3\] The assumption $(A_5)$ covers, for example, the Gaussian and the exponential distributions but it does not cover the Cauchy or other distribution with heavy tails. It will be necessary for the one dimensional result of theorem \[1dloc\] below. More specifically, this requirement will imply a moment condition which will be needed to relate the Green’s function to the Lyapunov exponent, see sections \[1dideas\] and \[detailslowerbound\].
\[remark1\] The specific bound on $\zeta(\nu)$ is necessary to ensure that the Combes-Thomas bound $|G(m,n;E+i\eta)|\leq \frac{2}{\eta}e^{-\nu|m-n|}$ holds [@A-W-B Theorem 10.5], where $G(m,n;z)$ denotes $\mel{m}{(H-z)^{-1}}{n}$, whenever this quantity is defined.
Define the operator $H_{\mathrm{Hub}}$, acting on $\ell^{2}\left(\mathbb{Z}^d\right)\oplus \ell^{2}\left(\mathbb{Z}^d\right)$ by $$\label{Hubbarddefmain}H_{\mathrm{Hub}}=\begin{pmatrix}
\,H_{\uparrow}(\omega) & 0 \\
0 & H_{\downarrow}(\omega)\,\\
\end{pmatrix}
:=
\begin{pmatrix}
\,H_0+\lambda V_{\omega}+gV_{\uparrow}(\omega) & 0 \\
0 & H_0+\lambda V_{\omega}+gV_{\downarrow}(\omega)\,\\
\end{pmatrix}$$ where the operators $H_{\uparrow}(\omega)$ and $H_{\downarrow}(\omega)$ act on $\ell^{2}\left(\mathbb{Z}^d\right)$ and the so-called effective potentials are defined via $$\label{eff}
\begin{pmatrix}
\,V_{\uparrow}(\omega)(n) \\
V_{\downarrow}(\omega)(n)\,\\
\end{pmatrix}
=\begin{pmatrix}
\,\mel{n}{F(H_{\downarrow})}{n} \\
\mel{n}{F(H_{\uparrow})}{n}\,\\
\end{pmatrix}.$$ Note that the above equations only define $H_{\uparrow}(\omega)$ and $H_{\downarrow}(\omega)$ implicitly. Existence and uniqueness of $V_{\uparrow}$ and $V_{\downarrow}$ will be shown in section \[existHubb\] via a fixed point argument. The model (\[Hubbarddefmain\]) is usually referred to as the Hartree approximation, due to the absence of exchange terms. In section \[motivation\] below we will show that the terminology Hartree-Fock approximation is justified when $g<0$, which represents a repulsive interaction.
The Hubbard model is schematically represented in the following picture. The black (horizontal) edges represent hopping between sites and the red (vertical) edges represent the effective interaction between the two layers, which are identical copies of $\mathbb{Z}^d$.
(-1,1) circle (3pt); (-1,2) circle (3pt);
(0,1) circle (3pt); (0,2) circle (3pt);
(1,1) circle (3pt); (1,2) circle (3pt);
(2,1) circle (3pt); (2,2) circle (3pt);
(3,1) circle (3pt); (3,2) circle (3pt);
(4,1) circle (3pt); (4,2) circle (3pt);
(5,1) circle (3pt); (5,2) circle (3pt);
at (2.5,0) [(0,0)]{}; at (6.5,1) [$n_{\downarrow}$]{}; at (6.5,2) [$n_{\uparrow}$]{}; (5,1)–(5,2); (4,1)–(4,2); (3,1)–(3,2); (2,1)–(2,2); (1,1)–(1,2); (0,1)–(0,2); (-1,1)–(-1,2); (-1,2)–(0,2) –(1,2) – (2,2) – (3,2) –(4,2) – (5,2); (-1,1)–(0,1) –(1,1) – (2,1) – (3,1) –(4,1) – (5,1);
Main Theorems
-------------
Fix an interval $I\subset \mathbb{R}$ and define the eigenfunction correlator through $$Q_{I}(m,n):=\sup_{|\varphi|\leq 1}\left(|\mel{m}{\varphi(H_{\uparrow})}{n}|+|\mel{m}{\varphi(H_{\downarrow})}{n}|\right).$$ The operators $H_{\uparrow}$ and $ H_{\downarrow}$ and defined as in (\[Hubbarddefmain\]) and the supremum being taken over Borel measurable functions bounded by one and supported on the interval $I$. In case $I=\mathbb{R}$ we simply write $Q(m,n)$. Our first result is the following:
\[1dloc\] In dimension $d=1$, let $H_0=-\Delta$ and assume that the conditions $(A_1)-(A_5)$ hold. For any $\lambda>0$ and any closed interval $I\subset \mathbb{R}$ , there is a constant $g_1>0$ such that whenever $|g|<g_1$ we have $$\label{dynloc1d}
\mathbb{E}\left(Q_I(m,n)\right)\leq Ce^{-\mu_{1}|m-n|}.$$ for any $m,n\in \mathbb{Z}^d$ and positive constants $\mu_1=\mu_1(\lambda,\nu,\eta,I)$, $C(\eta,g,\lambda,\|F\|_{\infty},I)$.
\[main\] Suppose that the conditions $(A_1)-(A_4)$ hold. For any dimension $d\geq 1$, there exists a constant $g_d=g(d,\eta,\|F\|_{\infty},\nu)$ such that, whenever $|g|<g_d$, there is a positive constant $\lambda_0(g)$ for which $$\label{dynloc}
\mathbb{E}\left(Q(m,n)\right)\leq Ce^{-\mu_{d}|m-n|}.$$ holds for $\lambda >\lambda_0(g)$, any $m,n\in \mathbb{Z}^d$ and some positive constants $\mu_{d}=\mu(d,\lambda,g,\nu,\eta)$, $C(\eta,\nu,d,g,\lambda,\|F\|_{\infty})$.
It will follow from the proof that the constant $g_d$ in theorem \[main\] can be taken proportional to $\frac{\eta\left(1-e^{-\nu}\right)^d}{\|F\|_{\infty}}$.
The constant $g_1$ in theorem \[1dloc\] can be taken equal to be the minimum among a factor proportional to $\frac{\eta\left(1-e^{-\nu}\right)}{\|F\|_{\infty}}$ and the upper bound obtained in corollary \[uniformpos\], which also depends on the lower bound for the Lyapunov exponent of the Anderson model on $\ell^2\left(\mathbb{Z}\right)$.
Recall the definition of the integrated density of states for an ergodic operator $H$ : $$N_H(E)=\lim_{|\Lambda|\to \infty}\frac{\mathrm{Tr}P_{(-\infty,E)}(\mathds{1}_{\Lambda}H\mathds{1}_{\Lambda})}{|\Lambda|}.$$ For the definition of ergodic operator one may consult [@A-W-B Definition 3.4]. In what follows, we denote by $N_0(E)$ the corresponding quantity for the free operator $H_0$ defined above, which is assumed to be ergodic for the result below, where we shall be concerned with the small disorder regime and aim for bounds which do not depend upon $\lambda$ as $\lambda \to 0$.
\[thmids\] Assume that $(A_1)-(A_2)$ hold with $x^2\rho(x)$ bounded and that $g^2<\lambda$ . Fix a interval $I$ where $E\mapsto N_0(E)$ is $\alpha_0$-Hölder continuous and a bounded interval $J\subset \mathbb{R}$. The integrated density of states $N_{\lambda,g}(E)$ of $H_{\mathrm{Hub}}$ is Hölder continuous with respect to $E$ and with respect to the pair $(\lambda,g)$. More precisely:
1. [ For $E,E' \in I$ $$\label{IDSenergy}
|N_{\lambda,g}(E)-N_{\lambda,g}(E')|\leq C(\alpha,I,g)|E-E'|^{\alpha}$$ for $\alpha\in [0,\frac{\alpha_0}{2+\alpha_0}]$ and $C(\alpha,I,g)$ independent of $\lambda$. ]{}
2. [\[IDSdisorder\] If $\lambda,\lambda'\in J$, we have that, for any $E\in I$, $\alpha\in [0,\frac{\alpha_0}{2+\alpha_0}]$ and $\beta\in [0,\frac{2}{\alpha+3d+4}]$, $$|N_{\lambda,g}(E)-N_{\lambda',g'}(E)|\leq C(\alpha_0.d,I)\left(|\lambda-\lambda'|^{\beta}+|g-g'|^{\beta}\right).$$]{}
Motivation
==========
We shall explain the motivation for the above choice of the effective potential. We are only going to outline the derivation of the self-consistent equations as this is a standard topic, see, for instance, [@Kurig Chapter 3].
Let $\Lambda \subset \mathbb{Z}^d$ be a finite set. Following the notation of [@B-Lieb-S], we use $\Gamma$ to denote a one particle density matrix, i.e, a $2\times 2$ matrix whose entries are operators on $\ell^2\left(\Lambda\right)$ and which satisfies $0\leq \Gamma \leq \mathds{1}$. We then write $$\Gamma
:=
\begin{pmatrix}
\,\Gamma_{\uparrow} & \Gamma_{\uparrow \downarrow} \\
\Gamma_{\downarrow \uparrow} &\Gamma_{\uparrow}\,\\
\end{pmatrix}$$ where $\Gamma_{\downarrow \uparrow}=\Gamma^{\dag}_{\uparrow \downarrow}$.
As in [@B-Lieb-S Equation 3a.8], the pressure functional $\mathcal{P}(\Gamma)$ is defined as $$-\mathcal{P}(\Gamma)=\mathcal{E}(\Gamma)-\beta^{-1}\mathcal{S}(\Gamma).$$ The energy functional is $$\label{energy}
\mathcal{E}(\Gamma)=
\mathrm{Tr}\left( H_0-\kappa+\lambda V_{\omega}\right)\Gamma
+g\sum_{n}\mel{n}{\Gamma_{\uparrow}}{n}\mel{n}{\Gamma_{\downarrow}}{n},$$ where we have identified $H_0-\kappa+\lambda V_{\omega}$ with $ \begin{pmatrix}
\,H_0-\kappa+\lambda V_{\omega} &0 \\
0 &H_0-\kappa+\lambda V_{\omega}\,\\
\end{pmatrix}$.
The entropy is given by $$\mathcal{S}(\Gamma)=-\mathrm{Tr}\left(\Gamma \log \Gamma+(1-\Gamma)\log (1-\Gamma)\right)
.$$ Generally, the choice of energy functional (\[energy\]) is referred to as Hartree approximation as exchange terms are neglected. However, in the case of a repulsive interaction among the particles, it is easy to prove that such exchange terms do not affect the choice of minimizer for $-\mathcal{P}(\Gamma)$ and the process may be referred to as the Hartree-Fock approximation. Indeed, the Hartree-Fock energy for the repulsive interaction would incorporate the term $-g|\mel{n}{\Gamma _{\uparrow \downarrow}}{n}|^2$, which is non-negative when $g<0$. Thus, for repusive interactions, off-diagonal terms can be disregarded for minimization purposes, see the analogue discussion in [@B-Lieb-S Section 4a]. The minimizer $\Gamma$ of $-\mathcal{P}(\Gamma)$ exists since $\Lambda$ is a finite set. Moreover, it satisfies $$\label{densityHubbard}
\mel{n}{\Gamma_{\uparrow}}{n}=\mel{n}{\frac{1}{1+e^{\beta(H_0-\kappa+\lambda V_{\omega}+\mathrm{Diag}\left(\Gamma_{\downarrow})\right)}}}{n}.$$ $$\mel{n}{\Gamma_{\downarrow}}{n}=\mel{n}{\frac{1}{1+e^{\beta(H_0-\kappa+\lambda V_{\omega}+\mathrm{Diag}\left(\Gamma_{\uparrow})\right)}}}{n}.$$
Thus, the effective Hamiltonian on $\ell^2\left(\Lambda\right)\oplus \ell^2\left(\Lambda\right)$ is determined by
$$H^{\Lambda}_{\omega}
:=
\begin{pmatrix}
\,H_0+\lambda \omega(n)+gV^{\Lambda}_{\uparrow}(n) & 0 \\
0 & H_0+\lambda \omega(n)+gV^{\Lambda}_{\downarrow}(n)\,\\
\end{pmatrix}$$
$$\label{potentialHubbard1}
V^{\Lambda}_{\uparrow}(\omega)(n):=\mel{n}{\frac{1}{1+e^{\beta(H_0-\kappa+\lambda \omega+gV_{\downarrow})}}}{n}$$
$$\label{potentialHubbard2}
V^{\Lambda}_{\downarrow}(\omega)(n):=\mel{n}{\frac{1}{1+e^{\beta(H_0-\kappa+\lambda \omega+gV_{\uparrow})}}}{n}.$$
It will follow from arguments given below that if $\Lambda_R$ is an increasing sequence with $\cup_{R\in \mathbb{N}}\Lambda_R=\mathbb{Z}^d$ then, for fixed $m \in \mathbb{Z}^d$, $$\lim_{R\to \infty} V^{\Lambda_R}_{\mathrm{eff}}(m)=V_{\mathrm{eff}}(m)$$ and this fact ensures that, for localization purposes in the Hubbard model, it suffices to study $H_{\mathrm{Hub}}$ and its finite volume restrictions.
Outline of the Proof of theorem \[main\]
=========================================
We now want to outline the proof of the theorem \[main\] in the related model where $H_{\mathrm{Hub}}$ is replaced by the operator $$\label{toymodel}
H=H_0+\lambda\omega(n)+gV_{\mathrm{eff}}(n)$$ acting on $\ell^2\left(\mathbb{Z}^d\right)$ with $$\label{potentialHubbard2}
V_{\mathrm{eff}}(n)=\mel{n}{\frac{1}{1+e^{\beta(H_0+\lambda \omega+gV_{\mathrm{eff}})}}}{n}.$$ In this case, the correlator is defined as $$Q_{I}(m,n):=\sup_{|\varphi|\leq 1}|\mel{m}{\varphi(H)}{n}|.$$ where $\varphi$ is Borel measurable and supported on $I$.
The above operator exhibits the main mathematical features of the Hubbard model, namely: the effective potential is defined self-consistently as a non-local and non-linear function of $H$. Thus, it is natural to first illustrate our methods here. For now let’s assume the existence and uniqueness of $V_{\mathrm{eff}}$ are proven as well as its regularity with respect to $\{\omega(n)\}_{n\in \mathbb{Z}^d}$. Combined with estimates on the derivatives of $V_{\mathrm{eff}}$, the above facts form a significant portion of the proof which is developed in sections \[existencesection\] and \[regularitysection\]. The, somewhat straightforward, extension of the proof to $H_{\mathrm{Hub}}$ will be explained in section \[hubbardext\]. A feature which theorem \[1dloc\] and theorem \[main\] have in common is that the eigenfunction correlator decay will be achieved via the Green’s function of $H^{\Lambda}=\mathds{1}_{\Lambda}H\mathds{1}_{\Lambda}$, which is $H$ restricted to finite sets $\Lambda \subset \mathbb{Z}^d$. Let $$\label{greensdef}
G^{\Lambda}(m,n,z)=\mel{m}{(H^{\Lambda}-z)^{-1}}{n}.$$ Using the basics of the fractional moment method, which dates back to [@A-M] and [@Aiz], we aim at showing that, for some $s\in(0,1)$, $$\label{Greendecay}\mathbb{E}\left(\Big|G^{\Lambda}(m,n;z)\Big|^s\right)\leq Ce^{-\mu_d|m-n|}$$ holds uniformly in $z\in \mathbb{C}^{+}$, with positive constants $C=C(d,s,g,\lambda,\nu,\eta,\|F\|_{\infty})$ and $\mu(d,s,g,\lambda,\nu,\eta,\|F\|_{\infty})$ independent of the volume $|\Lambda|$. In this context, the Green’s function decay expressed by equation (\[Greendecay\]) implies $$\label{eigencorr}
\mathbb{E}\left(Q(m,n)\right)\leq C'e^{-\mu_{d}'|m-n|}$$ for some exponent $\mu'_d=\mu'(d,s,g,\lambda,\nu,\eta,\|F\|_{\infty})>0$ and $C'=C'(\eta,\nu,d,g,\lambda,s,\|F\|_{\infty})$. This is well known and explained in great generality, for instance, in [@A-S-F-H Theorem A.1].
Another aspect which is shared by the proofs of theorems \[1dloc\] and \[main\] is that the starting point to obtain (\[Greendecay\]) will be the following *a-priori* bound.
\[apriori\] Given a finite set $\Lambda \subset \mathbb{Z}^d$, there exist a constant $C_{\mathrm{AP}}=C_{\mathrm{AP}}(\eta,\nu,d,g,\lambda,s,\|F\|_{\infty})$, independent of $\Lambda$, such that $$\mathbb{E}\left(\Big|G^{\Lambda}(m,n;z)\Big|^s\right)\leq C_{\mathrm{AP}}$$ holds for any $m,n\in \Lambda$.
The proof of lemma \[apriori\] will follow from lemma \[bdddensity\] below. Let $$\label{U}
U_{\omega}(n)=\omega(n)+\frac{g}{\lambda}V_{\mathrm{eff}}(n,\omega).$$ be the “full” potential at site $n$. From now on, to keep the notation simple, we drop the dependence on $\omega$ in the new variables $\{U(n)\}_{n\in \Lambda}$. Note that $U(n)$ and $U(m)$ are correlated for all values of $m$ and $n$. The strategy is to show that, for $g$ sufficiently small, they still behave as if they were independent in the following sense:
\[bdddensity\] Fix $\Lambda \subset \mathbb{Z}^d$ finite and $n_0\in \Lambda$. The conditional distribution of $U(n_0)=u$ at specified values of $\{U(n)\}_{n\in {\Lambda \setminus\{n_0\}}}$ has density ${\rho}^{\Lambda}_{n_0}$. Moreover, under assumptions $(A_1)-(A_4)$ we have that $$\sup_{\Lambda}\sup_{n_0\in \Lambda} \sup_{u\in \mathbb{R}}{\rho}^{\Lambda}_{n_0}(u)<\infty.$$ If, additionally, assumption \[momentassumption\] holds then ${\rho}^{\Lambda}_{n_0}(u)\in L^{1}\left(\mathbb{R},|x|^{\varepsilon}dx\right)$.
The proof of the above result is detailed in section \[proofoflemma\]; it requires exponential decay of $|\frac{\partial V_{\mathrm{eff}}(n)}{\partial \omega(m)}|$ and $|\frac{\partial^2 V_{\mathrm{eff}}(n)}{\partial \omega(m)\omega(l)}|$ with respect to $|m-n|$ and $|m-n|+|l-n|$, respectively. The need for this decay is the main reason to require $\beta>0$ or, in other words, to require analiticity of $F$ on a strip. The intuitive explanation for lemma \[bdddensity\] is that the random variables $U(n)$ and $U(n_0)$ decorrelate in a strong fashion as $|n-n_0|$ becomes large. Lemma \[bdddensity\] implies (\[Greendecay\]) for any $0<s<1$ as long as $\lambda$ is taken sufficiently large, see [@A-W-B Theorem 10.2].
The proof of theorem \[1dloc\] will require additional efforts involving tools which are specific to one dimension, which we shall comment on below.
One dimensional aspects: strategy of the proof of theorem \[1dloc\] {#1dideas}
===================================================================
The argument for proving theorem \[1dloc\] follows closely the approach in the proof of theorem 12.11 in [@A-W-B], which we now recall.
Main ideas in the i.i.d case
----------------------------
In the reference [@A-W-B Chapter 12] Green’s function decay is described in terms of the moment generating function, defined by $$\label{momentgen}
\varphi(s,z) =\lim_{|n|\to \infty}\frac{\ln \mathbb{E}{\left(|G(0,n;z)|^s\right)}}{|n|}.$$ The existence of the above quantity for all $z\in \mathbb{C}^{+}$ and $s\in (0,1)$ and its relationship to the Lyapunov exponent are a consequence of Fekete’s lemma:
[\[Feketeclass\]]{} Let $\{a_n\}_{n\in \mathbb{N}}$ be a sequence of real numbers such that, for every pair $(m,n)$ of natural numbers, $$\label{subadc}
a_{n+m}\leq a_n+a_m$$ Then, $\alpha=\lim_{n\to \infty} \frac{a_n}{n}$ exists and equals $\inf_{n\in \mathbb{N}}\frac{a_n}{n}$.
It is an elementary observation that if, instead, the sequence $\{a_n\}_{n\in \mathbb{N}}$ satisfies $a_{n+m}\leq a_n+a_m+C$ then the above result applies to $b_n:=a_n+C$ and that an analogous statement holds for superadditive sequences, which satisfy (\[subadc\]) with the inequality reversed. In the i.i.d. context, the sequence $a_n=\ln \mathbb{E}\left(|G(0,n;z)|^s\right)$ is shown to be both subbaditive and superadditive, meaning that there exist constants $C_{-}(s,z)$ and $C_{+}(s,z)$ for which $$\label{upperandloweradd}
a_n+a_m+C_{-} \leq a_{n+m}\leq a_n+a_m+C_{+}.$$ holds for all $m,n \in \mathbb{N}$, see [@A-W-B Lemma 12.10]. A consequence of this fact, together with a precise control of the arising constants, is stated below.
\[greenmoment\] [@A-W-B Theorem 12.8] For any $z\in \mathbb{C}^{+}$, there are $c_s(z),C_s(z)\in (0,\infty)$ such that for all $n\in \mathbb{Z}$ $$c^{-1}_{s}(z)e^{\varphi(s,z)|n|}\leq \mathbb{E}\left(|G_{\mathrm{And}}(0,n;z)|^s\right)\leq C_s(z)e^{\varphi(s,z)|n|}.$$ Moreover, for any compact set $K\subset \mathbb{R}$ and $S\subset [-1,1)$, we have the local uniform bound $$\sup_{s\in S}\sup_{z\in K+i(0,1]}\max\{c_s(z),C_s(z)\}<\infty$$ and the same result holds with $z$ replaced by its boundary value $E+i0$ for Lebesgue almost every $E$.
On the other hand, for fixed $z\in \mathbb{C}^{+}$, $\varphi(s,z)$ is shown to be convex function of $s$ and non-increasing in $[-1,+\infty)$, with its derivative at $s=0$ satisfying $\frac{\partial \varphi(0,z)}{\partial s}=-\mathcal{L}(z)$. It is a consequence of these facts that for almost every $E\in \mathbb{R}$ there exists a value $s=s(E)\in (0,1)$ such that $$\label{boundmomentgeniid}
\varphi(s,E)\leq -\frac{s}{2}\mathcal{L}(E).$$ The above is the content of [@A-W-B Equation (12.86)]. Dynamical localization is shown to hold locally as a consequence of the inequality (\[upperandloweradd\]) along with lemma \[greenmoment\], the inequality (\[boundmomentgeniid\]) and Kotani theory, which establishes that $\mathcal{L}(E)$ is positive for almost every $E\in \mathbb{R}$.
Modifications
-------------
In this section we will outline the proof theorem \[1dloc\] with $H_{\mathrm{Hub}}$ again replaced by the operator $H$ on $\ell^2\left(\mathbb{Z}^d\right)$ defined in (\[toymodel\]). For simplicity we set $\lambda=1$ since the disorder strenght does not play an important role in theorem \[1dloc\]. Let $H_{+}=H_{[0,\infty)\cap\mathbb{Z}}$ be the restriction of $H$ to $\ell^2\left(\mathbb{Z}^{+}\right)$ and denote by $G^{+}(m,n;z)$ the Green’s function of $H^{+}$. Recall the definition of the Lyapunov exponent: initially, for $z\in \mathbb{C}^{+}$, we let $$\mathcal{L}(z)=-\mathbb{E}\left(\ln|G^{+}(0,0;z)|\right).$$ By Herglotz theory (see, for instance, [@A-W-B Appendix B] and references therein) it is seen that, for Lebesgue almost every $E\in \mathbb{R}$, $\mathcal{L}(E)$ is well defined as $\lim_{\delta \to 0^{+}} \mathcal{L}(E+i\delta)$. Finally, recall the uniform positivity of the Lyapunov exponent for the Anderson model on $\ell^2\left( \mathbb{Z}\right)$: $$\mathrm{ess}\inf_{E\in \mathbb{R}}\mathcal{L}_{\mathrm{And}}(E)>\mathcal{L}_{\mathrm{And}}$$ for some $\mathcal{L}_{\mathrm{And}}>0$. The first step towards Green’s function decay (\[Greendecay\]) will be showing uniform positivity of $\mathcal{L}(E)$, which is accomplished by the following.
\[uniformposh\] There exists a constant $C_{\mathrm{Lyap}}(s,\eta,g,\|F\|_{\infty})>0$ such that $$|\mathcal{L}(z)-\mathcal{L}_{\mathrm{And}}(z)|\leq C_{\mathrm{Lyap}}|g|^s$$ for all $z\in \mathbb{C}^{+}$.
From the resolvent identity we obtain $$\frac{|G^{+}(0,0;z)|}{|G^{+}_{\mathrm{And}}(0,0;z)|}\leq 1+|g|\|F\|_{\infty}\sum_{n}|G^{+}(0,n;z)|\frac{|G^{+}_{\mathrm{And}}(n,0;z)|}{|G^{+}_{\mathrm{And}}(0,0;z)|}$$
$$\frac{|G^{+}_{\mathrm{And}}(0,0;z)|}{|G^{+}(0,0;z)|}\leq 1+|g|\|F\|_{\infty}\sum_{n}|G^{+}_{\mathrm{And}}(0,n;z)|\frac{|G^{+}(n,0;z)|}{|G^{+}(0,0;z)|}$$
Using the bound $\ln(1+x)\leq \frac{x^{s}}{s}$ for $0<s<1$ and $x>0$ we reach, for $0<s<1/2$, $$\ln\left(\frac{|G^{+}(0,0;z)|}{|G^{+}_{\mathrm{And}}(0,0;z)|}\right)\leq \frac{|g|^{s}}{s}\|F\|^{s}_{\infty}\sum_{n}|G^{+}(0,n;z)|^s\frac{|G^{+}_{\mathrm{And}}(n,0;z)|^s}{|G^{+}_{\mathrm{And}}(0,0;z)|^s}.$$ Taking expectations, using the definition of the Lyapunov exponents and the Cauchy-Schwarz inequality $$\mathcal{L}_{\mathrm{And}}(z)-\mathcal{L}(z)\leq \frac{|g|^{s}}{s}\|F\|^{s}_{\infty} \sup_n \mathbb{E}\left(|G^{+}(0,n;z)|^{2s}\right)^{1/2}\sum_n\mathbb{E}\left(\frac{|G^{+}_{\mathrm{And}}(n,0;z)|^{2s}}{|G^{+}_{\mathrm{And}}(0,0;z)|^{2s}}\right)^{1/2}:=C_{\mathrm{Lyap}}(s,\eta,\nu,\|F\|_{\infty})|g|^{s}.$$ The fact that $C_{\mathrm{Lyap}}$ is a finite quantity follows from a couple of remarks. Firstly, by Feenberg’s expansion [@A-W-B Theorem 6.2] we have the identity $$\label{feenberg}
|G^{+}_{\mathrm{And}}(n,0;z)|=|G^{+}_{\mathrm{And}}(0,0;z)||G^{+}_{\mathrm{And}}(1,n;z)|$$ where $G^{+}_{\mathrm{And}}(1,n;z)$ denotes the Green’s function of $H_{\mathrm{And}}$ restricted to $\ell^{2}\left(\mathbb{Z} \right)\cap[1,\infty)$. From the *a-priori* fractional moment bound on lemma \[apriori\] combined with the Green’s function decay for dimensional Anderson model $$\mathbb{E}\left(|G^{+}_{\mathrm{And}}(1,n;z)|^{2s}\right)<C(s)e^{-\mu_{\mathrm{And}}|n|}$$ we conclude that that $C_{\mathrm{Lyap}}<\infty$. The estimate for $\mathcal{L}(z)-\mathcal{L}_{\mathrm{And}}(z)$ is similar.
In principle one might worry that the pre-factor $C_{\mathrm{Lyap}}$ on the above bound will depend on $g$. However, it is easy to see from the arguments in the proof of lemma \[apriori\], that $C_{\mathrm{AP}}$ converges to a finite quantity as $g\to 0$, thus we shall disregard its dependence on $g$.
\[uniformpos\] Whenever $|g|<\left(\frac{\mathcal{L}_{\mathrm{And}}}{C_{\mathrm{Lyap}}}\right)^{1/s}$ holds for some $s\in (0,1/2)$, we have $$\mathcal{L}_{0}:=\mathrm{ess}\inf_{E\in \mathbb{R}}\mathcal{L}(E)>0.$$
We can now proceed to the second step of the proof of theorem \[1dloc\], which consists of establishing Green’s function decay from corollary \[uniformpos\]. For that purpose, an important detail to keep in mind is that, in the correlated context, if we choose $a_n=\log\mathbb{E}\left(|G(0,n;z)|^s\right)$, the condition (\[upperandloweradd\]) will not be fulfilled for all pairs $(m,n)$ due to the lack of independence between the potentials. This means that Fekete’s lemma is not applicable. Moreover, its well-studied modifications (for instance by P. Erdös and N. G. de Bruijn [@Erdos]) do not seem to suffice either.
To the best of our knowledge the result given below is new. Its formulation takes into account the strong decorrelation between the potentials in the Hubbard model and introduces a notion of approximate subbaditivity.
[\[Fekete\]]{}
Let $\delta>0$ be given and $\{a_n\}_{n\in \mathbb{N}}$ be a sequence of real numbers such that, for every triplet $m, n, r$ of natural numbers with $r\geq \delta\max\{\log m,\log n\}$, the inequality $$\label{subad}
a_{n+m+r}\leq a_n+a_m+C$$ holds with a constant $C$ independent of $m,n$ and $r$. Then, $$\alpha=\lim_{n\to \infty} \frac{a_n}{n}$$ exists and equals $\inf_{n\in \mathbb{N}}\frac{a_n+C}{n}$ . Moreover, $\alpha\in [-\infty,0]$.
Note that, as a consequence, we have $$\label{relatemomgren}
a_n\geq n\alpha -C$$ for all $n\in \mathbb{N}$, where $C$ is the same constant as in (\[subad\]). The following decoupling estimate guarantees the applicability of the above lemma with the choice $a_n=\log\mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right)$, where $\hat G(0,n;z)=\mel{0}{(H_{[0,n]}-z)^{-1}}{n}$ is the Green’s function of the operator $H$ restricted to $\ell^2\left([0,n]\cap \mathbb{Z}\right)$ and $\mathbb{E}_{[0,n]}$ denotes the expectation with respect to $U(0),...U(n)$.
\[mixinglem\]\[Strong mixing decoupling\] There exist positive numbers $C_{\mathrm{Dec}}(s,\nu,\eta,g,\|F\|_{\infty})$ and $\delta=\delta(\eta,\nu,g,\|F\|_{\infty})$ such that the inequality $$\mathbb{E}_{[0,n+m+r]}\left(|\hat G(0,n+m+r;z)|^s \right)\leq C_{\mathrm{Dec}}
\mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right) \mathbb{E}_{[0,m]}\left(|\hat G(0,m;z)|^s \right)$$ holds whenever $r\geq \delta \log\max\{m,n\}$.
A combination of lemmas \[mixinglem\], \[Fekete\] and equation (\[relatemomgren\]) yields the lower bound $$\label{lowerbound}
C^{-1}_{\mathrm{Dec}}e^{\varphi(s,z)n}\leq \mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right)\,\,\,\,\,\mathrm{for\,\, all}\,\,\,n\in \mathbb{N}.$$ As we shall see in section \[detailslowerbound\] below, an application of the lower bound (\[lowerbound\]) in combination with the superadditive version of lemma \[Fekete\] applied to the sequence $b_n=-\varphi(s,E)n+\log\mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right)$ is enough to establish an upper bound $$\label{upperbound}
\mathbb{E}\left(|\hat G(0,n;z)|^s \right)\leq C(s,z)e^{\varphi(s,z)n}\,\,\,\,\,\mathrm{for\,\, all}\,\,\,n\in \mathbb{N}.$$ where the constant $C(s)$ is locally uniform in $(s,z)\in (0,1)\times \mathbb{C}^{+}$.
After obtaining an analogue of lemma \[greenmoment\] , the final step will be to relate the moment-generating function to the Lyapunov exponent through an inequality of the type $$\label{boundmomentgen}
\varphi(s,E)\leq -\frac{s}{2}\mathcal{L}_{0}.$$ In reference [@A-W-B], the bound (\[boundmomentgen\]) is stated with $\mathcal{L}_{0}$ replaced by $\mathcal{L}(E)$ and with $s$ depending on $E$. However, it is easy to see from the arguments given there that $s$ can be chosen locally uniformly in $E$, see [@A-W-B Equations (12.79) and (12.80)]. Moreover, by making use uniform positivity of the Lyapunov exponent obtained in corollary \[uniformpos\] we reach the inequality (\[boundmomentgen\]). The Green’s function decay follows from the bounds (\[upperbound\]) and (\[boundmomentgen\]).
The remainder of the paper is organized as follows. Sections \[existencesection\], \[regularitysection\] and \[decaysection\] establish existence, regularity and decay properties of the effective potential in the model (\[toymodel\]). Section \[proofoflemma\] contains the proof of lemma \[bdddensity\]. Section \[hubbardext\] explains the required modifications for the Hubbard model. The proof of theorem \[main\] is finished combining sections \[proofoflemma\] and \[hubbardext\]. Section \[details1d\] covers the remaining details of the proof of theorem \[1dloc\] and section \[idssection\] contains the proof of theorem \[thmids\].
Existence of the effective potential {#existencesection}
====================================
To justify the definition of the effective potential in (\[toymodel\]), let $\Phi(V):\ell^{\infty}(\mathbb{Z}^d)\rightarrow \ell^{\infty}(\mathbb{Z}^d)$ be given by $\Phi(V)(n):=\mel{n}{F(T+\lambda V_{\omega}+gV)}{n}.$ Recall that $F$ is analytic, bounded on the strip $S=\{|\mathrm{Im} z|<\eta\}$ and continuous up to the boundary of $S$. Our goal is to check that $\Phi$ is a contraction in $\ell^{\infty}\left( \mathbb{Z}^d\right)$, meaning that $$\label{contractiondef}\|\Phi(V)-\Phi(W)\|_{\ell^{\infty}(\mathbb{Z}^d)}<c\|V-W\|_{\ell^{\infty}(\mathbb{Z}^d)}$$ holds for some $c<1$ and all $V,W \in \ell^{\infty}(\mathbb{Z}^d)$. Using the analiticity of $F$ we have the following representation [@A-Graf Equation (D.2)]
$$F(T+\lambda V_{\omega}+gV)=\frac{1}{2\pi i}\int^{\infty}_{-\infty}\left(\frac{1}{T+\lambda V_{\omega}+gV-i\eta+t}-\frac{1}{T+\lambda V_{\omega}+gV+i\eta+t}\right) f(t)\,dt$$
for all $V\in \ell^{\infty}(\mathbb{Z}^d)$, where $ f=F_{+} + F_{-} + D\ast F $ for $F_{\pm}(u)=F(u\pm i\eta)$ and $D(u)=\frac{\eta}{\pi\left( \eta^2+u^2\right)}$ is the Poisson kernel. It follows immediately that $\|f\|_{\infty}\leq 3\|F\|_{\infty}$. This is a prelude for the following fixed point argument, where the operator $T$ will be assumed to satisfy $$\label{offdecayssump}
\sup_{n}\sum_m|T(m,n)|\left(e^{\nu|m-n|}-1\right)<\frac{\eta}{2}.$$
\[Contraction\]
1. [ For any self-adjoint operator $T$ on $\ell^2\left(\mathbb{Z}^d\right)$ satisfying (\[offdecayssump\]) and bounded potentials $V,W$, we have, for any $\nu'\in (0,\nu)$, that $$\label{contraction1} \Big|\mel{m}{\left(F(T+V)-F(T+W)\right)}{n}\Big|\leq \frac{72\sqrt{2}e^{-\nu'|m-n|}}{\eta\left(1-e^{\nu'-\nu}\right)^d}\|F\|_{\infty}\|V-W\|_{\infty}.$$]{}
2. [ For any self-adjoint operator $T$ on $\ell^2\left(\mathbb{Z}^d\right)\oplus \ell^2\left(\mathbb{Z}^d\right)$ satisfying (\[offdecayssump\]) and bounded potentials $V,W$ on $\ell^2\left(\mathbb{Z}^d\right)\oplus \ell^2\left(\mathbb{Z}^d\right)$ we have, for any $\nu'\in (0,\nu)$, that $$\label{contraction2} \Big|\mel{m}{\left(F(T+V)-F(T+W)\right)}{n}\Big|\leq \frac{144\sqrt{2}e^{-\nu'|m-n|}}{\eta\left(1-e^{\nu'-\nu}\right)^{d}}\|F\|_{\infty}\|V-W\|_{\infty}$$ ]{}
3. [ For any $m,n,j \in\mathbb{Z}^d$, the matrix elements $\mel{m}{F(T+gV)}{n}$ are differentiable with respect to $V(j)$ and $$\label{contraction3}\Big|\frac{\partial\mel{m}{F(T+gV)}{n}}{\partial V(j)}\Big|\leq |g|\frac{72\sqrt{2}e^{-\nu\left(|m-j|+|n-j|\right)}}{\eta}\|F\|_{\infty}\|V\|_{\infty}.$$]{}
The resolvent identity gives $$\begin{aligned}
& \mel{m}{\frac{1}{T+V-t-i\eta}-\frac{1}{T+W-t-i\eta}}{n}+\mel{m} {\frac{1}{T+W-t+i\eta}-\frac{1}{T+V-t+i\eta}}{n}\\
&=\mel{m}{(\frac{1}{T+V-t-i\eta}-\frac{1}{T+V-t+i\eta})(W-V)\frac{1}{T+W-t-i\eta}}{n}-\\
&\mel{m}{ \left(\frac{1}{T+W-t+i\eta}-\frac{1}{T+W-t-i\eta}\right)(W-V)\frac{1}{T+V-t+i\eta}}{n}.\\\end{aligned}$$ Taking absolute values in the first term on the right-hand side we obtain
$$\begin{aligned}
&\Big|\mel{m}{\left(\frac{1}{T+V-t-i\eta}-\frac{1}{T+V-t+i\eta}\right)(W-V)\frac{1}{T+W-t-i\eta}}{n}\Big| \\
&\leq \sum_{l\in\mathbb{Z}^d}|G^{V}(m,l;t+i\eta)-G^{V}(m,l;t-i\eta)||(W-V)(l)|G^{W}(l,n;t+i\eta)|\\
&\leq 24\sum_{l}|(V-W)(l)|e^{-\nu\left(|l-n|+|l-m|\right)} \mel{m}{\frac{1}{(T+V-t)^2+{\eta^2}/2}}{m}^{1/2}
\mel{l}{\frac{1}{(T+V-E)^2+{\eta^2}{/2}}}{l}^{1/2}.\\\end{aligned}$$
In the last step we made use of the Combes-Thomas bound $|G^{W}(m,n;t+i\eta)|\leq \frac{2}{\eta}e^{-\nu|m-n|}$ as well as lemma 3 in [@A-Graf appendix D] to estimate the difference between the Green functions as $$\begin{aligned}
|G^{V}(m,l;t+i\eta)-G^{V}(m,l;t-i\eta)|\leq 12\eta e^{-\nu|m-l|} \mel{m}{\frac{1}{(T+V-t)^2+{\eta^2}/2}}{m}^{1/2}
\mel{l}{\frac{1}{(T+V-E)^2+{\eta^2}{/2}}}{l}^{1/2}.\end{aligned}$$ Integrating over $t$ we conclude, using Cauchy-Schwarz and the spectral measure representation, that $$\begin{aligned}
\label{resolventdiff}&\int^{\infty}_{-\infty}\left|\mel{m}{ \left(\frac{1}{T+V-t-i\eta}-\frac{1}{T+V-t+i\eta}\right)(W-V)\frac{1}{T+W-E-i\eta}}{n}\right|\,dt\\
&\leq
\frac{24\sqrt{2}\pi}{\eta}\sum_{l}|(V-W)(l)|e^{-\nu\left(|l-n|+|l-m|\right)}.\end{aligned}$$ The above implies that $$\begin{aligned}
&\frac{1}{2\pi}\int^{\infty}_{-\infty}\Big| \mel{m}{ \left(\frac{1}{T+V-t-i\eta}-\frac{1}{T+V-t+i\eta}\right)(W-V)\frac{1}{T+W-t-i\eta}}{n}\Big|\,dt\\
&\leq \frac{12\sqrt{2}}{\eta}\|V-W\|_{\infty}e^{-\nu'|m-n|}\sum_{l\in \mathbb{Z}^d}e^{(\nu'-\nu)|l-n|}\\
&=\frac{12\sqrt{2}}{\eta}\|V-W\|_{\infty}e^{-\nu'|m-n|}\frac{1}{\left(1-e^{\nu'-\nu}\right)^d}.\\
\end{aligned}$$ As a similar bound holds for $\frac{1}{2\pi}\int^{\infty}_{-\infty}\Big|\mel{m}{\left(\frac{1}{H_0+W-t+i\eta}-\frac{1}{H_0+W-t-i\eta}\right)(V-W)\frac{1}{H_0+V-t+i\eta}}{n}\Big|\,dt$ , we conclude the proof of the inequality (\[contraction1\]) by recalling that $\|f\|_{\infty}\leq 3 \|F\|_{\infty}$. The inequality (\[contraction2\]) in the statement of proposition \[Contraction\] follows from the same argument with the only difference that one has to sum two geometric series, hence the modification on the upper bound. The bound (\[contraction3\]) is proven similarly: note that $h\frac{12\sqrt{2}\pi}{\eta}e^{-\nu|j-n|} e^{-\nu|m-j|}$ is an upper bound for the left-hand side of equation (\[resolventdiff\]) with $V$ replaced by $gV$ and $W=g(V+hP_{j})$, where $P_j$ denotes the projection onto $\mathrm{Span}\{\delta_j\}$. We also observe that this time there will be no summation over $l$, hence the introduction of the $\nu'$ is unnecessary. We then conclude that $$\label{differencequotient}
\Big|\frac{\mel{m}{F(T+gV+hP_j)}{n}-\mel{m}{F(T+gV)}{n}}{h}\Big|\leq \frac{72\sqrt{2}\pi}{\eta}e^{-\nu|j-n|} e^{-\nu|m-j|}.$$ Letting $h\to 0$ finishes the proof.
Taking $m=n$, as a consequence of the above proposition, (\[contractiondef\]) holds whenever $$\label{smallg}
|g|<\frac{\eta\left(1-e^{-\nu}\right)^d }{72\sqrt{2}\|F\|_{\infty}}.$$ This observation yields the following.
\[existencepot\] Let $g_d=\frac{\eta\left(1-e^{-\nu}\right)^d}{72\sqrt{2}\|F\|_{\infty}}$. Then, for $|g|<g_d$, there is a unique effective potential $V_{\mathrm{eff}}\in \ell^{\infty}\left(\mathbb{Z}^d\right)$ satisfying $$\label{toy}
V_{\mathrm{eff}}(n)=\mel{n}{F(H_0+\lambda \omega+gV_{\mathrm{eff}}}{n}.$$ Moreover, for $\Lambda \subset \mathbb{Z}^d$, there is a unique $V^{\Lambda}_{\mathrm{eff}}$ in $ \ell^2\left(\Lambda\right)$ satisfying (\[toy\]) with $H$ replaced by $H^{\Lambda}=\mathds{1}_{\Lambda}H\mathds{1}_{\Lambda}$.
Replacing $H_0$ by $H_0-\kappa I$ we can incorporate a chemical potential in our results. For simplicity, we shall make no further reference to $\kappa$ during the proofs and assume it was already incorporated to $H_0$.
Regularity of the effective potential {#regularitysection}
=====================================
Our goal in this section is to conclude that, for a fixed finite subset $\Lambda \subset \mathbb{Z}^d$ with $|\Lambda|=n$, the effective potential $V_{\mathrm{eff}}$ is a smooth function of $\{\omega(j)\}_{j\in \Lambda}$. This will be of relevance for several resampling arguments later in the note. For that purpose, define a map $\xi:\ell^{\infty}\left(\mathbb{Z}^d \right)\times \mathbb{R}^{n}\rightarrow \ell^{\infty}\left(\mathbb{Z}^d \right)$ by $$\xi(V,\omega)(j)=V(j)-\mel{j}{F(H_0+\lambda\omega+gV)}{j}$$ Then, $V_{\mathrm{eff}}$ is the unique solution of $\xi(V,\omega)=0$. Thus, its regularity can inferred via the implicit function theorem once we check that the derivative $D\xi(\cdot,\omega)$ is non-singular. Note that $$\frac{\partial \xi(V,\omega)(j)}{\partial V(l)}=
\delta_{jl}-\frac{\partial \mel{j}{F(H_0+\lambda\omega+gV)}{j}}{\partial V(l)}.$$ Using lemma \[Contraction\], we have that $$\Big|\frac{\partial \mel{j}{F(H_0+\lambda\omega+gV)}{j}}{\partial V(l)}\Big|\leq
|g|\frac{72\sqrt{2}e^{-2\nu|j-l|}}{\eta}\|F\|_{\infty}.$$
In particular, whenever $|g|\frac{72\sqrt{2}\|F\|_{\infty}}{\eta(1-e^{-2\nu})^{d}}<1$ we have that the operator $D\xi(\omega,.):\ell^{\infty}\left(\Lambda\right)\rightarrow \ell^{\infty}\left(\Lambda \right)$ is invertible since it has the form $I+gM$ where $gM$ has operator norm less than one. Note the smallness condition on $g$ is independent of $\Lambda\subset \mathbb{Z}^d$. It is a consequence of the implicit function theorem that $V$ is a smooth function of $(\omega(1),...,\omega(n))$.
Decay estimates for the effective potential {#decaysection}
===========================================
We start this section with the following lemma, which will be useful to formulate the decay of correlations between $U(n)$ and $U(m)$ as $|m-n|\to \infty$.
\[decay1\] Whenever $\frac{72\sqrt{2}|g|\|F\|_{\infty}}{\eta\left(1-e^{-\nu}\right)^d}<1$, there exist constants $C_1(d,\lambda,g,\eta,\|F\|_{\infty},\nu)$ and $C_2(d,\lambda,g,\eta,\|F\|_{\infty},\nu)$ such that $$\label{lemmapotential1}\max\Big\{\sum_{m}e^{\nu|n-m|}\Big|\frac{\partial V_{\mathrm{eff}}(n)}{\partial \omega(m)}\Big|,\sum_{n}e^{\nu|n-m|}\Big|\frac{\partial V_{\mathrm{eff}}(n)}{\partial \omega(m)}\Big|\,\Big\}\leq C_1$$
$$\label{lemmapotential2}
\sum_{l,m,n}e^{\nu\left(|l-n|+|n-m|+|l-m|\right)}\Big|\frac{\partial^2V_{\mathrm{eff}}(n)}{\partial \omega(m)\partial \omega(l)}\Big|\leq C_2.$$
Moreover $C_1$ and $C_2$ can be bounded from above by a constant of the form $\frac{\lambda D}{1-g\theta}$ with $D$ and $\theta$ independent of $g$ and these constants are explicit in the proof.
For convenience we denote $V_{\mathrm{eff}}=V$. As in section \[existencesection\] we write $F(H)=\frac{1}{2\pi i}\int^{\infty}_{-\infty}\left(\frac{1}{H+t-i\eta}-\frac{1}{H+t+i\eta}\right) f(t)\,dt$ where $f$ is bounded by $3\|F\|_{\infty}$. Thus $$\label{intrep}
V(n,\omega)=\frac{1}{2\pi i}\int^{\infty}_{-\infty} K(n,t,\omega)f(t)\,dt$$ where $K(n,t,\omega)=G(n,n;t-i\eta)-G(n,n;t+i\eta)$. Denote by $P_{m}$ the projection mapping $\ell^2\left(\mathbb{Z}^d\right)$ onto $\ell^2\left(\mathrm{Span}\{\delta_m\}\right)$. Using difference quotients, it is easy to check $$\label{ResDeriv}
\frac{\partial}{\partial \omega(m)}\frac{1}{H-z}+ g \frac{1}{H-z}\frac{\partial V}{\partial \omega(m)}\frac{1}{H-z}=- \lambda\frac{1}{H-z}P_{m}\frac{1}{H-z}.$$ Taking matrix elements we obtain
$$\frac{\partial K(n,t,\omega)}{\partial \omega(m)}=-g\sum_{l}\tilde{G}(l,n)\frac{\partial V(l)}{\partial \omega(m)}+
\lambda r(m,n).$$ $$\tilde{G}(l,n):=G(l,n;t+i\eta)G(n,l;t+i\eta)-G(l,n;t-i\eta)G(n,l;t-i\eta).$$ $$r(m,n):=G(n,m;t+i\eta)G(m,n;t+i\eta)-G(n,m;t-i\eta)G(m,n;t-i\eta).$$ Note $$\label{rewriteG}\tilde{G}(l,n)=\left(G(l,n;t+i\eta)-G(l,n;t-i\eta)\right)G(n,l;t+i\eta)+\left(G(n,l;t+i\eta)-G(n,l;t-i\eta)\right)G(l,n;t-i\eta).$$ We now make use of [@A-Graf Lemma 3]: $$\label{cancellemma}|G(l,n;t+i\eta)-G(l,n;t-i\eta)|\leq 12\eta e^{-\nu|l-n|}\mel{n}{\frac{1}{(H-t)^2+\eta^2/2}}{n}^{1/2}\mel{l}{\frac{1}{(H-t)^2+\eta^2/2}}{l}^{1/2}.$$ This, together with the Combes-Thomas bound $|G(l,n,t\pm i\eta)|\leq \frac{2}{\eta}e^{-\nu|l-n|}$ and (\[rewriteG\]) implies $$|\tilde{G}(l,n)|\leq 48e^{-2\nu|l-n|}\mel{n}{\frac{1}{(H-t)^2+\eta^2/2}}{n}^{1/2}\mel{l}{\frac{1}{(H-t)^2+\eta^2/2}}{l}^{1/2}.$$ $$|r(m,n)|\leq 48e^{-2\nu|m-n|}\mel{m}{\frac{1}{(H-t)^2+\eta^2/2}}{m}^{1/2}\mel{n}{\frac{1}{(H-t)^2+\eta^2/2}}{n}^{1/2}.$$ Thus $$\tilde{K}(l,n):=\int^{\infty}_{-\infty}|\tilde{G}(l,n)|\,dt\leq\frac{48\sqrt2 \pi}{\eta}e^{-2\nu|l-n|}$$ $$\tilde{r}(m,n):=\int^{\infty}_{-\infty} |r(m,n)|\,dt\leq\frac{48\sqrt2 \pi}{\eta}e^{-2\nu|m-n|}.$$ To summarize, we have shown the following inequality $$\Big|\frac{\partial V(n)}{\partial \omega(m)}\Big |\leq \frac{3\|F\|_{\infty}}{2\pi}\left(|g|\sum_{l}\tilde{K}(l,n)\Big|\frac{\partial V(l)}{\partial \omega(m)}\Big|+
\lambda\tilde{r}(m,n)\right).$$ Whenever $\frac{72\sqrt{2}|g|}{\eta\left(1-e^{-2\nu}\right)^d}<1$ we have that $$\label{smallkernel}
|g|\|\tilde{K}\|_{\infty,\infty}<1$$ where
$$\label{Ksum}\|\tilde{K}\|_{\infty,\infty}=\sup_{l}\sum_{m}\tilde{K}(l,m).$$
Considering the weight $W(n):=e^{\nu|m-n|}$ we let $$\theta:=\frac{3\|F\|_{\infty}}{2\pi}\sup_{n}\sum_{l}\frac{W(n)}{W(l)}\tilde{K}(n,l).$$ By the triangle inequality, $$\begin{aligned}
\\
\theta&\leq \frac{3\|F\|_{\infty}}{2\pi}\sup_{n}\sum_{l}e^{\nu|n-l|}\tilde{K}(n,l)\\
&\leq \frac{72\sqrt2 \|F\|_{\infty}}{\eta\left(1-e^{-\nu}\right)^d}.\\\end{aligned}$$ hence, whenever $\frac{72\sqrt{2}|g|}{\eta\left(1-e^{-\nu}\right)^d}<1$, we have that $$\label{smallkernel2} |g|\theta<1.$$ Moreover, with the choice $$D_1:=\sum_{n}W(n)\tilde{r}(m,n)$$ we have $$\label{smallkernel3}
D_1\leq \frac{72\sqrt2 \|F\|_{\infty}}{\eta\left(1-e^{-\nu}\right)^d}.$$
After conditions (\[smallkernel\]), (\[smallkernel2\]) and (\[smallkernel3\]) have been verified, the general result [@A-W-B Theorem 9.2] applies, yielding $$\sum_{m}e^{\nu|n-m|}\Big|\frac{\partial V(n)}{\partial \omega(m)}\Big|<\frac{\lambda D_1}{1-g\theta}:=C_1(d,\|F\|_{\infty},\lambda,g,\eta,\nu).$$
Differentiating (\[ResDeriv\]) with respect to $\omega(l)$, $$\begin{aligned}
\frac{\partial^2}{\partial \omega(m)\partial \omega(l)}\frac{1}{H-z}&+ g \left(\frac{\partial}{\partial \omega(l)}\frac{1}{H-z}\right)\frac{\partial V}{\partial \omega(m)}\frac{1}{H-z}
+g \frac{1}{H-z}\frac{\partial V}{\partial \omega(m)}\left(\frac{\partial}{\partial \omega(l)}\frac{1}{H-z}\right)\\
+g\frac{1}{H-z}\frac{\partial^2 V}{\partial \omega(m)\partial \omega(l)}\frac{1}{H-z}
&=-\lambda\left(\frac{\partial}{\partial \omega(l)} \frac{1}{H-z}\right)P_{m}\frac{1}{H-z}-\lambda\frac{1}{H-z}P_{m}\left(\frac{\partial}{\partial \omega(l)}\frac{1}{H-z}\right)\\\end{aligned}$$
Repeating the previous argument and using the established decay of $\frac{\partial V(n)}{\partial \omega(m)}$ we reach (\[lemmapotential2\]), finishing the proof.
Given a finite set $\Lambda \subset \mathbb{Z}^d$, let us define $\mathcal{T}:\mathbb{R}^{|\Lambda|} \rightarrow \mathbb{R}^{|\Lambda|}$ by $$\left(\mathcal{T}\omega\right)(n)=\omega(n)+\frac{g}{\lambda}V_{\mathrm{eff}}(n).$$ Let $U(n):=\left(\mathcal{T}\omega\right)(n)$ be the new coordinates in the probability space. The bound (\[lemmapotential1\]) implies that, for $|g|$ sufficiently small, $\mathcal{T}$ is a differentiable perturbation of the identity by an operator with norm less than one hence $\mathcal{T}^{-1}$ is well defined.
Fix $n_0 \in \Lambda$ and denote by $U_{\alpha}=U+\left(\alpha-U(n_0)\right)\delta_{n_0}$ the new potential obtained from $U$ by setting its value at $n_0$ equal to $\alpha$. Let $\omega_{\alpha}(n)=\left(\mathcal{T}^{-1}U_{\alpha}\right)(n)$. The variables $\omega_{\alpha}(n)$ correspond to the change in $\omega(n)$ when a resampling argument is applied to the new probability space at the point $n_0$. Intuitively, the exponential decay guarantees that this change is not too large if $n$ and $n_0$ are far away. This is the content of the lemma below.
\[resamp\] For all $\alpha\in \mathbb{R}$ and $|g|<\lambda C^{-1}_1$, we have $$\sum_{n\neq n_0} e^{\nu |n-n_0|}\big|\omega_\alpha(n)-\omega(n)\big|\leq
\frac{C_1|g|}{\lambda}\frac{\left(|\alpha-U(n_0)|+2\frac{|g|\|F\|_{\infty}}{\lambda}\right)}{\left(1-\frac{|g|}{\lambda}C_1\right)}.$$ where $C_1$ is the upper bound on equation (\[lemmapotential1\]).
Using the given definitions and the mean value inequality we obtain, for $n\neq n_0$, $$\begin{aligned}
|\omega(n)-\omega_{\alpha}(n)|&\leq \frac{|g|}{\lambda}|V(n,\omega)-V(n,\omega_{\alpha})|\\
&\leq\frac{|g|}{\lambda}\sum_{l\in \mathbb{Z}^d} \Big|\frac{\partial V_{\mathrm{eff}}(n,\hat{\omega}_{\alpha})}{\partial \omega(l)}\Big|\Big|\omega_{\alpha}(l)-\omega(l)\Big|\\
&\leq \frac{|g|}{\lambda}\Big|\frac{\partial V_{\mathrm{eff}}(n,\hat{\omega}_{\alpha})}{\partial \omega(n_0)}\Big|\left(|\alpha-U(n_0)|+2\frac{|g|\|F\|_{\infty}}{\lambda}\right)+\frac{|g|}{\lambda}\sum_{l\neq n_0} \Big|\frac{\partial V_{\mathrm{eff}}(n,\hat{\omega}_{\alpha})}{\partial \omega(l)}\Big|\Big|\omega_{\alpha}(l)-\omega(l)\Big|.\end{aligned}$$ Where $\hat{\omega}_{\alpha}$ denotes some configuration with $\hat{\omega}_{\alpha}(l)$ in the interval connecting $\omega(l)$ to $\omega_{\alpha}(l)$. Let $W(n)=e^{\nu|n-n_0|}$. According to (\[lemmapotential1\]), $$\begin{aligned}
\sup_n\sum_l \frac{W(n)}{W(l)}\Big|\frac{\partial V_{\mathrm{eff}}(n,\hat{\omega}_{\alpha})}{\partial \omega(l)}\Big|&\leq\sup_n\sum_l e^{\nu|n-l|}\Big|\frac{\partial V_{\mathrm{eff}}(n,\hat{\omega}_{\alpha})}{\partial \omega(l)}\Big|\\
&\leq C_1.\\\end{aligned}$$ Once again, the conditions of [@A-W-B Theorem 9.2] are satisfied for $|g|<\lambda C^{-1}_1$ therefore $$\sum_{n\neq n_0} e^{\nu|n-n_0|}\big|\omega_{\alpha}(n)-\omega(n)\big|\leq \frac{C_1|g|}{\lambda}\frac{\left(|\alpha-U(n_0)|+2\frac{|g|\|F\|_{\infty}}{\lambda}\right)}{\left(1-\frac{|g|}{\lambda}C_1\right)}.\qedhere$$
Since another application of the mean value theorem gives, after a possible correction on $\hat{\omega}_{\alpha}$ that $$\begin{aligned}
\Big|\frac{\partial V_{\mathrm{eff}}(n,\omega)}{\partial \omega(m)}-\frac{\partial V_{\mathrm{eff}}(n,\omega_{\alpha})}{\partial \omega(m)}\Big|&\leq
\sum_{l\in \mathbb{Z}^d}\Big| \frac{\partial^2 V_{\mathrm{eff}}(n,\hat{\omega}_{\alpha})}{\partial \omega(l)\partial \omega(m)}\Big|\Big|\omega(l)-\omega_{\alpha}(l)\Big|\\\end{aligned}$$ we obtain, for any $\nu'\in(0,\nu)$, $$\begin{aligned}
\Big|\frac{\partial V_{\mathrm{eff}}(n,\omega)}{\partial \omega(m)}-\frac{\partial V_{\mathrm{eff}}(n,\omega_{\alpha})}{\partial \omega(m)}\Big|\leq&\frac{C_2|g|}{\lambda}\left(C_1\frac{\left(|\alpha-U(n_0)|+2\frac{|g|\|F\|_{\infty}}{\lambda}\right)}{\left(1-\frac{|g|}{\lambda}C_1\right)(1-e^{\nu'-\nu})^d}+2\|F\|_{\infty} \right)e^{-\nu'\left(|m-n|+|n-n_0|+|m-n_0|\right)}.\end{aligned}$$ where $C_2$ is the constant in (\[lemmapotential2\]).
In particular, letting $\nu'=\nu/2$, if $A=\frac{g}{\lambda}\left(\frac{\partial V_{\mathrm{eff}}(n_i,\omega_{\alpha})}{\partial \omega(n_j)}\right)_{|\Lambda|\times |\Lambda|}$ and $B=\frac{g}{\lambda}\left(\frac{\partial V_{\mathrm{eff}}(n_i,\omega)}{\partial \omega(n_j)}\right)_{|\Lambda|\times |\Lambda|}$ we have $$\label{explicitboundsum}\sum_{(m,n)\in \Lambda \times \Lambda}|(A-B)_{m,n}|\leq \frac{C_2|g|^2}{\lambda^2}\left(C_1\frac{\left(|\alpha-U(n_0)|+2\frac{|g|\|F\|_{\infty}}{\lambda}\right)}{\left(1-\frac{|g|}{\lambda}C_1\right)(1-e^{-\nu/2})^{3d}}+\frac{2\|F\|_{\infty}}{(1-e^{-\nu/2})^{2d}} \right).$$ We summarize the above observation as a lemma.
\[bddtrace\] Let $A=\frac{g}{\lambda}\left(\frac{\partial V_{\mathrm{eff}}(n_i,\omega_{\alpha})}{\partial \omega(n_j)}\right)_{|\Lambda|\times |\Lambda|}$ and $B=\frac{g}{\lambda}\left(\frac{\partial V_{\mathrm{eff}}(n_i,\omega)}{\partial \omega(n_j)}\right)_{|\Lambda|\times |\Lambda|}$. Whenever $ \frac{|g|}{\lambda}C_1<1$ we have $$\sum_{(m,n)\in \Lambda \times \Lambda}|(A-B)_{m,n}|\leq |g|^2\left(C_3|\alpha-U(n_0)|+C_4\right).$$ Moreover, the constant $C_3$ can be chosen independent of $\lambda$ and $C_4$ is proportional to $\frac{1}{\lambda}$.
Finally, we analyze how the effective potential varies with respect to disorder and interaction. This will be relevant to the Integrated Density of States regularity. More precisely
\[comparepotentials\] For a fixed $\omega \in \Omega$ $$|V_{\lambda,g}(n)-V_{\lambda',g'}(n)|\leq \frac{C_5(d,\|F\|_{\infty},g,\eta,\nu,\omega)}{1-gC_6(d,\|F\|_{\infty},g,\eta,\nu)}|\lambda-\lambda'| +C_7(d,\|F\|_{\infty},g,\eta,\nu)|g-g'|.$$
Note when $\lambda\neq \lambda'$ the bound depends on $\omega$ through the constant $C_5$.
Let $R_{\lambda,g}(z)=\frac{1}{H_0+\lambda \omega+gV_{\lambda,g}-z}$ and $R_{\lambda',g'}(z)=\frac{1}{H_0+\lambda' \omega+g'V_{\lambda',g'}-z}$ for $z=t+i\eta$. Similarly as in the above proofs, it is immediate to check that $$R_{\lambda,g}(z)-R_{\lambda',g'}(z)=(\lambda-\lambda')R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)+(g-g')R_{\lambda,g}(z)V_{\lambda',g'}R_{\lambda',g'}(z)
-gR_{\lambda,g}(z)\left(V_{\lambda,g}-V_{\lambda',g'}\right)R_{\lambda',g'}(z).$$ Replacing $z$ by $\bar{z}$ and subtracting the resulting equations: $$\begin{aligned}
\left(R_{\lambda,g}(z)-R_{\lambda,g}(\bar{z})\right)-\left(R_{\lambda',g'}(z) -R_{\lambda',g'}(\bar{z})\right)=&
\left(R_{\lambda,g}(z)-R_{\lambda,g}(\bar{z})\right)\left((\lambda-\lambda')V_{\omega}+(g-g')V_{\lambda',g'}\right)R_{\lambda',g'}(z) \\
&+R_{\lambda,g}(z)\left((\lambda-\lambda')V_{\omega}+(g-g')V_{\lambda',g'}\right)\left(R_{\lambda',g'}(z)-R_{\lambda',g'}(\bar{z})\right)\\
&-gR_{\lambda,g}(z)\left(V_{\lambda,g}-V_{\lambda',g'}\right)\left(R_{\lambda',g'}(z)-R_{\lambda',g'}(\bar{z})\right).
\end{aligned}$$ Taking matrix elements, multiplying by $f(t)$, integrating with respect to $t$ and taking absolute values we can read from the representation (\[intrep\]) that, denoting $$\label{differencekernel}
K_{\lambda,g}(n,l)=|G_{\lambda,g}(n,l;z)-G_{\lambda,g}(n,l;\bar{z})|,$$ $$\begin{aligned}
|V_{\lambda,g}(n)-V_{\lambda',g'}(n)|\leq&\frac{3\|F\|_{\infty}}{2\pi}|\lambda-\lambda'|\sum_{l\in \mathbb{Z}^d}|\omega(l)|\int^{\infty}_{-\infty}\left({{K}}_{\lambda,g}(n,l){{K}}_{\lambda',g'}(l,n)+|G_{\lambda,g}(n,l)|{\tilde{K}}_{\lambda',g'}(l,n)\right)\,dt\\
&+\frac{3\|F\|^2_{\infty}}{2\pi}|g-g'|\sum_{l\in \mathbb{Z}^d}\int^{\infty}_{-\infty}\left(|G_{\lambda,g}(n,l)|{{K}}_{\lambda',g'}(l,n)+G_{\lambda',g'}(n,l)|{{K}}_{\lambda,g}(l,n) \right)\,dt.\\
&+g\sum_{l\in \mathbb{Z}^d}\int^{\infty}_{-\infty}|G_{\lambda,g}(n,l)||V_{\lambda,g}(l)-V_{\lambda',g'}(l)|{{K}}_{\lambda',g'}(l,n)\,dt.
\end{aligned}$$ Using equation \[cancellemma\] together with [@A-W-B Theorem 9.2] we conclude the proof.
Improvements {#improvesec}
------------
We will now improve upon the previous bounds. Specifically, we need robust bounds which also reflect the decay of the derivatives of $V_{\mathrm{eff}}(n)$ when the local potential $\omega(n)$ is large. The improvements on this section will be important for a general fluctuation analysis on section \[proofoflemma\] and for localization in the one dimensional setting. Before stating the main result of the section we start with the following deterministic estimate, which incorporates ideas from [@A-Graf Lemma 3].
\[adaptAG\] $$|G(m,l;t+i\eta)|\leq \sqrt{2}\mel{l}{\frac{1}{(H-t)^2+\eta^2}}{l}^{1/2}e^{-\nu|m-l|}$$
To keep the notation simple, we set $t=0$ without loss of generality. Let $H_{f}=e^{f}He^{-f}$ where $f(n)=\nu\min\{|n-l|,R\}$ for a fixed $l\in \mathbb{Z}^d$ and $R>0$. By choosing $R$ sufficiently large, we may assume that $|m-l|<R$. We then have $$\label{trick}e^{\nu|m-l|}G(m,l;i\eta)=\mel{m}{(H_f-i\eta)^{-1}}{l}.$$ We claim that $$\label{claimAG}
||(H_f-i\eta)^{-1}(H^2+\eta^2)^{1/2}||\leq \sqrt{2}.$$ Indeed, $$\begin{aligned}
||(H_f-i\eta)^{-1}(H^2+\frac{\eta^2}{2})^{1/2}||^2&=||(H^2+\frac{\eta^2}{2})^{1/2}(H^{\ast}_f+i\eta)^{-1}(H_f-i\eta)^{-1}(H^2+\frac{\eta^2}{2})^{1/2}||\\
&=||(H^2+\frac{\eta^2}{2})^{1/2}\frac{1}{(H_f-i\eta)(H^{\ast}_f-i\eta)}(H^2+\frac{\eta^2}{2})^{1/2}||
\end{aligned}$$ where by [@A-Graf Eq D.9] (with $f$ replaced by $-f$) we have $$(H_f-i\eta)(H^{\ast}_f-i\eta)\geq \frac{1}{2}\left( H^2+\frac{\eta^2}{2}\right)$$ showing the claim in (\[claimAG\]). Equation (\[adaptAG\]) will now follow from $$\begin{aligned}
|\mel{m}{(H_f-i\eta)^{-1}}{l}|&\leq \|(H_f-i\eta)^{-1}(H^2+\frac{\eta^2}{2})^{1/2}\|\,|(H^2+\frac{\eta^2}{2})^{-1/2}\delta_l|\\
&\leq\sqrt{2}\mel{l}{(H^2+\frac{\eta^2}{2})^{-1}}{l}^{1/2}.\qedhere\end{aligned}$$
\[improvedct\] There exists $C_7(\lambda,\eta,d,g,\|F\|_{\infty},\nu)>0$ such that, for $m\neq n$, $$\label{improvedct1}
\max\{\,|\omega(n)|,|\omega(m)|\,\}\Big|\frac{\partial V(n)}{\partial \omega(m)}\Big|\leq C_7e^{-2\nu|m-n|}$$
and, for $n\neq n_0$, $$\label{improvedct2}\big|\omega(n)(\omega_{\alpha}(n)-\omega(n))\big|\leq
\frac{C_7|g|}{\lambda-|g|C_1}\left(|\alpha-U(n_0)|+2\frac{|g|\|F\|_{\infty}}{\lambda}+\frac{1}{\left(1-e^{-\nu}\right)^d}\right)e^{-\nu|n-n_0|}.$$ Moreover, whenever $\frac{|g|}{\lambda}C_1<1$, $C_7$ can be chosen to be uniformly bounded as a function of the parameters $\lambda$ and $g$.
Recall that $U(n)=\omega(n)+\frac{g}{\lambda}V_{\mathrm{eff}}(n)$ denotes the “full" potential at site $n$. We split the proof in two cases.
1. Case one: $U(n)\geq 0$.
Let us start by noting that lemma \[adaptAG\] implies that for $n,l \in \mathbb{Z}^d$ $$\int^{\infty}_{-\infty}|G(n,l;t+i\eta)G(l,n;t+i\eta)|\,dt\leq\frac{2\sqrt{2}\pi}{\eta}e^{-2\nu|n-l|}.$$
From the previous section we already know that $$\label{repderv}
\frac{\partial V(n)}{\partial \omega(m)}=\int^{\infty}_{-\infty}\left( -g\sum_{l}r(n,l)\frac{\partial V(l)}{\partial \omega(m)}+
\lambda r(m,n)\right)f(t)\,dt$$ where $f(t)=F_{+}(t+i\eta)+F_{-}(t-i\eta)-D\ast F(t)$ and $$r(m,n)=G(n,m;t+i\eta)G(m,n;t+i\eta)-G(n,m;t-i\eta)G(m,n;t-i\eta).$$ Observe that, for $z=t+i\eta$ and $n\neq m$, $$\begin{aligned}
\lambda|U(n)G(n,m;t+i\eta)G(m,n;t+i\eta)|
&=\frac{\lambda|U(n)|}{|\lambda U(n)-z|}\sum_{l} |H_{0}(n,l)G(l,m;t+i\eta)G(m,n,t+i\eta)|\\
&\leq \left(1+\frac{|z|}{|\lambda U(n)-z|}\right)\sum_{l}
|H_{0}(n,l)G(l,m;t+i\eta)G(m,n,t+i\eta)|\end{aligned}$$ where we made use of the identity
$$\label{depleted}
(\lambda U(m)-z)G(n,m;z)=\delta_{mn}-\sum_{l}H_{0}(n,l) G(l,m;z).$$
Note that if $U(n)\geq 0$ and $t=\mathrm{Re}z<0$, then $$\frac{|z|}{|\lambda U(n)-z|}\leq 1.$$ Using the fact that $tf(t)$ goes to zero as $t\to \infty$ we conclude that $$\int^{\infty}_{-\infty}|U(n)G(n,m;t+i\eta)G(m,n;t+i\eta)||f(t)|\,dt\leq \frac{C(\nu,\eta,\|F\|_{\infty})}{\lambda}e^{-2\nu|m-n|}.$$ Since a similar equation holds with $m$ replaced by $l$, we can proceed as in the previous section and, using the exponential decay of $\frac{\partial V(n)}{\partial \omega(m)}$, conclude the proof.
2. Case two: $U(n)<0$.
In this case, the argument given above must be modified to take into account that the inequality $$\frac{|z|}{|\lambda U(n)-z|}\leq 1.$$ is satisfied when $t=\mathrm{Re}>0$. In this case, the use of (\[repderv\]) would result in a problem as $tf(t)$ is unbounded as $t\to -\infty$. This can be addressed by observing that the Fermi-Dirac function $F(z)$ has the following symmetry $$\frac{1}{1+e^{\beta(z-\mu)}}=1-\frac{1}{1+e^{\beta(-z+\mu)}}.$$ Hence we can make use of the representation (\[repderv\]) corresponding to $$-\frac{1}{1+e^{\beta(-z+\mu)}}:=F^{\ast}_{\mu}(z)$$ since, for $m\neq n$, the constant term does not affect the calculation of $\frac{\partial V(n)}{\partial \omega(m)}$. Denoting by $$f^{\ast}(t)=F^{\ast}_{+}(t+i\eta)+F^{\ast}_{-}(t-i\eta)-D\ast F^{\ast}(t)$$ we reach $$\label{repderv2}
\frac{\partial V(n)}{\partial \omega(m)}=\int^{\infty}_{-\infty}\left( -g\sum_{l}r(n,l)\frac{\partial V(l)}{\partial \omega(m)}+
r(m,n)\right)f^{\ast}(t)\,dt$$ where now $tf^{\ast}(t)\to 0$ as $t\to -\infty$. Proceeding as in the first case the proof is finished, showing (\[improvedct1\]). Following the proof of lemma \[resamp\] and using (\[improvedct1\]) we conclude (\[improvedct2\])
\[localbeh\] Let $\Lambda_1$ and $\Lambda_2$ be subsets of $\mathbb{Z}^d$ with $dist(\Lambda_1,\Lambda_2)\geq r$. Let $V^{c}_{2}$ be the effective potential defined by $$V^{c}_{2}(n)=\mel{n}{F(H^{\Lambda^{c}_{2}})}{n}\,\,\,n\in \mathbb{Z}^d.$$ where $H^{\Lambda^{c}_{2}}$ denotes the restriction of $H$ to the complement of $\Lambda_2$.
Then, for any $n\in \Lambda_1$ $$\Big|\frac{\partial V(n)}{\partial \omega(m)}-\frac{\partial V^{c}_2(n)}{\partial \omega(m)}\Big|\leq C(\eta,d,\lambda,g,\|F\|_{\infty},\nu)e^{-\nu(|m-n|+r)}$$
The proof follows the same steps as in the previous results. The only modification which is required comes when comparing the quantities $r(m,n)$ and $r^{\Lambda_1}(m,n)$ given by $$r(m,n)=G(n,m;t+i\eta)G(m,n;t+i\eta)-G(n,m;t-i\eta)G(m,n;t-i\eta)$$ $$r^{\Lambda^{c}_{2}}(m,n)=G^{\Lambda^{c}_{2}}(n,m;t+i\eta)G^{\Lambda^{c}_{2}}(m,n;t+i\eta)-G^{\Lambda^{c}_{2}}(n,m;t-i\eta)G^{\Lambda^{c}_{2}}(m,n;t-i\eta).$$ We observe that $$\begin{aligned}
&G(n,m;z)G(m,n;z)-G^{\Lambda^{c}_{2}}(n,m;z)G^{\Lambda^{c}_{2}}(m,n;z)=\\
&G(n,m;z)\left(G(m,n;z)-G^{\Lambda^{c}_{2}}(m,n;z)\right)+\left(G(n,m;z)-G^{\Lambda^{c}_{2}}(n,m;z)\right)G^{\Lambda^{c}_{2}}(m,n;z).\end{aligned}$$ Moreover, $$G(m,n;z)-G^{\Lambda^{c}_{2}}(m,n;z)=-\lambda\sum_{l\in {\Lambda_{2}}}G(m,l;z)U(l)G^{\Lambda^{c}_{2}}(l,n;z).$$ The proof is now finished using arguments identical to the proof of lemma \[decay1\] and the improvement on lemma \[improvedct\].
Proof of lemma \[bdddensity\] {#proofoflemma}
==============================
In this section we show the existence of the effective density $\rho_{\mathrm{eff}}$. Fix $\Lambda \subset \mathbb{Z}^d$ finite. Recall that we defined $$\label{U}
U(n,\omega):=\omega(n)+\frac{g}{\lambda}V_{\mathrm{eff}}(n,\omega).$$ Until the end of this section we suppress the $\omega$ dependence on $U(n)$ and $V_{\mathrm{eff}}$. Note that, for $m,n\in \Lambda$, $$\label{DerU}
\frac{\partial U(m)}{\partial \omega(n)}=\delta_{mn}+\frac{g}{\lambda} \frac{\partial V_{\mathrm{eff}}(m)}{\partial \omega(n)}.$$ We have denoted the above change of variables by $\mathcal{T}:\mathbb{R}^{|\Lambda|} \rightarrow \mathbb{R}^{|\Lambda|}$, which reads $$\label{DefT} \mathcal{T}(\omega(n_1),...,\omega(n_{|\Lambda|}))=(U(n_1),...,U(n_{|\Lambda|}))$$
We can now compute the joint distribution of the $\{U(n)\}$. Using the fact that the random variables $\{\omega(n)\}_{n\in \mathbb{Z}^d}$ have a common density $\rho$ we conclude that for all Borel sets $I_1,...,I_N$ in $\mathbb{R}$: $$\begin{aligned}
\mathbb{P}\left( U(n_1)\in I_1,\,...\,,U(n_{|\Lambda|}),\in I_{|\Lambda|}\right)&=\int_{{\mathcal{T}}^{-1}\left(I_1\times...\times I_{|\Lambda|}\right)}\prod^{|\Lambda|}_{k=1}\rho(\omega(n_k))\,d\omega(n_k)\\
&=\int_{I_1\times...\times I_{|\Lambda|}}|\det J_{{\mathcal{T}}^{-1}}|\prod^{|\Lambda|}_{k=1}\rho\left({\mathcal{T}}^{-1}U(n_k)\right)\,dU(n_1)...dU(n_{|\Lambda|})\\
&=\int_{I_1\times...\times I_{|\Lambda|}}\big|\det\Big{(}I+\frac{g}{\lambda}\frac{\partial V_{\mathrm{eff}}(n_i,{\mathcal{T}}^{-1}U)}{\partial U(n_j)}\Big{)}\big|\prod^{|\Lambda|}_{k=1}\rho\left(U(n_k)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_k,{\mathcal{T}}^{-1}U)\right)\,dU(n_k).\end{aligned}$$
Therefore the joint distribution of $\{U(n_k)\}^{|\Lambda|}_{k=1}$ is given by the measure $$\label{densU}\Big|\det\Big{(}I+\frac{g}{\lambda}\frac{\partial V_{\mathrm{eff}}(n_i,{\mathcal{T}}^{-1}U)}{\partial U(n_j)}\Big{)}\Big|\prod^{|\Lambda|}_{k=1}\rho\left(U(n_k)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_k,\mathcal{T}^{-1}U) \right)\,dU(n_1)...dU(n_{|\Lambda|}).$$ It follows that for each $n_0\in \Lambda$ the conditional expectation of $U(n_0)$ at specified values of $\{U(n)\}_{n\neq n_0}$ has a density given by $$\label{cdensU}{\rho}^{\Lambda}_{n_0}=\frac{\big|\det\Big{(}I+\frac{g}{\lambda}\frac{\partial V_{\mathrm{eff}}(n_i,{\mathcal{T}}^{-1}U)}{\partial U(n_j)}\Big{)}\big|\prod^{|\Lambda|}_{k=1}\rho\left(U(n_k)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_k,{\mathcal{T}}^{-1}U)\right)}{\int^{\infty}_{-\infty}\big|\det\Big{(}I+\frac{g}{\lambda}\frac{\partial V_{\mathrm{eff}}(n_i,{\mathcal{T}}^{-1}U^{\alpha})}{\partial U(n_j)}\Big{)}\big|\prod^{|\Lambda|}_{k=1}\rho\left(U^{\alpha}(n_k)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_k,{\mathcal{T}}^{-1}U^{\alpha}) \right)\,d\alpha}$$ Where $U^{\alpha}(n):=U(n)+\left(\alpha-U(n_0)\right)\delta_{n=n_0}$. This strategy naturally leads to the analysis of ratios of determinants. A sufficient condition for an upper bound to the right-hand side of (\[cdensU\]) is to obtain positive constants $C=C_{\mathrm{fluct}}(U(n_0))$ and $D=D(\alpha)$ which are independent of $|\Lambda|$ and such that the following estimates hold true $$\label{fluctuation2}\frac{\big|\det\Big{(}I+\frac{g}{\lambda}\frac{\partial V_{\mathrm{eff}}(n_i,{\mathcal{T}}^{-1}U^{\alpha})}{\partial U(n_j)}\Big{)}\big|}{\big|\det\Big{(}I+\frac{g}{\lambda}\frac{\partial V_{\mathrm{eff}}(n_i,{\mathcal{T}}^{-1}U)}{\partial U(n_j)}\Big{)}\big|}\geq D(\alpha).$$ $$\label{fluctuation1} \int^{\infty}_{-\infty}D(\alpha)\rho\left(\alpha-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U^{\alpha})\right)\prod_{n\in |\Lambda|\setminus\{n_0\}}\frac{\rho\left(U^{\alpha}(n)-\frac{g}{\lambda}V_{\mathrm{eff}}(n,{\mathcal{T}}^{-1}U^{\alpha})\right)}{\rho\left(U(n)-\frac{g}{\lambda}V_{\mathrm{eff}}(n,{\mathcal{T}}^{-1}U)\right)} \,d\alpha\geq C_{\mathrm{fluct}}(U(n_0)).$$
The bounds (\[fluctuation1\]) and (\[fluctuation2\]) readily imply that, setting $U(n_0)=u$ $$\label{conditionalbound}{\rho}^{\Lambda}_{n_0}(u)\leq \frac{\rho\left(u-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U)\right)}{C_{\mathrm{fluct}}(u)}.$$ Lemma \[bdddensity\] will follow from a precise control of the right-hand side of equation (\[conditionalbound\]).
We now execute the strategy which was outlined above. The ratio of determinants can be controlled through the following bound, where $\|M\|_1$ denotes the trace norm of a matrix $M$.
\[detabsbd\] Let $A,B$ be square matrices with $I+B$ invertible. Then, $$\left|\frac{\det\left(I+A\right)}{\det\left(I+B\right)}\right|\leq e^{\|(A-B)(I+B)^{-1}\|_1}.$$
We make use of the elementary identities $$\frac{\det(I+A)}{\det(I+B)}=\det(I+A)(I+B)^{-1}$$ and $$(I+A)(1+B)^{-1}=I+(A-B)(I+B)^{-1}.$$ The proof is now finished making use of the inequality $$\left|\det(1+M)\right|\leq e^{\|M\|_{1}}$$ which holds on the general setting of trace class operators, see [@trace Lemma 3.3]
The triangle inequality for the trace norm implies the following.
\[ratiolbound\]Under the above conditions $$\Big|\frac{\det(I+B)}{\det(I+A)}\Big|\geq e^{-\sum_{m,n}|\left((A-B)(I+B)^{-1}\right)_{mn}|}$$
Letting $A=\frac{g}{\lambda}\left(\frac{\partial V(n_i,\omega)}{\partial U(n_j)}\right)_{|\Lambda|\times |\Lambda|}$ and $B=\frac{g}{\lambda}\left(\frac{\partial V(n_i,\omega_{\alpha})}{\partial U(n_j)}\right)_{|\Lambda|\times |\Lambda|}$ and using lemma \[decay1\] we see that, for $|g|<\lambda C_1^{-1}$, $(1+B)^{-1}$ has uniformly bounded operator norm. Using lemma \[bddtrace\] and corollary \[ratiolbound\] we conclude that (\[fluctuation2\]) holds with $D(\alpha)=e^{-|g|^2C_3\left(|\alpha-U(n_0)|+C_4\right)}$.
We now check that equation (\[fluctuation1\]) holds when $\rho$ satisfies the fluctuation bound (\[flucassump\]). We divide the proof in two cases:
1. Suppose that $c_2(\rho)>0$.
Let $c_{\rho}=\max\{c_1(\rho),c_2(\rho)\}$. The left-hand side of (\[fluctuation1\]) is bounded from below by $$\int^{\infty}_{-\infty}D(\alpha)\rho\left(\alpha-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U^{\alpha})\right)\prod_{n\in \Lambda \setminus\{n_0\}}e^{-c_{\rho}|\omega(n)-\omega_{\alpha}(n)|(1+|\omega(n)|+|\omega_{\alpha}(n)|)}\,d\alpha$$ which equals $$\label{densityint}
\int^{\infty}_{-\infty}D(\alpha)\rho\left(\alpha-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U^{\alpha})\right) e^{\sum_{n\in \Lambda \setminus\{n_0\}}-c_{\rho}|\omega(n)-\omega_{\alpha}(n)|(1+|\omega(n)|+|\omega_{\alpha}(n)|\,)}\,d\alpha.$$
Due to the triangle inequality and lemmas \[resamp\] and \[improvedct\], we conclude that there is a positive constant $C=C(d,\|F\|_{\infty},g,\eta,\nu)$ with $\lim_{g\to 0}C(d,\|F\|_{\infty},g,\eta,\nu)<\infty$ such that for $n\neq n_0$ $$\begin{aligned}
-c_{\rho}|\omega(n)-\omega_{\alpha}(n)|\left(1+|\omega(n)|+|\omega_{\alpha}(n)|\right)&\geq -c_{\rho}|\omega(n)-\omega_{\alpha}(n)|\left(1+2|\omega(n)|+|\omega_{\alpha}(n)-\omega(n)|\right)\\
&\geq -\frac{|g|}{\lambda}c_{\rho}e^{-\nu|n-n_0|}\left(C^{2}|\alpha-U(n_0)|^2+2C|\alpha-U(n_0)| \right).\\\end{aligned}$$
Therefore, $$\begin{aligned}
-c_{\rho}\sum_{n\in \Lambda\setminus\{n_0\}}|\omega(n)-\omega_{\alpha}(n)|\left(1+|\omega(n)|+|\omega_{\alpha}(n)|\right)&\geq -\frac{|g|}{\lambda}\frac{2c_{\rho}}{\left(1-e^{-\nu}\right)^d}\left(C^{2}|\alpha-U(n_0)|^2+2C|\alpha-U(n_0)| \right)).\\\end{aligned}$$
Thus, by choosing $\frac{|g|}{\lambda}$ sufficiently small so that $\frac{|g|}{\lambda}\frac{4c_{\rho}}{\left(1-e^{-\nu}\right)^d}C^{2}<{\overline{c}}_{\rho}$ and using the assumption \[fluctintegral1\] we obtain $$0<\int^{\infty}_{-\infty}D(\alpha)\frac{\rho\left(\alpha-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U^{\alpha})\right)}{\rho\left(U(n_0)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U)\right)} \prod_{n\in \Lambda \setminus\{n_0\}}e^{-c_{\rho}|\omega(n)-\omega_{\alpha}(n)|(1+|\omega(n)|+|\omega_{\alpha}(n)|)}\,d\alpha<\infty.$$ this, together with (\[conditionalbound\]), verifies lemma \[bdddensity\] when $c_2(\rho)>0$. If $\rho$ satisfies the assumption \[momentassumption\] the above argument yields ${\rho}^{\Lambda}_{n_0}(u)\in L^{1}\left(\mathbb{R},|x|^{\varepsilon}dx\right)$.
2. Assume that $c_2(\rho)=0$:
Similarly to the above argument, the left-hand side of (\[fluctuation1\]) is bounded from below by $$\int^{\infty}_{-\infty}D(\alpha)\frac{\rho\left(\alpha-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U^{\alpha})\right)}{\rho\left(U(n_0)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U)\right)}e^{-\frac{|g|}{\lambda}\frac{c_{1}({\rho})}{\left(1-e^{-\nu}\right)^d}C|\alpha-U(n_0)|}.$$ Where, from (\[flucassump\]) $$\frac{\rho\left(\alpha-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U^{\alpha})\right)}{\rho\left(U(n_0)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U)\right)}\geq e^{-c_1(\rho)\left(|\alpha-U(n_0)|+2\frac{|g|}{\lambda}\right)}.$$ Again, choosing $|g|$ sufficiently small we conclude that $$0<\int^{\infty}_{-\infty}D(\alpha)\frac{\rho\left(\alpha-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U^{\alpha})\right)}{\rho\left(U(n_0)-\frac{g}{\lambda}V_{\mathrm{eff}}(n_0,{\mathcal{T}}^{-1}U)\right)} \prod_{n\in \Lambda \setminus\{n_0\}}e^{-c_{1}({\rho})|\omega(n)-\omega_{\alpha}(n)|}\,d\alpha<\infty.$$ finishing the proof.
The Hartree approximation for the Hubbard model {#hubbardext}
===============================================
Let us now adapt the techniques from the previous sections to the Hubbard model. Recall that $H_{\mathrm{Hub}}$ is defined as
$$\begin{pmatrix}
\,H_{\uparrow}(\omega) & 0 \\
0 & H_{\downarrow}(\omega)\,\\
\end{pmatrix}
:=
\begin{pmatrix}
\, H_0+\lambda V_{\omega}+gV_{\uparrow}(\omega) & 0 \\
0 & H_0+\lambda V_{\omega}+gV_{\downarrow}(\omega)\,\\
\end{pmatrix}$$ acting on $\ell^2\left(\mathbb{Z}^d\right)\oplus \ell^2\left(\mathbb{Z}^d\right)$. The operators $H_0$ and $V_{\omega}$ are defined as before, i.e; $H_0+\lambda V_{\omega}$ is the standard Anderson model acting on $\ell^2\left(\mathbb{Z}^d\right)$. The effective potentials are defined as $$\label{potentialHubbard}
\begin{pmatrix}
\,V_{\uparrow}(\omega)(n) \\
V_{\downarrow}(\omega)(n)\,\\
\end{pmatrix}
=\begin{pmatrix}
\,\mel{n}{F(H_{\downarrow})}{n} \\
\mel{n}{F(H_{\uparrow})}{n}\,\\
\end{pmatrix}.$$
Mathematically, the treatment of the above model is very similar to the the proof of theorem \[main\] above, therefore some details are skipped and we just indicate the required modifications.
Existence of the Effective potential {#existHubb}
------------------------------------
Let $\Phi(X,Y):\ell^{\infty}\left(\mathbb{Z}^d\right)\oplus \ell^{\infty}\left(\mathbb{Z}^d \right)\rightarrow \ell^{\infty}\left(\mathbb{Z}^d\right)\oplus \ell^{\infty}\left(\mathbb{Z}^d \right)$ be given by $$\Phi(X,Y)(m,n):=\left(\mel{n}{F(H_0+V_{\omega}+gY)}{n},\mel{m}{F(H_0+V_{\omega}+gX)}{m}\right).$$ using proposition \[Contraction\], we immediately reach $$\|\Phi(X_1,Y_1)-\Phi(X_2,Y_2)\|_{\ell^{\infty}\left(\mathbb{Z}^d\right)\oplus \ell^{\infty}\left(\mathbb{Z}^d \right)}\leq |g|\frac{72\sqrt{2}}{\eta\left(1-e^{\nu'-\nu}\right)^d}\|F\|_{\infty}\left(\|X_1-X_2\|_{\ell^{\infty}\left(\mathbb{Z}^d\right)}+\|Y_1-Y_2\|_{\ell^{\infty}\left(\mathbb{Z}^d\right)}\right).$$ Therefore, if $|g|\frac{72\sqrt{2}}{\eta\left(1-e^{\nu'-\nu}\right)^d}\|F\|_{\infty}<1$ we conclude $\Phi$ has a unique fixed point $V_{\mathrm{eff}}=\left(V_{\uparrow},V_{\downarrow}\right)$ belonging to $\ell^{\infty}\left(\mathbb{Z}^d\right)\oplus \ell^{\infty}\left(\mathbb{Z}^d \right) $.
Regularity of the effective potential {#regularity-of-the-effective-potential}
-------------------------------------
Fix $\Lambda \subset \mathbb{Z}^d$ finite and define functions $\xi:\left(\ell^{\infty}\left(\Lambda\right)\oplus \ell^{\infty}\left(\Lambda \right)\right)\times \mathbb{R}^{n}\rightarrow \ell^{\infty}\left(\Lambda\right)\oplus \ell^{\infty}\left(\Lambda \right)$ through $$\xi^{\uparrow}(V,\omega)(j)=V^{\uparrow}(j)-\mel{j}{F(H_0+\lambda\omega+gV_{\downarrow})}{j}.$$ $$\xi^{\downarrow}(V,\omega)(j)=V^{\downarrow}(j)-\mel{j}{F(H_0+\lambda\omega+gV_{\uparrow})}{j}.$$ Our goal is to conclude $V^{\uparrow}$,$V^{\downarrow}$ are smooth functions of an arbitrary, but finite, list $(\omega(1),...,\omega(n))$. Again, this can be done via implicit function theorem once we check that the derivative $$\frac{\partial \xi(\omega, V)(j)}{\partial V(l)}=
\delta_{jl}-\frac{\partial \mel{j}{F(H_0+\lambda\omega+gV)}{j}}{\partial V(l)}.$$ is non-singular. Using lemma \[Contraction\], we have that for $\sharp\in \{\uparrow,\downarrow\}$ $$\Big|\frac{\partial \mel{j}{F(H_0+\lambda\omega+gV_{\sharp})}{j}}{\partial V(l)}\Big|\leq
|g|\frac{72\sqrt{2}e^{-2\nu|j-l|}}{\eta}\|F\|_{\infty}.$$
In particular, whenever $|g|\frac{144\sqrt{2}\|F\|_{\infty}}{\eta(1-e^{-2\nu})^{d}}<1$ we have that the operator $D\xi(\omega,.):\ell^{\infty}\left(\Lambda \right)\oplus \ell^{\infty}\left(\Lambda \right)\rightarrow \ell^{\infty}\left(\Lambda \right)\oplus \ell^{\infty}\left(\Lambda \right)$ has an inverse. From the implicit function theorem it follows that $V$ is a smooth function of $(\omega(1),...,\omega(n))$ for $n=|\Lambda|$.
Decay estimates
---------------
The decay rate in the case of the Hubbard model is dictated by $$\label{Hubbarddecayup} \Big|\frac{\partial V_{\uparrow}(n)}{\partial \omega(m)}\Big |\leq 3|g|\|F\|_{\infty}\sum_{l}\tilde{K_{\downarrow}}(l,m)\Big|\frac{\partial V_{\downarrow}(l)}{\partial \omega(m)}\Big|+
\tilde{r_{\downarrow}}(n).$$
$$\label{Hubbarddecaydown} \Big|\frac{\partial V_{\downarrow}(n)}{\partial \omega(m)}\Big |\leq 3|g|\|F\|_{\infty}\sum_{l}\tilde{K_{\uparrow}}(l,m)\Big|\frac{\partial V_{\uparrow}(l)}{\partial \omega(m)}\Big|+
\tilde{r_{\uparrow}}(n).$$
where, for $\sharp\in\{\uparrow,\downarrow\}$
$$\tilde{G_{\sharp}}(l,m):=G_{\sharp}(l,n;t+i\eta)G_{\sharp}(n,l;t+i\eta)-G_{\sharp}(l,n;t-i\eta)G_{\sharp}(n,l;t-i\eta).$$ $$r_{\sharp}(m,n):=G_{\sharp}(n,m;t+i\eta)G_{\sharp}(m,n;t+i\eta)-G_{\sharp}(n,m;t-i\eta)G_{\sharp}(m,n;t-i\eta).$$
$$\tilde{K_{\sharp}}(l,m):=\int^{\infty}_{-\infty}|\tilde{G_{\sharp}}(l,m)|\,dt.$$ $$\tilde{r_{\sharp}}(n):=\int^{\infty}_{-\infty} |r_{\sharp}(n)|\,dt.$$
In particular, $$\label{Hubbardfulldecay}
\Big|\frac{\partial V_{\uparrow}(n)}{\partial \omega(m)}\Big|+\Big|\frac{\partial V_{\downarrow}(n)}{\partial \omega(m)}\Big |\leq 3|g|\|F\|_{\infty}\sum_{l}\left(\tilde{K_{\uparrow}}(l,m)+\tilde{K_{\downarrow}}(l,m)\right)\left(\Big|\frac{\partial V_{\uparrow}(l)}{\partial \omega(m)}\Big|+\Big|\frac{\partial V_{\downarrow}(l)}{\partial \omega(m)}\Big|\right)
+\left(\tilde{r_{\uparrow}}(n,m)+\tilde{r_{\downarrow}}(n,m)\right).$$
The analysis from the previous sections applies and we obtain lemmas \[decay1\],\[resamp\],\[bddtrace\] and \[improvedct\] with $|.|$ being replaced by the matrix norm $|M|=|M_{11}|+|M_{21}|$ for $M=
\begin{pmatrix}
M_{11}\\
M_{21}\,\\
\end{pmatrix}$. The effective potential and its derivatives are to be interpreted as follows:
$$V_{\mathrm{eff}}(n)=
\begin{pmatrix}
V^{\uparrow}_{\mathrm{eff}}(n)\\
V^{\downarrow}_{\mathrm{eff}}(n)\,\\
\end{pmatrix},
\,\,\,\,\,\,
\frac{\partial V_{\mathrm{eff}}(n)}{\partial \omega(m)}=
\begin{pmatrix}
\frac{\partial V^{\uparrow}_{\mathrm{eff}}(n)}{\partial \omega(m)}\\
\frac{\partial V^{\downarrow}_{\mathrm{eff}}(n)}{\partial \omega(m)}\,\\
\end{pmatrix}
\,\,\,\mathrm{and}\,\,\,
\frac{\partial^2 V_{\mathrm{eff}}(n)}{\partial \omega(m)\partial \omega(l)}=
\begin{pmatrix}
\frac{\partial^2 V^{\uparrow}_{\mathrm{eff}}(n)}{\partial \omega(m)\omega(l)}\\
\frac{\partial^2 V^{\downarrow}_{\mathrm{eff}}(n)}{\partial \omega(m)\omega(l)}\,\\
\end{pmatrix}.$$
One dimensional Aspects:proof of theorem \[1dloc\] {#details1d}
==================================================
In this section we will prove theorem \[1dloc\]. We let $H_{+}=H_{[0,\infty)\cap\mathbb{Z}}$ and denote by $G^{+}(m,n;z)$ the Green’s function of $H^{+}$. Recall the definition of the Lyapunov exponent: $$\mathcal{L}(z)=-\mathbb{E}\left(\ln|G^{+}(0,0;z)|\right)$$
$$\mathcal{L}_{\mathrm{And}}(z)=-\mathbb{E}\left(\ln|G^{+}_{\mathrm{And}}(0,0;z)|\right).$$
Recall in this case $H_0=-\Delta$ hence, we define $H_{\mathrm{Hub}}$ acting on $\left(\ell^{2}\left(\mathbb{Z}\right)\oplus \ell^{2}\left(\mathbb{Z} \right)\right)$ by $$H_{\mathrm{Hub}}=\begin{pmatrix}
\,H_{\uparrow}(\omega) & 0 \\
0 & H_{\downarrow}(\omega)\,\\
\end{pmatrix}$$ where, denoting by $H_{\mathrm{And}}$ the standard Anderson model $-\Delta+V_{\omega}$ on $\ell^2\left( \mathbb{Z}\right)$, $$\label{Hubbarddefonedim}\begin{pmatrix}
\,H_{\uparrow}(\omega) & 0 \\
0 & H_{\downarrow}(\omega)\,\\
\end{pmatrix}
:=
\begin{pmatrix}
\,H_{\mathrm{And}}+gV_{\uparrow}(\omega) & 0 \\
0 & H_{\mathrm{And}}+gV_{\downarrow}(\omega)\,\\
\end{pmatrix}.$$ The effective potentials are defined as (\[eff\]). In the theorem below, we will use an abbreviation and $\mathcal{L}(z)$ will refer to the Lyapunov exponent of either $H_{\uparrow}$ or $H_{\downarrow}$ whereas $\mathcal{L}_{\mathrm{And}}(z)$ will denote the Lyapunov exponent of the Anderson model on $\ell^2\left( \mathbb{Z}\right)$.
Proof of Lemma \[Fekete\]
-------------------------
For simplicity we set $C=0$. The general statement will follow by considering the related sequence $b_n:=a_n+C$. Given integers $L$ and $\ell$ with $L>>\ell$, our goal is to bound $\frac{a_{L}}{L}$ from above in terms of $\frac{a_{\ell}}{\ell}$. As a initial step, observe that by (\[subad\]) we have $$a_L\leq a_{L-\delta \log L-\ell}+a_{\ell}.$$ Iterating the above procedure $k+1$ times for $$k=k_{\ell,L}:=\lfloor \frac{L-2\ell-\delta \log L}{\delta \log L+\ell}\rfloor$$ we obtain $$a_L\leq \left(k+2\right) a_{\ell}$$ In the above iteration we have made use of the fact that in the assumption (\[subad\]) the remainder $r$ can be adjusted as long as it satisfies the inequality given there. Thus, $$\label{startscal}
\frac{a_L}{L}\leq \frac{\ell(k+2)}{L}\frac{a_{\ell}}{\ell}$$ Before proceeding with the proof, a few remarks are in order. Firstly, nothing is achieved by holding $\ell$ fixed and letting $L\to \infty$ directly on equation (\[startscal\]) since this only yields the upper bound of zero. A second attempt would be showing that letting $\ell \to \infty$ (hence, $L \to \infty$ as well) implies that the ratio $\frac{k\ell}{L}$ converges to one. However, as $$q_{\ell,L}-\frac{\ell}{L}\leq \frac{k\ell}{L}\leq q_{\ell,L}$$ for the choice $$\label{qdef}
q_{\ell,L}=\frac{1-2\frac{\ell}{L}-\delta \frac{\log L}{L}}{1+\frac{\delta \log L}{\ell}}$$ we see that $\frac{k\ell}{L}$ converges to one as $\ell \to \infty$ only along a subsequence where $$\label{initialreq}
\frac{\ell}{L}\to 0\,\,\,\,\mathrm{and}\,\,\,\, \frac{\log L}{\ell}\to 0.$$ Taking this into account, let $\varepsilon>0$ be given and $\ell_1$ be an initial scale to be determined. Let $L>>\ell_1$ be a positive integer to be interpreted as a larger scale. Iterating equation (\[startscal\]) throughout a sequence of scales $$\ell_1<\ell_2<...<\ell_{N_L} \leq L<\ell_{N_L+1}<...$$ satisfying, for some $p>0$, $$p\log \left(\ell_j\right)\leq \log \ell_{j+1} < p^2\log \left(\ell_j\right).$$ and $$\label{summability}
\sum^{\infty}_{j=1}\frac{\ell_{j}}{\ell_{j+1}}<\infty$$
we reach, for $q_{\ell,L}$ defined in (\[qdef\]), $$\frac{a_L}{L}\leq \left(q_{\ell_{N_L},L}+\frac{2\ell_{N_L}}{L}\right)\prod^{N_{L}-1}_{j=1} \left(q_{\ell_{j},\ell_{j+1}}+\frac{2\ell_j}{\ell_{j+1}}\right)
\frac{a_{\ell_1}}{\ell_1}.$$ Since $q_{\ell_{j},\ell_{j+1}}\to \infty$ as $j\to \infty$ Due to (\[summability\]), we conclude that the value of $\ell_1$ can be chosen(independently of $L$) so that $$\label{tails}
\sum^{N_{L}-1}_{j=1} \log\left(q_{\ell_{j},\ell_{j+1}}+\frac{2\ell_j}{\ell_{j+1}}\right)+\log\left(q_{\ell_{N_L},L}+\frac{2\ell_{N_L}}{L}\right)<\varepsilon.$$ Thus
$$\label{conclusionseq}
\frac{a_L}{L}\leq e^{\varepsilon}\frac{a_{\ell_1}}{\ell_1}.$$
Moreover, the above conclusion holds for any integer $\ell_1$ sufficiently large, as long and $L>>\ell_1$. In particular, we can also require that $$\label{infimumseq}
\frac{a_{\ell_1}}{\ell_1}\leq \inf_{n}\frac{a_n}{n}+\varepsilon.$$ Combining equations (\[conclusionseq\]) and (\[infimumseq\]) the proof is finished letting $\varepsilon \to 0$.
Proof of lemma \[mixinglem\]
----------------------------
We will show that the following inequality holds $$\label{upperdecoupling}
\mathbb{E}_{[0,n+m+r]}\left(|\hat G(0,n+m+r;z)|^s \right)\leq C_{\mathrm{AP}}e^{C(\eta,g,\|F\|_{\infty}) e^{-\nu'r}(m^2+n^2)}\mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right) \mathbb{E}_{[0,m]}\left(|\hat G(0,m;z)|^s \right)$$ where $\mathbb{E}_{[0,n]}$ denotes the expectation with respect to the variables $U(0),...,U(n)$ and $C_{\mathrm{AP}}$ is, up to a multiplication by a constant independent of $m,n$ and $r$, the constant obtained on the *a priori* from lemma \[apriori\]. Let us change variables according to $$\label{change1d}
\left(\omega(1),...,\omega(n+1),\omega(n+r),...,\omega(n+r+m)\right)\mapsto
\left(U(1),..., U(n+1), U(n+r),...,U(n+r+m)\right).$$ We remark that the variables $\omega(n+2),...,\omega(n+r-1)$ are fixed in this process.
Note that by lemma \[apriori\] and a geometric resolvent expansion we have
$$\label{firstapriori}
\mathbb{E}\left(\hat G(0,n+m+r;z)|^s \right)\leq
C_{\mathrm{AP}}\mathbb{E}_{\neq n+1,n+r}\left(|\hat G(0,n;z)|^s |\hat G(n+r+1,n+r+m;z)|^s \right).$$
where $\mathbb{E}_{\neq n+1,n+r}$ indicates that the variables $\omega(n+1)$ and $\omega(n+r)$ were integrated out. Observe that the corresponding Jacobian has the structure $$\label{jacobian}
\mathcal{J}=\begin{pmatrix}
\mathcal{A}_{n\times n} & \mathcal{B}_{n\times(m+r)}\\
\mathcal{C}_{(m+r) \times n} & \mathcal{D}_{(m+r)\times(m+r)}\\
\end{pmatrix}$$ where $$\mathcal{A}_{jk}=\delta_{jk}+g\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(k)},\,\,\, \mathcal{B}_{jk}=g\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(n+k)},\,\,\, \mathcal{C}_{jk}=g\frac{\partial V_{\mathrm{eff}}(n+j)}{\partial U(k)}.$$ Moreover, $$\mathcal{D}=\begin{pmatrix}
\mathcal{I}_{r\times r} & \mathcal{Q}_{r\times m}\\
\mathcal{O}_{m \times r} & \mathcal{P}_{m\times m}\\
\end{pmatrix}$$ where $\mathcal{I}_{r\times r}$ is the identity matrix and $$\mathcal{P}_{jk}=\delta_{jk}+g\frac{\partial V_{\mathrm{eff}}(n+r+j)}{\partial U(n+r+k)},\,\,\, \mathcal{Q}_{jk}=g\frac{\partial V_{\mathrm{eff}}(n+j)}{\partial U(n+r+k)},\,\,\, \mathcal{O}_{jk}=g\frac{\partial V_{\mathrm{eff}}(n+r+j)}{\partial U(n+k)}.$$
By the Schur complement formula $$\label{determinantschur}\det \mathcal{J}=\det \mathcal{A}\det \mathcal{P} \det\left(\mathcal{I}_{{m+r}\times{m+r}}-{\mathcal{D}}^{-1}
\mathcal{C}{\mathcal{A}}^{-1}\mathcal{B}\right)\det\left(\mathcal{I}_{{m}\times{m}}-{\mathcal{P}}^{-1}
\mathcal{O}\mathcal{Q}\right).$$ where, according to the estimate \[lemmapotential1\], the matrices $\mathcal{B}$ and $\mathcal{Q}$ have entries which decay exponentially away from their lower-left corner. Likewise, the entries of $\mathcal{C}$, $\mathcal{O}$ decay exponentially away from their upper-right corner. It readily follows from lemma that \[detabsbd\], $$\begin{aligned}
\det\left(\mathcal{I}_{{m+r}\times{m+r}}-{\mathcal{D}}^{-1}
\mathcal{C}{\mathcal{A}}^{-1}\mathcal{B}\right)\det\left(\mathcal{I}_{{m}\times{m}}-{\mathcal{P}}^{-1}
\mathcal{O}\mathcal{Q}\right)\leq C(\eta,\nu,g,\|F\|_{\infty}).
\end{aligned}$$ therefore, for $C$ as above, $$\det \mathcal{J}\leq C\det \mathcal{A}\det \mathcal{P}.$$ Let $\rho(l)=\rho\left(U(l)-gV_{\mathrm{eff}}(l)\right)$. We obtain a decoupling estimate by observing that, setting $U(l)=0$ for $l\geq n+r+1$ would only alter $$V_{\mathrm{eff}}(j)\,\,\mathrm{and}\,\,\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(k)}$$ by at most a factor which decays as $C(\eta,g,\|F\|_{\infty})e^{-\nu(r+|j-k|)}$ for $1\leq j,k\leq n$. This follows from the exponential decay on lemmas \[decay1\], \[improvedct\] and \[localbeh\]. Similarly, we can set $U(l)=0$ for $l\leq n$ and this only changes $$V_{\mathrm{eff}}(j)\,\,\mathrm{and}\,\,\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(k)}$$ by at most a factor bounded which decays as $C(\eta,\lambda,g,\|F\|_{\infty})e^{-\nu(r+|j-k|)}$ for $n+r\leq j,k\leq m+n+r$. The above process yields two independent measures $${\det}_{0}\left (I+g\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(k)}\right)_{ [0,n]} \prod_{0\leq l\leq n}\rho^{0}(l)\,dU(l)$$ and $${\det}_{0}\left ( I+g\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(k)}\right)_{ [n+r+1,n+r+m]}\prod_{n+r+1\leq n+r+m}\rho^{0}(l)\,dU(l).$$
Using lemma \[detabsbd\] we then arrive at an inequality of the type $$\begin{aligned}
\mathbb{E}_{[0,m+n+r]}&\left(|\hat G(0,n+m+r;z)|^s \right)\\
&\leq C_{\mathrm{AP}}e^{ C(\eta,g,\|F\|_{\infty})e^{-\nu r}(m^2+n^2)}\int |\hat G(0,n;z)|^s {\det}_{0}\left (I+g\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(k)}\right)_{ [0,n]} \prod_{0\leq l\leq n}\rho^{0}(l)\,dU(l)\\
&\times | G_{+}(n+r,n+r+m;z)|^s {\det}_{0}\left ( I+g\frac{\partial V_{\mathrm{eff}}(j)}{\partial U(k)}\right)_{ [n+r+1,n+r+m]}\prod_{n+r+1\leq n+r+m}\rho^{0}(p)\,dU(p).\\\end{aligned}$$ Rewriting the above conclusion in terms of expectations we obtain $$\mathbb{E}_{[0,m+n+r]}\left(|\hat G(0,n+m+r;z)|^s \right)\leq C_{\mathrm{AP}}e^{ C(\eta,g,\|F\|_{\infty})e^{-\nu r}(m^2+n^2)} \mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s\right)| \mathbb{E}_{[n+r+1,n+m+r]}\left(G_{+}(n+r,n+r+m;z)|^s\right).$$ At the expense of increasing $C_{\mathrm{AP}}$ the above expectations can be replaced by expectations over the full probability space, which yields the desired conclusion by translation invariance.
Proof of (\[upperbound\]) {#detailslowerbound}
-------------------------
We shall modify the proof of lemma \[mixinglem\] to obtain an “super-additive" estimate of the form $$\label{goallowerdec}
\mathbb{E}_{[0,n+m+r]}\left(|\hat G(0,n+m+r;z)|^s \right)\geq \underline{C}(s,E)e^{\varphi(s,z)r}e^{-C(\eta,g,\|F\|_{\infty}) e^{-\nu'r}(m^2+n^2)}\mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right) \mathbb{E}_{[0,m]}\left(|\hat G(0,m;z)|^s \right)$$ where the constant $\underline{C}(s,z)$ can be chosen locally uniform in $z$ and $s\in (0,1)$. Since the argument is very similar to the one in the proof of (\[upperdecoupling\]), we only explain the key modification which consists in obtaining a lower bound for $\mathbb{E}_{[n+1,n+r]}\left(|\hat G(n+1,n+r;z)|^s \right)$ as follows. We start by writing $$\label{bottleneck}|\hat G(n+1,n+r;z)|^s=|\hat G(n+1,n+1;z)|^s|\hat G(n+2,n+r;z)|^s.$$ Using Jensen’s inequality we have that, for any $\varepsilon\in (0,s)$, $$\label{jensen}
\mathbb{E}_{n+1}\left(|\hat G(n+1,n+1;z)|^s\right)\geq \mathbb{E}_{n+1}\left(|\hat G(n+1,n+1;z)|^{-\varepsilon}\right)^{\frac{-s}{\varepsilon}}.$$ where $\mathbb{E}_{n+1}$ denotes the conditional expectation with respect to $U(n+1)=\omega(n+1)+gV_{\mathrm{eff}}(n+1)$. Making use of the discrete Riccati equation [@A-W-B Proposition 12.1] we obtain $$\label{riccati}
\mathbb{E}_{n+1}\left(|\hat G(n+1,n+1;z)|^{-\varepsilon}\right)=\mathbb{E}_{n+1}\left(|U(n+1)-z-\hat G(n+2,n+2;z)|^{\varepsilon}\right).$$ Equations (\[bottleneck\]), (\[jensen\]) and (\[riccati\]) together with lemma \[bdddensity\] yield a lower bound $$\mathbb{E}_{[n+1,n+r]}\left(|\hat G(n+1,n+r;z)|^s\right)\geq \underline{C}(z,s)\mathbb{E}_{[n+2,n+r]}\left(|\hat G(n+2,n+r;z)|^s\right)$$ which in combination with (\[lowerbound\]) implies that, after a suitable adjustment of the constant $\underline{C}(s,z)$, $$\mathbb{E}_{[n+1,n+r]}\left(|\hat G(n+1,n+r;z)|^s\right)\geq \underline{C}(z,s)e^{\varphi(s,z)r}.$$ Equation (\[goallowerdec\]) follows from the above inequality combined with a decoupling estimate analogous to the one in the proof of (\[upperdecoupling\]). Again, choosing $r$ comparable to $\max\{\log m,\log n\}$ we obtain $$\mathbb{E}_{[0,n+m+r]}\left(|\hat G(0,n+m+r;z)|^s \right)\geq \underline{C}(s,z)e^{\varphi(s,z)r}\mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right) \mathbb{E}_{[0,m]}\left(|\hat G(0,m;z)|^s \right)$$ multiplying both sides of the above inequality by $e^{-\varphi(s,z)(m+n)}$ and taking logarithms we conclude that the sequence $b_n=\log e^{-\varphi(s,z)n}\left( \mathbb{E}_{[0,n]}\left(|\hat G(0,n;z)|^s \right)\right)$ satisfies $$b_{n+m+r}\geq \log(\underline{C}(s,z))+b_n+b_m.$$ The bound (\[upperbound\]) now follows from an application of the supperaditive version of lemma \[Fekete\].
Hölder Continuity for the integrated density of states at weak interaction {#idssection}
==========================================================================
In this section we shall address the problem of Hölder continuity for the integrated density of states for the Hubbard model with respect to energy, disorder and interaction. Our results follow from modifications of the methods in [@H-K-S] and references therein after we have established the existence of a suitable conditional density as in lemma \[bdddensity\].
Let’s now prove theorem \[thmids\], starting from Hölder continuity with respect to energy, equation (\[IDSenergy\]). We proceed as in [@H-K-S Section 2]. For simplicity, we replace $H_{\mathrm{Hub}}$ by $H$ defined in (\[toymodel\]). The arguments given below will apply directly to $H_{\uparrow}$ and $H_{\downarrow}$ and, therefore, suffice to show the same result for $H_{\mathrm{Hub}}$.
Fix an energy interval $I$ of length $\varepsilon>0$ centered at $E\in \mathbb{R}$. The idea is to use the Hölder continuity of $N_0$ and the resolvent identity to reach the following inequality for $\varepsilon<<1$ and $|I|=\varepsilon$, where we denote by $P_{\Lambda}(I)$ the spectral projection of $H^{\Lambda}$ on the interval $I$. $$\label{goalids}(1-\mathrm{o}(\varepsilon))\mathbb{E}\left( \mathrm{Tr}P_{\Lambda}(I)\right)\leq C(I,\rho)\varepsilon^{\alpha}|\Lambda|.$$ Dividing both sides of (\[goalids\]) by $|\Lambda|$ and letting $|\Lambda|\to \infty$ gives (\[IDSenergy\]). To obtain (\[goalids\]) we fix an interval $J$ containing $I$ with $|J|$ to be determined. We then write, with $P_{0,\Lambda}(J)=P\left(H_{0}^{\Lambda}\right)(J)$, $$\label{eq0}
\mathrm{Tr}(P_{\Lambda}(I))=\mathrm{Tr}(P_{\Lambda}(I)P_{0,\Lambda}(J))+\mathrm{Tr}(P_{\Lambda}(I)P_{0,\Lambda}(J^c)).$$ Note $$\mathrm{Tr}(P_{\Lambda}(I)P_{0,\Lambda}(J))\leq \mathrm{Tr}(P_{0,\Lambda}(J)).$$ The above inequality combined with to the Hölder continuity of $N_0$ with respect to $E\in \mathbb{R}$ $$|N_0(E)-N_0(E')|\leq C(I,d)|E-E'|^{\alpha_0}.$$ yields, for $|\Lambda|$ sufficiently large depending only on $J$, $$\label{1st}
\mathrm{Tr}(P_{\Lambda}(I)P_{0,\Lambda}(J))\leq C(J,d)|J|^{\alpha_0}|\Lambda|.$$ We now estimate the second term on the left-hand side of equation (\[eq0\]). By the resolvent identity, $$\label{remainingterm}
\mathrm{Tr}\left(P_{\Lambda}(I)P_{0,\Lambda}(J^c)\right)=\mathrm{Tr}\left(P_{\Lambda}(I)(H-E)P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-1}\right)-\lambda \mathrm{Tr}\left(P_{\Lambda}(I)U^{\Lambda}P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-1}\right).$$ Where we have written $U=V_{\omega}+\frac{g}{\lambda}V_{\mathrm{eff}}$. Moreover, using using functional calculus and that $E$ is the center of $I$, we estimate the first term on the left-hand side of equation (\[remainingterm\]) by $$\label{2nd}
\mathrm{Tr}\left((P_{\Lambda}(I))(H^{\Lambda}-E)P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-1}\right)\leq \frac{|I|}{|J|-|I|}\mathrm{Tr}(P_{\Lambda}(I)).$$ Now, the second term in in equation (\[remainingterm\]) can be controlled by means of $$\begin{aligned}
-\lambda \mathrm{Tr}\left(P_{\Lambda}(I)U^{\Lambda}P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-1}\right)=&-\lambda \mathrm{Tr}\left((H^{\Lambda}-E)(P_{\Lambda}(I))U^{\Lambda}P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-2}\right)\\
&+\lambda^2\mathrm{Tr}\left(U^{\Lambda}(P_{\Lambda}(I))U^{\Lambda}P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-2}\right).\\
&=A+B\end{aligned}$$ Now, because $U^{\Lambda}$ is unbounded, we continue a slight modification of the argument in [@H-K-S]. The only difference is that we bound term (A) above (which corresponds to [@H-K-S (iii) in equation (2.6)] as $$\label{iii}
|\mathrm{Tr}\left((H^{\Lambda}-E)(P_{\Lambda}(I))U^{\Lambda}P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-2}\right)|\leq \frac{|I|}{(|J|-|I|)^2}|\mathrm{Tr}\left(P_{\Lambda}(I)U^{\Lambda}\right)|.$$ At this point, with an estimate analogous to the one in the proof of Proposition 3.2 in [@C-H-K] we reach $$\mathbb{E}\left(|\mathrm{Tr}P_{\Lambda}(I)V_{\omega}|\right)\leq \lambda^{-1} \sup_{m\in \mathbb{N}}\Big\{\int^{(m+1)\varepsilon}_{m\varepsilon}\omega_j\rho(\omega_j)\,d\omega_j\Big\}|\Lambda| \,\,\,\,\varepsilon=|I|.$$ Thus, with $M_1(\varepsilon):=\sup_{m\in \mathbb{N}}\Big\{\int^{(m+1)\varepsilon}_{m\varepsilon}\omega_j\rho(\omega_j)\,d\omega_j\Big\}$, $$\label{3rd}
\lambda|\mathrm{Tr}(H_{\Lambda}-E)P_{\Lambda}(I)U^{\Lambda}P_{0,\Lambda}(J^c)(H^{\Lambda}_{0}-E)^{-2}|\leq \frac{\lambda|I|}{(|J|-|I|)^2}(\frac{M_1(\varepsilon)}{\lambda}+\frac{g\|F\|_{\infty}}{\lambda})|\Lambda|.$$ Similarly, with $M_2(\varepsilon):=\sup_{m\in \mathbb{N}}\Big\{\int^{(m+1)\varepsilon}_{m\varepsilon}\omega^2_j\rho(\omega_j)\,d\omega_j\Big\}$, we estimate term (B) through $$\label{4th}
\lambda^2|\mathrm{Tr}U^{\Lambda}(P_{\Lambda}(I))U^{\Lambda}P_{0,\Lambda}(J^c)(H_{0,\Lambda}-E)^{-2}|\leq \frac{4\lambda^2}{(|J|-|I|)^2}\left(\frac{M_2(\varepsilon)}{\lambda}|\Lambda|+\frac{g^2}{\lambda^2}\mathrm{Tr}(P_{\Lambda}(I))\right).$$ Due lemma \[bdddensity\] and the Wegner estimate (see [@A-W-B theorem 4.1]) we conclude that $$\label{wegner}
\mathrm{Tr}(P_{\Lambda}(I))\leq \frac{C}{\lambda}|I||\Lambda|.$$
Choosing the interval $J$ such that $|J|=\varepsilon^{\delta}$ for $\delta<1$, keeping in mind the assumption $g^2<\lambda$ and combining the bounds (\[1st\]), (\[2nd\]), (\[3rd\]), (\[4th\]), (\[wegner\]) and optimizing over $\delta$ gives $\delta=\frac{1}{2+\alpha_0}$ therefore we reach (\[goalids\]) for $\alpha\in [0,\frac{\alpha_0}{2+\alpha_0}]$ and (\[IDSenergy\]) is proven.
To show \[IDSdisorder\] we follow the proof of theorem 1.2 in [@H-K-S]. We fix $\lambda,\lambda'\in J$ and $E\in I$. As explained in [@H-K-S], using Hölder continuity with respect to energy given by equation (\[IDSenergy\]), trace identities and ergodicity of $H_{\lambda,g}$ and $H_{\lambda',g'}$, it suffices to estimate $\mathbb{E}\left( \mathrm{Tr} P_0\varphi(H_{\lambda,g})(\varphi(H_{\lambda,g})-\varphi(H_{\lambda',g'}))P_0\right)$ where $\varphi$ is a smooth function such that
$$\left\{
\begin{array}{lll}
\varphi\equiv 1\; \mathrm{on} \;(-\infty,E], \\
\varphi\equiv 0 \;\mathrm{on} \; (-\infty,E+|\lambda-\lambda'|^{\delta}+|g-g'|^{\delta})^c , \\
\|\varphi^{(j)}\|_{\infty}\leq C\left(|\lambda-\lambda'|^{\delta}+|g-g'|^{\delta}\right)^{-j},\,j=1,2...,3d+4 \;\;\;
\end{array}
\right.$$
with $\delta>0$ to be determined. The need for a high regularity of $\varphi$ is due to the fact that the random potential $V_{\omega}$ may be unbounded. Let $\tilde{\varphi}$ be an almost analytic extension of $\varphi$ of order $3+3d$. In particular, $\tilde{\varphi}$ is defined in a complex neighborhood of the support of $\varphi$ and if $z=E+i\eta$ we have that $$\label{order3}|\partial_{\overline{z}}\tilde{\varphi}(z)|\leq |\eta|^{3d+3}|\varphi^{(3d+4)}(E)|.$$ By the Helffer-Sjöstrand formula, $$\begin{aligned}
\mathrm{Tr}\left( P_0\varphi(H_{\lambda,g})(\varphi(H_{\lambda,g})-\varphi(H_{\lambda',g'}))P_0\right)&=\frac{1}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)\left(\lambda 'U_{\lambda',g'}-\lambda U_{\lambda,g}\right)R_{\lambda',g'}(z)P_0\,d^2z\\
&=\frac{(\lambda'-\lambda)}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)P_0\,d^2z\\
&+\frac{(g'-g)}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\mathrm{eff},\lambda}(g)R_{\lambda',g'}(z)P_0\,d^2z\\
&+\frac{g'}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)\left(V_{\mathrm{eff},\lambda'}(g')-V_{\mathrm{eff},\lambda}(g)\right)R_{\lambda',g'}(z)P_0\,d^2z.\\\end{aligned}$$ Since the last two terms enjoy a better modulus of Höldr continuity (since they do not involve $V_{\omega}$) and can be treated as in [@H-K-S], we shall only estimate the first of the above integrals. By the resolvent identity, $$\begin{aligned}
R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)=&R_{\lambda,g}(z)V_{\omega}R_{\lambda,g}(z)+(\lambda'-\lambda)R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)V_{\omega}R_{\lambda,g}(z)+(g'-g)R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)V_{\mathrm{eff},\lambda'}(g')R_{\lambda,g}(z)\\
&+gR_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)(V_{\mathrm{eff},\lambda'}(g')-V_{\mathrm{eff},\lambda}(g))R_{\lambda,g}(z).\end{aligned}$$
The above considerations lead to a perturbative expansion of $\frac{(\lambda'-\lambda)}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)P_0\,d^2z$ into four terms. We will show below that each of them can be bounded in terms of powers of either $|\lambda-\lambda'|$ or $|g-g'|$.
We start by estimating $\mathbb{E}\left( \Big|\frac{(\lambda-\lambda')^2}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)V_{\omega}R_{\lambda,g}(z)P_0\,d^2z\Big|\right)$ with a slight modification of equation (3.15) in [@H-K-S] since $V_{\omega}$ is unbounded. By the Combes-Thomas bound, equation (\[order3\]) and the choice of $\varphi$ $$\begin{aligned}
\mathbb{E}\left( \Big|\frac{(\lambda-\lambda')^2}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)V_{\omega}R_{\lambda,g}(z)P_0\,d^2z\Big|\right)\leq C(d)\left(1+\mathbb{E}^2(|V_{\omega}|)\right)\frac{|\lambda-\lambda'|^{2}}{(|\lambda-\lambda'|^{\delta}+|g-g'|^{\delta})^{3d+4}}.\end{aligned}$$ Similarly, $$\begin{aligned}
\mathbb{E}&\left( \Big|\frac{(\lambda-\lambda')(g-g')}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\mathrm{eff},\lambda'}(g')R_{\lambda',g'}(z)V_{\omega}R_{\lambda,g}(z)P_0\,d^2z\Big|\right)\\
&\leq C(d)\left(1+\mathbb{E}(|V_{\omega}|)\right)\frac{|\lambda-\lambda'||g-g'|}{(|\lambda-\lambda'|^{\delta}+|g-g'|^{\delta})^{3d+4}}.\end{aligned}$$ Moreover, using lemma \[comparepotentials\] with the the explicit dependence on $\omega$ given there, we obtain $$\begin{aligned}
\mathbb{E}&\left( \Big|\frac{g(\lambda-\lambda')}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)(V_{\mathrm{eff},\lambda'}(g')-V_{\mathrm{eff},\lambda}(g))R_{\lambda,g}(z)P_0\,d^2z\Big|\right)\leq\\
&C(d)\left(1+\mathbb{E}(|V_{\omega}|)\right)\frac{|g||\lambda-\lambda'|(|g-g'|+|\lambda-\lambda'|)}{(|\lambda-\lambda'|^{\delta}+|g-g'|^{\delta})^{3+3d}}.\\\end{aligned}$$ Using the same arguments as in[@H-K-S Equations 3.17 and 3.18] we see that $$\Big|\frac{(\lambda'-\lambda)}{\pi}\int_{\mathbb{C}}\partial_{\overline{z}}\tilde{\varphi}\,\mathrm{Tr}P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\omega}R_{\lambda',g'}(z)P_0\,d^2z\Big|$$ can be bounded from above by $$|\lambda-\lambda'||\mathbb{E}\left( \mathrm{Tr}(P_0\varphi(H_{\lambda,g})R_{\lambda,g}(z)V_{\omega}R_{\lambda,g}(z)P_0)\right)|\leq \frac{C|\lambda-\lambda'|\mathbb{E}\left(|V_{\omega}|\right)}{(|\lambda-\lambda'|^{\delta}+|g-g'|^{\delta})}.$$ Finally, we conclude that $|N_{\lambda,g}(E)-N_{\lambda',g'}(E)|$ is bounded from above by $$\begin{aligned}
C(\alpha_0,d,I)\left(|\lambda-\lambda'|^{\delta \alpha}+|g-g'|^{\delta \alpha}+|\lambda-\lambda'|^{2-(3d+4)\delta}+|g-g'|^{2-(3d+4)\delta}+|\lambda-\lambda'|^{1-\delta}+|g-g'|^{1-\delta}\right).\end{aligned}$$ Choosing $\delta=\frac{2}{\alpha+3d+4}$ we obtain, for any $\beta\in [0,\frac{2}{\alpha+3d+4}]$, $$|N_{\lambda,g}(E)-N_{\lambda',g'}(E)|\leq C(\alpha_0,I)\left(|\lambda-\lambda'|^{\beta}+|g-g'|^{\beta}\right)$$ finishing the proof of theorem \[thmids\].
[10]{}
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abstract: 'We propose a novel algorithm - Multifractal Cross-Correlation Analysis (MFCCA) - that constitutes a consistent extension of the Detrended Cross-Correlation Analysis (DCCA) and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods like MF-DXA have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time, and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter $\lambda_q$. This relation provides information about character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.'
author:
- Paweł Oświȩcimka
- Stanisław Drożdż
- Marcin Forczek
- Stanisław Jadach
- Jarosław Kwapień
title: 'Detrended Cross-Correlation Analysis Consistently Extended to Multifractality'
---
Introduction
============
Analysis of time series with nonlinear long-range correlations is often grounded on a study of their multifractal structure [@halsey86; @mandelbrot96; @muzy94; @kwapien12; @struzik02; @oswiecimka05; @drozdz09; @mandelbrot89]. Existing algorithms used in such an analysis allow for determining generalized fractal dimensions or Hölder exponents based either on statistical properties of time series [@kantelhardt02; @grech13] or on time-frequency information [@muzy94; @oswiecimka06]. Because of implementation simplicity and their utility, these algorithms have already been applied to characterize correlation structure of data in various areas of science like physics [@muzy08; @subramaniam08], biology [@ivanov99; @makowiec09; @rosas02], chemistry [@stanley88; @udovichenko02], geophysics [@witt13; @telesca05], economics [@ausloos02; @calvet02; @turiel05; @drozdz10; @oswiecimka08; @zhou09; @bogachev09; @su09], hydrology [@koscielny06], atmospheric physics [@kantelhardt06], quantitative linguistics [@ausloos12; @grabska12], music [@jafari07; @oswiecimka11], and human communications [@perello08]. As an important step towards quantifying complexity, in recent years algorithms designed for investigation of fractal cross-correlations were proposed [@podobnik08; @podobnik09] followed by the new statistical cross-correlation tests [@zebende11; @podobnik11]. These developments are based on the Detrended Cross-Correlation Analysis (DCCA) which constitutes a straightforward generalization of the fractal auto-correlation (DFA) [@kantelhardt01] on the case of fractally cross-correlated signals. In that case, the cross-correlation scaling exponent $\lambda$ can be obtained. However, literature still lacks comprehensive interpretation of this quantity.
Subsequently, the multifractal extension (MF-DXA) of the DCCA method was proposed [@zhou08]. Other closely related methods to deal with multifractal cross-correlations have also been introduced [@kristoufek11]. However, these extensions naturally involve computation of arbitrary powers of cross-covariances and this leads to serious limitations since such cross-covariances may, in general, become negative. In such a case the net result, expressed in terms of the usual fluctuation functions, thus becomes complex-valued which does not allow to determine the scaling exponents by conventional means. A simplistic resolution, so far available in the literature, to this difficulty is based on taking modulus [@jiang11; @he11; @li12; @wang12] of the cross-covariance function in order to get rid of its negative signs. In most realistic cases, as our analysis below shows, this however seriously distorts or even spuriously amplifies the multifractal cross-correlation measures. Our motivation therefore is to elaborate an algorithm that we call Multifractal Cross-Correlation Analysis (MFCCA), such that for any two signals it allows to compute their arbitrary-order covariance function and at the same time it properly takes care of the relative signs in the signals.
The proposed method allows us to calculate the spectrum of the exponents $\lambda_q$, which characterize multifractal properties of the cross-covariance. However, unlike the method proposed ealier, in our procedure, the scaling properties of the $q$th order cross-covariance function are estimated with respect to the original sign of the cross-covariance. This procedure makes the method both more sensitive to cross-correlation structure and free from limitations of other algorithms. It also turns out that the proposed method is a more natural generalization of the monofractal DCCA than is MF-DXA. The robustness of our algorithm makes it applicable to different data types in various fields of science.
Description of the MFCCA algorithm \[Method\]
=============================================
Multifractal Cross-Correlation Analysis consists of several steps which are described in detail below. As it was mentioned above, MFCCA has been developed based on the DCCA procedure [@podobnik08], therefore the initial steps are the same.
Consider two time series $x_i$, $y_i$ where $i=1,2...N$. At first, the signal profile has to be calculated for each of them: $$X\left(j\right) =\sum_{i=1}^j[x_{i}-\langle x\rangle] ,\quad
Y\left(j\right) =\sum_{i=1}^j[y_{i}-\langle y\rangle].$$ Here, $\langle \rangle$ denotes averaging over entire time series. Then, both signal profiles are divided into $M_s=N/s$ disjoint segments $\nu$ of length $s$. For each box $\nu$, the assumed trend is estimated by fitting a polynomial of order $m$ ($P^{(m)}_{X,\nu}$ for $X$ and $P^{(m)}_{Y,\nu}$ for $Y$). Based on our own experience [@oswiecimka06], as optimal we use a polynomial of order $m=2$ throughout this paper but the proposed procedure is not restricted to this particular order and can be used for much larger one when needed (as, for instance, in signals involving a highly periodic component [@horvatic11; @ludescher11]). Next, the trend is subtracted from the data and the detrended cross-covariance within each box is calculated: $$\begin{gathered}
F_{xy}^{2}(\nu,s)=\frac{1}{s}\Sigma_{k=1}^{s}\lbrace
(X((\nu-1)s+k)-P^{(m)}_{X,\nu}(k)) \times \\ \times
(Y((\nu-1)s+k)-P^{(m)}_{Y,\nu}(k))\rbrace
\label{Fxy2}\end{gathered}$$ In contrast to the detrended variance calculated in the MFDFA procedure [@kantelhardt02], in the present case, $F_{xy}^{2}(\nu,s)$ can take both positive and negative values (for an example see Sec. \[stocks\] Fig.\[fig12\]). Therefore, gradual investigation of scaling properties from small to large fluctuations through their covariances of increasing order should take into account also sign of $F_{xy}^{2}(\nu,s)$. Accordingly, the most natural form of the $q$th order covariance function is postulated by the following equation: $$F_{xy}^{q}(s)=\frac{1}{M_s}\Sigma_{\nu=1}^{M_s} {\rm
sign}(F_{xy}^{2}(\nu,s))|F_{xy}^{2}(\nu,s)|^{q/2} ,
\label{Fq}$$ where ${\rm sign}(F_{xy}^{2}(\nu,s))$ denotes the sign of $F_{xy}^{2}(\nu,s)$. The parameter $q$ can take any real number except zero. However, for $q=0$, the logarithmic version of Eq. (\[Fq\]) can be employed [@kantelhardt02]: $$F_{xy}^{0}(s)=\frac{1}{M_s}\Sigma_{\nu=1}^{M_s} {\rm sign}(F_{xy}^{2}(\nu,s))
\ln|F_{xy}^{2}(\nu,s)| .$$ As we can see in Eq. (\[Fq\]), for negative values of $q$, small values of the covariance function $F_{xy}^{2}(\nu,s)$ are amplified, while for large $q>0$, its large values dominate. Moreover, the formula for calculating $F_{xy}^{q}(s)$ respects the genuine signs of the amplified (or supressed) fluctuations of the detrended cross-covariance function (Eq. (\[Fxy2\])) and, at the same time, it allows to avoid complex numbers associated with the arbitrary powers of negative fluctuations. The above described steps of MFCCA should be repeated for different scales $s$. If the so-obtained function $F_{xy}^{q}(s)$ does not develop scaling, by for instance fluctuating around zero, there is no fractal cross-correlation between the time series under study for the considered value of $q$. Multifractal cross-correlation is expected to manifest itself in the power-law dependence of $F_{xy}^{q}(s)$ (if the $q$th order covariance function is negative for every $s$, we may take $F_{xy}^{q}(s)\longrightarrow -F_{xy}^{q}(s)$ [@podobnik08]) and the following relation is fulfilled: $$F_{xy}^{q}(s)^{1/q}=F_{xy}(q,s) \sim s^{\lambda _q}
\label{Fxy}$$ (or $\exp(F_{xy}^{0}(s)) = F_{xy}(0,s) \sim s^{\lambda_0} $ for $q=0$), where $\lambda_q$ is an exponent that quantitatively characterizes fractal properties of the cross-covariance. For the monofractal cross-correlation, the exponents $\lambda _q$ are independent of $q$ and equal to $\lambda$ as obtained from the DCCA method. In the case of multifractal cross-correlation, however, $\lambda _q$ varies with $q$, with $\lambda$ retrieved for $q=2$. The minimum and maximum scales ($s_{min}$ and $s_{max}$, respectively) depend on the length $N$ of the time series under study. In practice, it is reasonable to take $s_{max} < N/5$.
Analysis of examplary models and stock market data
==================================================
In order to verify the usefulness of MFCCA algorithm, we test it by using both artificially generated cross-correlated time series and real-world signals. In order to avoid divergent moments due to fat tails in the distribution of fluctuations, we restrict $q$ to $\langle -4,4 \rangle$ with a step $0.2$ throughout this paper. In the case of computer-generated signals, results for each process are averaged over its 20 independent realizations.
ARFIMA processes
----------------
We start our study from an analysis of the well-known ARFIMA processes [@hosking81], which are examples of monofractal, long-range correlated signals. In Ref. [@podobnik08], such processes were used to show usefulness of the DCCA algorithm. Our goal is to show the cross-correlation structure of the above-mentioned processes more completely. To generate a pair ($x_i,y_i$) of the cross-correlated ARFIMA processes, we use the following equations: $$x_i=\Sigma_{j=1}^{\infty}a_j(d_x)x_{i-j}+\epsilon_i \,
\label{x}$$ $$y_i=\Sigma_{j=1}^{\infty}a_j(d_y)y_{i-j}+\epsilon_i
\label{y}$$ where $d_x$ and $d_y$ are parameters characterizing linear long-range autocorrelations of the times series. These quantities can be related to the Hurst exponents [@kantelhardt01] by the relation $H=1/2+d_{x(y)}$, ($-1/2<d_{x(y)}<1/2$). Positively correlated (persistent) time series are characterized by $H>0.5$, whereas negative autocorrelation (antipersistent signal) is characterized by $H<0.5$; H=0.5 means no linear autocorrelation. The quantity $a_j(d_{x(y)})$ is called weight and is defined by $a_j(d_{x(y)})
= \Gamma (j-d_{x(y)})/[\Gamma(-d_{x(y)})\Gamma(1+j)]$, where $\Gamma()$ stands for Gamma function. $\epsilon_i$ is an i.i.d. Gaussian random variable. The processes $x_i$ and $y_i$ are cross-correlated, because the same noise component $\epsilon_i$ is used in both Eq.(\[x\]) and Eq.(\[y\]). We generate three pairs of cross-correlated signals: ($H_1$=0.5, $H_2$=0.6), ($H_1$=0.5, $H_2$=0.7), and ($H_1$=0.5, $H_2$=0.9), where $H_1$ and $H_2$ characterize long-range autocorrelation of the first and the second time series, respectively. In order to obtain statistically significant results, we generate time series of lengh $N=100,000$ points each.
In the left panels of Fig. \[fig1\], we present the calculated $F_{xy}(q,s)$ for all the signal pairs. Each line corresponds to a different value of $q$. As it can be seen, in all the cases, $F_{xy}(q,s)$ is a power function of scale $s$. This indicates the fractal nature of the cross-correlations. Moreover, for all types of signals, the functions $F_{xy}(q,s)$ are almost parallel to each other implying largely homogeneous character of the corresponding cross-correlations. Indeed, as shown in the right panels of the Fig. \[fig1\], the difference between the extreme values of $\lambda_q$ expressed by $\Delta \lambda_q=\max(\lambda_q)-\min(\lambda_q)$ is approximately 0.005, 0.007, and 0.011 for the top, middle, and the bottom panel, respectively. These narrow ranges of $\lambda_q$ indicate that the ARFIMA processes reveal correlations that are monofractal regardless of the types of linear autocorrelation of signals.
In literature, the estimated fractal cross-correlations are often related to the fractal properties of the individual signals themselves [@podobnik08; @siqueira10; @shadkhoo09]. Therefore, in Fig. \[fig1\], we also show the average of the generalized Hurst exponents [@kantelhardt02]: $$h_{xy}(q)=(h_x(q)+h_y(q))/2,$$ where $h_x(q)$ and $h_y(q)$ refer to fractal properies of individual time series, respectively and, for $q=2$, they correspond to the Hurst exponent $H$. It is worth noticing that relation between $\lambda_q$ and $h_{xy}(q)$ depends on temporal organization of the signals as determined by their Hurst exponents. For two signals whose Hurst exponents $H$ are alike, their multifractal cross-correlation characteristics described by $\lambda_q$ and $h_{xy}(q)$ are almost identical, while the divergence between $\lambda_q$ and $h_{xy}(q)$ becames more sizeble for time series with more significant differences in autocorrelation (different Hurst exponent $H$).
This result means that, in the case of the ARFIMA processes, the relation $\lambda\approx (h_x(2)+h_y(2))/2$ introduced in Ref. [@podobnik08] applies only to a situation when differences between $h_x$ and $h_y$ are negligible.
Markov-switching multifractal model \[Markov\]
----------------------------------------------
As an example of multfractal process, we consider the Markov-switching multifractal model (MSM) [@liu07]. MSM is an iterative model, which is able to replicate hierarchical, multiplicative structure of real data and, thus, insures multifractal properties of the generated time series. Because of its properties, MSM is commonly used in finance, where multifractality of price fluctuations is one of the main stylized facts [@liu08; @kwapien05]. Equally well this model can be used to simulate many other multifractal time series representing natural phenomena as it is able to generate the volatility clustering responsible for the underlying nonlinear temporal correlations [@drozdz09]. Below, we present the main stages of the model’s construction.
In MSM, evolution of an observable $r_t$ in time $t$ is modeled by the formula [@liu08]: $$r_t=\sigma_t\cdot u_t ,
\label{msm}$$ where $u_t$ stands for a Gaussian random variable and $\sigma_t$ (multifractal process) stands for the instantaneous volatility component. The volatility $\sigma_t$ is a product of $k$ multipliers $M_1(t),M_2(t),...,M_k(t)$ such that $$\sigma^2_t=\sigma^2\prod _{i=1}^k M_i(t) ,$$ where $\sigma^2$ is a constant factor. A common version of the model assumes that the multipliers $M_i(t)$ are drawn from the binomial or from the log-normal distribution. Here, we use the binomial one with $M_i(t) \sim
\{m_0,2-m_0\}$, $1 \leq m_0<2$. Any change of a multiplier in the hierarchical structure of volatility is determined by the transition probabilities [@liu07]: $$\gamma _i = 1-(1-\gamma _k)^{b^{i-k}} ,\quad i=1,2...k.$$ Thus, a multiplier $M_i(t)$ is renewed with probability $\gamma_i$ and remains unchanged with probability $1-\gamma_i$. The parameter $\gamma_k$ is taken from the range $(0,1)$, and $b > 1$. We put $\gamma_k=0.5$ and $b=2$, which leads to the relation: $$\gamma _i = 1-(0.5)^{2^{i-k}} ,\quad i=1,2...k.$$ Thus, for the initial stages of the cascade, a renewal of the multipliers $M_i(t)$ occurs with relatively small probability, while the largest $\gamma_i=0.5$ appears for $i=k$.
### Unsigned version of the MSM model
For the purpose of this analysis, we generate a set of multifractal time series ($\sigma_t$) of length $131, 072$ points each. However, in all realizations of the model, we conserve the hierarchical structure of the multipliers, since the renewals of $M_i(t)$ appear for the same $i$ and $t$ in each generated series. This procedure insures cross-correlations between series with different $m_0$.
In Fig. \[fig2\], we present sample results of MFCCA obtained for two pairs of MSM series with the parameters $m_{0}^{(1)}=1.2$ and $m_{0}^{(2)}=1.35$ (top panels) and $m_{0}^{(1)}=1.2$ and $m_{0}^{(2)}=1.6$ (bottom panels). The $q$th-order covariance functions $F_{xy}(q,s)$ (left hand side of this Figure) display a clear multifractal scaling within the whole range $(-4,4)$ of the $q$ values. The resulting $\lambda_q$ is a decreasing function of $q$, which is a hallmark of multifractality. Moreover, the rate of decrease of $\lambda_q$ depends on the values of mutlipliers. For the first pair of signals (with $m_{0}^{(2)}=1.35$), the exponents $\lambda_q$ are contained in the range (0.81,1.25), while for the second pair (with $m_{0}^{(2)}=1.6$), $0.75 \le \lambda_q \le 1.7$. In the same Fig. \[fig2\], we also show the average of the generalized Hurst exponents calculated for each time series independently (red squares).
Interestingly, for the signals with a relatively small difference $\Delta
m_{12}=m_{0}^{(2)}-m_{0}^{(1)}$ - in other words, for similar multifractals - $\lambda_q$ approximately equals the average of $h_{x}(q)$ and $h_y(q)$. A tiny difference between $h_{xy}(q)$ and $\lambda_q$ is here visible only for $q>0$. This effect is depicted more quantitatively in the insets of Fig. \[fig2\], where $h_{xy}(q)-\lambda_q$ is presented as a function of $q$. The maximum deviation from zero can be seen for $q=2.2$, reaching a value of $0.02$. For the second pair of signals, the difference $h_{xy}(q)-\lambda_q$ is more pronounced and concerns both negative and positive $q$’s. In this case, the largest difference of $h_{xy}(q)$ and $\lambda_q$ is for $q=-2$ and equals 0.07.
### Relation between $\lambda_q$ and $h_{xy}$
To have some insight into the relation between $\lambda_q$ and $h_{xy}(q)$, we perform a systematic MFCCA study for the set of time series pairs, such that one of them is generated with $m_{0}^{(1)}=1.2$ and the other one with $m_{0}^{(2)}$ from the range $\langle 1.25, 1.9 \rangle$ (the step is 0.05). However, the multifractal characteristics were possible to estimate only for $\Delta m_{12}<0.6$. In the case of $\Delta m_{12}>0.6$, $F_{xy}^q$ takes both positive and negative values and Eq.(\[Fxy\]) is not satisfied. At first, we focus on the relationship between $\lambda_2=\lambda$ and the average Hurst exponent $h_{xy}(2)$. In the inset of Fig. \[fig3\], we present these quantities as a function of $\Delta m_{12}$. It is clearly visible that both these quantities are monotonically decreasing and they take approximately the same values for small $\Delta m_{12}$. However, for $\Delta m_{12}>0.25$, $\lambda(\Delta m_{12})$ decreases slower than the Hurst index (thus $\lambda>H$) and the statistics diverge. To highlight this result, we calculate also the difference between these two quantities which is shown in Fig.\[fig3\]. As one can see, $\lambda-h_{xy}(2)$ is an increasing function of $\Delta m_{12}$. This result indicates that the difference between $\lambda$ and the average Hurst exponent becomes larger for time series whose multifractal characteristics depart more from each other, while the opposite is observed when these characteristics are alike, which at the same time results in stronger cross-correlations.
To better understand this effect, we analyze a covariance function $F_{xy}(2,s) \sim s^\lambda$ and an expression based on fluctuation functions [@kantelhardt02]: $\sqrt{F_{xx}(2,s)F_{yy}(2,s)} \sim
s^{\frac{h_x(2)+h_y(2)}{2}} = s^{h_{xy}(2)}$. In Fig. \[fig4\], we show these functions calculated for different values of $\Delta m_{12}$. It is easy to notice that the presented functions are almost identical to each other for small $\Delta m_{12}$. However, the larger $\Delta m_{12}$ is, the more visible is a departure between the analyzed statistics. In all cases, the values of $F_{xy}(2,s)$ are at most equal to $\sqrt{F_{xx}(2,s)F_{yy}(2,s)}$ and estimated $\lambda$ is larger than $h_{xy}(2)$. These numerical results are in accord with the following relation: $$F_{xy} (2,s)\leq \sqrt{F_{xx}(2,s)F_{yy}(2,s)} ,
\label{FxyF}$$ which straightforwardly results from the definitions of these quantities considered in terms of the scalar products of vectors formed from the underlying time series [@he11]. In order to more clearly see the relationship between $\lambda$ and $h_{xy}(2)$, we can reformulate Eq. (\[FxyF\]) in the case when the relations $F_{xy}
(2,s)=a_{xy}s^\lambda$, $F_{xx}(2,s)=a_xs^{h_x(2)}$, and $F_{yy}(2,s)=a_ys^{h_y(2)}$ apply, to obtain: $$a_{xy}s^{\lambda}\leq (a_xa_y)^{1/2}s^{\frac{h_x(2)+h_y(2)}{2}} .$$ This leads to: $$\lambda \leq \log_s(\frac{(a_xa_y)^{1/2}}{a_{xy}}) + \frac{h_x(2)+h_y(2)}{2}.$$ For two identical time series, the equality in Eq. (\[FxyF\]) holds leading to obvious $\lambda = \frac{h_x(2)+h_y(2)}{2}$. In general, however, $$A_r = \frac{(a_xa_y)^{1/2}}{a_{xy}}\neq 1,$$ and thus a difference between $\lambda$ and $h_{xy}(2)$ in either direction is allowed or even forced, depending on a sign of $\log_s(A_r)$. For negative values of this quantity, $\lambda$ has to be smaller than $h_{xy}(2)$, while for positive values it can become larger. An example demonstrating the rate of changes of $\ln(A_r)$ as a function of $\Delta m_{12}$ for $q=2$ is shown in Fig. \[fig5\]. In this case, $\ln(A_r)$ is positive and quickly increases with $\Delta m_{12}$, thus with the degree of dissimilarity between the two series. The related dependencies are even more involved and appear to strongly vary with the parameter $q$ as it is more systematically shown in Fig. \[fig6\]. The $\ln(A_r)$ is seen to be positive for $q > 0$ with an increasing value at maximum with increasing $\Delta m_{12}$, and a larger amplitude of changes with increasing $q$. Similar, but reversed in sign and with an even larger amplitude of changes, is the situation for $q < 0$. These results nicely coincide - and thus point to their origin - with those presented in Fig. \[fig2\], where $\lambda_q$ is larger than $h_{xy}(q)$ for positive values of $q$ and smaller for negative ones. Even the maxima of these differences occur for those values of $q$, where they are seen in Fig. \[fig6\] and they are larger on the negative side of $q$. Of course, they are also larger for larger $\Delta m_{12}$.
The difference between $\lambda_q$ and $h_{xy}(q)$ has its reflection - also consistent with the findings presented in Figs. \[fig2\] and \[fig6\] - in another popular multifractal measure, namely in the range of scaling exponents. In Fig. \[fig7\], we display $\Delta \lambda_q$ as a function of $\Delta m_{12}$ for the two ranges of $q$: $-2 \le q \le 2$ and $-4 \le q \le
4$. For comparison, in the same Figure, we show $\Delta
h_{xy}=\max(h_{xy})-\min( h_{xy})$ calculated for the same ranges of $q$. The $\Delta h_{xy}(q)$ and $\Delta \lambda_q$ are seen to be monotonically increasing functions of $\Delta m_{12}$ in all the cases. However, for $-4 \le
q \le 4$ these characteristics are almost the same, while for $-2 \le q \le 2$ the difference between $\Delta h_{xy}$ and $\Delta \lambda_q$ systematically increases with $\Delta m_{12}$. This suggests that for relatively large values of $|q|$ (magnifying the largest and the smallest fluctuations of instantaneous volatility components) the fractal character of the considered processes is similar, which may reflect the effect of preserving the same hierarchical structure of multipliers for all generated multifractals, where only relative changes of volatility are possible. The above results thus indicate that the difference between $h_{xy}(q)$ and $\lambda_q$ is to be considered an important ingredient of measure of the fractal cross-correlation between two time series.
### Performance comparison of MFCCA and MF-DXA
In the next stage of our study of the MSM generated $\sigma_t$ time series, we analyze output of MFCCA if one time series of a pair is gradually being shifted in time with respect to the other one. Then the correlations, especially their fractal character, should undergo an obvious weakening. This test is aimed at further verifying performance of the algorithm. As an input, we use a time series with $m_0=1.2$ and the same one, but shifted by a certain number of points. We notice that the larger is the relative shift between the time series, the shorter is the scaling range of $F_{xy}$. However, in all cases, the estimated $\lambda$ is equal to the generalized Hurst exponent calculated for a single series. This shortening of the range of scaling is not symmetric from both sides of the scale range, but gradually arrives entirely from the small scale side. The shift dependence of the lower bound of the scaling regime that can be used to determine $\lambda_q$ is shown in Fig. \[fig8\]. As expected, lifting of this lower bound is seen to be almost linear. In the same Figure, we also present the result of an analogous analysis, but performed by means of the common variant of the MF-DXA procedure that, in order to resolve the sign problem, makes use of the absolute values of the fluctuation functions [@jiang11; @he11; @li12; @wang12]. In this case, the procedure is seen not to be sensitive to this type of surrogate and, thus, evidently generates spurious cross-correlations.
In order to elaborate more in detail on this last issue, we generate an example of a pair of the MSM time series with $m_0=1.2$ drawn independently, i.e., with no taking care about preserving the hierarchical structure of the multipliers. Even though individually both such series are multifractal with the same multifractality characteristics, there is no reason to expect them to be multifractally cross-correlated. Indeed, in the present case the corresponding $q$th order covariance functions determined through the Eq. (\[Fq\]) do not scale and for small and moderate scales they even assume the negative values by fluctuating around zero. An example of $F^2_{xy}(s)$ demonstrating this behavior is shown in the left panel of Fig. \[fig9\]. In fact, this dependence closely resembles the DCCA result (Fig. 1b in ref. [@podobnik08]) obtained in an analogous situation of the two uncorrelated series. This correspondence thus provides an additional argument that it is MFCCA proposed here that constitutes a natural and correct multifractal generalization of DCCA. Application of a previously postulated [@zhou08] extension of DCCA in the present example would lead to complex-valued $q$th-order covariances. As already mentioned in Introduction, a commonly adopted resolution to this difficulty is based on taking modulus of the cross-covariance before computing its $q$th order. The result of such a procedure applied to our example of two independently generated MSM time series with $m_0=1.2$ is shown in the right panel of Fig. \[fig9\] and it clearly indicates a convincing multifractal scaling. This, of course, is however a false signal as these series are not expected to be multifractally correlated.
### Signed version of the MSM model
The time series of $\sigma_t$ considered above represent unsigned fluctuations (volatility in financial terms) and therefore their individual Hurst exponents are significantly larger than 0.5. By incorporating the Gaussian random variable $u_t$ drawn from $N(0,1)$ through the Eq. (\[msm\]), one obtains the signed time series $r_t$ with the Hurst exponents close to 0.5 (as for the financial returns, for instance). Very similar effect one obtains when multiplying the original unsigned fluctuations simply by randomly drawn either $+1$ or $-1$. The influence of such procedures on the generalized Hurst exponents $h(q)$ is shown in Fig. \[fig10\] for the same pairs of the MSM time series as before, i.e., with $m_0=1.2, m_0=1.35$ and $m_0=1.2, m_0=1.6$. Circles indicate $h(q)$ for the original unsigned series while squares and triangles indicate the series signed by the Gaussian random variable and by the pure random sign, respectively. Introducing sign clearly shifts the lines down relative to the unsigned case, such that the usual Hurst exponent $H=h(2)$ assumes value of $0.5$ for all the signed series. The $q$-dependence of $h(q)$, naturally stronger for larger $m_0$, remains however essentially preserved after introducing the sign, which reflects the fact that such an operation influences primarily the linear temporal correlations in the series leaving the nonlinear ones, related to the volatility clustering [@drozdz09], preserved. As far as multifractal cross-correlations between such series are concerned, more care is needed. Drawing the term $u_t$ in Eq. (\[msm\]) independently for the two series destroys their original (unsigned) cross-correlations and the corresponding $q$th-order covariances calculated through the Eq. (\[Fq\]) develop similar fluctuations as those in the left panel of Fig. \[fig9\]. One most straightforward way to preserve multifractal cross-correlations is to use the same $u_t$ for the two series under consideration. Examples of the so-prepared pairs of series, for the same combination of the parameters $m_0$ as before for the unsigned series, i.e., $m_0=1.2$ versus $m_0=1.35$ and $m_0=1.2$ versus $m_0=1.6$, are analyzed in Fig. \[fig11\] in terms of $\lambda_q$ and $h_{xy}(q)$. For the first of these pairs, irrespective of the sign adding variant, the multifractal cross-correlations are seen to remain essentially on the same level of strength as those for the corresponding unsigned signals shown in the upper panel of Fig \[fig2\]. The departures between $\lambda_q$ and $h_{xy}(q)$ for the other pair ($m_0=1.2$ and $m_0=1.6$) of the signed series can be seen to be somewhat larger relative to their unsigned counterparts, which signals slight weakening of their multifractal cross-correlations. This in fact is consistent with the generalized Hurst exponents $h(q)$ seen in Fig. \[fig10\]. When sign is applied to the series, the distance between the corresponding $h(q)$ routes increases, especially on the negative $q$ side, as compared to the unsigned case.
Examples of stock market data \[stocks\]
----------------------------------------
The financial fluctuations can be considered a physical process which constitutes one of the most complex generalizations of the conventional Brownian motion carrying at the same time convincing traces of nontrivial fractality [@mandelbrot97; @bogachev07; @ludescher11b]. They therefore offer a very demanding territory to test the related concepts and algorithms. For this reason, as final examples of utility of the MFCCA method, we present an analysis of empirical data coming from the German stock exchange. Furthermore, since multifractal analysis of financial data is one of the most informative methods of investigating such complex systems [@oswiecimka05; @drozdz10; @zhou09; @kwapien05; @lopez03], we believe that MFCCA will be very useful in this field as well. We consider logarithmic price increments $g(i)$ and linear time increments $\Delta t(i)$ representing dynamics of a sample German stocks - E.ON (ticker: EOA) and Deutsche Bank (ticker: DBK) (from the same database as used before [@oswiecimka05]) being part of the DAX30 index. These quantities are obtained according to the formulas: $$g(i)=\ln(p(i+1))-\ln(p(i)), \ \$$ $$\Delta t(i)=t(i+1)-t(i),$$ where $p(i), i=1,...,N$ is a time series of price quotes taken in discrete transaction time $t(i)$. As it has been shown previously [@oswiecimka05], both $g(i)$ and $\Delta t(i)$ are processes with self-similar structure and could be analyzed by the multifractal methods. Quantifying the character of cross-correlations just between these two characteristics of the financial dynamics is also of particular importance for forecasting volatility within models such as the Multifractal Model of Asset Returns [@mandelbrot96; @mandelbrot97; @lux03; @oswiecimka06b; @eisler04]. Our analysis is performed on time series comprising the period between Nov. 28, 1997 and Dec. 31, 1999. The time series consists of $T=294,862$ and $T=497,513$ points for EOA and DBK, respectively. Therefore, the time series are long enough to bring statistically significant results.
In Fig. \[fig12\], we show one of the first steps of our algorithm, i.e., the detrended cross-covariance function $F^2_{xy}(\nu,500)$ (for the scale $s=500$) as a function of the box number $\nu$ (Eq. (\[Fxy2\])). For comparison, in the same Fig. \[fig12\], we depicted the detrended variance function $F_{xx}^2(\nu,500)$ and $F_{yy}^2(\nu,500)$ (obtained from MFDFA) calculated for individual time series of price increments and waiting times, respectively. It is easy to notice that the detrended variance calculated for individual time series takes only positive values, whereas the detrended cross-covariance function $F_{xy}^2(\nu,s)$ takes both negative and positive values. This constitutes the already-mentioned principal problem in straightforward calculation of the $q$th-order cross-covariance function $F_{xy}(q,s)$ for odd $q$s that results in complex values of this function (see Eq. (\[Fxy\])). It is worth stressing that this difficulty does not affect the fractal analysis of individual time series (MFDFA), because then the detrended variance function $F_{xx}^2(\nu,s)$ may only be positive. It follows that proper handling of the sign of $F_{xy}^2(\nu,s)$ is of crucial importance for a consistent extension of DCCA to treat the multifractally correlated signals. At present, a solution of this problem is offered only by the MFCCA algorithm proposed in Sec. \[Method\].
In order to characterize the cross-correlations in the present case, the function $F_{xy}(q,s)$ is calculated. As far as the multifractal scaling is concerned, the situation is significantly more subtle than in the previous model cases. It turns out that the scaling property of $F_{xy}(q,s)$ applies only selectively. First of all, for the negative $q$s, $F_{xy}^q(s)$ fluctuates around zero and Eq. (\[Fxy\]) is not satisfied. For positive values of $q$, the function $F_{xy}^q(s)$ assumes positive values but clear scaling of $F_{xy}(q,s)$ begins with $q=1$ upwards. For $q < 1$, these functions develop increasing fluctuations when $q$ moves towards zero. This effect is especially strong for DBK. Furthermore, the lower limit of scales where $F_{xy}(q,s)$ develops the convincing power-law behavior varies and it takes place at the higher values of $s$ for DBK than for EOA, which signals a weaker form of multifractal cross-correlation in the former case. The corresponding characteristics are shown in the upper panels of Fig. \[fig13\] with the scaling bounds both in $q$ and in $s$ indicated by the dashed lines.
The calculated $\lambda_q$ and $h_{xy}(q)$ are shown in the bottom panels of Fig. \[fig13\]. It is clearly visible that, for EOA, both functions converge to each other for large values of $q$, while $\lambda_q$ is significantly larger than $h_{xy}(q)$ for smaller values of $q$. These results imply that the scaling properties of $F_{xy}(q,s)$ strongly depend on the considered time span and they cannot be fully quantified by a unique exponent $\lambda$. Moreover, based on our results for the MSM model, we can infer that the analyzed processes are ruled by the similar fractal dynamics only in periods with relatively large $F_{xy}^2(\nu,s)$ (associated with large $q$). For smaller $q$’s, the difference between $\lambda_q$ and $h_{xy}(q)$ is more evident, which suggests that dynamics of these processes is significantly different, but still cross-correlative. It is worth to mention that large values of $F_{xy}^2(\nu,s)$ can be a consequence of cross-correlation both in the signs and the amplitudes of the signals. However, the waiting times are unsigned and the price increments are signed, but the sign is uncorrelated. This means that, in our case, the amplitude of $F_{xy}^2(\nu,s)$ is only a result of the cross-correlation of the observed amplitude. The strong cross-correlation of volatility (modulus of time series) is confirmed by Fig.\[fig14\], where the cross-correlation function for the waiting times and absolute values of the price increments is depicted. Therefore, we conclude that large fluctuations are much more strongly cross-correlated than the smaller ones. Complexity of the multifractal cross-correlation is expressed by the range of $\lambda_q$ that is approximately $0.32$ in this case.
As may be anticipated already from the structure of $F_{xy}(q,s)$ for DBK, the behavior of $\lambda_q$ is slightly different and the difference between $h_{xy}(q)$ and $\lambda_q$ is substantial both for small and for large values of $q$ (Fig. \[fig13\], right panel). Also $\Delta \lambda_q$ for DBK is smaller than in the case of EOA and takes a value of $0.22$. This suggest that although structure of the cross-correlation between the inter-transaction times and the price increments for DBK is multifractal, its heterogeneity is poorer than in the case of EOA. Moreover, similarity between the fractal dynamics of large fluctuations is not so evident than in the former case. These results are also confirmed by Fig. \[fig14\], where a difference between the strength of volatility cross-correlations for both considered stocks is easily visible.
The results presented here indicate that the multifractal cross-correlation characterizes only relatively large fluctuations of the signals under study. Smaller fluctuations that are filtered out by $q<1$, from the perspective of multifractal cross-correlation, may be considered mutually independent.
In connection with the present example we also wish to mention - but without showing the results explicitly in order not to confuse the reader - that taking absolute values of the fluctuation functions to get rid of the sign problem (as recently often done in literature [@he11; @li12; @wang12]), in the present financial data case, would result in a convincing but apparent multifractal scaling for all values of $q$, similar to one that we have already seen for the MSM model in Fig. \[fig9\]. Also, the so determined $\lambda_q$ equals $h_{xy}(q)$ as in the MSM model. This way one, however, does not extract genuine correlations, but only measures the averaged multifractal properties of individual time series.
Another type of correlations that are of theoretical as well as of practical interest are the correlations among stock returns [@kwapien12; @mantegna00]. These are typically quantified in terms of the Pearson correlation coefficients or, more generally, in terms of the correlation matrix. This way of quantifying correlations is, however, restricted to their linear component only. The present formalism of studying the multifractal cross-correlations allows one to reveal some of their potential nonlinear components. As an example, we therefore use the same two stocks as above (EOA and DBK) and, in addition, Commerzbank (CBK) from the same, German stock exchange over the same period, and perform a similar analysis as above for two pairs of time series (CBK-DBK and DBK-EON) representing the corresponding 1 min returns. Over the period considered, this yields 267,241 data points. The results in the same representation as before are presented in Fig. \[fig15\].
For $q < 0$, the $F_{xy}(q,s)$ are not drawn since the corresponding $F_{xy}^q(s)$ functions fluctuate around zero. As we go to the positive $q$ values, however, they start developing a convincing scaling already for $q=0.6$ (as indicated by the dashed lines) for both pairs and for all the scales considered. This scaling is clearly multifractal and the resulting $\lambda_q$ and $h_{xy}(q)$, shown in the lower panels of Fig. \[fig15\], are somewhat closer to each other than the ones previously considered for correlations between the price increments and the inter-transaction times. Slight differences in relation between the present two pairs of time series are also visible, however. For DBK-EON, the departures between $\lambda_q$ and $h_{xy}(q)$ are largely independent on $q$ in the region where scaling applies, while for CBK-DBK it starts from larger values for the smallest $q$-values (0.6), but it converges to even smaller values with an increasing $q$. This can be interpreted as an indication that multifractal character of cross-correlations resembles more each other for CBK and DBK on the level of large fluctuations and weakens for the smaller ones, while, within the pair DBK-EON, they are of similar strength in the comparable range of fluctuation size. Of course, in both cases this kind of cross-correlations disappears on the level of small fluctuations that are filtered out by the negative values of $q$, and this seems quite a natural effect in the financial context.
As a final example indicating possible applications of the MFCCA method introduced in this paper, we study the cross-correlations between the two world leading stock market indices, the Dow Jones Industrial Average (DJIA) and the Deutscher Aktienindex (DAX), based on their daily returns. The period considered for both these indices begins on January 12, 1990 and ends on October 12, 2013. This results in time series of length of 5881 data points. Due to different time zones which the two indices are traded in and in order to test potential applicability of the present algorithm in detecting possible time-lags or asymmetry effects in correlations, we study three possible variants of positioning the time series relative to each other. The first variant is most natural, i.e., data points in the two time series meet each other at the same date they are recorded. The other two variants are such that the time series are shifted by one day relative to each other, either DAX is advanced by one day or DJIA is. The corresponding $F_{xy}(q,s)$ functions are displayed in the upper panel of Fig. \[fig16\]. Unlike the high-frequency recordings discussed above, the significantly shorter time series in the present case restrict us to cover a smaller scale range. Nevertheless, evident multifractal scaling can still be identified in this case as indicated by the dashed lines in Fig. \[fig16\], provided the range of $q$ is restricted as well. Similarly to the situation with the high frequency cross-correlations within the German stocks, here $F_{xy}^q(s)$ also fluctuates around zero for negative $q$s, and therefore the corresponding functions $F_{xy}(q,s)$ are not shown. Interestingly, the lower bound in $s$ where scaling starts visibly lifts up as we move from the same date, through the situation described as ’DJIA leads’, and becomes the shortest in the situation ’DAX leads’. Accordingly, the departures between $h_{xy}(q)$, which, of course, remains invariant with respect to such relative shifts of the time series, and $\lambda_{q}$ increase as we go through the above three relative locations of DJIA versus DAX (lower panel of Fig. \[fig16\]). The strongest DJIA-DAX multifractal cross-correlation is detected when the series are originally arranged relative to each other. Their relative 1-day shifts reveal an effect of asymmetry, however. The situation ’DJIA leads’ preserves significantly more of such cross-correlations than the opposite "DAX leads’ one. This result can be interpreted as an indication that the DJIA close has more influence on the DAX close next day than the DAX close has on the DJIA close next day. In fact, as verified additionaly, splitting the time series considered here into two halves shows that this effect is more evident in 1990’s than more recently. Such an asymmetry in information transfer between these two stock markets is understandable in economic terms, and in fact it is also consistent with the previous study [@drozdz01] based on the correlation matrix formalism. Finally, we wish to mention that more distant relative shifts of the two present time series quickly deteriorate the multifractal cross-correlation, while at the same time the modulus-based MF-DXA approach leaves them unchanged.
Summary and conclusions
=======================
We proposed an algorithm, which we called Multifractal Cross-Correlation Analysis, that allows for quantitative description of multiscale cross-correlations between two time series and that is free of limitations the other existing algorithms, like MF-DXA, suffer from. The key point that distinguishes MFCCA from other related methods is construction of the $q$th-order cross-covariance function $F_{xy}^q(s)$ in Eq.(\[Fq\]), which preserves the sign of the cross-covariance fluctuation function $F_{xy}^2(\nu,s)$ after its modulus has been raised to a power of $q/2$. This step has two immediate consequences: (1) it eliminates the risk of appearance of complex values that might lead to problems with their correct interpretation, and (2) it prohibits losing information that is stored in the negative cross-covariance. It follows that, as we showed in Sec. \[Markov\] regarding known model data, the results obtained with MFCCA are more logical and better coincide with intuition than do the parallel results of MF-DXA. This was true both for the signed and the unsigned, volatility-like processes. On this ground we concluded that MFCCA provides us with the most complete information about fractal cross-correlations possible as compared to the other related methods existing so far. Having realized this, we applied MFCCA to sample real-world data from the stock markets. We found that both the cross-stock correlations and the lagged inter-market correlations of returns, as well as the correlations between price movements and the corresponding transaction time intervals are clearly multifractal. Moreover, we showed that carriers of these cross-correlations are predominantly the large fluctuations in both signals, while the smaller fluctuations contribute rather little. This outcome may suggest that an important ingredient of financial complexity, which manifests itself here as multifractality, might be temporal relations between large events.
Apart from the introduction of MFCCA, we also focused our attention on the relation between the $q$th-order scaling exponent $\lambda_q$ and the averaged generalized Hurst exponents $h_{xy}(q)$. Both these measures are equally important if one intends to comprehend fractal structure of the data under study. This is because their spectra analyzed in parallel for each signal separately contain information about similarity of their fractal structure. For example, based on model data, we found that the larger is the difference between $\lambda_q$ and $h_{xy}(q)$, the more different are the considered (multi)fractals. We thus strongly recommend investigation of both these quantities in parallel.
We believe that our approach presented here will allow for a wider application of multifractal cross-correlation analysis to empirical data in different areas of science.
Acknowledgements
================
The calculations were done at the Academic Computer Centre CYFRONET AGH, Kraków, Poland (Zeus Supercomputer) using Matlab environment.
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{
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---
abstract: |
We consider the problem of metric learning subject to a set of constraints on relative-distance comparisons between the data items. Such constraints are meant to reflect side-information that is not expressed directly in the feature vectors of the data items. The relative-distance constraints used in this work are particularly effective in expressing structures at finer level of detail than must-link (ML) and cannot-link (CL) constraints, which are most commonly used for semi-supervised clustering. Relative-distance constraints are thus useful in settings where providing an ML or a CL constraint is difficult because the granularity of the true clustering is unknown.
Our main contribution is an efficient algorithm for learning a kernel matrix using the log determinant divergence — a variant of the Bregman divergence — subject to a set of relative-distance constraints. The learned kernel matrix can then be employed by many different kernel methods in a wide range of applications. In our experimental evaluations, we consider a semi-supervised clustering setting and show empirically that kernels found by our algorithm yield clusterings of higher quality than existing approaches that either use ML/CL constraints or a different means to implement the supervision using relative comparisons.
author:
- |
Ehsan Amid\
Department of Computer Science, UC - Santa Cruz\
Santa Cruz, CA, 95064\
`[email protected]`\
Aristides Gionis\
Department of Computer Science, Aalto University\
Helsinki Institute for Information Technology\
02150 Espoo, Finland\
`[email protected] `\
Antti Ukkonen\
Finnish Institute of Occupational Health\
00290 Helsinki, Finland\
`[email protected] `\
bibliography:
- 'refs-short.bib'
title: |
Semi-supervised Kernel Metric Learning\
Using Relative Comparisons
---
Introduction {#sec:intro}
============
[*Metric learning*]{} is the task of finding an appropriate metric (distance function) between a set of items. In many cases, a feature representation of the items is provided as input and distances between the data items can be calculated using a proper norm on the corresponding feature vectors. However, it is often the case that the feature-vector representation of the items alone is not sufficient to describe intricate and refined relations in the data. For instance, when clustering images it may be necessary to make use of semantic information about the image content, in addition to some standard image-processing features.
[*Semi-supervised metric learning*]{} provides a principled framework for combining feature vectors with other external information that can help capturing more refined relations in the data. Such external information is usually given as labels about [*pair-wise distances*]{} between a few data items. Such labels may be obtained by crowd-sourcing, or provided by the data analyst, or anyone interacting with the clustering application, and they reflect properties of the data that are not expressed directly from the data features. The metric-learning problem then corresponds to finding a linear transformation of the initial features such that the constraints imposed by the external information are satisfied.
There are two commonly used ways to formalize such side information. The first are [*must-link*]{} (ML) and [*cannot-link*]{} (CL) constraints. An ML (CL) constraint between data items $i$ and $j$ suggests that the two items are similar (dissimilar), and should thus be placed close to (far away from) each other. These types of constraints can be generated using a subset of labeled items where an ML (CL) constraint represents a pair of items from the same class (different classes). The second way to express pair-wise similarities are [*relative distance comparisons*]{}. These are statements that specify how the distances between some data items relate to each other. The most common relative distance comparison task asks the data analyst to specify which of the items $i$ and $j$ is closer to a third item $k$. Note that unlike the ML/CL constraints, the relative comparisons do not as such say anything about the clustering structure.
Given a number of distance constraints between few data items, provided as examples, the objective of metric learning is to devise a new distance function, which takes into account both the supplied features and the additional distance constraints, and which expresses better the semantics of the application. Approaches using both types of constraints discussed above, ML/CL constraints and relative distance comparisons, have been studied in the literature, and a lot is known about the problem.
The method we discuss in this paper is [*a combination of metric-learning and relative distance comparisons*]{}. We deviate from existing literature by eliciting every constraint with the question
> *“Given three items $i$, $j$, and $k$, which one is the least similar to the other two?”*
The labeler should thus select one of the items as an [*outlier*]{}. Such tasks have been used e.g., by @crowdmedian and @UkkonenDH15. Notably, we also allow the labeler to leave the answer as [*unspecified*]{}. The main practical novelty of this approach is in the [*capability to gain information also from comparisons where the labeler has not been able to give a concrete solution*]{}. Some sets of three items can be all very similar (or dissimilar) to each other, so that picking one item as an obvious outlier is difficult. In those cases that the labeler gives a “don’t know” answer, it is beneficial to use this answer in the metric-learning process as it provides a valuable cue, namely, that the three displayed data items are roughly equidistant.
We cast the metric-learning problem as a [*kernel-learning problem*]{}. The learned kernel can be used to easily compute distances between data items, even between data items that did not participate in the metric-learning training phase, and only their feature vectors are available. The use of relative comparisons, instead of hard ML/CL constraints, leads to learning a more accurate metric that captures relations between data items at different scales. The learned metric can be used for many data-analysis tasks such as classification, clustering, information retrieval, etc. In this paper, we evaluate our proposed method in a multi-level clustering framework. However, the same method can be applied in many other settings, without loss of generality.
On the technical side, we start with an initial kernel ${\ensuremath{\mathbf{K}}}_0$, computed using only the feature vectors of the data items. We then formulate the kernel-learning task as an optimization problem: the goal is to find the kernel matrix ${\ensuremath{\mathbf{K}}}$ that is the closest to ${\ensuremath{\mathbf{K}}}_0$ and satisfies the constraints induced by the relative-comparison labellings. To solve this optimization task we use known efficient techniques, which we adapt for the case of relative comparisons. Figure \[fig:example\] illustrates an example of the steps of the algorithm.
More concretely, we make the following contributions:
1. We design a kernel-learning method that can also use unspecified relative distance comparisons. This is done by extending the method of @skms, which works with ML and CL constraints.
2. We introduce a soft formulation that can handle inconsistent constraints. The new formulation relies on the use of slack variables. To solve the resulting optimization problem we develop a new iterative method.
3. We perform an extensive experimental validation of our approach and show that the proposed labeling is indeed more flexible, and can lead to a substantial improvement in the clustering accuracy. We experimentally demonstrate the effectiveness and robustness of the new soft-margin formulation in handling noisy and inconsistent constraints.
An earlier version of this work appeared in conference proceedings [@sklr]. In this paper, we have extended the previous version of the paper by introducing the soft-margin formulation and by providing additional experimental results.
The rest of this paper is organized as follows. We start by reviewing the related literature in Section \[sec:related\]. In Section \[sec:prob\_form\] we introduce our setting and formally define our problem, and in Section \[sec:semi\_kernel\_learning\] we present our solution. In Section \[sec:experiments\] we discuss our empirical evaluation, and Section \[sec:conclusion\] is a short conclusion.
Related Work {#sec:related}
============
The idea of semi-supervised clustering was initially introduced by @wagstaff2000, and since then a large number of different problem variants and methods have been proposed, the first ones being [COP-Kmeans]{} [@wagstaff2001] and [CCL]{} [@klein2002]. Some of the later methods handle the constraints in a probabilistic framework. For instance, the ML and CL constraints can be imposed in the form of a Markov random field prior over the data items [@basu2004b; @basu2004; @lu2005]. Alternatively, @lu2007 generalizes the standard Gaussian process to include the preferences imposed by the ML and CL constraints. Recently, @pei2014discriminative propose a discriminative clustering model that uses relative comparisons and, like our method, can also make use of unspecified comparisons.
The semi-supervising clustering setting has also been studied in the context of spectral clustering, and many spectral clustering algorithms have been extended to incorporate pairwise constraints [@lu2010; @lu2008]. More generally, these methods employ techniques for semi-supervised graph partitioning and kernel $k$-means algorithms [@dhillon2005]. For instance, @kulis2009 present a unified framework for semi-supervised vector and graph clustering using spectral clustering and kernel learning.
As stated in the introduction, our work is based on [*metric learning*]{}. Most of the metric-learning literature, starting by the work of @XingNJR02, aims at finding a Mahalanobis matrix subject to either ML/CL or relative distance constraints. @XingNJR02 use ML/CL constraints, while @SchultzJ03 present a similar approach to handle relative comparisons. Metric learning often requires solving a semidefinite optimization problem. For instance, [@heim2015relkernel] propose an online kernel learning method using stochastic gradient descent. However, the method requires an additional projection step onto the positive semidefinite cone. This problem becomes easier if Bregman divergence, in particular the log det divergence, is used to formulate the optimization problem. Such an approach was first used for metric learning by @davis2007information with ML/CL constraints, and subsequently by @LiuMTLL10 likewise with ML/CL constraints, as well as by @LiuGZJW12 with relative comparisons. Our algorithm also uses the log det divergence, and we extend the technique of @davis2007information to handle relative comparisons.
The metric-learning approaches can also be more directly combined with a clustering algorithm. The [MPCK-Means]{} algorithm by @bilenko2004 is one of the first to combine metric learning with semi-supervised clustering and ML/CL constraints. @xiang2008learning use metric learning, as well, to implement ML/CL constraints in a clustering and classification framework, while @KumarK08 follow a similar approach using relative comparisons. Recently, @skms use a kernel-transformation approach to adapt the mean-shift algorithm [@mean_shift] to incorporate ML and CL constraints. This algorithm, called semi-supervised kernel mean shift clustering ([SKMS]{}), starts with an initial kernel matrix of the data points and generates a transformed matrix by minimizing the log det divergence using an approach based on the work by @lowrank. Our paper is largely inspired by the [SKMS]{} algorithm. Our main contribution is to extend the [SKMS]{} algorithm so that it handles relative distance comparisons.
Kernel Learning with Relative Distances {#sec:prob_form}
=======================================
In this section we introduce the notation used throughout the paper and formally define the problem we address.
Basic Definitions
-----------------
Let ${\ensuremath{\mathcal{D}}}= \{ 1, \ldots, n \}$ denote a set of [*data items*]{} and let ${\ensuremath{\mathcal{X}}}= \{\mathbf{x}_i\}_{i=1}^n$, with $\mathbf{x}_i \in \mathbb{R}^d$, denote the set of vector representation of these items in a $d$ dimensional Euclidean space; one vector for every item in [$\mathcal{D}$]{}. The vector set [$\mathcal{X}$]{} is the [*initial feature representation*]{} of the items in [$\mathcal{D}$]{}. We are also given the set [$\mathcal{C}$]{} of [*relative distance comparisons*]{} between data items in [$\mathcal{D}$]{}. These distance comparisons are given in terms of some [*unknown distance function*]{} ${\ensuremath{\delta}}: {\ensuremath{\mathcal{D}}}\times {\ensuremath{\mathcal{D}}}\rightarrow \mathbb{R}$. We assume that [$\delta$]{} reflects certain domain knowledge that is difficult to quantify precisely, and can not be directly captured by the features in [$\mathcal{X}$]{}. Thus, the set of distance comparisons [$\mathcal{C}$]{}*augments our knowledge* about the data items in [$\mathcal{D}$]{}, in addition to the feature vectors in [$\mathcal{X}$]{}. The comparisons in [$\mathcal{C}$]{} are given by human evaluators, or they may come from some other source. Given ${\ensuremath{\mathcal{X}}}$ and ${\ensuremath{\mathcal{C}}}$, our objective is to find a kernel matrix [$\mathbf{K}$]{}that captures more accurately the distance between data items. Such a kernel matrix can be used for a number of different purposes. In this paper, we focus on using the kernel matrix for clustering the data in [$\mathcal{D}$]{}. The kernel matrix [$\mathbf{K}$]{} is computed by considering both the similarities between the points in [$\mathcal{X}$]{}as well as the user-supplied constraints induced by the comparisons in [$\mathcal{C}$]{}.
In a nutshell, we compute the kernel matrix [$\mathbf{K}$]{}by first computing an initial kernel matrix ${\ensuremath{\mathbf{K}}}_0$ using only the vectors in [$\mathcal{X}$]{}. The matrix ${\ensuremath{\mathbf{K}}}_0$ is computed by applying a Gaussian kernel on the vectors in [$\mathcal{X}$]{}. We then solve an optimization problem in order to find the kernel matrix ${\ensuremath{\mathbf{K}}}$ that is the closest to ${\ensuremath{\mathbf{K}}}_0$ and satisfies the constraints in [$\mathcal{C}$]{}.
Relative-distance Constraints
-----------------------------
The constraints in [$\mathcal{C}$]{}express information about distances between items in [$\mathcal{D}$]{}in terms of the distance function [$\delta$]{}. However, we do not need to know the absolute distances between any two items $i,j\in {\ensuremath{\mathcal{D}}}$. Instead we consider constraints that express information of the type ${\ensuremath{\delta}}(i,j) < {\ensuremath{\delta}}(i,k)$ for some $i,j,k \in {\ensuremath{\mathcal{D}}}$.
In particular, every constraint $C \in {\ensuremath{\mathcal{C}}}$ is a statement about the relative distances between [*three*]{} items in [$\mathcal{D}$]{}. We consider two types of constraints, i.e., [$\mathcal{C}$]{} can be partitioned into two sets [${\ensuremath{\mathcal{C}}}_{\operatorname{neq}}$]{} and [${\ensuremath{\mathcal{C}}}_{\operatorname{eq}}$]{}. The set [${\ensuremath{\mathcal{C}}}_{\operatorname{neq}}$]{} contains constraints where one of the three items has been singled out as an “outlier.” That is, the distance of the outlying item to the two others is clearly larger than the distance between the two other items. The set [${\ensuremath{\mathcal{C}}}_{\operatorname{eq}}$]{} contains constraints where no item appears to be an obvious outlier. The distances between all three items are then assumed to be approximately the same.
More formally, we define [${\ensuremath{\mathcal{C}}}_{\operatorname{neq}}$]{} to be a set of tuples of the form [$({i},{j}\mid{k})$]{}, where every tuple is interpreted as “item $k$ is an outlier among the three items $i$, $j$ and $k$.” We assume that the item $k$ is an outlier if its distance from $i$ and $j$ is at least [$\gamma$]{} times larger than the distance ${\ensuremath{\delta}}(i,j)$, for some ${\ensuremath{\gamma}}>1$. This is because we assume small differences in the distances to be indistinguishable by the evaluators, and only such cases end up in [${\ensuremath{\mathcal{C}}}_{\operatorname{neq}}$]{}where there is no ambiguity between the distances. Here [[$\gamma$]{}]{} is a parameter that must be set in advance by the user. As a result each triple [$({i},{j}\mid{k})$]{} in [${\ensuremath{\mathcal{C}}}_{\operatorname{neq}}$]{} implies the following two inequalities $$\begin{aligned}
\label{eq:outlier1}
{\ensuremath{{i}{j}{k}}}:\quad {\ensuremath{\gamma}}^2{\ensuremath{\delta}}(i,j) &\leq& {\ensuremath{\delta}}(i,k) \;\; \hbox{and}\\
\label{eq:outlier2}
{\ensuremath{{j}{i}{k}}}:\quad {\ensuremath{\gamma}}^2{\ensuremath{\delta}}(j,i) &\leq &{\ensuremath{\delta}}(j,k).\end{aligned}$$ We denote by ${\ensuremath{\mathcal{I}}}$ the set of all pairs of inequalities imposed by the tuples ${\ensuremath{({i},{j}\mid{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}$.
Likewise, we define [${\ensuremath{\mathcal{C}}}_{\operatorname{eq}}$]{} to be a set of tuples of the form [$({i},{j},{k})$]{} that translates to “the distances between items $i$, $j$ and $k$ are equal.” In terms of the distance function [$\delta$]{}, each triple [$({i},{j},{k})$]{} in [${\ensuremath{\mathcal{C}}}_{\operatorname{eq}}$]{} implies $$\label{eq:dontknow}
{\ensuremath{\delta}}(i,j) = {\ensuremath{\delta}}(j,k) = {\ensuremath{\delta}}(i,k).$$ Similarly, we denote by ${\ensuremath{\mathcal{E}}}$ the set of all pairs of equalities imposed by the tuples ${\ensuremath{({i},{j},{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$.
Extension to a Kernel Space
---------------------------
As mentioned above, the first step of our approach is forming the initial kernel ${\ensuremath{\mathbf{K}}}_0$ using the feature vectors [$\mathcal{X}$]{}. We do this using a standard Gaussian kernel. Details are provided in Section \[section:algorithm-wrapup\].
Next we show how the constraints implied by the distance comparison sets [${\ensuremath{\mathcal{C}}}_{\operatorname{neq}}$]{} and [${\ensuremath{\mathcal{C}}}_{\operatorname{eq}}$]{} extend to a kernel space, obtained by a mapping $\Phi: {\ensuremath{\mathcal{D}}}\rightarrow {\ensuremath{\mathbb{R}^m}}$. As usual, we assume that an inner product $\Phi(i){\ensuremath{^\top}}\Phi(j)$ between items $i$ and $j$ in [$\mathcal{D}$]{} can be expressed by a symmetric kernel matrix ${\ensuremath{\mathbf{K}}}$, that is, ${\ensuremath{\mathbf{K}}}_{ij} = \Phi(i){\ensuremath{^\top}}\Phi(j)$. Moreover, we assume that the kernel ${\ensuremath{\mathbf{K}}}$ (and the mapping $\Phi$) is connected to the unknown distance function [[$\delta$]{}]{} via the equation $$\label{eq:dfunkernel}
{\ensuremath{\delta}}(i,j) = \| \Phi(i) - \Phi(j) \|^2 = {\ensuremath{\mathbf{K}}}_{ii} - 2{\ensuremath{\mathbf{K}}}_{ij} + {\ensuremath{\mathbf{K}}}_{jj}.$$ In other words, we explicitly assume that the distance function [[$\delta$]{}]{} is in fact the Euclidean distance in some unknown vector space. This is equivalent to assuming that the evaluators base their distance-comparison decisions on some implicit features, even if they might not be able to quantify these explicitly. Next, we discuss the constraint inequalities and equalities (Equations (\[eq:outlier1\]), (\[eq:outlier2\]), and (\[eq:dontknow\])) in the kernel space. Let ${\ensuremath{\mathbf{e}}}_i$ denote the vector of all zeros with the value $1$ at position $i$. The distance function ${\ensuremath{\delta}}(i,j)$ given in Equation (\[eq:dfunkernel\]) can be expressed in matrix form as follows: $${\ensuremath{\delta}}(i,j) =
{\ensuremath{\mathbf{K}}}_{ii} - 2{\ensuremath{\mathbf{K}}}_{ij} + {\ensuremath{\mathbf{K}}}_{jj}
= ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j){\ensuremath{^\top}}{\ensuremath{\mathbf{K}}}({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j)
= \operatorname{tr}({\ensuremath{\mathbf{K}}}({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j){\ensuremath{^\top}}),$$ where $\operatorname{tr}(\mathbf{A})$ denotes the trace of the matrix $\mathbf{A}$ and we use the fact that ${\ensuremath{\mathbf{K}}}= {\ensuremath{\mathbf{K}}}^\top$. Using the previous equation we can write Inequality (\[eq:outlier1\]) as $$\begin{aligned}
\gamma^2 \operatorname{tr}\left({\ensuremath{\mathbf{K}}}({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j){\ensuremath{^\top}}\right) -
\operatorname{tr}\left({\ensuremath{\mathbf{K}}}({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k){\ensuremath{^\top}}\right) &\leq& 0 \\
\operatorname{tr}\left( {\ensuremath{\mathbf{K}}}\gamma^2 ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j){\ensuremath{^\top}}-
{\ensuremath{\mathbf{K}}}({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k){\ensuremath{^\top}}\right) &\leq& 0 \\
\operatorname{tr}\left( {\ensuremath{\mathbf{K}}}(\gamma^2 ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j){\ensuremath{^\top}}-
({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k){\ensuremath{^\top}}) \right) &\leq& 0 \\
\operatorname{tr}\left( {\ensuremath{\mathbf{K}}}\, {\ensuremath{\mathbf{C}}}_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}\right) &\leq& 0,\end{aligned}$$ where ${\ensuremath{\mathbf{C}}}_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}} =
\gamma^2 ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j){\ensuremath{^\top}}- ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k){\ensuremath{^\top}}$ is defined to be a matrix that represents the constraint ${\ensuremath{{i}{j}{k}}}$. The matrix ${\ensuremath{\mathbf{C}}}_{{\ensuremath{\scriptscriptstyle{j}{i}{k}}}}$ defined to represent the constraint ${\ensuremath{{j}{i}{k}}}$ for Inequality (\[eq:outlier2\]) is formed in exactly the same manner. Note that unless we set ${\ensuremath{\gamma}}> 1$ the Inequalities (\[eq:outlier1\]) and (\[eq:outlier2\]) can be satisfied trivially for a small difference between the longer and the shorter distance and thus, the constraint becomes inactive. Setting ${\ensuremath{\gamma}}> 1$ helps avoiding such solutions.
We use a similar technique to represent the constraints in the set [${\ensuremath{\mathcal{C}}}_{\operatorname{eq}}$]{}. Recall that the constraint ${\ensuremath{({i},{j},{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$ implies that the items $i$, $j$, and $k$ are equidistant. This yields three equations on the pairwise distances between the items: ${\ensuremath{{ \sbox{\myboxA}{$\m@th{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.7\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}:{\ensuremath{\delta}}(i,j) = {\ensuremath{\delta}}(i,k)$, ${\ensuremath{{ \sbox{\myboxA}{$\m@th{j}{i}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.7\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}:{\ensuremath{\delta}}(j,i) = {\ensuremath{\delta}}(j,k)$, and ${\ensuremath{{ \sbox{\myboxA}{$\m@th{k}{i}{j}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.7\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}:{\ensuremath{\delta}}(k,i) = {\ensuremath{\delta}}(k,j)$. Reasoning as above, we let ${\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}} =
({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_j){\ensuremath{^\top}}- ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k) ({\ensuremath{\mathbf{e}}}_i - {\ensuremath{\mathbf{e}}}_k){\ensuremath{^\top}}$, and can thus write the first equation for the constraint ${\ensuremath{({i},{j},{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$ as $$\operatorname{tr}({\ensuremath{\mathbf{K}}}\, {\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = 0.$$ The two other equations are written in a similar manner.
Log Determinant Divergence for Kernel Learning
----------------------------------------------
Recall that our objective is to find the kernel matrix ${\ensuremath{\mathbf{K}}}$ that is close to the initial kernel ${\ensuremath{\mathbf{K}}}_0$. Assume that ${\ensuremath{\mathbf{K}}}$ and ${\ensuremath{\mathbf{K}}}_0$ are both positive semidefinite matrices. We will use the [*log determinant divergence*]{} to compute the similarity between ${\ensuremath{\mathbf{K}}}_0$ and ${\ensuremath{\mathbf{K}}}$. This is a variant of the [*Bregman divergence*]{} [@bregman].
The Bregman divergence between two matrices ${\ensuremath{\mathbf{K}}}$ and ${\ensuremath{\mathbf{K}}}_0$ is defined as $$\label{eq:basic_bregman}
{\ensuremath{\operatorname{D}}}_{\phi}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_0) = \phi({\ensuremath{\mathbf{K}}}) - \phi({\ensuremath{\mathbf{K}}}_0) - \operatorname{tr}(\nabla\phi({\ensuremath{\mathbf{K}}}_0)^\top ({\ensuremath{\mathbf{K}}}- {\ensuremath{\mathbf{K}}}_0)),$$ where $\phi$ is a strictly-convex real-valued function, and $\nabla\phi({\ensuremath{\mathbf{K}}}_0)$ denotes the gradient evaluated at ${\ensuremath{\mathbf{K}}}_0$. Many well-known distance measures are special cases of the Bregman divergence. These distance measures can be instantiated by selecting the function $\phi$ appropriately. For instance, $\phi({\ensuremath{\mathbf{K}}}) = \sum_{ij} K_{ij}^2$ gives the squared Frobenius norm ${\ensuremath{\operatorname{D}}}_{\phi}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_0) = \Vert{\ensuremath{\mathbf{K}}}- {\ensuremath{\mathbf{K}}}_0\Vert_F^2$.
For our application in kernel learning, we are interested in one particular case; setting $\phi({\ensuremath{\mathbf{K}}}) = -\log\det({\ensuremath{\mathbf{K}}})$. This yields the log determinant (log det) matrix divergence: $$\label{eq:logdet}
{\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_0) = \operatorname{tr}({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{K}}}_0^{-1}) - \log\det({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{K}}}_0^{-1}) - n\, .$$ The log det divergence has many interesting properties, which make it ideal for kernel learning. As a general result of Bregman divergences, log det divergence is [*convex*]{} with respect to the first argument. Moreover, it can be evaluated using the eigenvalues and eigenvectors of the matrices ${\ensuremath{\mathbf{K}}}$ and ${\ensuremath{\mathbf{K}}}_0$. This property can be used to extend log det divergence to handle rank-deficient matrices [@lowrank], and we will make use of this in our algorithm described in Section \[sec:semi\_kernel\_learning\].
Problem Definition
------------------
We now have the necessary ingredients to formulate our semi-supervised kernel learning problem. Given the set of constraints ${\ensuremath{\mathcal{C}}}= {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}\cup {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$, the parameter ${\ensuremath{\gamma}}$, and the initial kernel matrix ${\ensuremath{\mathbf{K}}}_0$, we aim to find a new kernel matrix [[$\mathbf{K}$]{}]{}, which is as close as possible to ${\ensuremath{\mathbf{K}}}_0$ while satisfying the constraints in [$\mathcal{C}$]{}. This objective can be formulated as the following constrained minimization problem: $$\label{eq:minimize}
\begin{aligned}
& \underset{{\ensuremath{\mathbf{K}}}}{\text{minimize}}\quad {\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_0)
& &\\
& \text{subject to} & &\\
& \quad \operatorname{tr}\left( {\ensuremath{\mathbf{K}}}\, {\ensuremath{\mathbf{C}}}_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}\right) \leq 0,
\; \operatorname{tr}\left( {\ensuremath{\mathbf{K}}}\, {\ensuremath{\mathbf{C}}}_{{\ensuremath{\scriptscriptstyle{j}{i}{k}}}}\right) \leq 0, \quad\mbox{for all }{\ensuremath{({i},{j}\mid{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}& \\
& \quad \operatorname{tr}({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = \operatorname{tr}({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{j}{i}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = \operatorname{tr}({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{k}{i}{j}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = 0,\quad \mbox{for all }{\ensuremath{({i},{j},{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}\\
&\quad {\ensuremath{\mathbf{K}}}\succeq 0, & &
\end{aligned}$$ where ${\ensuremath{\mathbf{K}}}\succeq 0$ constrains ${\ensuremath{\mathbf{K}}}$ to be a positive semidefinite matrix.
Relaxation of the Constraints
-----------------------------
The optimization problem (\[eq:minimize\]) does not have a solution if the set intersection of the inequality and equality constraints and the positive semidefinite cone is empty. This can happen if there exist a subset of inconsistent relative-comparison constraints. In order to be able to handle inconsistent constraints and improve the robustness of our approach, we formulate a *soft-margin* version of the problem by introducing a set of slack variables $\Xi = \{\xi_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}, \xi_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}\}$. In more detail, the soft-margin problem formulation asks to $$\label{eq:minimize_soft}
\begin{aligned}
& \underset{{\ensuremath{\mathbf{K}}},\, \Xi}{\text{minimize}}\quad {\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_0) + \frac{1}{2}\,\lambda_{\text{neq}} \sum_{ {\ensuremath{\scriptscriptstyle{i}{j}{k}}} \in {\ensuremath{\mathcal{I}}}} \xi^2_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}} + \frac{1}{2}\,\lambda_{\text{eq}} \sum_{ {\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}} \in {\ensuremath{\mathcal{E}}}} \xi^2_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}},
& &\\
& \text{subject to} & &\\
& \quad \operatorname{tr}\left( {\ensuremath{\mathbf{K}}}\, {\ensuremath{\mathbf{C}}}_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}\right) \leq \xi_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}},
\; \operatorname{tr}\left( {\ensuremath{\mathbf{K}}}\, {\ensuremath{\mathbf{C}}}_{{\ensuremath{\scriptscriptstyle{j}{i}{k}}}}\right) \leq \xi_{{\ensuremath{\scriptscriptstyle{j}{i}{k}}}},
\;\mbox{for all }{\ensuremath{({i},{j}\mid{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}& & \\
& \quad \operatorname{tr}({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = \xi_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}},
\; \operatorname{tr}({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{j}{i}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = \xi_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{j}{i}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}},\;
\operatorname{tr}({\ensuremath{\mathbf{K}}}\,{\ensuremath{\mathbf{C}}}_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{k}{i}{j}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = \xi_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{k}{i}{j}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}},\; \mbox{for all }{\ensuremath{({i},{j},{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}& &\\
&\quad {\ensuremath{\mathbf{K}}}\succeq 0, & &
\end{aligned}$$ in which, $\xi_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}$ and $\xi_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}$ are the slack variables associated with the inequality constraint ${\ensuremath{{i}{j}{k}}}$ and the equality constraint ${\ensuremath{{ \sbox{\myboxA}{$\m@th{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.7\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}$, respectively. Note that the inequality slack variables $\xi_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}$ must be non-negative,[^1] while no such non-negativity condition is required for the equality slack variables $\xi_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}$. Finally, $\lambda_{\text{eq}}$ and $\lambda_{\text{neq}}$ are regularization parameters that control the trade-off between minimizing the divergence and minimizing the magnitude of the slack variables.
Semi-supervised Kernel Learning {#sec:semi_kernel_learning}
===============================
We now focus on the optimization problems defined above, Problems (\[eq:minimize\]) and (\[eq:minimize\_soft\]). It can be shown that in order to have a finite value for the log det divergence, the rank of the matrices must remain equal [@lowrank]. This property along with the fact that the domain of the log det divergence is the positive-semidefinite matrices, allow us to perform the optimization without explicitly restraining the solution to the positive-semidefinite cone nor checking for the rank of the solution. This is in contrast with performing the optimization using, say, the Frobenius norm, where the projection to the positive semidefinite cone must be explicitly imposed.
Bregman Projections for Constrained Optimization {#subsec:bregman_proj}
------------------------------------------------
In solving the optimization Problem (\[eq:minimize\]), the aim is to minimize the divergence while satisfying the set of constraints imposed by ${\ensuremath{\mathcal{C}}}= {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}\cup {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$. In other words, we seek for a kernel matrix ${\ensuremath{\mathbf{K}}}$ by projecting the initial kernel matrix ${\ensuremath{\mathbf{K}}}_0$ onto the convex set obtained from the intersection of the set of constraints. The optimization Problem (\[eq:minimize\]) can be solved using the method of *Bregman projections* [@lowrank; @bregman; @tsuda2005]. The idea is to consider one unsatisfied constraint at a time and project the matrix so that the constraint gets satisfied. Note that the projections are not orthogonal and thus, a previously satisfied constraint might become unsatisfied. However, as stated before, the objective function in Problem (\[eq:minimize\]) is convex and the method is guaranteed to converge to the global minimum if all the constraints are met infinitely often (randomly or following a more structured procedure).
Let us consider the update rule for an unsatisfied constraint from [${\ensuremath{\mathcal{C}}}_{\operatorname{neq}}$]{}. We first consider the case of full-rank symmetric positive semidefinite matrices. Let ${\ensuremath{\mathbf{K}}}_t$ be the value of the kernel matrix at step $t$. For an unsatisfied inequality constraint ${\ensuremath{\mathbf{C}}}$, the optimization problem becomes[^2] $$\label{eq:sub_minimize}
\begin{aligned}
& {\ensuremath{\mathbf{K}}}_{t+1} = \underset{{\ensuremath{\mathbf{K}}}}{\text{arg\,min }} {\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_t), \\
\text{subject to}\;\;
& \langle {\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{C}}}\rangle = \operatorname{tr}({\ensuremath{\mathbf{K}}}{\ensuremath{\mathbf{C}}}) \leq 0.
\end{aligned}$$ Using a Lagrange multiplier $\alpha \geq 0$, we can write $$\label{eq:lagrange_form}
{\ensuremath{\mathbf{K}}}_{t+1} =
\underset{{\ensuremath{\mathbf{K}}}}{\text{arg\,min }} {\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_t) + \alpha \operatorname{tr}({\ensuremath{\mathbf{K}}}{\ensuremath{\mathbf{C}}}).$$ Taking the derivative of Equation with respect to ${\ensuremath{\mathbf{K}}}$ and setting it to zero, we have the update $$\label{eq:kernel_opt}
{\ensuremath{\mathbf{K}}}_{t+1} = ({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}.$$
In order to determine the value of $\alpha$, we substitute the update Equation into and form the conjugate dual optimization problem with respect to $\alpha$. This simplifies to the following optimization problem (see Appendix A) $$\label{eq:alpha_const}
\begin{aligned}
& \alpha^* = \arg\max_{\alpha} \log\det(\mathbf{I}_n + \alpha\,{\ensuremath{\mathbf{K}}}_t\,{\ensuremath{\mathbf{C}}}), \\
\text{subject to}\;\;
& \alpha \geq 0.
\end{aligned}$$ in which $\mathbf{I}_n$ is the $n\times n$ identity matrix.
Equation (\[eq:alpha\_const\]) does not have a closed form solution for $\alpha$, in general. However, we exploit the fact that both types of our constraints, the matrix [$\mathbf{C}$]{} has rank 2, i.e., $\text{rank}({\ensuremath{\mathbf{C}}}) = 2$. Let $\eta_1$ and $\eta_2$ be the eigenvalues of the matrix product ${\ensuremath{\mathbf{K}}}_t {\ensuremath{\mathbf{C}}}$. It can be shown that $\eta_2\leq 0 \leq \eta_1$ and $|\eta_2| \leq |\eta_1|$. Thus, Equation can be written as follows $$\label{eq:alpha_const_eig_form}
\begin{aligned}
& \alpha^* = \arg\max_{\alpha}\, \log(1+\alpha\eta_1)(1+\alpha\eta_2), \\
\text{subject to}\;\;
& \alpha \geq 0.
\end{aligned}$$ Solving Equation for $\alpha^*$ gives $$\label{eq:alpha_eig}
\frac{\eta_1}{1+\alpha^*\eta_1} + \frac{\eta_2}{1+\alpha^*\eta_2} = 0 ,$$ and $$\label{eq:alpha_sol}
\alpha^* = -\frac{1}{2}\frac{\eta_1 + \eta_2}{\eta_1\eta_2}\geq 0 .$$ Finally, substituting for Equation (\[eq:alpha\_sol\]) for $\alpha$ into (\[eq:kernel\_opt\]), yields the update equation for the kernel matrix. The update Equation can be simplified further using some matrix algebra. Let ${\ensuremath{\mathbf{C}}}= \mathbf{UV}^\top$ where $\mathbf{U,V}$ are $n\times 2$ matrices of rank-$2$ $$\mathbf{U} = [\gamma(\mathbf{e}_i - \mathbf{e}_j),\, (\mathbf{e}_i - \mathbf{e}_k)],\quad \mathbf{V} = [\gamma(\mathbf{e}_i - \mathbf{e}_j),\, -(\mathbf{e}_i - \mathbf{e}_k)]$$
Let ${\ensuremath{\mathbf{K}}}_t = \mathbf{G}_t\,\mathbf{G}_t^\top$ be the Cholesky decomposition of the kernel matrix. Using Sherman-Morrison-Woodbury formula ([@sherman; @golub]), we can write (\[eq:kernel\_opt\]) as $$\label{eq:kernel_update_modified}
\begin{split}
{\ensuremath{\mathbf{K}}}_{t+1} &= {\ensuremath{\mathbf{K}}}_t - \alpha^*\,{\ensuremath{\mathbf{K}}}_t\, \mathbf{U}\, (\mathbf{I} + \alpha^* \mathbf{V}^\top\, {\ensuremath{\mathbf{K}}}_t\, \mathbf{U} )^{-1}\, \mathbf{V}^\top\, {\ensuremath{\mathbf{K}}}_t\\
&= \mathbf{G}_t\, \left(\mathbf{I}_n - \alpha^*\,\mathbf{G}_t^\top\, \mathbf{U}\, (\mathbf{I} + \alpha^* \mathbf{V}^\top\, {\ensuremath{\mathbf{K}}}_t\, \mathbf{U} )^{-1}\, \mathbf{V}^\top\, \mathbf{G}_t\right)\, \mathbf{G}_t^\top\\
& = \mathbf{G}_t\, \mathbf{W}\,\mathbf{W}^\top\, \mathbf{G}_t^\top\\
& = \mathbf{G}_{t+1} \mathbf{G}_{t+1}^\top
\end{split}$$ where $\mathbf{W}\,\mathbf{W}^\top$ is the Cholesky decomposition of the diagonal plus rank-2 update term which can be efficiently calculated in $\mathcal{O}(n)$ time (see Appendix C). Thus, the computational bottleneck is the calculation of the matrix product $\mathbf{G}_{t+1} = \mathbf{G}_t\, \mathbf{W}$. For an equality constraint [${\ensuremath{\mathcal{C}}}_{\operatorname{eq}}$]{}, $\alpha^*$ must satisfy the following constraint $$\label{eq:alpha_eq}
\operatorname{tr}(({\ensuremath{\mathbf{K}}}_t^{-1} + \alpha^* {\ensuremath{\mathbf{C}}})^{-1} {\ensuremath{\mathbf{C}}}) = 0.$$ The value of $\alpha^*$ satisfying Equation can be written as the stationary point of the following function $$-\log\det({\ensuremath{\mathbf{K}}}_t^{-1} + \alpha\, {\ensuremath{\mathbf{C}}}) = -\log\det(\mathbf{I}_n + \alpha\, {\ensuremath{\mathbf{K}}}_t\,{\ensuremath{\mathbf{C}}}) + \text{const.}$$ which is the same as stationary point of Equation . Thus, $\alpha^*$ for an equality constraint is calculated similar to a inequality constraint using Equation . In oder words, the Bregman projection for an inequality constraint projects the kernel matrix onto the boundary of the convex set of matrices that satisfy the given constraint.
For a rank-deficient kernel matrix ${\ensuremath{\mathbf{K}}}_0$ with $\text{rank}({\ensuremath{\mathbf{K}}}_0)=r$, we employ the results of @lowrank, which state that for any column-orthogonal matrix $\mathbf{Q}$ with $\text{range}({\ensuremath{\mathbf{K}}}_0) \subseteq \text{range}(\mathbf{Q})$ (e.g., obtained by singular value decomposition of ${\ensuremath{\mathbf{K}}}_0$), we first apply the transformation $$\mathbf{M} \rightarrow \mathbf{\hat{M}} = \mathbf{Q}^\top\, \mathbf{M\, Q},$$ on all the matrices, and after finding the kernel matrix $\mathbf{\hat{K}}$ satisfying all the transformed constraints, we can obtain the final kernel matrix using the inverse transformation $${\ensuremath{\mathbf{K}}}= \mathbf{Q\,\hat{K}\, Q^\top}.$$ Note that since log det preserves the matrix rank, the mapping is one-to-one and invertible.
Bregman Projections for Soft Margin Formulation {#subsec:bregman_proj_soft}
-----------------------------------------------
We now consider solving the soft-margin formulation defined as Problem (\[eq:minimize\_soft\]). Similar to the previous case, the method of Bregman projections can be applied iteratively by considering one unsatisfied constraint at a time and performing the update. In our exposition below, we consider the update rule for an inequality constraint. The update rule for an equality constraint applies in a similar manner.
Let us denote by ${\ensuremath{\mathbf{K}}}_t$ the value of the kernel matrix at step $t$. Considering an unsatisfied inequality constraint ${\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}$, now the optimization becomes $$\label{eq:sub_minimize_soft}
\begin{aligned}
& \underset{{\ensuremath{\mathbf{K}}},\, \xi}{\text{minimize}}
&&{\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_t) + \frac{1}{2}\, \lambda\, \xi^2, \\
& \text{subject to}\;\;
&& \langle {\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{C}}}\rangle = \operatorname{tr}({\ensuremath{\mathbf{K}}}^\top {\ensuremath{\mathbf{C}}}) \leq \xi, \\
\end{aligned}$$ where $\xi$ is the non-negative slack variable associated with ${\ensuremath{\mathbf{C}}}$ and $\lambda$ is the regularization factor. Using the Lagrange multiplier method, the optimization Problem (\[eq:sub\_minimize\_soft\]) can be written so as the expression $$\label{eq:lagrange_form_soft}
{\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_t) + \frac{1}{2}\,\lambda\, \xi^2 + \alpha\, \left(\operatorname{tr}({\ensuremath{\mathbf{K}}}^\top {\ensuremath{\mathbf{C}}})-\xi\right)$$ is minimized with respect to ${\ensuremath{\mathbf{K}}}$ and $\xi$, and maximized with respect to $\alpha$. Lagrange multiplier $\alpha$ must also satisfy the set of Karush-Kuhn-Tucker (KKT) conditions $$\begin{aligned}
\alpha\, \left(\operatorname{tr}({\ensuremath{\mathbf{K}}}^\top {\ensuremath{\mathbf{C}}})-\xi \right) &=& 0\label{eq:alpha_kkt1}\\
\alpha &\geq & 0\,.\label{eq:alpha_kkt2}
$$ Setting the derivative of the expression (\[eq:lagrange\_form\_soft\]) equal to zero, with respect to ${\ensuremath{\mathbf{K}}}$ and $\xi$, yields the following two equations $$\begin{aligned}
{\ensuremath{\mathbf{K}}}_{t+1} &=& ({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\label{eq:kernel_opt_soft} \;\mbox{ and}\\
\lambda\,\xi &=& \alpha \label{eq:xi_alpha_mu}\,.\end{aligned}$$ Substituting for ${\ensuremath{\mathbf{K}}}_{t+1}$ and $\xi$ in Equation using Equations and , respectively, we have the following dual optimization problem over $\alpha$ (see Appendix B) $$\label{eq:alpha_dual_soft}
\begin{aligned}
& \alpha^* = \arg\max_{\alpha} \log\det(\mathbf{I}_n + \alpha\,{\ensuremath{\mathbf{K}}}_t\,{\ensuremath{\mathbf{C}}}) - \frac{1}{2} \lambda\, \alpha^2, \\
\text{subject to}\;\;
& \alpha \geq 0.
\end{aligned}$$
Equation (\[eq:alpha\_dual\_soft\]) can be solved similar to Equation (\[eq:alpha\_const\_eig\_form\]) using the eigenvalues of the product matrix ${\ensuremath{\mathbf{K}}}_t\, {\ensuremath{\mathbf{C}}}$, that is $$\label{eq:alpha_eig_soft}
\frac{\eta_1}{1+\alpha\eta_1} + \frac{\eta_2}{1+\alpha\eta_2} - \lambda\alpha = 0$$ Equation (\[eq:alpha\_eig\_soft\]) yields to a cubic polynomial equation $$\label{eq:poly}
\alpha^3 + \frac{\eta_1+\eta_2}{\eta_1\eta_2}\, \alpha^2 + \left(\frac{1}{\eta_1\eta_2}-2\lambda\right)\, \alpha - \frac{\eta_1\eta_2}{\eta_1+\eta_2} = 0\, .$$ The polynomial Equation (\[eq:poly\]) has two positive roots and one negative root since the sum of the roots (the coefficient of the quadratic term) is negative and the multiplication of the roots (the constant term) is positive. To choose the appropriate positive root of the polynomial, we note that Equation (\[eq:alpha\_eig\_soft\]) is consistent with Equation (\[eq:alpha\_eig\]) when the constraints are forced to be exactly satisfied, i.e., when the penalty $\lambda$ for an unsatisfied constraint is very large. Thus, when $\frac{\alpha}{\lambda} \ll 1$ holds for large values of $\lambda$, the solution of Equation (\[eq:alpha\_eig\_soft\]) can be seen as a perturbation on the value of $\alpha^*$, obtained using Equation (\[eq:alpha\_sol\]). Using the first-order expansion near $\alpha^*$, we can write the solution of Equation (\[eq:alpha\_eig\_soft\]) as $$\label{eq:alpha_sol_pert}
\hat{\alpha}^* = \alpha^* - \delta\alpha + \mathcal{O}(({\scriptstyle \frac{\alpha^*}{\lambda}})^2),\;\; \delta\alpha = \frac{1}{8}\frac{(\eta_1-\eta_2)^2}{(\eta_1\eta_2)^2}\, \frac{\alpha^*}{\lambda} \geq 0.$$ Note that from Equation (\[eq:alpha\_eig\_soft\]), for large values of $\vert\alpha\eta_i \vert\gg 1,\, i = 1, 2$, the roots of the polynomial approach the value $\pm\sqrt{2\lambda}$. Thus, the smaller positive root of the polynomial corresponds the desired $\alpha^*$ for the update. The update for an equality constraint follows similarly, as in the previous case.
After obtaining the proper value of $\alpha^*$ from solving Equation (\[eq:poly\]), the update rule for the kernel matrix can be applied similarly, using Equation (\[eq:kernel\_update\_modified\]). We repeat the procedure until the values of $\alpha$ for all constraints stabilize (the values of $\xi$ converge).
Relations to Metric Learning and Generalization to Out-of-sample Data
----------------------------------------------------------------------
In semi-supervised kernel-learning, the set of constrained data points may be significantly smaller than the set of all data points. Thus, applying the kernel updates on the full kernel matrix may become computationally expensive. Additionally, the learned kernel function, satisfying the set of constraints, may need to be generalized to a set of new (unseen) data points. Our kernel-learning method can handle both cases by bridging the kernel-learning problem with an equivalent metric-learning problem in a transformed kernel space.
@jain demonstrated the equivalence between the kernel-learning problem (Problem (\[eq:minimize\])) and the following metric-learning problem using a linear transformation in the feature space corresponding to the mapping $\Phi_0: {\ensuremath{\mathcal{D}}}\rightarrow {\ensuremath{\mathbb{R}^m}}$ of the initial kernel space
$$\label{eq:metric_learn}
\begin{aligned}
& \underset{{\ensuremath{\mathbf{W}}}}{\text{minimize}}\quad {\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{W}}},\mathbf{I})
& &\\
& \text{subject to} & &\\
& \quad \operatorname{tr}( {\ensuremath{\mathbf{W}}}\, {\ensuremath{\mathbf{C}}}'_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}) \leq 0,
\; \operatorname{tr}( {\ensuremath{\mathbf{W}}}\, {\ensuremath{\mathbf{C}}}'_{{\ensuremath{\scriptscriptstyle{j}{i}{k}}}}) \leq 0, \;\mbox{ for all }{\ensuremath{({i},{j}\mid{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}& & \\
& \quad \operatorname{tr}({\ensuremath{\mathbf{W}}}\,{\ensuremath{\mathbf{C}}}'_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{i}{j}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = 0,
\; \operatorname{tr}({\ensuremath{\mathbf{W}}}\,{\ensuremath{\mathbf{C}}}'_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{j}{i}{k}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = 0,\; \operatorname{tr}({\ensuremath{\mathbf{W}}}\,{\ensuremath{\mathbf{C}}}'_{{\ensuremath{{ \sbox{\myboxA}{$\m@th\scriptscriptstyle{k}{i}{j}$} \setbox\myboxB\null \ht\myboxB=\ht\myboxA \dp\myboxB=\dp\myboxA \wd\myboxB=0.6\wd\myboxA \sbox\myboxB{$\m@th\overline{\copy\myboxB}$} \setlength\mylenA{\the\wd\myboxA} \addtolength\mylenA{-\the\wd\myboxB} \ifdim\wd\myboxB<\wd\myboxA \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA} \else
\hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB} \fi}}}}) = 0,\;\mbox{ for all }{\ensuremath{({i},{j},{k})}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}\\
&\quad {\ensuremath{\mathbf{W}}}\succeq 0, & &
\end{aligned}$$
where ${\ensuremath{\mathbf{C}}}'_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}}$ is defined as $$\label{eq:metric_learn_const}
{\ensuremath{\mathbf{C}}}'_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}} = \gamma \left(\Phi_0(i) - \Phi_0(j)\right)\left(\Phi_0(i) - \Phi_0(j)\right)^\top - \left(\Phi_0(i) - \Phi_0(k)\right)\left(\Phi_0(i) - \Phi_0(k)\right)^\top .$$ The soft-margin formulation using the slack variables in Problem (\[eq:minimize\_soft\]) can be written in a similar manner. Note that the Problem (\[eq:metric\_learn\]) is defined in the feature space imposed by $\Phi_0$, and thus, the matrix ${\ensuremath{\mathbf{W}}}$ can be of infinite dimension. However, the problem can be solved implicitly by solving the kernel-learning Problem (\[eq:minimize\]), in which ${\ensuremath{\mathbf{K}}}_0 = \boldsymbol{\Phi}_0^\top \boldsymbol{\Phi}_0$, and where $\boldsymbol{\Phi}_0$ is the feature matrix whose $i$-th column is equal to $\Phi_0(i)$ and the optimal solutions are related by the following equations $$\begin{aligned}
{\ensuremath{\mathbf{K}}}&= \boldsymbol{\Phi}_0^\top {\ensuremath{\mathbf{W}}}\boldsymbol{\Phi}_0 ,\label{eq:kernel_w}\\
{\ensuremath{\mathbf{W}}}&= \mathbf{I} + \boldsymbol{\Phi}_0 {\ensuremath{\mathbf{K}}}_0^{-1} ({\ensuremath{\mathbf{K}}}- {\ensuremath{\mathbf{K}}}_0) {\ensuremath{\mathbf{K}}}_0^{-1} \boldsymbol{\Phi}_0 .\label{eq:w_kernel}\end{aligned}$$ Note that for the identity mapping $\Phi_0(i) = \mathbf{x}_i$, Problem (\[eq:metric\_learn\]) reduces to a linear transformation of the vector in the original space.
As a result, the kernel matrix learned by minimizing the log det divergence subject to the set of constraints ${\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}\cup {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$ can be also extended to handle *out-of-sample* data points, i.e., data points that were not present when learning the kernel matrix. Using and , the inner product between a pair of out-of-sample data points $\mathbf{x}, \mathbf{y} \in \mathbb{R}^d$ in the transformed kernel space can written as $$\label{eq:out_of_sample}
k(\mathbf{x},\mathbf{y})
= k_0(\mathbf{x},\mathbf{y})
+ \mathbf{k_x}^\top ({\ensuremath{\mathbf{K}}}_0^\dagger\, ({\ensuremath{\mathbf{K}}}-{\ensuremath{\mathbf{K}}}_0)\,{\ensuremath{\mathbf{K}}}_0^\dagger)\,\mathbf{k_y} ,$$ where the value of $k_0(\mathbf{x},\mathbf{y})$ and the vectors $\mathbf{k_x} = [k_0(\mathbf{x},\mathbf{x}_1), \ldots, k_0(\mathbf{x},\mathbf{x}_n)]^\top$ and $\mathbf{k_y} = [k_0(\mathbf{y},\mathbf{x}_1), \ldots, k_0(\mathbf{y},\mathbf{x}_n)]^\top$ are computed using the initial kernel function. This observation allows learning the kernel matrix by using only the subset of constrained data points and thus, avoiding the computational overhead in large datasets.
Semi-supervised Kernel Learning with Relative Comparisons {#section:algorithm-wrapup}
---------------------------------------------------------
In this section, we summarize the proposed approach, which we name [SKLR]{}, for Semi-supervised Kernel-Learning with Relative comparisons. We refer to the soft formulation of the algorithm as [soft SKLR]{} (or [sSKLR]{}, for short). The pseudo-code of the [SKLR]{} method is shown in Algorithm \[alg:main\].[^3] As already discussed, the main ingredients of the method are the following.
### Selecting the Bandwidth Parameter for $k_0$.
We consider an adaptive approach to select the bandwidth parameter of the Gaussian kernel function. First, we set $\sigma_i$ equal to the distance between point $\mathbf{x}_i$ and its $k$-th nearest neighbor. Next, we set the kernel between $\mathbf{x}_i$ and $\mathbf{x}_j$ to $$\label{eq:adaptive_kernel}
k_0(\mathbf{x}_i,\mathbf{x}_j)
= \exp \left( -\frac{\Vert\mathbf{x}_i-\mathbf{x}_j\Vert^2}{\sigma_{ij}^2}\right),$$ where, $\sigma_{ij}^2 = \sigma_{i}\sigma_{i}$. This process ensures a large bandwidth for sparse regions and a small bandwidth for dense regions.
### Semi-supervised Kernel learning with Relative Comparisons.
After finding the low-rank approximation of the initial kernel matrix ${\ensuremath{\mathbf{K}}}_0$ and transforming all the matrices by a proper matrix $\mathbf{Q}$, as discussed in Section \[subsec:bregman\_proj\], the algorithm proceeds by randomly considering one unsatisfied constrained at a time and performing the Bregman projections (\[eq:kernel\_update\_modified\]) until all the constraints are satisfied (or alternatively, the slack variables converge).
### Clustering Method.
After obtaining the kernel matrix ${\ensuremath{\mathbf{K}}}$ satisfying the set of all relative and undetermined constraints, we can obtain the final clustering of the points by applying any standard kernelized clustering method. In this paper, we consider the kernel $k$-means because of its simplicity and good performance. Generalization of the method to other clustering techniques such as kernel mean-shift is straightforward.
### Computational Complexity
The computation complexity of the algorithm is dominated by the Bregman projection step, given in Equation (\[eq:kernel\_update\_modified\]). This step requires $\mathcal{O}(n^2)$ flops since both $\mathbf{U}$ and $\mathbf{V}$ are $n \times 2$ matrices. On the other hand, working with the rank-deficient matrices, we can reduce the complexity to $\mathcal{O}(r^2)$ where $r \leq n$. It is also possible to generalize the approach in ([@lowrank]) to work on a factorized form of the update equation (\[eq:kernel\_update\_modified\]) by factoring the kernel matrix ${\ensuremath{\mathbf{K}}}$ into $\mathbf{GG}^\top$. However, due to the cross terms in rank-$2 $ updates, it is not possible to directly obtain the right multiplication of the update term and the low-rank factor (see @lowrank, Algorithm 1). As the final remark, we note that it is possible to consider only the subset of the constrained data points in the kernel learning step and then, generalize the learned kernel to the out-of-sample data points using Equation (\[eq:out\_of\_sample\]). This can significantly reduce the size of the kernel learning problem and thus, result in a major speed-up.
initial $n\times n$ kernel matrix ${\ensuremath{\mathbf{K}}}_0$, set of relative comparisons ${\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}$ and ${\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$, constant distance factor $\gamma$ kernel matrix ${\ensuremath{\mathbf{K}}}$
$\bullet$ **Find low-rank representation:**
- Compute the $n\times n$ low-rank kernel matrix $\mathbf{\tilde{K}}_0$ such that $\text{rank}(\mathbf{\tilde{K}}_0) = r \leq n$ using singular value decomposition such that $\frac{\Vert\mathbf{\tilde{K}}_0\Vert_F}{\Vert{\ensuremath{\mathbf{K}}}_0\Vert_F} \geq 0.9$\
- Set ${\ensuremath{\mathbf{K}}}_0 \gets \mathbf{\tilde{K}}_0$\
- Find $n\times r$ column orthogonal matrix $\mathbf{Q}$ such that $\text{range}({\ensuremath{\mathbf{K}}}_0) \subseteq \text{range}(\mathbf{Q})$\
- Apply the transformation $\mathbf{\hat{M}} \gets \mathbf{Q}^\top\,\mathbf{M\,Q}$ on all matrices
$\bullet$ **Initialize the kernel matrix**
- Set $\mathbf{\hat{K}} \gets \mathbf{\hat{K}}_0$
$\bullet$ **Repeat**
- \(1) Select an unsatisfied constraint $\mathbf{\hat{C}} \in {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{neq}}}}\cup {\ensuremath{{\ensuremath{\mathcal{C}}}_{\operatorname{eq}}}}$\
- \(2) Apply Bregman projection (\[eq:kernel\_update\_modified\])\
**Until** all the constraints are satisfied $\bullet$ **Return** ${\ensuremath{\mathbf{K}}}\gets \mathbf{Q\, \hat{K}\, Q^\top}$
Experimental Results {#sec:experiments}
====================
In this section, we evaluate the performance of the proposed kernel-learning method, [SKLR]{} along with its soft margin counterpart, [sSKLR]{}. As the under-the-hood clustering method required by [SKLR]{}, we use the standard kernel $k$-means with Gaussian kernel and without any supervision (Equation (\[eq:adaptive\_kernel\])). We compare [SKLR]{} and [sSKLR]{} to three different semi-supervised metric-learning algorithms, namely, [ITML]{} [@davis2007information], [SK$k$m]{} [@skms] (a variant of [SKMS]{} with kernel $k$-means in the final stage), and [LSML]{} [@LiuGZJW12]. We select the [SK$k$m]{} variant as @skms have shown that [SK$k$m]{} tends to produce more accurate results than other semi-supervised clustering methods. Two of the baselines, [ITML]{} and [SK$k$m]{}, are based on pairwise ML/CL constraints, while [LSML]{} uses relative comparisons. For [ITML]{} and [LSML]{} we apply $k$-means on the transformed feature vectors to find the final clustering, while for [SK$k$m]{}, [SKLR]{}, and [sSKLR]{}, we apply kernel $k$-means on the transformed kernel matrices.
To assess the quality of the resulting clusterings, we use the Adjusted Rand (AR) index [@AR]. Each experiment is repeated $30$ times and the average over all trials is reported. For the parameter [[$\gamma$]{}]{} required by [SKLR]{} and [sSKLR]{} we use ${\ensuremath{\gamma}}^2 = 2$. We also use $\lambda_{\text{neq}} = \lambda_{\text{eq}} = 10^5$ for the regularization parameters required by the [sSKLR]{} method[^4]. In the following, all experiments are conducted with noise-free constraints, with the exception of the one discussed in Section \[sect:noise\]. Our implementation of [SKLR]{} and [sSKLR]{} is in MATLAB and the code is publicly available.[^5] For the other three methods we use publicly available implementations.[^6][^7][^8]
Datasets
--------
We conduct the experiments on three different real-world datasets.\
**Vehicle:[^9]** The dataset contains $846$ instances from $4$ different classes and is available on the LIBSVM repository.\
**MIT Scene:[^10]** The dataset contains $2688$ outdoor images, each sized $256\times 256$, from $8$ different categories: $4$ natural and $4$ man-made. We use the GIST descriptors [@gist] as the feature vectors.\
**USPS Digits:[^11]** The dataset contains $16\times 16$ grayscale images of handwritten digits. It contains $1100$ instances from each class. The columns of each images are concatenated to form a $256$ dimensional feature vector.
Dataset [K$k$m]{} [LSML]{} [ITML]{} [SK$k$m]{} [SKLR]{} [sSKLR]{}
--------- ----------- ---------- ---------- ------------ ---------------- ----------------
-0.0504 0.0748 0.0938 0.2312 [**0.4706**]{} 0.3633
0.0099 0.4466 0.4792 0.5502 [**0.8198**]{} 0.7998
0.0194 0.2864 0.3385 0.4049 [**0.7147**]{} 0.7054
0.0541 0.0789 0.0780 0.1338 [**0.2582**]{} 0.2167
0.3363 0.4494 0.3681 0.4062 0.4609 [**0.5029**]{}
0.3028 0.3377 0.4072 0.4821 0.4304 [**0.5283**]{}
: Adjusted Rand (AR) index for binary and multiclass clustering of [**Vehicle**]{}, [**MIT Scene**]{}, and [**USPS Digits**]{} datasets using $80$, $160$, and $200$ constraints, respectively.[]{data-label="tab:smallest_const"}
Relative Constraints vs. Pairwise Constraints
---------------------------------------------
We first demonstrate the performance of the different methods using relative and pairwise constraints. For each dataset, we consider two different experiments: ($i$) *binary* in which each dataset is clustered into [*two groups*]{}, based on some predefined criterion, and ($ii$) *multi-class* where for each dataset the clustering is performed with number of clusters being equal to [*number of classes*]{}. In the binary experiment, we aim to find a crude partitioning of the data, while in the multi-class experiment we seek a clustering at a finer granularity. The 2-class partitionings of our datasets required for the binary experiment are defined as follows: For the [**vehicle**]{} dataset, we consider class $4$ as one group and the rest of the classes as the second group (an arbitrary choice). For the [**MIT Scene**]{} dataset, we perform a partitioning of the data into natural vs. man-made scenes. Finally, for the [**USPS Digits**]{}, we divide the data instances into even vs. odd digits.
To generate the pairwise constraints for each dataset, we vary the number of labeled instances from each class (from $5$ to $19$ with step-size of $2$) and form all possible ML constraints. We then consider the same number of CL constraints. Note that for the binary case, we only have two classes for each dataset. To compare with the methods that use relative comparisons, we consider an equal number of relative comparisons and generate them by sampling two random points from the same class and one point (outlier) from one of the other classes. Note that for the relative comparisons, there is no need to restrict the points to the labeled samples, as the comparison is made in a relative manner.
One should note the fundamental difference between the pairwise and relative constraints when comparing the different methods using different types of constraints. At first glance, the number of independent pairwise comparisons for detecting the outlier among three given items might seem larger than the one in a single pairwise comparison task (ML or CL). However, one should also consider the contrast imposed by the existence of the two similar items which facilitates the decision making task for the user. In other words, we argue that the user does not perform three independent pairwise comparisons for detecting the outlier. Additionally, we allow the user to skip the difficult cases but use the unspecified answers to improve the final results. In short, we acknowledge that it might not be totally plausible to compare different methods that use different types of distance constraints. However, the comparison provides insightful intuition to assess the performance of these methods in different settings.
Finally, in these experiments, we consider a subsample of both [**MIT Scene**]{} and [**USPS Digits**]{} datasets by randomly selecting $100$ data points from each class, yielding $800$ and $1000$ data points, respectively.
\
\
The results for the binary and multi-class experiments are shown in Figures \[fig:binary\] and \[fig:multiclass\], respectively. We see that all methods perform equally with no constraints. As constraints or relative comparisons are introduced the accuracy of all methods improves very rapidly. The only surprising behavior is the one of [ITML]{} in the multi-class setting, whose accuracy drops as the number of constraints increases. Table \[tab:smallest\_const\] shows the performance of different methods for binary and multiclass clustering of the datasets after introducing constraints computed from five labeled examples per class. (This is the least amount of supervision we envision is feasible in practical settings.) As can be seen, the increase in performance is much larger for both [SKLR]{} and [sSKLR]{} methods in all cases. From the figures we see that [SKLR]{} outperforms all competing methods by a large margin, for all three datasets and in both settings. Additionally, the overall performance of [sSKLR]{} is only slightly lower than [SKLR]{}, however, it still outperforms all the other methods.
\
\
Multi-resolution Analysis
-------------------------
As discussed earlier, one of the main advantages of kernel learning with relative comparisons is the feasibility of multi-resolution clustering using a single kernel matrix. To validate this claim, we repeat the [*binary*]{} and [*multi-class*]{} experiments described above. However, this time, we mix the binary and multi-class constraints and use the same set of constraints in both experimental conditions. We evaluate the results by performing binary and multi-class clustering, as before.
Figures \[fig:binary\_mixed\] and \[fig:multiclass\_mixed\] illustrate the performance of different algorithms using the mixed set of constraints. Again, [SKLR]{} produces more accurate clusterings, especially in the multi-class setting. In fact, two of the methods, [SK$k$m]{} and [ITML]{}, perform worse than the kernel $k$-means baseline in the multi-class setting. On the other hand all methods outperform the baseline in the binary setting. The reason is that most of the constraints in the multi-class setting are also relevant to the binary setting, but the converse does not hold. Note that in the multi-class setting, [sSKLR]{} fixes the early drop in the accuracy of [SKLR]{} by handling the irrelevant constraints more efficiently.
Figure \[fig:usps\_vis\] shows a visualization of the [**USPS Digits**]{} dataset using the SNE method [@sne] in the original space, and the spaces induced by [SK$k$m]{} and [SKLR]{}. We see that [SKLR]{} provides an excellent separation of the clusters that correspond to even/odd digits as well as the sub-clusters that correspond to individual digits.
Generalization Performance
--------------------------
We now evaluate the generalization performance of the different methods to out-of-sample data on the [**MIT Scene**]{} and [**USPS Digits**]{} datasets (recall that we do not subsample the [**Vehicles**]{} dataset). For the baseline kernel $k$-means algorithm, we run the algorithm on the whole datasets. For [ITML]{} and [LSML]{}, we apply the learned transformation matrix on the new out-of-sample data points. For [SK$k$m]{}, [SKLR]{}, and [sSKLR]{}, we use Equation (\[eq:out\_of\_sample\]) to find the transformed kernel matrix of the whole datasets. The results of this experiment are shown in Figure \[fig:generalization\_results\]. As can be seen from the figure, also in this case, when generalizing to out-of-sample data, [SKLR]{} and [sSKLR]{} produce significantly more accurate clusterings.
Effect of Noise {#sect:noise}
---------------
To evaluate the effect of noise, we first generate a fixed number of pairwise or relative constraints for each method ($360$, $720$, and $900$ constraints for [**Vehicle**]{}, [**MIT Scene**]{}, and [**USPS Digits**]{}, respectively) and then corrupt up to $0.6$ of the constraints with step size of $0.1$. We corrupt the ML and CL constraints by simply flipping the type of the constraints. For the relative constraints, we randomly swap the outlier with one of inlier points. The results on the different datasets are shown in Figure \[fig:noise\_results\]. As can be seen, the performance of the [SLKR]{} drops immediately as the noise is introduced. This behavior is expected as the method is not tailored to handle noise. The [sSKLR]{} method, on the other hand, degrades more gradually as the level of noise increases. Clearly, [sSKLR]{} outperforms all other methods until around $0.4$ noise level and performs comparably good as the other methods for higher levels of noise.
Effect of Equality Constraints
------------------------------
To evaluate the effect of equality constraints on the clustering, we consider a multi-class clustering scheme. For all datasets, we first generate a fixed number of relative comparisons ($360$, $720$, and $900$ relative comparisons for [**Vehicle**]{}, [**MIT Scene**]{}, and [**USPS Digits**]{}, respectively) and then we add some additional equality constraints (up to $200$). The equality constraints are generated by randomly selecting three data points, all from the same class, or each from a different class. The results are shown in Figure \[fig:equality\_results\]. As can be seen, considering the equality constraint also improves the performance, especially on the [**MIT Scene**]{} and [**USPS Digits**]{} datasets. On the [**Vehicle**]{} dataset and using the [SKLR]{} method, the performance starts to drop after around $20$ constraints. The reason can be that after this point, many [*unrelated*]{} equality constraints, i.e., the ones with data points all coming from the same class, start to appear more and thus, reduce the performance. [sSKLR]{} method, on the other hand, can handle the unrelated constraints. Finally, note that none of the other methods that we consider can handle equality constraints.
Running Time
------------
We note that in in this paper, we do not report running-time results, since the implementations of the different methods are provided in different programming languages. However, all tested methods have comparable running times. In particular, the computational overhead of our method can be limited by leveraging the fact that the algorithm has to work with low-rank $r\times r$ matrices. For all our datasets, we have $r \approx \frac{1}{2} n$. As an example, on a machine with a $2.6$ GHz processor and $16$ GB of RAM, performing $10$ passes over all constraints on the subset of $1000$ data points from the [**USPS Digits**]{} dataset with number of constraints equal to $500$ takes around $670$ seconds. A higher speed-up can be obtained by further reducing the rank of the approximation (possibly sacrificing the performance).
Conclusion {#sec:conclusion}
==========
We have devised a semi-supervised kernel-learning algorithm that can incorporate relative distance constraints, and used the resulting kernels for clustering. In particular, our method incorporates relative distance constraints among triples of items, where labelers are asked to select one of the items as an outlier, while they may provide a “don’t know” answer. The metric-learning problem is formulated as a kernel-learning problem, where the goal is to find the kernel matrix ${\ensuremath{\mathbf{K}}}$ that is the closest to an initial kernel ${\ensuremath{\mathbf{K}}}_0$ and satisfies the constraints induced by the relative distance constraints. We have also introduced a soft formulation that can handle inconsistent constraints, and thus, reduce the robustness of the approach.
Our experiments show that our method outperforms by a large margin other competing methods, which either use ML/CL constraints or use relative constraints but different metric-learning approaches. Our method is compatible with existing kernel-learning techniques [@skms] in the sense that if ML and CL constraints are available, they can be used together with relative comparisons.
We have also proposed to interpret an “unsolved” distance comparison so that the interpoint distances are roughly equal. Our experiments suggest that incorporating such equality constraints to the kernel learning task can be advantageous, especially in settings where it is costly to collect constraints.
Appendix A. Bregman Projection for the Rank-2 Inequality Constraint {#sec:app-A .unnumbered}
===================================================================
We derive the dual of the optimization problem given in Equation
$$ {\ensuremath{\mathbf{K}}}_{t+1} =
\underset{{\ensuremath{\mathbf{K}}}}{\text{arg\,min }} {\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_t) + \alpha \operatorname{tr}({\ensuremath{\mathbf{K}}}{\ensuremath{\mathbf{C}}}).$$
Setting the derivative of above to zero yields the following update for ${\ensuremath{\mathbf{K}}}_{t+1}$ $${\ensuremath{\mathbf{K}}}_{t+1} = ({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}.$$ Now, substituting for ${\ensuremath{\mathbf{K}}}$ and using , we have the dual problem $$\begin{aligned}
& \operatorname{tr}(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\, {\ensuremath{\mathbf{K}}}_t^{-1}) - \log \det(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\,{\ensuremath{\mathbf{K}}}_t^{-1} ) - n + \alpha \operatorname{tr}(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\, {\ensuremath{\mathbf{C}}})\\
= & \operatorname{tr}(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\, ({\ensuremath{\mathbf{K}}}_t^{-1} + \alpha{\ensuremath{\mathbf{C}}})) - \log \det(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\,{\ensuremath{\mathbf{K}}}_t^{-1} ) - n\\
= & n - \log \det(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\,{\ensuremath{\mathbf{K}}}_t^{-1} ) - n\\
= & \log \det({\ensuremath{\mathbf{K}}}_t\, ({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}}))\\
= & \log \det(\mathbf{I}_n + \alpha{\ensuremath{\mathbf{K}}}_{t} {\ensuremath{\mathbf{C}}}))\end{aligned}$$ which is maximized w.r.t. $\alpha$.
Appendix B. Bregman Projection for the Soft Margin Rank-2 Inequality Constraint {#sec:app-B .unnumbered}
===============================================================================
Similarly, for the soft margin case, we have the Lagrangian form $${\ensuremath{{\ensuremath{\operatorname{D}}}_{\operatorname{ld}}}}({\ensuremath{\mathbf{K}}},{\ensuremath{\mathbf{K}}}_t) + \frac{1}{2}\,\lambda\, \xi^2 + \alpha\, \left(\operatorname{tr}({\ensuremath{\mathbf{K}}}^\top {\ensuremath{\mathbf{C}}})-\xi\right)$$ We substitute for ${\ensuremath{\mathbf{K}}}$ as before and we set $\xi = \alpha/\lambda$ for the slack variable. This yields the following dual maximization problem w.r.t. $\alpha$ $$\begin{aligned}
{2}
& \operatorname{tr}(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\, {\ensuremath{\mathbf{K}}}_t^{-1}) - \log \det(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\,{\ensuremath{\mathbf{K}}}_t^{-1} ) - n && \\
&\hspace{4.7cm} + \alpha( \operatorname{tr}(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\, {\ensuremath{\mathbf{C}}}) - \frac{\alpha}{\lambda}) && + \frac{1}{2} \lambda (\frac{\alpha}{\lambda})^2\\
= & \operatorname{tr}(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\, ({\ensuremath{\mathbf{K}}}_t^{-1} + \alpha{\ensuremath{\mathbf{C}}})) - \log \det(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\,{\ensuremath{\mathbf{K}}}_t^{-1} ) - n && - \frac{1}{2} \frac{\alpha^2}{\lambda} \\
= &- \log \det(({\ensuremath{\mathbf{K}}}_{t}^{-1} + \alpha {\ensuremath{\mathbf{C}}})^{-1}\,{\ensuremath{\mathbf{K}}}_t^{-1} ) - \frac{1}{2} \frac{\alpha^2}{\lambda} && \\
= & \log \det(\mathbf{I}_n + \alpha{\ensuremath{\mathbf{K}}}_{t} {\ensuremath{\mathbf{C}}})) - \frac{1}{2} \frac{\alpha^2}{\lambda} && \end{aligned}$$
Appendix C. Cholesky Decomposition of Identity Plus Rank-2 Matrix {#sec:app-C .unnumbered}
=================================================================
We provide an algorithm to calculate the Cholesky decomposition of symmetric identity plus rank-2 matrix of the form $\mathbf{A} = \mathbf{I}_n + \lambda_1 \mathbf{u\,v}^\top + \lambda_2 \mathbf{w\,z}^\top$. Note that in order for $\mathbf{A}$ to be symmetric, $\mathbf{w}$ and $\mathbf{z}$ must be linearly dependent on $\mathbf{u}$ and $\mathbf{v}$. However, for simplicity, we omit this dependence and derive the update in terms of all vectors. In order to find the lower-triangular matrix $\mathbf{L}$ such that $\mathbf{L\, L}^\top = \mathbf{A}$, we perform the decomposition $$\mathbf{A} = \begin{bmatrix}
\sqrt{\tau} & 0\\
\lambda_1 v_1 \mathbf{u}_{2:n} + \lambda_2 z_1 \mathbf{w}_{2:n} & \mathbf{I}_{n-1}
\end{bmatrix}
\begin{bmatrix}
1 & 0\\
0 & \tilde{\mathbf{A}}_{2,2}
\end{bmatrix}
\begin{bmatrix}
\sqrt{\tau} & \lambda_1 u_1 \mathbf{v}_{2:n}^\top + \lambda_2 w_1 \mathbf{z}_{2:n}^\top\\
0 & \mathbf{I}_{n-1}
\end{bmatrix}$$ where $\tau = 1 + \lambda_1 u_1 v_1 + \lambda_2 w_1 z_1$. We can also write $\tilde{\mathbf{A}}_{2,2}$ in a diagonal plus rank-2 update form as follows. $$\tilde{\mathbf{A}}_{2,2} = \mathbf{I}_{n-1} + \lambda_1' \mathbf{u}_{2:n} (\mathbf{v}_{2:n} + \lambda_2 w_1 (z_1\mathbf{v}_{2:n} - v_1\mathbf{z}_{2:n}))^\top + \lambda_2' \mathbf{w}_{2:n} (\mathbf{z}_{2:n} - \lambda_1 u_1 (z_1\mathbf{v}_{2:n} - v_1\mathbf{z}_{2:n}))^\top$$ in which, $\lambda_1' = \lambda_1/\tau$ and $\lambda_2' = \lambda_2/\tau$. Thus, we can recursively apply the above procedure and solve for the Cholesky decomposition of matrix $\tilde{\mathbf{A}}_{2,2}$.
0.2in
[^1]: However, we do not need to impose this constraint implicitly since a solution with $\xi_{{\ensuremath{\scriptscriptstyle{i}{j}{k}}}} < 0$ cannot be optimum.
[^2]: We skip the subscript for notational simplicity.
[^3]: Note that the algorithms for [SKLR]{} and [sSKLR]{} essentially differ only on the value of $\alpha$ for update.
[^4]: We did not observe significant difference in the performance when varying $\lambda$ is the range $[10^3,10^5]$.
[^5]: <https://github.com/eamid/sklr>
[^6]: <http://www.cs.utexas.edu/~pjain/itml>
[^7]: <https://github.com/all-umass/metric_learn>
[^8]: <https://www.iiitd.edu.in/~anands/files/code/skms.zip>
[^9]: <http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/>
[^10]: <http://people.csail.mit.edu/torralba/code/spatialenvelope/>
[^11]: <http://cs.nyu.edu/~roweis/data.html>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Graph games of infinite length are a natural model for open reactive processes: one player represents the controller, trying to ensure a given specification, and the other represents a hostile environment. The evolution of the system depends on the decisions of both players, supplemented by chance.
In this work, we focus on the notion of *randomised* strategy. More specifically, we show that three natural definitions may lead to very different results: in the most general cases, an almost-surely winning situation may become almost-surely losing if the player is only allowed to use a weaker notion of strategy. In more reasonable settings, translations exist, but they require infinite memory, even in simple cases. Finally, some traditional problems becomes undecidable for the strongest type of strategies.
author:
- Julien Cristau
- Claire David
- Florian Horn
nocite: '[@Dic55]'
title: 'How do we remember the past in randomised strategies?'
---
*“You can’t have a strategy against telepaths: you have to act randomly. You have to not know what you’re going to do next. You have to shut your eyes and run blindly. The problem is: how can you randomise your strategy, yet move purposefully towards your goal?”*
Solar Lottery
Philip K. Dick
Introduction
============
Since their introduction to verification in the late eighties, graph games have emerged as the model of choice for problems about *open systems*, where a controller (Eve) must interact with an *a priori* hostile environment (Adam) [@PR89-POPL]. In such games, an *arena* —[*i.e.*]{}a graph— models the system and its evolution: at the beginning, a token is laid on one of the vertices, and its moves are determined by the actions of the players, supplemented by chance. The infinite sequence of vertices that ensues constitutes a *play* of the game, whose winner is defined by some predetermined specification, often given as a regular condition on infinite words [@MP92].
This model has been declined in a multiplicity of variants, in terms of both arenas and objectives. However, the questions are nearly always the same: Is there a winning strategy? For which player? How complex is it, in terms of memory and randomisation? Memory is the quantity of information that one is allowed to remember from the past: in general, the whole history is available, but it is often enough to remember a finite quantity of information. In addition, a strategy is *pure* if it proposes only one possible action after any given sequence of observations. The notion of *randomised* strategy, and its relation to memory, is the subject of this paper.
In verification, “randomised strategy” usually refers to a function from the history to a probability distribution over the actions. In other domains, such strategies are called “*behavioural* strategies”, as opposed to two other models of randomised strategies: a *mixed* strategy is a measure over pure strategies, and a *general* strategy is a measure over behavioural strategies. These models are also relevant in computer science. Indeed, the IPv6 “Stateless Address Autoconfiguration” protocol, which only uses randomisation at the beginning to generate a new I.P. address [@TNJ07-RFC], can be accurately described as a mixed strategy. Likewise, the secure shell protocol (ssh) is a general strategy, since a new session key has to be randomly[^1] generated every hour or gigabyte [@YL06-RFC].
In this paper, we propose definitions for mixed and general strategies, with or without memory, in the framework of graph games for verification. We expose several situations in which their analyses differ significantly from the behavioural model. In the most general case, the same game can be almost-surely losing or almost-surely winning depending on the type of strategies we consider. In other situations, we conjecture that the values are the same, but we show that memory needs vary (from two to infinity). Altogether, we hope to ask more questions than we give answers: our main objective is to describe these three models for randomised strategies in graph games and to point out that many problems which are solved for behavioural strategies are still open in the mixed and general cases.
The paper is organised as follows. In Section \[section:definitions\], we recall the classical notions about graph games in verification, in a very general framework which subsumes a large part of the literature. Section \[section:randomisedstrategies\] presents our definitions for behavioural, mixed, and general strategies in graph games, and stresses the fundamental differences between the three notions. Section \[section:memory\] focuses on memory-related issues: it exhibits variations in the elementary cases of concurrent safety games and simple Muller games. In Section \[section:discussion\], we sum up our observations and results, and propose some open problems.
Definitions {#section:definitions}
===========
### Notation {#notation .unnumbered}
For a finite or countable set ${{\cal S}}$, we denote by ${{\cal D}}({{\cal S}})$ the set of probability distributions over ${{\cal S}}$, [*i.e.*]{}the set of functions from ${{\cal S}}$ to positive real numbers that sums up to one.
### Arenas and plays {#arenas-and-plays .unnumbered}
An *arena* ${{{\cal A}}}$ is a tuple $({{{\cal V}}}, {{{\cal X}}}, {{{\cal Y}}}, {\delta}, {\Gamma}, {\gamma}, {\Phi}, {{\varphi}}, {\Psi}, {\psi})$ where ${{{\cal V}}}$ is the set of vertices in the graph, ${{{\cal X}}}$ is the set of actions of Eve, ${{{\cal Y}}}$ is the set of actions of Adam, ${\delta}: {{{\cal V}}}\times {{{\cal X}}}\times {{{\cal Y}}}\rightarrow {{\cal D}}({{{\cal V}}})$ is the transition function, ${\Gamma}$ is the set of colours, ${\gamma}: {{{\cal V}}}\rightharpoonup {\Gamma}$ is the colouring function, ${\Phi}$ is the set of signals of Eve, ${{\varphi}}: {{{\cal V}}}\cup {{{\cal X}}}\rightharpoonup {\Phi}$ is her observation function, ${\Psi}$ is the set of signals of Adam, and ${\psi}: {{{\cal V}}}\cup {{{\cal Y}}}\rightharpoonup {\Psi}$ is his observation function. Many results about graph games for verification consider only restricted arenas, such as:
[**Synchronous:**]{} an arena is *synchronous* if ${{\varphi}}$ and ${\psi}$ are total.
[**Observable actions:**]{} a synchronous arena has *observable actions* if the restrictions of ${{\varphi}}$ to ${{{\cal X}}}$ and ${\psi}$ to ${{{\cal Y}}}$ are one-to-one.
[**Perfect information:**]{} a synchronous arena has *perfect information* (or is *concurrent*) if the restrictions of ${{\varphi}}$ and ${\psi}$ to ${{{\cal V}}}$ are one-to-one.
[**Simple:**]{} a concurrent arena is *simple* (or *turn-based*) if for each vertex $q$, ${\delta}(q,x,y)$ depends either on $x$ or on $y$, but not both.
A *play* on the arena ${{{\cal A}}}$ is a (possibly infinite) sequence ${\pi}= {\pi}_0 {\pi}_1 \ldots$ of states such that $\forall i < |{\pi}|\! -\! 1,\exists x \in {{{\cal X}}}, y \in {{{\cal Y}}}, {\delta}({\pi}_i,x,y)({\pi}_{i+1}) > 0$. The set of plays is usually denoted ${\Omega}$, and the set of plays starting with the vertex $q$ by ${{\Omega}_{q}}$.
### Pure strategies and measures {#pure-strategies-and-measures .unnumbered}
A *pure strategy $\sigma$* for Eve (resp. $\tau$ for Adam) on the arena ${{{\cal A}}}$ associates an action to each finite sequence of observations: $\sigma : {\Phi}^* \rightarrow {{{\cal X}}}$ (resp. $\tau : {\Psi}^* \rightarrow {{{\cal Y}}}$) . A play ${\pi}$ is *consistent with a strategy $\sigma$ for Eve* (resp. *$\tau$ for Adam*) [if and only if]{}at each step $i$, there is an action $y \in {{{\cal Y}}}$ for Adam (resp. $x \in {{{\cal X}}}$) such that ${\delta}({\pi}_i,\sigma({{\varphi}}({\pi}_{1..i})),y)({\pi}_{i+1}) > 0$ (resp. ${\delta}({\pi}_i,x,\tau({\psi}({\pi}_{1..i})))({\pi}_{i+1}) > 0$). Notice that, in the case of an asynchronous arena, actions can only change with new observations: otherwise, the same argument leads to the same result over and over. The set of plays consistent with $\sigma$ (resp. $\tau$; $\sigma$ and $\tau$) is denoted by ${{\Omega}^{\sigma}}$ (resp. ${{\Omega}^{\tau}}$; ${{\Omega}^{\sigma,\tau}}$). Once an initial vertex $q$ and two strategies $\sigma$ and $\tau$ have been fixed, ${{\Omega}^{\sigma,\tau}_q}$ can naturally be made into a measurable space $({{\Omega}^{\sigma,\tau}_q},{{\cal O}})$, where ${{\cal O}}$ is the $\sigma$-field generated by the cones $\{{{\cal O}}_w \mid w \in {{{\cal V}}}^*\}$: ${\pi}\in {{\cal O}}_w$ [if and only if]{}$w$ is a prefix of ${\pi}$. The probability measure ${{{{\mathbb P}}^{\sigma,\tau}_{q}}}$ is recursively defined by ${{{{\mathbb P}}^{\sigma,\tau}_{q}}}({{\cal O}}_q) = 1$ and: $$\forall w \in {{{\cal V}}}^*, (r,s) \in {{{\cal V}}}^2, {{{{\mathbb P}}^{\sigma,\tau}_{q}}}({{\cal O}}_{wrs}) = {{{{\mathbb P}}^{\sigma,\tau}_{q}}}({{\cal O}}_{wr}) \cdot {\delta}(r,\sigma({{\varphi}}(wr)),\tau({\psi}(wr)))(s) .$$ Carathéodory’s extension theorem allows us to extend ${{{{\mathbb P}}^{\sigma,\tau}_{q}}}$ to the Borel sets of $({{\Omega}^{\sigma,\tau}_q},{{\cal O}})$ [@Wil91].
### Winning conditions and values {#winning-conditions-and-values .unnumbered}
A *winning condition ${{\cal W}}$ on a set of colours ${\Gamma}$* is a Borel subset of ${\Gamma}^\infty$. A play ${\pi}$ in an arena ${{{\cal A}}}$ on ${\Gamma}$ is *winning for Eve* in the game $({{{\cal A}}},{\Gamma})$ if ${\gamma}({\pi}) \in {{\cal W}}$, and *winning for Adam* otherwise. In a game ${{{\cal G}}}= ({{{\cal A}}},{{\cal W}})$, the *pure value of a state $q$ under the strategies $\sigma$ and $\tau$*, denoted ${v_{\sigma,\tau}}(q)$, is the measure of ${{\cal W}}$ under ${{{{\mathbb P}}^{\sigma,\tau}_{q}}}$. The *value for Eve of a state $q$* is the supremum of the values that she can ensure from $q$ against any strategy of Adam. Symmetrically, the *value for Adam of a state $q$* is the infimum of the values he can defend against any strategy of Eve. In simple stochastic games, these two values coincide and are usually called the *value of $q$* [@Mar98-JSL; @MS98-IJGT][^2]. $${\mathbf{v}}(q) = \sup_\sigma {v_{\sigma}}(q) = \inf_\tau {v_{\tau}}(q) \enspace .$$
### Winning criteria {#winning-criteria .unnumbered}
Following de Alfaro and Henzinger [@dAH00-LICS], we consider several notions of *winning strategies* and *winning regions*, depending on the chances Eve has to win. In decreasing order of difficulty, and from an initial vertex $q$, a strategy $\sigma$ for Eve:
- is *sure* if any play consistent with $\sigma$ is winning for Eve;
- is *almost-sure* if for any strategy $\tau$ for Adam, ${{v_{\sigma,\tau}}(q)}= 1$;
- *ensures ${\varepsilon}$* if for any strategy $\tau$ for Adam, ${{v_{\sigma,\tau}}(q)}\ge {\varepsilon}$;
- is *positive* if for any strategy $\tau$ for Adam, ${{v_{\sigma,\tau}}(q)}> 0$;
- is *heroic* if for any strategy $\tau$, there is a play consistent with $\sigma$ and $\tau$ which is winning for Eve.
The *sure region* (resp. *almost-sure region of Eve*, *positive region*, *heroic region*) of Eve is the set of vertices from which she has a sure (resp. almost-sure, positive, heroic) strategy. Furthermore, the *bounded region* is the set of vertices from which Eve has a strategy ensuring a positive ${\varepsilon}$ and the *limit-one region* is the set of vertices from which Eve has a strategy ensuring ${\varepsilon}$ for any ${\varepsilon}< 1$. The same concepts are defined accordingly for Adam, except that we say that a strategy $\tau$ for Adam *defends ${\varepsilon}$* if it guarantees that, for any strategy $\sigma$ for Eve, ${{v_{\sigma,\tau}}(q)}\le {\varepsilon}$.
Behavioural, mixed, and general strategies {#section:randomisedstrategies}
==========================================
As soon as we deal with concurrent arenas, we cannot rely only on pure strategies to make meaningful analyses. In the classical game of “Janken”, any pure strategy is surely beaten by the appropriate counter-strategy ($\mathtt{paper}$ against $\mathtt{rock}$, $\mathtt{scissors}$ against $\mathtt{paper}$, and $\mathtt{rock}$ against $\mathtt{paper}$), but a strategy which plays each action with probability $\frac{1}{3}$ eventually wins with probability $\frac{1}{2}$. The main point of this paper is that there are several possible definitions for the notion of “randomized strategy”.
- A *behavioural strategy* returns at each step a distribution over the actions: ${\Phi}^* \rightarrow {{\cal D}}({{{\cal X}}})$;
- a *mixed strategy* is a measure over pure strategies: ${{\cal D}}({\Phi}^* \rightarrow {{{\cal X}}})$;
- a *general strategy* is a measure over behavioural strategies: ${{\cal D}}({\Phi}^* \rightarrow {{\cal D}}({{{\cal X}}}))$.
As we show in this paper the expressive powers of these models are quite different. Intuitively, a behavioural strategy does not know in advance what it will play next, so its actions can change when its decisions do not (even when there are no observations). Mixed strategies use randomization to get hidden information at the beginning of the play, which can later be used to correlate undistinguishable actions, [*e.g.*]{}playing $aa$ or $bb$ with probability $\frac{1}{2}$. General strategies subsume both, so they can, in particular, generate hidden information on the fly. These distinctions have mostly been overlooked in verification (apart from a few remarks, [*e.g.*]{}[@dAHK98-FOCS; @DHR08-IJFCS]). One reason is that the games we consider are usually synchronous, with observable actions. On synchronous arenas, mixed strategies can simulate behavioural strategies: as each action can be uniquely identified beforehand by its position in the play, it is possible to define a measure which somehow makes all the random draws at the beginning of the play. If furthermore, the actions are observable, Kuhn’s theorem states that mixed and behavioural (and thus general) strategies have the same expressive power [@Aum64-AMST].
These hypotheses have been inconspicuously challenged in recent papers. In this regard, the comparison between [@BGG09-LICS] and [@GS09-ICALP] makes for an enlightening example. At first glance, these two papers look very similar: they both ponder the problem of the existence of almost-sure strategies in games where both players have (asymmetric) imperfect information. A closer examination reveals the differences: Bertrand, Genest, and Gimbert use general strategies, while Gripon and Serre use behavioural strategies; furthermore, in the latter paper, the players cannot observe their own actions. As a consequence, there are cases where the answer to the synthesis problem depends on which model is used. Consider for example the synchronous arena depicted in Figure \[figure:valuefails\], where Eve cannot distinguish vertices nor actions in the dashed area, ${\otimes}$ is a losing sink state, and $\circledcirc$ is her “target”, for either a reachability or a Büchi condition.
=1.35mm
(70,42)(0,0) (i)(35,15) (delay)(35,27) (a)(10,21)[$\alpha$]{} (b)(59,21)[$\beta$]{} (top)(35,39) (bottom)(35,3) (36.45,4.45)(33.55,1.55) (36.45,1.55)(33.55,4.45) (i,delay)[$aB$]{} (i,delay)[$bA$]{} (i,a)[$aA$]{} (i,b)[$bB$]{} (delay,i)[$b*$]{} (delay,i)[$a*$]{} (a,top)[$a*$]{} (a,bottom)[$b*$]{} (b,top)[$b*$]{} (b,bottom)[$a*$]{} (top,delay) (5,25)(5,17)(65,17)(65,25)
With a behavioural strategy, Eve’s strategy can only depend on the length of the play. At any even move, if her strategy is to play $a$ with probability $p$ and $b$ with probability $1-p$, Adam can answer by playing $A$ with probability $1-p$ and $B$ with probability $p$, so the odds of the token going to $\alpha$ or to $\beta$ are equal (they are worth $p \cdot (1-p)$ each). In the next step, no matter what $\sigma$ advocates, the odds of the token going to ${\circledcirc}$ or to ${\otimes}$ will again be equal. In the reachability game, this limits Eve’s prospects to half chances. In the Büchi game, the probability that she wins drops to $0$.
On the flip side, she has an almost-sure mixed strategy for both objectives: the natural “uniform” measure over the strategies of the form $(aa|bb)^\omega$ guarantees that each sequence of two moves starting in the initial vertex has a probability of $\frac{1}{2}$ to send the token to ${\circledcirc}$, and a probability of $\frac{1}{2}$ to send the token back to the initial vertex, no matter what Adam does. It cannot go to ${\otimes}$, as Eve never plays $ab$ or $ba$ from the initial vertex.
The arena of Figure \[figure:valuefails\] is synchronous, so any behavioural strategy can be emulated by a mixed one. If we remove this hypothesis, it is not always the case, as in the one-player game of Figure \[figure:drinkanddrive\], where Eve is unaware of any action or vertex.
=1.5mm
(42,18)(0,0) (first)(7,9) (second)(21,9) (exit)(35,3) (fail)(35,15) (36.45,16.45)(33.55,13.55) (36.45,13.55)(33.55,16.45) (first,second)[$\mathtt{stay}$]{} (second,fail)[$\mathtt{stay}$]{} (first,fail)[$\mathtt{leave}$]{} (second,exit)[$\mathtt{leave}$]{}
As Eve observes nothing, her strategy is completely determined by what she does on the empty word $\lambda$. She has only two pure strategies: $\lambda \rightarrow \mathtt{stay}$ and $\lambda \rightarrow \mathtt{leave}$. Both lead to ${\otimes}$, and so does any mixed strategy of the form $\{p \cdot (\lambda \rightarrow \mathtt{stay}), (1-p) \cdot (\lambda \rightarrow \mathtt{leave})\}$. The behavioural strategy $\lambda \rightarrow \{\frac{1}{2} \cdot \mathtt{stay}, \frac{1}{2} \cdot \mathtt{leave}\}$, on the other hand, yields one chance out of four to to reach ${\circledcirc}$.
The case of games with perfect information and invisible actions is still open: there are mixed strategies which cannot be imitated by any behavioural one, so we cannot hope for a “generic” translation. But that does not rule out the possibility of specific, objective-dependent constructions which would yield a different strategy with the same value.
Memory issues {#section:memory}
=============
A refinement of the synthesis problem asks that the controller uses only finite memory, as a natural requirement for implementability. Pure strategies with memory are defined in the following way:
\[definition:purememory\] A *pure strategy $\sigma$ with memory $M$* is a triple $({\sigma^{{\tt i}}}, {\sigma^{{\tt u}}}, {\sigma^{{\tt a}}})$ where ${\sigma^{{\tt i}}}\in M$ is the initial memory state; ${\sigma^{{\tt u}}}: ({\Phi}\times M) \rightarrow M$ is the *memory update* function, which maps a signal and a memory state to a new memory state and is called at each new observation of Eve; and ${\sigma^{{\tt a}}}: ({\Phi}\times M) \rightarrow {{{\cal X}}}$ is the *next-action* function, which maps a signal and a memory state to an action and is called at each step.
Notice that any pure strategy $\sigma$ can be represented as a strategy with memory ${\Phi}^*$, with ${\sigma^{{\tt i}}}= \lambda$, ${\sigma^{{\tt u}}}= \cdot$ and ${\sigma^{{\tt a}}}= \sigma$. A strategy has *finite memory* if $M$ is a finite set, and is *memoryless* if $M$ is a singleton.
Randomized strategies with (countable) memory are defined with similar tuples, except that some of their elements use randomization.
[**Behavioural:**]{} In a *behavioural strategy with memory $M$*, the next-action ${\sigma^{{\tt a}}}: ({\Phi}\times M) \rightarrow {{\cal D}}({{{\cal X}}})$ is randomized.
[**Mixed:**]{} In a *mixed strategy with memory $M$*, the initial memory ${\sigma^{{\tt i}}}\in {{\cal D}}(M)$ is randomised.
[**General:**]{} In a *general strategy with memory $M$*, the next-action, initial memory, and memory-update ${\sigma^{{\tt u}}}: ({\Phi}\times M) \rightarrow {{\cal D}}(M)$ are randomised.
The memory requirements can also depend of the type of strategy. In the game of Figure \[figure:valuefails\], for example, there is no almost-sure mixed strategy with finite memory (in the reachability game, there are ${\varepsilon}$-optimal strategies with finite (unbounded) memory; in the Büchi game, every mixed strategy with finite memory has value $0$). However, the strategy we described can be realized by a general strategy with four memory states $\mathtt{a_{even}}$, $\mathtt{a_{odd}}$, $\mathtt{b_{even}}$ and $\mathtt{b_{odd}}$: in the $\mathtt{even}$ memory state, she updates her memory at random to one of the $\mathtt{odd}$ states; in the $\mathtt{odd}$ states, she updates her memory to the corresponding $\mathtt{even}$ state; in all states, she plays the action corresponding to her memory state.
Concurrent safety games
-----------------------
In [@dAHK98-FOCS], de Alfaro, Henzinger, and Kupferman study the problem of concurrent reachability/safety games and establish the qualitative determinacy of these games, as well as several results on the nature (memory and randomization) of the strategies needed to achieve various objectives. In particular, they show that positive strategies for safety objectives require, in general, an infinite amount of memory. The proof is based on the famous “snowball game” of [@KS81-SJCO], which is pictured in Figure \[figure:snowball\].
= 3.50mm
(24,8)(0,0) (un)(2,3) (deux)(12,3) (trois)(22,3) (22.7,3.7)(21.3,2.3) (21.3,3.7)(22.7,2.3) (deux,un)[$\mathtt{throw}|\mathtt{run}$]{} (deux)[$\mathtt{wait}|\mathtt{hide}$]{} (deux,trois)[$\mathtt{wait}|\mathtt{run}$]{} (deux,trois)[$\mathtt{throw}|\mathtt{hide}$]{}
In this game, Adam loses if he never runs and Eve never throws, or if Eve happens to throw the snowball exactly at the moment he runs. It is clear that Adam has memoryless behavioural strategies with value arbitrarily close to one: if, at each step, he chooses to run with probability ${\varepsilon}$, he ensures a probability of winning of $1-{\varepsilon}$ (Eve’s best chance is to throw the ball right away). It is also clear that he cannot win almost-surely: if he has a positive probability of never running, Eve can keep the snowball forever; and if he has a positive probability of running at any step, Eve can thwart him by throwing the ball with probability $\frac{1}{2}$ at each step.
By the *qualitative determinacy* of concurrent regular games, Eve has a positive strategy, [*i.e.*]{}a unique strategy which prevents Adam from winning almost-surely with any strategy. De Alfaro, Henzinger, and Kupferman use behavioural strategies, and argue that Eve needs infinite memory: the sequence $(\sigma(\Circle^i)(\mathtt{throw}))_{i\in{{\mathbb N}}}$ must go to $0$ but never reach it. It is clear that there are no positive mixed strategies with finite memory, as pure strategies with $n$ memory states can only throw the snowball in the first $n$ steps.
On the other hand, there is a general strategy with only $2$ memory states: in the memory state $\mathtt{Never}$, Eve keeps the snowball with probability $1$, in the state $\mathtt{Eventually}$, she throws it with probability $\frac{1}{2}$; the memory never changes, and the initial memory state is chosen at random. This strategy prevents Adam from winning almost-surely, since he can never be sure that Eve is not in the memory state $\mathtt{Eventually}$. In fact, this is the case in every finite concurrent safety game:
\[theorem:concurrentsafetymixed\] In every finite concurrent safety game, Eve has a positive general strategy with memory $2$ from her positive region.
It follows from the analysis of the fix-points in [@dAHK98-FOCS] that there is a total preorder $\prec$ on the vertices such that:
- the minimal vertices belong to the almost-sure region of Adam;
- for each non-minimal vertex $q \in {{{\cal V}}}$, there is an action ${\mathtt{safe}}_q \in {{{\cal X}}}$ of Eve such that, for any action $y$ of Adam:
- either for any vertex $r \in {{{\cal V}}}$, ${\delta}(q,{\mathtt{safe}}_q,y)(r) > 0 \Rightarrow q \precsim r$,
- or there is an action $x \in {{{\cal X}}}$ of Eve and a vertex $r \in {{{\cal V}}}$ such that ${\delta}(q,x,y)(r) > 0 \wedge q \precnsim r$
Notice that always playing the ${\mathtt{safe}}$ action is a pure and positional sure strategy for Eve in the maximal vertices (unless they are also minimal). For the vertices in between, we claim that the following strategy with two memory states is positive for Eve:
- the memory states are called ${\mathtt{Sound}}$ and ${\mathtt{Chance}}$;
- each time the token goes to a new $\prec$-class, the memory state is updated to either ${\mathtt{Sound}}$ or ${\mathtt{Chance}}$ with equal probabilities; otherwise, the memory does not change;
- in the ${\mathtt{Sound}}$ state, Eve always plays the ${\mathtt{safe}}$ action of the current vertex;
- in the ${\mathtt{Chance}}$ state, Eve plays any action in ${{{\cal X}}}$ with equal probabilities.
The situation is roughly the same as in the snowball game: if Adam’s actions have no chance to go to a lower vertex against the ${\mathtt{Sound}}$ strategy, he will lose with probability $\frac{1}{2}$; if he takes a risk at any point, there is a positive probability that Eve was in the ${\mathtt{Chance}}$ memory state all along, so he could end up in a greater vertex.
In addition to the finite memory, the strategy described in the proof of Theorem \[theorem:concurrentsafetymixed\] is simple, generic, and uses only uniform probabilities. By comparison, the description of a positive behavioural strategy is in general very complex and uses probabilities of unbounded precision.
Memory bounds for Muller games
------------------------------
Even in the elementary case of simple Muller games, it is not clear that the memory needs are the same for behavioural and general strategies. Recall that a simple arena is an arena with turn-based moves and perfect information for both players, and a Muller condition is a condition depending only on the set of colours visited infinitely often:
A Muller condition on a set of colours ${\Gamma}$ is specified by a subset ${{\cal F}}$ of ${{{\cal P}}({\Gamma})}$. A play ${\pi}$ satisfies $\mathtt{Muller}({{\cal F}})$ [if and only if]{}the set of colours occurring infinitely often in ${\gamma}({\pi})$ belongs to ${{\cal F}}$.
In such games, both players have pure optimal strategies with finite memory [@BL69-TAMS]. A follow-up problem is to determine, for a given Muller condition ${{\cal F}}$ on a set of colours ${\Gamma}$, the necessary and sufficient amount of memory needed to define optimal pure strategies in *any* arena coloured by ${\Gamma}$. Gurevich and Harrington used the *latest appearance record* (LAR) structure of McNaughton to give a first upper bound of $|{\Gamma}|!$ [@GH82-STOC]. Zielonka refined the LAR into a tree, whose leaves could be used as memory [@Zie98-TCS]. Finally, Dziembowski, Jurdzinski, and Walukiewicz showed that each player needs only as much memory as the number of leaves in some particular sub-trees, establishing tight and asymmetrical bounds for pure strategies [@DJW97-LICS].
It is clear from their proof that mixing strategies does not help, since the other player can efficiently adapt their strategy in the witness arenas. This is not the case for behavioural strategies: Chatterjee, de Alfaro, and Henzinger observed that upward-closed winning conditions admitted memoryless strategies [@CdAH04-QEST], leading to smaller upper bounds for arbitrary Muller conditions [@Cha07-FoSSaCS]. Horn established even smaller tight bounds for general strategies [@Hor09-STACS] (see Figure \[figure:mullerbounds\] for a graphical representation of the three bounds on a Zielonka (sub-)tree). However, Horn’s upper-bound has not yet been proven (or refuted) for behavioural strategies.
= 1.35mm
Discussion {#section:discussion}
==========
We have compared three models of randomized strategies —behavioural, mixed, and general— in the context of graph games. Depending on the sub-case, we were able to expose variations in the amount of memory needed, the existence of finite-memory strategies, or even the values. In concurrent games with unobservable actions, the equivalence between the three models is still an open question.
In verification, the behavioural model has received most of the attention. Nevertheless, there is *a priori* nothing wrong with the other types of controllers. Furthermore, in several cases, general strategies can be much simpler than behavioural or mixed ones. On the other hand, general strategies are much less amenable to further analysis, as they introduce imperfect information. Even in simple safety games, one cannot compute the value of a general strategy —or even decide if it has positive value [@GO09-LaBRI].
Each model has strengths and weaknesses, and we do not favour one over the others. Our point is rather to stress the importance of this initial choice, and to note that many memory-related problems which have been solved for behavioural strategies are still open in the mixed and general frameworks.
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[^1]: except on Debian [@BB08-DEFCON].
[^2]: As a matter of fact, these papers shows the quantitative determinacy, in behavioural strategies, of Borel games on concurrent arenas. An inspection of the proof yields the same result for pure strategies in the case of simple arenas.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is well known that in a two-slit interference experiment, acquiring which-path information about the particle, leads to a degrading of the interference. It is argued that path-information has a meaning only when one can umabiguously tell which slit the particle went through. Using this idea, two duality relations are derived for the general case where the two paths may not be equally probable, and the two slits may be of unequal widths. These duality relations, which are inequalities in general, saturate for all pure states. Earlier known results are recovered in suitable limit.'
author:
- 'Keerthy K. Menon'
- Tabish Qureshi
title: Wave Particle Duality in Asymmetric Beam Interference
---
[^1]
Introduction
============
Wave-particle duality is an intriguing aspect of nature, which was first conceptualized by Neils Bohr in his principle of complementarity [@bohr]. It has been in debate right from the beginning when Einstein’s raised objections against it, proposing his famous recoiling slit experiment [@tqeinstein], and continues to be so even today [@liu]. The two-slit interference experiment has become a cornerstone for investigating such issues.
Bohr had emphasized that the wave and particle natures are mutually exclusive, revealing one, completely hides the other. Wooters and Zurek began by asking what happens if one probes the wave and particles natures at the same time, in a two-slit experiment [@wootters]. They found that it is indeed possible to partially reveal both the natures. This idea was later put on firm mathematical ground by Englert, in the form of a duality relation which puts a bound on how much of each nature can be revealed simultaneously: [@englert] $${\mathcal D}^2 + {\mathcal V}^2 \le 1,
\label{englert}$$ where ${\mathcal D}$ is path distinguishability, a measure of the particle nature, and ${\mathcal V}$ the visibility of interference, a measure of wave nature. Wave and particle natures are so fundamental to quantum objects that many prefer to call them [*quantons*]{} [@bunge; @levy]. A different kind of duality relations are also studied where one tries to [*predict*]{} the path information of the quanton based on the asymmetry of the two beams, without using any path detector [@greenberger; @vaidman].
Contemporary thinking is that, in a two-slit interference experiment, if one is able to tell which of the two slits the quanton went through, one has revealed the particle nature of the quanton. On ther other hand, if one obtains an interference patters, one has revealed the wave aspect of the quanton. The duality relation derived by Englert was for a symmetric two-slit experiment, in which the quanton is equally likely to go through both the slits. However, there can be situations where the setup is not symmetric, i.e., the state of the incident quanton is such that the probabilities to go through the two slits are not equal. Another possibility is that the two slits may not be of equal widths. This asymmetric case has not been probed in as much detail as the symmetic case [@englert]. There have been other studies on asymmetric two-path interference [@lili; @yliu], but none provides a tight duality relation for this case. The aim of this paper is obtain a general duality relation which also holds for asymmetric two-slit interference experiments.
Dealing with asymmetry
======================
We assume that the state of the quanton that emerges from the double-slit is given by the [*unnormalized*]{} state $$|\psi\rangle = \sqrt{p_1}|\psi_1\rangle + \sqrt{p_2}|\psi_2\rangle,
\label{asym}$$ where $p_1, p_2$ quantify the asymmetry of the incoming wave. In addition, we would like to take into account the effect of asymmetric slits, namely the situation where the two slits may have different widths. As different slit widths will also contribute to the probabilities of passing through the two slits, one should assume that the states $|\psi_1\rangle, |\psi_2\rangle$, corresponding to the quanton coming out of slit 1 and 2, respectively, are [*not normalized*]{}. One can see that for the case $p_1=p_2=1/2$, the probability of the quanton to pass through slit 1 and 2 is proportional to $\langle\psi_1|\psi_1\rangle$ and $\langle\psi_2|\psi_2\rangle$, respectively.
Given that the incoming quanton state is symmetric, the quanton is more likely to pass through the wider slit. The details of the effect of asymmetry due to the slits will be specfied while choosing the form of $|\psi_1\rangle, |\psi_2\rangle$. Needless to say, the state $|\psi\rangle$ as a whole should be normalized.
Getting which-way information
=============================
An experiment to find out which of the two paths a quanton has followed, would require some kind of detector which can retrieve and store information on which path a particular quanton took. We assume a fully quantum detector with states corresponding to the quanton taking one path or the other. Without going into the details of what this path detector should be like, we just use von Neumann’s criterion for a quantum measurement [@neumann]. According to von Neumann’s criterion, the basic requirement for a path detector to perform a which-path measurement is that it should interact with the quanton in such a way that its two states should get correlated with the two paths of the quanton. If the state of the quanton that emerges from the asymmetric double-slit is given by (\[asym\]), and the path-detector is in an initial state $|d_0\rangle$, the interaction between the two should be such that it evolves to the following: $$(\sqrt{p_1}|\psi_1\rangle + \sqrt{p_2}|\psi_2\rangle)|d_0\rangle \rightarrow
\sqrt{p_1}|\psi_1\rangle|d_1\rangle + \sqrt{p_2}|\psi_2\rangle|d_2\rangle$$ The quanton goes and registers on the screen, and the path-detector is left with the experimenter. If the experimenter finds that the state of the path-detector is $|d_1\rangle$, she can conclude that the quanton went through slit 1, else if the state of the path-detector is $|d_2\rangle$, it would imply that the quanton went through slit 2. The interaction between the quanton and the path-detector is designed by the experimenter, and thus the states $|d_1\rangle$, $|d_2\rangle$ are known. What is not known is, which of the two states one would get, for particular instance of quanton going through the double-slit.
The problem of finding which path the quanton followed then reduces to telling whether the state of the path-detector is $|d_1\rangle$ or $|d_2\rangle$. To solve this problem, Englert took the approach of calculating the optimum “likelihood for guessing the way (which of the two ways the quanton went) right". We take a somewhat different route. We believe that for any given instance of quanton passing through the double-slit, one can claim to have which-path knowledge only if one can tell [*for sure*]{} which of the two paths the quanton took. What we mean is, there should be no guessing involved. One should have an unambigous answer to the question which path the quanton took. In the path-detection scenaio discussed above, this would mean one should be able unambiguously tell which of the two states $|d_1\rangle$ or $|d_2\rangle$, is the given state of the path-detector. If $|d_1\rangle$ and $|d_2\rangle$ are orthogonal, one can measure an observable of the path-detector which has $|d_1\rangle$ and $|d_2\rangle$ as it’s two eigenstates, with distinct eigenvalues. Looking at the eigenvalue of the measurement, one would know [*for sure*]{} that the path-detector was (say) $|d_2\rangle$, and hence the quanton went through the lower slit. However, there are situations in which $|d_1\rangle$ and $|d_2\rangle$ are not orthogonal. There exists a method which allows for unambiguously distinguishing between two non-orthogonal states, and goes by the name of unambiguous quantum state discrimination (UQSD) [@uqsd; @dieks; @peres; @jaeger2]. A downside of this method is that there will be occasions where the method will fail to provide an answer, but the experimenter will know that it has failed. Thus, on the occasions on which UQSD succeeds, it can unambiguously distinguish between the two non-orthogonal states. The measurement method can be tuned to minimize the failure probability, and thus maximizing the probability of unambiguously distingishing between the two states.
Unambiguous path discrimination
===============================
The UQSD approach has been successfully used in defining a new distinguishability of paths, $\cal{D}_Q$, as the maximum probability of [*unambiguously*]{} distinguishing between the available quanton paths. This resulted in new duality relations for the symmetric two-slit interference [@3slit], three-slit interference [@3slit], and n-slit interference [@cd; @nslit]. Here we use it to study wave-particle duality in the case of interference involving asymmetric paths.
We begin at the instance the quanton emerges from the double-slit. The state of the quanton has to be a superposition of two localized parts, in front of slit 1 and 2, respectively. We assume the quanton is traveling along the positive y-axis, and the double-slit is the in the x-z plane, at $y=0$ (see FIG. \[2slit\]). For the purpose of interference, the motion along the y-axis is unimportant. It is the spread of the two emerging wave-packets along the x-axis and the overlap, which gives rise to interference. We neglect the dynamics along y-axis, and assume that the quanton is traveling along y-axis with an average momentum $p_0$, and that motion only serves to translate the quanton from the slit to the screen with time. For calculational simplicity, we assume the parts of the state emerging from the double-slit to be Gaussian wave-packets, localized in front of the two slits, namely at positions $x=x_0$ and $x=-x_0$. The state of the quanton at time $t=0$, is given by $$\langle x|\psi(0)\rangle = A\left( \sqrt{p_1}e^{-\frac{(x-x_0)^2}{\epsilon^2}}
+ \sqrt{p_2}e^{-\frac{(x+x_0)^2}{\xi^2\epsilon^2}}\right)$$ where $A = \left(\frac{2}{\pi\epsilon^2(p_1+\xi p_2)}\right)^{1/4}$, $d=2x_0$ is the separation between the slits and $\epsilon$ and $\xi\epsilon$ are the widths of the two Gaussians, and may loosely be considered the widths of the two slits. At the instant of emerging from the double-slit, the quanton interacts with a path-detector, and the combined state of the two should have the following form (as argued earlier): $$\langle x|\psi(0)\rangle = A \left( \sqrt{p_1}e^{-\frac{(x-x_0)^2}{\epsilon^2}}|d_1\rangle
+ \sqrt{p_2}e^{-\frac{(x+x_0)^2}{\xi^2\epsilon^2}}|d_2\rangle\right),
\label{istate}$$ where $|d_1\rangle, |d_2\rangle$ are the two states of the path-detector. The states $|d_1\rangle, |d_2\rangle$ are chosen to be normalized, although they are not orthogonal in general. It may be mentioned that choosing the probability amplitudes $\sqrt{p_1}, \sqrt{p_2}$ to be real and positive is not a loss of generality as $|d_1\rangle, |d_2\rangle$ may contain phases.
Now, the idea is to find out how much path information of the quanton can be retrieved from the path detector, [*in principle*]{}, given the state (\[istate\]). We would like to stress the point that a particular method of probing the path-detector may yield a certain amount of path information, but we are interested in best possible value that can be obtained in principle. UQSD works for the situation where the two states, $|d_1\rangle, |d_2\rangle$ occur randomly with different probability. If one is given one of the states and asked to tell which of the two it is, UQSD allows one to give the best possible answer. To use this method for the problem at hand, we should ascertain the probabilities with which $|d_1\rangle, |d_2\rangle$ occur. Looking at (\[istate\]) one may naively jump to the conclusion that the probabilities in question are $p_1$ and $p_2$. However, the different widths of the two slits would also contribute to the probability of the quanton passing through slit $k$ resulting the path-detector state $|d_k\rangle$. The probability amplitude for this possibility is given by $c_k = \frac{\langle\psi_k|\psi(0)\rangle}{\sqrt{\langle\psi_k|\psi_k\rangle
\langle\psi(0)|\psi(0)\rangle}}$, where $k=1,2$. Using the Gaussian form given in (\[istate\]), these probability amplitudes turn out to be $$\begin{aligned}
c_1 = {\sqrt{p_1}\over \sqrt{p_1+\xi p_2}} ~~~~
c_2 = {\sqrt{\xi p_2}\over\sqrt{p_1+\xi p_2}}.
\label{c1c2}\end{aligned}$$ As far as measurements on the path-detector are concerned, it can be assumed to randomly found in the state $|d_1\rangle$ with probability $c_1^2$, and in the state $|d_2\rangle$ with probability $c_2^2$. In addition, without loss of generality, we assume that $c_1 \ge c_2$.
In order to use UQSD, we assume that the Hilbert space of the path-detector is not two dimensional, but three dimensional, described by an orthonormal basis of states $|q_1\rangle, |q_\rangle, |q_3\rangle$. The reason for doing so will become clear in the following analysis. The basis is chosen in such a way that the detector states $|d_1\rangle,
|d_2\rangle$ can be represented as [@jaeger2] $$\begin{aligned}
|d_1\rangle &=& \alpha|q_1\rangle + \beta|q_3\rangle \nonumber\\
|d_2\rangle &=& \gamma|q_2\rangle + \delta|q_3\rangle ,
\label{d1d2}\end{aligned}$$ where $\alpha$ and $\gamma$ are real, and $\beta, \delta$ satisfy $$\begin{aligned}
|\beta| |\delta| &\ge& |\langle d_1|d_2\rangle|,\nonumber\\
|\beta|^2&=& \max\{|\langle d_1|d_2\rangle|c_2/c_1, |\langle d_1|d_2\rangle|^2\}
\label{betagamma}\end{aligned}$$ In the expanded Hilbert space, one can now measure an operator (say) $${\boldsymbol}{A} = |q_1\rangle\langle q_1| + 2|q_2\rangle\langle q_2|
+ 3|q_3\rangle\langle q_3|.$$ It is straightforward to see that getting eigenvalue 1 means the state was $|d_1\rangle$, getting eigenvalue 2 means the state was $|d_2\rangle$. However, there is also a finite probability of getting eigenvalue 3, in which case one cannot tell if the state was $|d_1\rangle$ or $|d_2\rangle$. One would like to minimize the probability of getting the eigenvalue 3, or failure of the state discrimination. It can be shown that chosen values of $\beta,\delta$ in (\[betagamma\]) are such that they minimize the probability of failure, and maximize the probability of successfully distinguishing between $|d_1\rangle$ and $|d_2\rangle$ [@jaeger2]. We will return to these in more detail later.
![A two-slit experiment with a path-detector in front of the double-slit. Slit separation is $d$ and the distance between the double-slit and the screen is $D$.[]{data-label="2slit"}](twoslit.pdf){width="8.0"}
Interference and fringe visibility
==================================
We now analyze what happens when the quanton reaches the screen. We assume that the quanton takes a time $t$ to travel along y-axis from the double-slit to the screen, a distance $D$ (see FIG. \[2slit\]). The time evolution depends on what is the nature of our quanton. It could be a photon traveling with the speed of light, or it could be a particle of mass $m$ under free time-evolution. One can write the time evolution of the state in a universal form $$|\psi(t)\rangle = {1\over 2\pi}\int_{-\infty}^{\infty}
|k\rangle \langle k|\psi(0)\rangle e^{-i\omega_kt} dk$$ where $|k\rangle$ are the momentum states. For photons $\omega_k = ck$ and for massive particles $\omega_k = \frac{\hbar k^2}{2m}$. The state of the quanton, after a time $t$ (after traveling a distance $D$ from the double-slit to the screen), can be worked out to be [@dillon] $$\langle x|\psi(t)\rangle = B\left(\tfrac{\sqrt{p_1}}{\sqrt[4]{\epsilon^4+\Gamma^2}}e^{-\frac{(x-x_0)^2}{\epsilon^2+i\Gamma}}|d_1\rangle
+ \tfrac{\sqrt{\xi p_2}}{\sqrt[4]{\xi^4\epsilon^4+\Gamma^2}}e^{-\frac{(x+x_0)^2}{\xi^2\epsilon^2+i\Gamma}}|d_2\rangle\right),
\label{fstate}$$ where $\Gamma = 2\hbar t/m = \lambda D/\pi$, if one defines $\lambda = h/p_0$, and $B = \sqrt[4]{\frac{2\epsilon^2}{\pi(p_1+\xi p_2)}}$. It can be shown that if the quanton is a photon, one gets the same expression with $\Gamma = \lambda D/\pi$, where $\lambda$ is the wavelength of the photon. Let us assume a phase factor associated with the detector states: $\langle d_1|d_2\rangle = |\langle d_1|d_2\rangle|e^{i\theta}$. The probability of the quanton to arrive at a position $x$ on the screen is then given by $$\begin{aligned}
|\langle x|\psi(t)|^2 &=& B^2\left(\tfrac{p_1}{\sqrt{\epsilon^4+\Gamma^2}}e^{-\frac{2\epsilon^2(x-x_0)^2}{\epsilon^4+\Gamma^2}}
+ \tfrac{\xi p_2}{\sqrt{\xi^4\epsilon^4+\Gamma^2}} e^{-\frac{2\xi^2\epsilon^2(x+x_0)^2}{\xi^4\epsilon^4+\Gamma^2}}\right.\nonumber\\
&+&\left. \tfrac{\sqrt{\xi p_1p_2}}{\sqrt[4]{\epsilon^4+\Gamma^2}\sqrt[4]{\xi^4\epsilon^4+\Gamma^2}} |\langle d_1|d_2\rangle|
e^{-\frac{(x-x_0)^2}{\epsilon^2-i\Gamma}}
e^{-\frac{(x+x_0)^2}{\xi^2\epsilon^2+i\Gamma}+i\theta}\right. \nonumber\\
&+&\left. \tfrac{\sqrt{\xi p_1p_2}}{\sqrt[4]{\epsilon^4+\Gamma^2}\sqrt[4]{\xi^4\epsilon^4+\Gamma^2}} |\langle d_1|d_2\rangle|
e^{-\frac{(x-x_0)^2}{\epsilon^2+i\Gamma}}
e^{-\frac{(x+x_0)^2}{\xi^2\epsilon^2-i\Gamma}-i\theta} \right).\nonumber\\\end{aligned}$$ In the Fraunhofer limit $\lambda D \gg \epsilon^2$, which implies $\Gamma^2 \gg \epsilon^4$, the above can be simplified to $$\begin{aligned}
|\langle x|\psi(t)|^2 &=& \tfrac{B^2}{\Gamma} \left[p_1e^{-\frac{2\epsilon^2(x-x_0)^2}{\Gamma^2}}
+ \xi p_2 e^{-\frac{2\epsilon^2(x+x_0)^2}{\Gamma^2}} \right.\nonumber\\
&&\left. +~ 2\sqrt{p_1p_2\xi} |\langle d_1|d_2\rangle|
e^{-\frac{\epsilon^2(x^2+x_0^2)(1+\xi^2)}{\Gamma^2}}
e^{\frac{\epsilon^2 2xx_0(1-\xi^2)}{\Gamma^2}}
\right. \nonumber\\
&&\times\left. \cos\left({\frac{4\pi xx_0}{\lambda D}+\theta}\right)\right].
\label{interf}\end{aligned}$$ Eqn. (\[interf\]) represents a two-slit interference pattern, with a fringe width $w = \lambda D/d$. We assume that tha intensity at position $x$ is given by $I(x) \propto |\langle x|\psi(t)|^2$. The maxima and minima of intensity occur at the values of $x$ where the cosine term is 1 and -1, respectively.
The visibility of the interference pattern is just the the contrast in intensities of neighbouring fringes [@born] $${\mathcal V} = \frac{I_{\rm{max}} - I_{\rm{min}}}{ I_{\rm{max}} + I_{\rm{min}} } ,$$ where $I_{\rm{max}}$ and $I_{\rm{min}}$ represent the maximum and minimum intensity in neighbouring fringes. The interference in (\[interf\]), ignoring the effect of a finite slit-width $\epsilon$, yields the following [*ideal*]{} fringe visibility: $${\mathcal V} = \frac{2\sqrt{p_1p_2\xi}}{p_1+\xi p_2} |\langle d_1|d_2\rangle|.
\label{V}$$ If $|d_1\rangle=|d_2\rangle$, which means that the path-detector is effectly absent, the fringe visibility reduces to ${\mathcal V}_0 = \frac{2\sqrt{p_1p_2\xi}}{p_1+\xi p_2}$, and is called the [*a priori*]{} fringe visibility. This means that even if no which-path information is extracted, the visibility will be less than 1 if either the incoming state is asymmetric, or the slits are of unequal widths.
Distinguishability & duality relations
======================================
Coming back to the issue of getting unambiguous path information about the quanton, notice that (\[betagamma\]) implies two cases: (a) $|\beta|^2 = |\langle d_1|d_2\rangle|c_2/c_1 \ge |\langle d_1|d_2\rangle|^2$ and (b) $|\beta|^2 = |\langle d_1|d_2\rangle|^2 > |\langle d_1|d_2\rangle|c_2/c_1$. These should be discussed separately. We define the distinguishability of two paths, ${\mathcal D}_Q$, as the maximum probability with which the two paths can be [*unambiguously*]{} distinguished. To get distinguishability, we first use (\[d1d2\]) to rewrite (\[fstate\]) as $$\begin{aligned}
|\psi(t)\rangle &=& c_1|\psi_1(t)\rangle|d_1\rangle
+ c_2|\psi_2(t)\rangle|d_2\rangle \nonumber\\
&=& c_1\alpha|\psi_1(t)\rangle|q_1\rangle
+ c_2\gamma|\psi_2(t)\rangle|q_2\rangle \nonumber\\
&& + (c_1\beta|\psi_1(t)\rangle
+ c_2\delta|\psi_2(t)\rangle)|q_3\rangle
\label{fnstate}\end{aligned}$$ where $|\psi_1(t)\rangle, |\psi_2(t)\rangle$ represent the wave-packets appearing in (\[fstate\]). From (\[fnstate\]) one can see that the unambiguous path discrimination fails when one gets the state $|q_3\rangle$ while measuring the operator ${\boldsymbol}{A}$. The probability of failure is just $|\langle q_3|\psi(t)\rangle|^2$ which turns out to be $c_1^2|\beta|^2+c_2^2|\delta|^2$, using the orthogonality of $|\psi_1(t)\rangle, |\psi_2(t)\rangle$. Subtracting that from 1, gives the [*optimal*]{} probability of unambiguously distinguishing between the two paths. Thus we can write $${\mathcal D}_Q = 1 - |\langle q_3|\psi(t)\rangle|^2,
\label{success1}$$ which is our general expression for path distinguishability. The distinguishability may also be calculated from the successful discrimination as $${\mathcal D}_Q = |\langle q_1|\psi(t)\rangle|^2+|\langle q_2|\psi(t)\rangle|^2,
\label{success2}$$ which would just be $c_1^2\alpha^2+c_2^2\gamma^2$.
Case: $|\langle d_1|d_2\rangle|c_2/c_1 \ge |\langle d_1|d_2\rangle|^2$
----------------------------------------------------------------------
This is the case when the orthogonality of $|d_1\rangle, |d_2\rangle$ is on the stronger side, and the asymmetry is not extreme. In this case the values of $\alpha, \gamma$, for optimal success, are given by [@jaeger2] $$\begin{aligned}
\alpha &=& \sqrt{1-|\langle d_1|d_2\rangle|c_2/c_1} \nonumber\\
\gamma &=& \sqrt{1-|\langle d_1|d_2\rangle|c_1/c_2}\end{aligned}$$ Using (\[success2\]), the distinguishability has the form $${\mathcal D}_Q = 1 - 2c_1c_2|\langle d_1|d_2\rangle|.
\label{DQ1}$$ Using (\[DQ1\]) and (\[V\]) we arrive at the following relation $${\mathcal V} = \frac{2\sqrt{p_1p_2\xi}}{p_1+\xi p_2} \frac{(1-{\mathcal D}_Q)}
{2 c_1c_2}
\label{preduality1}$$ Using (\[c1c2\]), the above equation reduces to a very simple duality relation $${\mathcal D}_Q + {\mathcal V} = 1.
\label{duality1}$$ This duality relation generalizes Englert’s relation (\[englert\]) to the case of asymmetric incoming quanton state, and is an equality, not an inequality for any pure state. The relation (\[duality1\]) implies that if one is able to unambiguously distinguish between the two paths with a probability P [*by any method*]{}, that P cannot exceed ${\mathcal D}_Q$, and the fringe visibility cannot exceed $1-{\mathcal D}_Q$.
If the state of the incoming quanton happens to be symmetric, i.e., $p_1=p_2=1/2$, and the two slits are of same width, i.e., $\xi=1$, we can define a new distinguishability ${\mathcal D}$ as $${\mathcal D}^2 \equiv {\mathcal D}_Q(2-{\mathcal D}_Q)
= 1 - |\langle d_1|d_2\rangle|^2,$$ which is precisely Englert’s distinguishability [@englert]. Using (\[V\]) we can write $${\mathcal D}^2 + {\mathcal V}^2 = 1,$$ which is just the saturated form of Englert’s duality relation (\[englert\]). So for the symmetric case, (\[duality1\]) is essentially the same as (\[englert\]).
Case $|\langle d_1|d_2\rangle|^2 > |\langle d_1|d_2\rangle|c_2/c_1$
-------------------------------------------------------------------
This is the case when the orthogonality of $|d_1\rangle, |d_2\rangle$ is on the lower side, and the asymmetry is large. In this case the values of the constants are as follows [@jaeger2] $$\begin{aligned}
\alpha &=& \sqrt{1-|\langle d_1|d_2\rangle|^2},~~~~
\beta=|\langle d_1|d_2\rangle| \nonumber\\
\gamma &=& 0,~~~~ |\delta| = 1.\end{aligned}$$ The expression for distinguishability can be obtained by using (\[success2\]): $${\mathcal D}_Q = c_1^2(1-|\langle d_1|d_2\rangle|^2).
\label{DQ2}$$ Combining (\[DQ2\]) and (\[V\]), we can write $$\frac{{\mathcal D}_Q}{c_1^2}
+ {\mathcal V}^2 \frac{(p_1+\xi p_2)^2}{4p_1p_2\xi} = 1,$$ which can be rewriten as a new duality relation for this specific case: $$\frac{{\mathcal D}_Q}{\tfrac{1}{2}(1+{\mathcal P}_0)}
+ \frac{{\mathcal V}^2}{{\mathcal V}_0^2} = 1,
\label{duality2}$$ where ${\mathcal V}_0$ is the [*a priori*]{} fringe visibility, and ${\mathcal P}_0$ is the [*a priori*]{} path-predictability defined as ${\mathcal P}_0 = \frac{|c_1|^2-|c_2|^2}{|c_1|^2+|c_2|^2}$ [@greenberger]. As a consistency check, we consider the case where $|d_1\rangle,|d_2\rangle$ are identical, and hence ${\mathcal D}_Q$ given by (\[DQ2\]) is zero. Here the visibility is reduced to the [*a priori*]{} fringe visibility, as it should when there is no path-detection. Another special case is when $p_1=1$, in which case ${\mathcal V}$ becomes zero, and (\[duality2\]) gives ${\mathcal D}_Q=1$. Notice that varying the widths of the slits affects the [*a priori*]{} fringe visibility and the [*a priori*]{} path-predictability, but the equality (\[duality2\]) continues to hold.
One might wonder if it is possible to have a single duality relation for both the cases. To address this question, we denote the distinguishability in the first case, i.e. (\[DQ1\]), by $\mathcal{D}_{Q1}$ and that in the second case (\[DQ2\]), by $\mathcal{D}_{Q2}$. Then, in the region $|\langle d_1|d_2\rangle|^2 > |\langle d_1|d_2\rangle|c_2/c_1$, one can show that $$\mathcal{D}_{Q1} - \mathcal{D}_{Q2} = c_1^2(|\langle d_1|d_2\rangle|
- c_2/c_1)^2,$$ which means that $\mathcal{D}_{Q2} \le \mathcal{D}_{Q1}$. This implies that the following inequality holds in all regions $${\mathcal D}_Q + {\mathcal V} \le 1,$$ but it cannot be saturated in the region $|\langle d_1|d_2\rangle|^2 > |\langle d_1|d_2\rangle|c_2/c_1$.
So we see that one cannot have a tight single duality relation for all asymmetric two-slit experiments. Depending on the asymmetry and the orthogonality of the path detector states, the duality relation has two distinct forms, (\[duality1\]) and (\[duality2\]).
The general case (pure/mixed)
-----------------------------
Till now we have been looking at the case where quanton and the path detector are in a pure state. However, there are effects of decoherence due to which there can be some loss of coherence, and it may become necessary to treat the quanton and path detector combine as a mixed state. In such a situation, the state of the quanton and path detector will be represented by a mixed state density matrix. The treatment of path-distinguishability will remain unchanged. For example, the path distinguishability given by (\[success1\]) will now be represented as ${\mathcal D}_Q = 1-\text{Trace}[\rho(t)|q_3\rangle\langle q_3|]$, and that given by (\[success2\]) will be represented as ${\mathcal D}_Q = \text{Trace}[\rho(t)|q_1\rangle\langle q_1|]
+ \text{Trace}[\rho(t)|q_2\rangle\langle q_2|]$.
Fringe visibility is a measure of quantum coherence in the system, and any mixedness will degrade the interference. This statement can be put on a strong footing as follows. Recently a new measure of quantum coherence was introduced, which, in a normalized form, can be written as ${\mathcal C} = {1\over n-1}\sum_{i\neq j} |\rho_{ij}|$. In our context, $\rho_{ij}$ are the elements of the density matrix of the quanton, $i,j$ corresponding to the two paths, and $n$ is the dimensionaility of the Hilbert space (in our case $n=2$ corresponding to the two paths). Using the final state of the quanton plus path detector as $|\psi(t)\rangle = c_1|\psi_1(t)\rangle|d_1\rangle
+ c_2|\psi_2(t)\rangle|d_2\rangle$, we first trace over the path detector states to get a reduced density matrix. The coherence ${\mathcal C}$ can then be evaluated, and turns out to be $${\mathcal C} = 2c_1c_2 |\langle d_1|d_2\rangle|.
\label{C}$$ We see that for the two-slit interference, coherence is the same as visibility. It has been shown that any incoherent operation on the system will lead to a decrease in coherence ${\mathcal C}$ [@coherence]. In our case it means, any mixedness introduced in the system will lead to a decrease in the visibility ${\mathcal V}$.
Consequently, the visibility will now be less than the maximum allowed by the amount of path information that has been acquired by the path detector. For the case $|\langle d_1|d_2\rangle| \le c_2/c_1$, it means ${\mathcal V} \le 1 - {\mathcal D}_Q $. Thus the duality relation becomes the inequality $${\mathcal D}_Q + {\mathcal V} \le 1.
\label{gduality1}$$ Similarly, for the case $|\langle d_1|d_2\rangle| > c_2/c_1$ too, the duality relation becomes $$\frac{{\mathcal D}_Q}{\tfrac{1}{2}(1+{\mathcal P}_0)}
+ \frac{{\mathcal V}^2}{{\mathcal V}_0^2} \le 1.
\label{gduality2}$$ The inequalities (\[gduality1\]) and (\[gduality2\]) quantify wave-particle duality for an asymmetric two slit interference. They are saturated for any pure state.
Particle or wave?
-----------------
The thought experiment in the preceding discussion, with an expanded Hilbert space, was introduced to get an upper bound on the probability with which the two paths can be unambiguously distinguished. However, if one were to actually carry out this experiment with an observable ${\boldsymbol}{A}$ of the path detector giving three measured values, an interesting possibility emerges. Suppose each quanton is detected on the screen in coincidence with measurement of the observable ${\boldsymbol}{A}$. Once the path detector is in place, the interference does not depend on what observable of the path detector we choose to measure. Everytime we get the measured value 1, we know the quanton went through slit 1, and everytime we get the value 2, we know that the particular quanton went through slit 2. In these two situations, the quanton behaves like a particle, choosing one of the two available paths. However, when the measurment of ${\boldsymbol}{A}$ yields the value 3, we conclude that the quanton went through both the slits at the same time, behaving like a spreadout wave. In fact, this can be experimentally verified by separating the detected quantons into two groups, one where ${\boldsymbol}{A}$ gave value 1 or 2, and two where ${\boldsymbol}{A}$ gave value 3. The first group of quantons will show no interference, since path information for each of them is stored in the path detector. The second group of quantons will show full interference.
The state of the quantons, for which measurement of ${\boldsymbol}{A}$ gives value 3, can be written using (\[fnstate\]) as $$\begin{aligned}
\langle q_3|\psi(t)\rangle &=&
c_1\alpha|\psi_1(t)\rangle \langle q_3|q_1\rangle
+ c_2\gamma|\psi_2(t)\rangle \langle q_3|q_2\rangle \nonumber\\
&& + (c_1\beta|\psi_1(t)\rangle
+ c_2\delta|\psi_2(t)\rangle) \langle q_3|q_3\rangle \nonumber\\
&=& c_1\beta|\psi_1(t)\rangle + c_2\delta|\psi_2(t)\rangle
\label{wstate}\end{aligned}$$ It is obvious that the above state will produce interference. This leads us to conclude that in a two-slit experiment with an [*imperfect*]{} path detector in place, each quanton can be thought of as randomly choosing to behave like a particle or a wave. This behaviour, obviously, is forced by the prescence of the path detector, in agreement with the philosophy behind Bohr’s principle of complementarity [@bohr].
Conclusion
==========
In conclusion, we have analyzed the issue of wave-particle duality in a two-slit experiment. For symmetric beams and equal slit widths, wave-particle duality can be captured by the well-known inequality (\[englert\]), which was derived using the ideas of minimum error discrimination of states [@englert]. The same relation can be derived by defining the distinguishability using the ideas of UQSD [@3slit]. This latter method has proved to be very useful in describing wave-particle duality in multi-slit interference [@3slit; @cd; @nslit]. For two-slit experiments where the two beams are asymmetric, and the slits may be of unequal widths, a result as strong as (\[englert\]) was lacking. We have used this new approach to study wave-particle duality in this asymmetric case.
We argue that in a two-slit experiment, getting path information should mean, being able to tell unambiguously for each quanton, which of the two slits it went through. Using this premise, we use a thought experiment to get which path information about the quantons using UQSD. We define the path distinguishability as the maximum probability with which one can unambguously tell which slit the quanton went through, [*in principle*]{}. Using it we derive two duality relations for intrference where the two paths may not be equally probable or the two slits may not be of equal widths. The two duality relations correspond to two difference ranges of asymmetry. Unlike the well studied symmetric case, a single tight duality relation is not possible for the asymmetric case. Additionally, if the thought experiment is actually performed, one can tell for each quanton if it went through slit 1 or slit 2 like a particle or through both the slits like a wave.
Acknowledgement {#acknowledgement .unnumbered}
===============
Keerthy Menon is thankful to the Centre for Theoretical Physics, Jamia Millia Islamia for providing the facilities of the Centre during the course of this work.
[0]{}
N. Bohr, “The quantum postulate and the recent development of atomic theory," Nature (London) 121, 580-591 (1928).
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[^1]: Published: **Phys. Rev. A 98, 022130 (2018)**
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this investigation the boundary value problem of light propagation in the gravitational field of one arbitrarily moving body with monopole structure is considered in the second post-Newtonian approximation. The solution of the boundary value problem comprises a set of altogether three transformations: ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$\sigma$}}$, ${\mbox{\boldmath$\sigma$}} \rightarrow {\mbox{\boldmath$n$}}$, and ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$n$}}$. Analytical solutions of these transformations are given and the upper limit of each individual term is determined. Based on these results, simplified transformations are obtained by keeping only those terms relevant for the given goal accuracy of $1$nano-arcsecond in light deflection. Like in case of light propagation in the gravitational field of one body at rest, there are so-called enhanced terms which are of second post-Newtonian order but contain one and the same typical large numerical factor. Finally, the impact of enhanced terms beyond 2PN approximation is considered. It is found that enhanced 3PN terms are relevant for astrometry on the level of $1$nano-arcsecond in light deflection, while enhanced 4PN terms are negligible, except for grazing rays at the Sun.'
address: 'Institute of Planetary Geodesy - Lohrmann Observatory, Dresden Technical University, Helmholtzstrasse 10, D-01069 Dresden, Germany'
author:
- Sven Zschocke
title: 'Light propagation in 2PN approximation in the field of one moving monopole II. Boundary value problem'
---
Introduction {#Section0}
============
The new era of space-based astrometry
-------------------------------------
While advancement in astrometry has always been benefited from ground-based telescope improvements, the new era of space-based astrometry missions has initiated unprecedented accuracies in positional measurements of celestial objects, like Solar System objects, stars, galaxies, and quasars [@History_Astrometry1; @History_Astrometry2; @Kovalevsky]. Most notably, the astrometry missions Hipparcos and Gaia of the European Space Agency (ESA) have opened this new age in astronomy. These missions have (i) adapted from wide-field astrometry realized by optical instruments which are designed to measure large angles on the sky simultaneously, (ii) utilized the most modern technologies in the optical design of scanning satellite, (iii) taken advantage of appreciable developments in theoretical astrometry and applied gravitational physics.
Approved by ESA in 1980 and launched on 8 August 1989, Hipparcos was the first ever astrometric satellite to precisely measure the positions and proper motions of stars in the vicinity of the Sun. The completion of the Hipparcos mission has led to the creation of three highly accurate catalogues of stellar positions, namely the star catalogues Hipparcos and Tycho in 1997 [@Hipparcos; @Hipparcos1] and Tycho-2 in 2000 [@Hipparcos2]. In particular, the Hipparcos final catalogue [@Hipparcos] provides astrometric positions and stellar motions up to $1$ milli-arcsecond (${\rm mas}$) in angular accuracies for about $120$ thousand stars. The catalogues Tycho [@Hipparcos1] and Tycho-2 [@Hipparcos2] contain positions of about $1\,{\rm million}$ and $2.5\,{\rm million}$ stars, respectively, with an accuracy of up to $20\,{\rm mas}$ in angular resolution which still represents an unprecedented accuracy at that time, also in view of such huge number of individual stars. These catalogues set the precedent on stellar positions and are continuously used in space science research and for spacecraft navigation.
Gaia is the second space-based mission ever and will provide fundamental data for many fields of astronomy. The Gaia mission was approved in 2000 by ESA as cornerstone mission and is aiming at precisions up to a few micro-arcseconds ($\mu{\rm as}$) in determining positions and proper motions of stellar objects [@GAIA], which is about $200$ times more accurately than the predecessor Hipparcos. Launched on 19 December 2013, the Gaia’s main goal is to create an extraordinarily precise three-dimensional map of more than $1300$ million stars of our galaxy, in order to determine the structure and dynamics of the Milky way. The observational data of Gaia comprise not only astrometry but also spectro-photometry. For the brightest subset of targets, spectra will be acquired to obtain radial velocities of stellar objects by means of the Doppler effect which is essential for the understanding of the kinematics of our Galaxy [@GAIA1].
The highly precise measurements of the astrometry mission Gaia are of fundamental importance to all the other fields of astronomy, specifically they will have a tremendous impact on stellar astrophysics and galaxy evolution, solar-system and extra-solar planet science, extra-galactic astrophysics, and fundamental physics like dark matter and dark energy physics, highly-precise determination of natural constants, testing equivalence principle, determination of Nordtvedt parameter, possible temporal variation of the gravitational constant, and last but not least testing alternative theories of gravity. Another aspect of highly-precise astrometric data concerns the essential fact that not only more accurate but also qualitatively new tests of general relativity become possible [@Theory_Experiment; @Kopeikin_Efroimsky_Kaplan; @Klioner2003b; @Book_Clifford_Will].
Preliminary results of the Gaia mission have been published in September 2016 by Gaia Data Release 1 (Gaia DR1), providing astrometric data which are more precise than those in any of the former star catalogues [@GAIA1; @GAIA_DR1_1; @GAIA_DR1_2]. The five-parameter astrometric solution (positions, proper motions, parallaxes) for about $2$ million stars in common between the Tycho-2 Catalogue and Gaia is contained in Gaia DR1.
The results of Gaia Data Release 2 (Gaia DR2) were published very recently in April 2018 by a series of articles. There are specific articles and processing papers which concern special scientific issues and which give technical details on the processing and calibration of the raw data. A comprehensive overview of Gaia DR2 is expounded in [@GAIA_DR2_1], while the full content of Gaia DR2 is available through the Gaia archive [@Gaia_Archive]. In particular, Gaia DR2 provides precise positions, proper motions, and parallaxes for more than $1300$ million stars. Furthermore, the Gaia DR2 contains positions for more than $550$ thousand quasars which allow for the definition of a new celestial reference frame fully based on optical observations of extra-galactic sources (Gaia-CRF2) [@Gaia_CRF2]. Based on these results the third realization of the International Celestial Reference Frame (ICRF-3) has recently been adopted by the XXXth General Assembly of the International Astronomical Union (IAU) in 2018 [@ICRF3_A], which is based on the accurate measurement of over $4000$ extragalactic radio sources. The ICRF-3 replaces ICRF-2 which was adopted at the XXVIIth General Assembly of IAU in 2009. These reference frames are of utmost importance for many branches is astronomy, like stellar catalogues, space navigations, or determination of the rotational motion of the Earth.
Of specific importance for our investigations here is the impressive advancement in astrometric accuracy of positional measurements arrived within the Gaia DR2. For parallaxes, uncertainties are typically around $30$ $\;$ for sources brighter than $V \!\! = \!\! 15\;{\rm mag}$, around $100$ $\;$ for sources with a magnitude about $V \!\! = \!\! 17\;{\rm mag}$, and around $700$ $\;$ for sources with about $V \!\! = \!\! 20\;{\rm mag}$ [@GAIA_DR2_1; @GAIA_DR2_2]. These results represent a giant advancement in astrometric science and comprise the fact that todays astrometry has reached the micro-arcsecond level of accuracy in astrometric measurements.
Another astrometric space mission aiming at the micro-arcsecond level of accuracy is JASMINE, an approved long-term project developed by the National Astronomical Observatory of Japan, and which consists of altogether three astrometry satellites, called Nano-JASMINE, Small-JASMINE, and (Medium-Sized) JASMINE [@Jasmine1], where the two last satellites shall observe in the infrared. The Nano-JASMINE (nominal mission: $2$ years) is a mission in the optical based on CCD (charge-coupled device) and the technical demonstrator of the entire JASMINE project, which represents the first space astrometry satellite mission in Japan and the third space-based astrometry mission ever following the ESA missions Hipparcos and Gaia. Meanwhile, the technical equipment of the satellite has fully been completed and the launch of Nano-JASMINE is expected within the very few next months. The launch of Small-JASMINE is expected around 2024, while there is no concrete plan for the launch of (Medium-Sized) JASMINE. Within the series of altogether three JASMINE missions, the target accuracy in the positional measurements of stellar objects will be increasing step-by-step, ranging from $3\,{\rm mas}$ by the Nano-JASMINE mission up to $10\,{\hbox{\rm $\mu$as}}$ within the (Medium-Sized) JASMINE mission.
Future astrometry on the sub-micro-arcsecond level
--------------------------------------------------
It is quite obvious that a long term goal of astrometric science is to arrive at the sub-micro-arcsecond (sub-$\mu{\rm as}$) or even the nano-arcsecond (nas) level of accuracy. The scientific objectives for such ultra-highly precise astrometry are overwhelming and it is almost impossible to enumerate all advances in science which astrometry on such scales would initiate. For instance, astrometry on sub-$\mu{\rm as}$ scale would make it possible to survey hundreds of thousands of stars up to a distance of about $100 \,{\rm pc}$ for detecting earth-like planets, would allow for much more stringent tests of General Relativity through light bending, would enable the measurement of the energy density of stochastic gravitational wave background, allows for precise mapping of dark matter from the areas beyond the Milky Way, enables direct distance measurement of various stellar standard candles up to the closest galaxy clusters, would allow for further tests of alternative theories of gravity with much better precision than in the weak-gravitational-field regime [@article_sub_micro_1; @article_sub_micro_2; @article_sub_micro_3; @article_sub_micro_4; @Conference_Cambridge]. Especially, the proposed mission Theia [@Theia] is primarily designed to study the local dark matter properties, the detection of Earth-like exoplanets in our nearest star systems and the physics of highly compact objects like white dwarfs, neutron stars, black holes. For a more comprehensive list of astronomical and astrophysical problems which can be solved by sub-$\mu{\rm as}$ astrometry we refer to the article [@Kopeikin_Gwinn].
Furthermore, as soon as the third (Gaia DR3) and final Gaia Data Release (Gaia Final DR), expected in the fall of 2020 and around the end of 2022, respectively, are achieved and analyzed, new questions will emerge, which will require new space-based astrometry missions, either in the form of a Gaia-like observer or in the form of satellites aiming at the sub-$\mu{\rm as}$ or even the nas level of accuracy. In fact, the impressive progress, made during the realization of the both ESA astrometry missions Hipparcos and Gaia, has already encouraged the astrometric science community to further proceed in such directions in foreseeable future. Among several astrometry missions suggested to ESA we mention the recent medium-sized (M-5) mission proposals Gaia-NIR [@Gaia_NIR], Theia [@Theia], and NEAT [@NEAT1; @NEAT2; @NEAT3], which in this order are aiming at the $\mu{\rm as}$, sub-$\mu{\rm as}$, and even the ${\rm nas}$ level of precision.
The envisaged advancement from -astrometry to sub--astrometry implies many subtle effects and new kind of challenges in technology and science such as: (a) determination of Solar System ephemerides precisely enough for sub--astrometry, (b) modeling the influence of interstellar medium on light propagation, (c) synchronization of atomic clocks between observer and ground stations on the sub-nano-second scale, (d) tracking the spacecraft’s worldline and velocity with sufficient accuracy for being able to account for aberrational effects, (e) development of new CCD-based technologies in the optical or infrared to achieve astrometric data on the sub--level, etc. Each of these and many other challenges have to be clarified before sub--astrometry becomes feasible. But it is clear that astrometric information is mainly carried by light signals of the celestial light sources, hence astrometric measurements are intrinsically related to the problem about how to trace a light ray detected by the observer back to the celestial light source. Therefore, the fundamental assignment in astrometry remains the precise description of the trajectory of the light signal as function of coordinate time. The foreseen progress in the accuracy of observations and new observational techniques necessitates to account for several relativistic effects in the theory of light propagation. A detailed review about the recent progress in the theory of light propagation has been given in text books [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991] as well as in several articles [@Klioner2003b; @Zschocke1; @Zschocke2; @KK1992; @Kopeikin1997; @KS1999; @KSGE; @Klioner2003a; @KopeikinMashhoon2002]. So in what follows an introduction of the theory of light propagation is just given to the extent that it proves necessary for our investigations.
The exact field equations of gravity
------------------------------------
According to the theory of general relativity [@Einstein1; @Einstein2] the space-time is not considered as rigidly given once and for all, but a differentiable manifold and subject to dynamical laws. Therefore, the determination of the (inner) geometry of space-time is the foundation for any measurement in relativistic astrometry. The (inner) geometry of the four-dimensional manifold is fully determined by the metric tensor $g_{\alpha\beta}$ whose components are identified with tensorial gravitational potentials generalizing the scalar gravitational potential of Newtonian theory of gravity. In compliance with Einstein’s field equations [@Einstein1; @Einstein2], the metric tensor $g_{\alpha\beta}$ is related to the stress-energy tensor $T_{\alpha\beta}$ of matter via a set of $10$ coupled non-linear partial differential equations given by [@Kopeikin_Efroimsky_Kaplan; @Einstein1; @Einstein2; @MTW; @Landau_Lifschitz; @Fock] (e.g. Sec. 17.1 in [@MTW]) $$\begin{aligned}
R_{\alpha\beta} - \frac{1}{2}\,g_{\alpha\beta}\,R &=& \frac{8\,\pi\,G}{c^4}\,T_{\alpha\beta}
\label{Field_Equations_5}\end{aligned}$$
where $R_{\alpha \beta} = \Gamma^{\rho}_{\alpha\beta,\rho} - \Gamma^{\rho}_{\alpha\rho,\beta}
+ \Gamma^{\rho}_{\sigma\rho}\,\Gamma^{\sigma}_{\alpha\beta}
- \Gamma^{\rho}_{\sigma\beta}\,\Gamma^{\sigma}_{\alpha\rho}\,$ is the Ricci tensor (cf. Eq. (8.47) in [@MTW]), $R = g^{\alpha\beta} R_{\alpha\beta}$ is the Ricci scalar, and $$\begin{aligned}
\Gamma^{\rho}_{\alpha\beta} &=& \frac{1}{2}\,g^{\rho\sigma}
\left(g_{\sigma\alpha , \beta} + g_{\sigma\beta , \alpha} - g_{\alpha\beta , \sigma} \right)
\label{Christoffel_Symbols}\end{aligned}$$
are the Christoffel symbols which are functions of the metric tensor.
The field equations of gravity (\[Field\_Equations\_5\]) are valid in any coordinate system. The final ambition in theoretical astrometry remains of course the determination of observables (scalars), which are, by definition, gauge-independent (coordinate-independent) quantities [@Observables]. There are three possibilities to get such observables [@Brumberg2010]:
1. performing the calculations solely in terms of coordinate-independent quantities.
2. using any coordinate system in the calculations.
3. adopting one coordinate system and determine observables in the final step.
The IAU has adopted the third way by recommending the use of harmonic coordinates in celestial mechanics and in the astrometric science [@IAU_Resolution1]. These harmonic coordinates considerably simplify the calculations in celestial mechanics and in the theory of light propagation. They are denoted by $x^{\mu} = \left(c t, {\mbox{\boldmath$x$}}\right)$, where $t$ is the coordinate time and ${\mbox{\boldmath$x$}} =\left(x^1,x^2,x^3\right)$ is a triplet of spatial coordinates. The harmonic coordinates are curvilinear and they are defined by the harmonic gauge condition [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991; @MTW], $$\begin{aligned}
\frac{\partial \left(\sqrt{- g}\,g^{\alpha \beta}\right)}{\partial x^{\beta}} = 0\,,
\label{harmonic_gauge_condition_1}\end{aligned}$$
where $g = {\rm det}\left(g^{\alpha\beta}\right)$ is the determinant of metric tensor. The condition (\[harmonic\_gauge\_condition\_1\]) is called de Donder gauge in honor of its inventor [@Donder], which was also found independently by Lanczos [@Lanczos]; we note that (\[harmonic\_gauge\_condition\_1\]) determines (a class of) concrete reference systems, hence it is not surprising that condition (\[harmonic\_gauge\_condition\_1\]) is not covariant. The harmonic coordinates can be treated like Cartesian coordinates besides that they are curvilinear [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991; @Thorne; @Poisson_Lecture_Notes; @Poisson_Will]; cf. text below Eq. (3.1.45) in [@Brumberg1991] or the statement above Eq. (1.1) in [@Thorne], while more detailed explanations for this fact are provided in Sections 1.5. and 1.6 in [@Poisson_Will].
In line with these statements, in practical calculations in celestial mechanics and astrometry it is very useful to express the exact field equations of gravity (\[Field\_Equations\_5\]) in terms of harmonic coordinates. In this so-called Landau-Lifschitz formulation of the field equations [@Landau_Lifschitz], the contravariant components of the gothic metric density are decomposed as follows $$\begin{aligned}
\sqrt{-g}\,g^{\alpha\beta} &=& \eta^{\alpha \beta} - \overline{h}^{\alpha \beta}\,,
\label{metric_20}\end{aligned}$$
which is especially useful in case of an asymptotically flat space-time. Here, $\overline{h}^{\alpha \beta}$ is the trace-reversed metric perturbation which describes the deviation of the gothic metric tensor density of curved space-time from the metric tensor of Minkowskian space-time.
The exact field equations (\[Field\_Equations\_5\]) in terms of harmonic coordinates can be written as follows (cf. Eq. (36.37) in [@MTW] or Eq. (5.2b) in [@Thorne]): $$\begin{aligned}
\opensquare\;
\overline{h}^{\alpha \beta} &=& - \frac{16\,\pi\,G}{c^4}\,\left(\tau^{\alpha \beta} + t^{\alpha \beta}\right),
\label{Field_Equations_10}\end{aligned}$$
where $\opensquare = \eta^{\mu\nu}\,\partial_{\mu}\,\partial_{\nu}$ is the (flat) d’Alembert operator and $$\begin{aligned}
\tau^{\alpha \beta} &=& \left( - g\right)\,T^{\alpha \beta}\,,
\label{metric_35}
\\
\nonumber\\
t^{\alpha \beta} &=& \left( - g\right)\,t_{\rm LL}^{\alpha \beta} + \frac{c^4}{16\,\pi\,G}\;
\left(\overline{h}^{\alpha\mu}_{\;\;\;,\;\nu}\;\overline{h}^{\beta \nu}_{\;\;\;,\;\mu}
- \overline{h}^{\alpha\beta}_{\;\;\;,\;\mu\nu}\;\overline{h}^{\mu \nu}\right),
\label{metric_40}\end{aligned}$$
where $t_{\rm LL}^{\alpha \beta}$ is the Landau-Lifschitz pseudotensor of gravitational field [@Landau_Lifschitz], which is symmetric in the indices and in explicit form given by Eq. (20.22) in [@MTW] or by Eqs. (3.503) - (3.505) in [@Kopeikin_Efroimsky_Kaplan]. We shall assume that the gravitational system is isolated, that means flatness of the metric at spatial infinity and the constraint of no-incoming gravitational radiation is imposed at past null infinity ${\cal J}^{-}$ (cf. notation in Section 34 in [@MTW] and Figure 34.2. in [@MTW]). These so-called Fock-Sommerfeld boundary conditions, for instance given by Eqs. (4.64) and (4.65) in [@Kopeikin_Efroimsky_Kaplan], have been adopted from classical electrodynamics [@Sommerfeld1; @Sommerfeld2] and later formulated for the general theory of gravity [@Fock]. By imposing the Fock-Sommerfeld boundary conditions, a formal solution of (\[Field\_Equations\_10\]) is then provided by the implicit integro-differential equation, $$\begin{aligned}
\overline{h}^{\alpha \beta} \left(t,{\mbox{\boldmath$x$}}\right) &=& \frac{4\,G}{c^4}\,
\int d^3 x^{\prime}\,
\frac{\tau^{\alpha\beta}\left(u, {\mbox{\boldmath$x$}}^{\prime}\right) + t^{\alpha\beta}\left(u, {\mbox{\boldmath$x$}}^{\prime}\right)}{\left| {\mbox{\boldmath$x$}} - {\mbox{\boldmath$x$}}^{\prime} \right|}\,,
\label{Introduction_2}\end{aligned}$$
where $$\begin{aligned}
u = t - \frac{\displaystyle \left| {\mbox{\boldmath$x$}} - {\mbox{\boldmath$x$}}^{\prime} \right|}{\displaystyle c}
\label{Introduction_2a}\end{aligned}$$
is the retarded time, which is associated with the finite speed of gravitational action and not with the finite speed of light, as one may recognize from the fact that electromagnetic fields are not necessarily involved in the stress-energy tensor on the r.h.s. of (\[Field\_Equations\_10\]) or (\[Introduction\_2\]). In order to deduce the formal solution (\[Introduction\_2\]) from the differential equation (\[Field\_Equations\_10\]) the Cartesian-like harmonic coordinates $\left(ct,{\mbox{\boldmath$x$}}\right)$ have been treated like Cartesian coordinates besides that they are curvilinear; cf. text below Eq. (36.38) in [@MTW]. The approach about how to solve (\[Introduction\_2\]) iteratively is described in some detail in [@Kopeikin_Efroimsky_Kaplan]; cf. Eqs. (3.530a) - (3.530d) in [@Kopeikin_Efroimsky_Kaplan]. In the first iteration (first post-Minkowskian approximation) the integral runs only over the three-dimensional volume of the matter source, while from the second iteration on (second post-Minkowskian approximation and higher) the integral (\[Introduction\_2\]) gets also support from the metric perturbation, hence runs over the entire three-dimensional space.
Four comments are in order about the exact field equations of gravity.
$\bullet$ First, the retarded time $u$, which is hidden in the exact field equations of gravity (\[Field\_Equations\_5\]), appears explicitly in the formal solution of the exact field equations (\[Introduction\_2\]), which states that a space-time point $\left(u,{\mbox{\boldmath$x$}}^{\prime}\right)$ (e.g. located inside the matter distribution) is in causal contact with a space-time point $\left(t,{\mbox{\boldmath$x$}}\right)$ (e.g. located outside the matter source).
$\bullet$ Second, one may consider the propagation of electromagnetic action in a curved space-time with background metric $g_{\alpha\beta}$. That means, the metric of the curved space-time is determined by some matter distribution $T_{\alpha\beta}$, while the impact of the electromagnetic field on the metric of space-time is neglected. The electromagnetic fields are generated by some electromagnetic four-current $j^{\mu} = \left(c \rho, {\mbox{\boldmath$j$}}\right)$ with $\rho$ and ${\mbox{\boldmath$j$}}$ being charge-density and current-density, respectively. The covariant field equations of Maxwell’s electrodynamics in curved space-time read $F^{\mu\nu}_{\quad\, ;\,\nu} = 4\,\pi\,j^{\mu}$ and $F_{\mu\nu\,;\,\rho} + F_{\nu\rho\,;\,\mu} + F_{\rho\mu\,;\,\nu} = 0$ (cf. Eqs. (22.17a) and (22.17b) in [@MTW]), where $F_{\mu\nu} = A_{\nu\, ; \,\mu} - A_{\mu \,;\, \nu}$ is the field-tensor of electromagnetic field (cf. Eq. (22.19a) in [@MTW]), the semicolon denotes covariant derivative, and $A^{\mu} =\left(\varphi/c, {\mbox{\boldmath$A$}}\right)$ is the four-potential, where $\varphi$ is the scalar potential and ${\mbox{\boldmath$A$}}$ is the vector potential. The Characteristics (also called characteristical surface) of the covariant Maxwell equations are governed by the following non-linear partial differential equation (non-linear PDE) of first order [@Fock; @Whittaker; @Article_Charaketristiken; @Sexl_Urbantke; @Iverno], $$\begin{aligned}
g^{\alpha \beta}\; \frac{\partial \phi}{\partial x^{\alpha}}\; \frac{\partial \phi}{\partial x^{\beta}} = 0 \,,
\label{Characteristics_ED}\end{aligned}$$
which is valid in the near-zone as well as in the far-zone of the four-current $j^{\alpha}\left(t,{\mbox{\boldmath$x$}}\right)$ and is valid in any reference system. The Characteristics are three-dimensional curved sub-manifolds, $\phi\left(x^0, x^1, x^2, x^3\right) = {\rm const}$, of the Riemannian space-time. In case of flat space-time, i.e. $g^{\alpha \beta} = \eta^{\alpha \beta}$, the characteristical surface at the event $\left(x^0_0,x^1_0,x^2_0,x^3_0\right)$ is given by the Minkowskian light-cone, $$\begin{aligned}
\fl \hspace{1.0cm}
\phi\left(x^0,x^1,x^2,x^3\right) = \left(x^0_0-x^0\right)^2 - \left(x^1_0-x^1\right)^2 - \left(x^2_0-x^2\right)^2 - \left(x^3_0-x^3\right)^2\,.
\label{Light_Cone_ED} \end{aligned}$$
That means, an electromagnetic discontinuity (abrupt electromagnetic signal) generated at $\left(x^0_0,x^1_0,x^2_0,x^3_0\right)$ propagates in the flat space-time along the light-cone (\[Light\_Cone\_ED\]). The generalization of the light-cone (\[Light\_Cone\_ED\]) in flat space-time is the light-conoid in curved space-time as governed by Eq. (\[Characteristics\_ED\]), which for the curved space-time of the Solar system can only be solved approximately, for instance by iteration. The PDE of the Characteristics (\[Characteristics\_ED\]) can be derived by means of the following consideration. Let $a^{\mu}$ be a continuous (smoothly changing) electromagnetic four-potential generated by some current $j^{\mu}$ somewhere located in the Riemannian space-time with metric $g^{\mu\nu}$. Now suppose that the four-current $j^{\mu}$ changes rapidly and generates an abrupt Theta-like discontinuity (perturbation) in the electromagnetic field with amplitude $u^{\mu}$, which propagates along some hypersurface $\phi$. Then, the entire electromagnetic four-potential $A^{\mu}$ is given by the following expression: $A^{\mu}\left(x^0,{\mbox{\boldmath$x$}}\right) = a^{\mu}\left(x^0,{\mbox{\boldmath$x$}}\right) + u^{\mu}\left(x^0,{\mbox{\boldmath$x$}}\right)\,\Theta\left(\phi\left(x^0,{\mbox{\boldmath$x$}}\right)\right)$ [@Sexl_Urbantke]. By inserting this ansatz into the covariant Maxwell equations one just obtains the equation (\[Characteristics\_ED\]) which governs the evolution of the hypersurface $\phi$ in the curved space-time on which any discontinuity of the electromagnetic field is located. Thus, the three-dimensional sub-manifolds $\phi\left(x^0, x^1, x^2, x^3\right)$ of the Riemannian space-time can be identified with the front of electromagnetic action (e.g. abrupt discontinuity in the near-zone of the four-current or wave-front of an electromagnetic wave in the far-zone of the four-current) caused by some rapid change in the electromagnetic four-current.
Furthermore, one may introduce a trajectory, $x^{\alpha}\left(\lambda\right)$ where $\lambda$ is an affine curve-parameter, which is orthogonal on the surface $\phi$ [@Fock; @Article_Charaketristiken; @Sexl_Urbantke; @Iverno], $$\begin{aligned}
\frac{d x^{\alpha}\left(\lambda\right)}{d \lambda} = g^{\alpha \beta} \frac{\partial \phi}{\partial x^{\beta}}\,,
\label{Biharacteristics_ED}\end{aligned}$$
that means is normal to the front of electromagnetic action; we will come back to that issue later, cf. text below Eqs. (\[Four\_Vector\_sigma\]) - (\[Four\_Vector\_r\_A\]). Such trajectories are called Bicharacteristics. The Bicharacteristics can be identified with the light rays, which propagate with the finite speed of light. Therefore, also the Characteristics, that is the surface of electromagnetic action, propagates with the finite speed of light. The light-conoid in curved space-time is built by all Bicharacteristics emanating from some (arbitrary) event. As mentioned above, besides that $c$ is defined as the fundamental speed of light in vacuum in the flat Minkowski space, it is clear that the retardation, that means the natural constant $c$ in the denominator on the r.h.s. in Eq. (\[Introduction\_2a\]), is caused by the finite speed of gravitational action and not due to the finite speed of light. Even in case the stress-energy tensor of matter would only consist of electromagnetic fields, $4\,\pi\,T^{\alpha\beta} = F^{\alpha \mu}\,F^{\beta}_{\;\;\;\mu} - \frac{1}{4}\,g^{\alpha\beta}\,F_{\mu\nu}\,F^{\mu\nu}$ [@MTW], then, nevertheless, the retardation would also originate from the finite speed of gravitational fields (in this case with the well-known property that the Ricci scalar vanishes but of course not the Ricci tensor) which, in this specific case, would entirely be generated by these electrodynamical fields.
$\bullet$ Third, let us now consider the non-linear PDE for the Characteristics of the exact field equations of gravity (\[Field\_Equations\_5\]), which is given by [@Fock; @Article_Charaketristiken; @Sexl_Urbantke; @Iverno], $$\begin{aligned}
g^{\alpha \beta}\; \frac{\partial \omega}{\partial x^{\alpha}}\; \frac{\partial \omega}{\partial x^{\beta}} = 0 \,,
\label{Characteristics_GR}\end{aligned}$$
which is valid in the near-zone as well as in the far-zone of the matter source $T_{\alpha\beta}\left(t,{\mbox{\boldmath$x$}}\right)$ and is valid in any reference system. The derivation of the PDE (\[Characteristics\_GR\]) for the Characteristics is similar to the above considerations in case of the covariant Maxwell equations. Consider a continuous (smoothly changing) background metric $g^{\mu\nu}_0$ which is generated by some matter $T^{\alpha\beta}$. Then assume a rapid acceleration of the matter which results in an abrupt Theta-like discontinuity (perturbation) with metric $h^{\alpha\beta}$. Hence, the entire metric is given by: $g^{\alpha\beta}\left(x^0,{\mbox{\boldmath$x$}}\right) = g^{\alpha\beta}_0\left(x^0,{\mbox{\boldmath$x$}}\right)
+ h^{\alpha\beta}\left(x^0,{\mbox{\boldmath$x$}}\right)\,\Theta\left(\omega\left(x^0,{\mbox{\boldmath$x$}}\right)\right)$ [@Sexl_Urbantke]. Now, if one wants to investigate how the gravitational discontinuity (non-analytic gravitational signal) propagates in space and time, one has to insert this ansatz into the exact Einstein equations, which yields the PDE (\[Characteristics\_GR\]). Thus, the Characteristics $\omega$ can be identified with the front of gravitational action (e.g. abrupt discontinuity in the near-zone of matter source or wave-front of a gravitational wave in the far-zone of matter source) caused by the matter source. The front of gravitational action $\omega$ is a curved three-dimensional sub-manifold, $\omega\left(x^0,x^1,x^2,x^3\right) = {\rm const}$, of the Riemannian space-time, that means a three-dimensional surface on which any discontinuities of the gravitational field must lie [@Fock; @Article_Charaketristiken; @Sexl_Urbantke; @Iverno]. In case of flat background metric, i.e. $g^{\alpha\beta}_0 = \eta^{\alpha\beta}$, the solution of the PDE (\[Characteristics\_GR\]) at the event $\left(x^0_0,x^1_0,x^2_0,x^3_0\right)$ is given by the null-cone, $$\begin{aligned}
\fl \hspace{1.0cm}
\omega\left(x^0,x^1,x^2,x^3\right) = \left(x^0_0-x^0\right)^2 - \left(x^1_0-x^1\right)^2 - \left(x^2_0-x^2\right)^2 - \left(x^3_0-x^3\right)^2\,.
\label{Null_Cone_GR} \end{aligned}$$
That means, a gravitational discontinuity (abrupt gravitational signal) generated at $\left(x^0_0,x^1_0,x^2_0,x^3_0\right)$ propagates in the flat space-time along the null-cone (\[Null\_Cone\_GR\]). The generalization of the null-cone (\[Null\_Cone\_GR\]) of gravitational action in flat background metric is the null-conoid in curved space-time as governed by (\[Characteristics\_GR\]), which for the curved space-time of the Solar system can only be solved approximately, for instance by iteration.
One may also introduce Bicharacteristics for the field equations of gravity, $z^{\alpha}\!\left(\rho\right)$ where $\rho$ is an affine curve-parameter, which are trajectories orthonormal on the hypersurface $\omega$ [@Fock; @Article_Charaketristiken; @Sexl_Urbantke; @Iverno], $$\begin{aligned}
\frac{d z^{\alpha}\!\left(\rho\right)}{d \rho} = g^{\alpha \beta} \frac{\partial \omega}{\partial x^{\beta}}\,,
\label{Biharacteristics_GR}\end{aligned}$$
that means is normal to the front of gravitational action; we will come back to that issue later, cf. text below Eqs. (\[Four\_Vector\_sigma\]) - (\[Four\_Vector\_r\_A\]). These Bicharacteristics can be considered as gravitational rays. Such an idealized picture is well justified for a gravitational wave when the wavelength is negligibly small in comparison with the spatial region of propagation of the wave. Such condition is satisfied in the far-zone of the Solar System, but not in the near-zone of the Solar System where the wavelength of gravitational radiation is larger than the boundary of the near-zone. That is why the Bicharacteristics in the near-zone should be considered as a mathematical concept of being normals onto the characteristic hypersurface $\omega$, while in the far-zone the Bicharacteristics can physically be interpreted as gravitational rays. But what is important here is the fact that the speed of gravity equals the speed of light, because the equations (\[Characteristics\_ED\]) and (\[Biharacteristics\_ED\]) are identical with (\[Characteristics\_GR\]) and (\[Biharacteristics\_GR\]), respectively; cf. Section 7.2 in [@Kopeikin_Efroimsky_Kaplan]. Therefore, as just mentioned above, the natural constant $c$ in the denominator on the r.h.s. in Eq. (\[Introduction\_2a\]) is related to the finite speed of gravity which equals the finite speed of light. The null-conoid (at some arbitrary event), can also be defined as the set of all Bicharacteristics emanating from that (arbitrary) event in the curved space-time.
$\bullet$ Fourth, as stated above, the equations for the Characteristics, Eq. (\[Characteristics\_ED\]) and Eq. (\[Characteristics\_GR\]), are fundamental consequences of the exact field equations of electrodynamics in curved space-time and the exact field equations of gravity, respectively. They state that there is no difference between the speed of light in curved space-time and the speed of gravitational action; cf. §53 in [@Fock]. Nevertheless, the propagation of electromagnetic action and the propagation of gravitational action are two different physical processes, and besides that their velocities are numerically equal, it does not mean that they can not be distinguished from each other. For instance, if the directions of electromagnetic wave propagation and propagation of gravitational action are different from each other, then one may distinguish between the directions of both these velocities; cf. Section 7.2 in [@Kopeikin_Efroimsky_Kaplan]. Furthermore, the important theoretical prediction of Einstein’s theory that both velocities are equal to each other, has recently been confirmed by the first detection of gravitational waves generated by the inspiral and merger of a binary neutron star and the determination of the location of the source by subsequent observations in the electromagnetic spectrum [@Ligo1; @Ligo2]. This measurement has constrained the difference between the speed of gravity and the speed of light to be between $ - 3 \times 10^{- 15}$ and $ + 7 \times 10^{- 16}$ times the speed of light [@Ligo3]. Needless to say that in this case both physical processes have clearly been separated, besides that the gravitational wave and the electromagnetic signal were parallel to each other.
A detailed description about how the finite speed of gravity in the near-zone of the Solar System could in principle be determined by means of Very Long Baseline Interferometry (VLBI) has been presented in [@Kopeikin_A]. The suggested approach is based on the increasing precision of VLBI facilities which allow to determine the impact of the orbital velocity ${\mbox{\boldmath$v$}}_A$ of a massive Solar System body on the Shapiro time-delay, which states that the total time of the propagation of a light signal from the four-coordinate of a light source $\left(c t_0,{\mbox{\boldmath$x$}}_0\right)$ to the four-coordinate of an observer $\left(c t_1,{\mbox{\boldmath$x$}}_1\right)$ is given by (e.g. Eq. (43) in [@KS1999]) $$\begin{aligned}
c\left(t_1 - t_0\right) = \left|{\mbox{\boldmath$x$}}_1 - {\mbox{\boldmath$x$}}_0\right| + c\, \Delta \left(t_1,t_0\right)\,,
\label{Shapiro_0}\end{aligned}$$
where $\left|{\mbox{\boldmath$x$}}_1 - {\mbox{\boldmath$x$}}_0\right|$ is the Euclidean distance between source and observer and $\Delta \left(t_1,t_0\right)$ is the time-delay of the light signal caused by the gravitational field of the massive body in motion.
In the first post-Minkowskian (1PM) approximation, which is exact up to terms to order ${\cal O}\left(G^2\right)$ and exact to all orders in the speed of the body, the time-delay is given by Eq. (51) in [@KS1999], which, by neglecting all terms proportional to the acceleration of the body (series expansion (\[Series\_A\]) is also employed), reads: $$\begin{aligned}
\fl \hspace{0.75cm} \Delta \left(t_1,t_0\right) = - 2\,\frac{G\,M_A}{c^3}\,\left(1 - \frac{{\mbox{\boldmath$k$}}\cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\right)
\ln \frac{r_A\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_1\right)}{r_A\left(s_0\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right)}
+ {\cal O}\left(G^2\right)\,,
\label{Shapiro_1}\end{aligned}$$
which is valid for light propagation in the field of one monopole in arbitrary motion, irrespective of the fact that acceleration terms of the body were neglected. The unit-vector ${\mbox{\boldmath$k$}}$ points from the light source towards the position of the observer, and the three-vectors ${\mbox{\boldmath$r$}}_A\left(s_0\right) = {\mbox{\boldmath$x$}}\left(t_0\right) - {\mbox{\boldmath$x$}}_A\left(s_0\right)$ and ${\mbox{\boldmath$r$}}_A\left(s_1\right) = {\mbox{\boldmath$x$}}\left(t_1\right) - {\mbox{\boldmath$x$}}_A\left(s_1\right)$, where ${\mbox{\boldmath$x$}}\left(t_0\right)$ and ${\mbox{\boldmath$x$}}\left(t_1\right)$ are the spatial coordinates of the light signal at source and observer, respectively, while ${\mbox{\boldmath$x$}}_A\left(s_0\right)$ and ${\mbox{\boldmath$x$}}_A\left(s_1\right)$ are the spatial position of the body at the retarded time $s_0$ and $s_1$, as defined in the below standing equations (\[retarded\_time\_s\_0\]) and (\[retarded\_time\_s\_1\]). Let us notice here that (\[Shapiro\_1\]) also agrees with Eqs. (146) - (148) in [@Zschocke2].
In the 1.5 post-Newtonian (1.5PN) approximation, which is exact up to terms to order ${\cal O}\left(c^{-4}\right)$ that means only exact to the first order in the speed of the body, the time-delay is given by Eq. (7) in [@Will_2003] and reads: $$\begin{aligned}
\fl \hspace{0.75cm} \Delta \left(t_1,t_0\right) = - 2\,\frac{G\,M_A}{c^3}\,\left(1 - \frac{{\mbox{\boldmath$k$}}\cdot {\mbox{\boldmath$v$}}_A}{c}\right)
\ln \frac{r_A\left(t_1\right) - {\mbox{\boldmath$K$}} \cdot {\mbox{\boldmath$r$}}_A\left(t_1\right)}{r_A\left(t_0\right) - {\mbox{\boldmath$K$}} \cdot {\mbox{\boldmath$r$}}_A\left(t_0\right)}
+ {\cal O}\left(c^{-4}\right)\,,
\label{Shapiro_2}\end{aligned}$$
which is valid for light propagation in the field of one monopole in uniform motion, that means all acceleration terms of the body are zero. The three-vectors ${\mbox{\boldmath$r$}}_A\left(t_0\right) = {\mbox{\boldmath$x$}}\left(t_0\right) - {\mbox{\boldmath$x$}}_A\left(t_0\right)$ and ${\mbox{\boldmath$r$}}_A\left(t_1\right) = {\mbox{\boldmath$x$}}\left(t_1\right) - {\mbox{\boldmath$x$}}_A\left(t_1\right)$, where ${\mbox{\boldmath$x$}}_A\left(t_0\right)$ and ${\mbox{\boldmath$x$}}_A\left(t_1\right)$ are the spatial positions of the body at time of emission $t_0$ and time of reception $t_1$ of the light signal. Furthermore, in Eq. (\[Shapiro\_2\]) the three-vector $\displaystyle {\mbox{\boldmath$K$}}={\mbox{\boldmath$k$}} - {\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A}{c} \times {\mbox{\boldmath$k$}}\right)$. Let us notice here that Eqs. (137) - (139) in [@Zschocke2] are valid for light propagation in the field of one arbitrarily moving body in slow motion, which in case of uniform motion coincide with (\[Shapiro\_2\]), as one may show by series expansion.
For grazing light rays or radio waves at massive bodies of the Solar System, the velocity dependent terms in (\[Shapiro\_1\]) or (\[Shapiro\_2\]) contribute of the order of a few picoseconds in time-delay; cf. Table II in [@Zschocke2] for grazing rays at Sun or giant planets. At this order of precision it becomes possible to measure such velocity-dependent terms in time-delay (\[Shapiro\_1\]) or (\[Shapiro\_2\]) by means of the most modern VLBI techniques. In fact, such a concrete experiment by VLBI facilities has been suggested in [@Kopeikin_A], and has finally been performed in 2002 with remarkable effort and precision [@Kopeikin_B]. In particular, in [@Kopeikin_B] the Shapiro time delay of a radio wave, emitted by the quasar ${\rm QSO}\;{\rm J}0842 + 1835$ and passing near Jupiter, has been determined with extremely high precision, in order to determine the finite speed of the gravity fields of that moving body. This experiment has, at the very first time, succeeded in determining the impact of the orbital velocity effects to order $v_A/c$ of Jupiter on the Shapiro time-delay. Subsequently, these results have initiated a controversial debate in the literature about the correct interpretation of this experiment [@Theory_Experiment; @Will_2003; @Kopeikin_E; @Samuel_1; @Faber; @Asada1; @Asada2; @Carlip; @Pascual; @Samuel_2; @Kopeikin_I; @Kopeikin_CQG; @Kopeikin_D; @Kopeikin_F; @Kopeikin_G; @Kopeikin_H]; further comments about the Kopeikin-Formalont experiment can be found in [@Kopeikin_Efroimsky_Kaplan; @Frittelli; @Mignard_Crosta; @Malkin; @Zhu]. While there is no doubt at all in the literature about the correctness of the expressions (\[Shapiro\_1\]) and (\[Shapiro\_2\]), a central topic of this conversion was about the correct physical meaning of the natural constant $c$ in the velocity-dependent terms in (\[Shapiro\_1\]) and (\[Shapiro\_2\]).
That remarkable debate had arisen just because of the above discussed fundamental prediction of general relativity that the speed of gravity and the speed of light are numerically equal. That is why it becomes a highly sophisticated assignment of a task to disentangle these both velocities in concrete astrometrical measurements. In [@KS1999] it was shown that the retarded instant of time $s_0$ and $s_1$ in (\[Shapiro\_1\]) are caused by the retarded time of the Liénard-Wiechert potential of the metric tensor (Eq. (10) in [@KS1999]), hence they are caused by the finite speed of gravity so that the natural constant $c$ is related to the finite speed of gravity. And due to the fact that (\[Shapiro\_2\]) can be deduced from (\[Shapiro\_1\]) by series expansion and by assuming a uniform motion of the body, one might be inclined to assume that the natural constant $c$ in (\[Shapiro\_2\]) is related to the finite speed of gravity. On the other side, in [@Will_2003] it was shown that the natural constant $c$ in (\[Shapiro\_2\]) is caused by the finite speed of light and is not related to the finite speed of gravity. So it might be that a unique interpretation of the experiment is impossible as long as one is restricted to terms of the first order in $v_A/c$. But it should be noticed that the controversy was not about the correctness of the theory of general relativity, but mainly about the question of whether the velocity-dependent term in the Shapiro time-delay is related to the finite speed of gravity (retardation of gravitational action) or to the finite speed of light (aberration of light).
In this context it should also be noticed that there is agreement among the participants of this controversy with respect to the following minimal set of issues:
1. the retarded time in (\[Introduction\_2a\]) is caused by the finite speed of gravity.
2. the finite speed of gravity has surely an impact on the Shapiro time-delay.
3. the impact of orbital velocity of Jupiter on time-delay has been detected in [@Kopeikin_B].
While in principle the experiment suggested in [@Kopeikin_A] is capable to measure the speed of gravity, there is no general consensus about the correct interpretation of the results of the concrete experiment in [@Kopeikin_B], as it was also formulated in [@Mignard_Crosta]. It seems that the fact that the retarded time $u$ in (\[Introduction\_2a\]) as well as the retarded time $s_1$ and $s_0$ in the below standing equations (\[retarded\_time\_s\_0\]) and (\[retarded\_time\_s\_1\]) are due to the finite speed of gravity might not necessarily be convincing for a unique and correct interpretation of these astrometrical VLBI measurements [@Will_2003]. Moreover, in [@Will_2003] it was argued that acceleration terms or terms of the order $v_A^2/c^2$ give the first level at which retardation effects due to the motion of the massive bodies occur. However, in order to determine the next higher order terms, that means terms proportional to the acceleration of the body or terms of the order $v_A^2/c^2$ in the Shapiro time-delay, one needs to improve the precision in time measurements by VLBI experiments by a factor of about $10^4$, which requires an ultra-high precision in the time-resolution of VLBI measurements of about $10^{-3}\,{\rm picoseconds}$, which is far out of reach of present-day VLBI facilities. Furthermore, the theoretical interpretation of the experiment might also depend on the generalized theoretical models beyond general relativity which allow to distinguish between the speed of light and speed of gravity [@Carlip]. Even the semantics in use could have an impact on the correct interpretation of these VLBI results [@Frittelli]. Here, also in view of the exceptional number of articles in the literature related to this subject, a detailed and correct interpretation of this famous experiment would be far beyond the intention of our investigation. For the moment being, it seems sensible to keep in mind the problem and to realize that further careful investigations and higher precisions in VLBI measurements are necessary in order to clarify such involved difficulties regarding the distinction between the speed of electromagnetic fields and the speed of gravitational action in the near-zone of the Solar System.
Finally, having said all that we emphasize again that the retarded instant of time (\[Introduction\_2a\]) originates from the finite speed of gravity which equals the speed of light, a fact that is in meanwhile sufficient for our considerations here; cf. also the comments in the text below Eqs. (\[retarded\_time\_s\]) and (\[retarded\_time\_s\_00\]) as well as in the text below Eqs. (\[retarded\_time\_s\_0\]) and (\[retarded\_time\_s\_1\]).
The exact geodesic equation for light propagation
-------------------------------------------------
Throughout the investigation the propagation of a light signal in vacuum is considered. The most simplest light tracking model presupposes a four-dimensional flat space-time with Minkowskian metric $\eta_{\alpha\beta} = {\rm diag}\left(-1,+1,+1,+1\right)$ which implicitly involves Cartesian coordinates, where the light ray propagates along a straight line. Then, a light signal emitted at some spatial point ${\mbox{\boldmath$x$}}_0$ at time $t_0$ propagates along it’s initial direction ${\mbox{\boldmath$\sigma$}}$, so that the light trajectory in the global system reads as follows, $$\begin{aligned}
{\mbox{\boldmath$x$}}_{\rm N}\left(t\right) = {\mbox{\boldmath$x$}}_0 + c \left(t-t_0\right) {\mbox{\boldmath$\sigma$}}\,,
\label{unperturbed_lightray_1} \end{aligned}$$
where suffix ${\rm N}$ labels Newtonian approximation. Such a simple light propagation model is not sufficient for todays precision of astrometric measurements which, as stated above, implicates a corresponding advancement in the theory of light propagation. Especially, relativistic astrometry has necessarily to account for the fact that the space-time is not flat but a four-dimensional curved manifold. Because the space-time is curved, a light signal propagates along a geodesic which is the generalization of the concept of a straight line because a geodesic is a curve that parallel-transports its own tangent vector. Consequently, a fundamental assignment in relativistic astrometry concerns the precise modeling of the time track of a light signal through the curved space-time of Solar System, that is to say the determination the trajectory of the light signal, ${\mbox{\boldmath$x$}}\left(t\right)$, in some reference system which covers the entire curved space-time (at least those part of the entire space-time which contains the light source and the observer) and, therefore, is called global coordinate system.
The trajectory of a light signal propagating in curved space-time is determined by the geodesic equation and isotropic condition, which in terms of coordinate time read as follows [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991; @MTW] (e.g. Eqs. (3.220) - (3.224) in [@Kopeikin_Efroimsky_Kaplan]): $$\begin{aligned}
\frac{\ddot{x}^{i}\left(t\right)}{c^2} + \Gamma^{i}_{\alpha\beta} \frac{\dot{x}^{\alpha}\left(t\right)}{c} \frac{\dot{x}^{\beta}\left(t\right)}{c}
- \Gamma^{0}_{\alpha\beta} \frac{\dot{x}^{\alpha}\left(t\right)}{c} \frac{\dot{x}^{\beta}\left(t\right)}{c} \frac{\dot{x}^{i}\left(t\right)}{c} = 0\;,
\label{Geodetic_Equation1}
\\
\nonumber\\
g_{\alpha\beta}\,\frac{\dot{x}^{\alpha}\left(t\right)}{c}\,\frac{\dot{x}^{\beta}\left(t\right)}{c} = 0\,,
\label{Null_Condition1}\end{aligned}$$
where a dot denotes total derivative with respect to coordinate time, hence $\dot{x}^{i}\left(t\right)$ are the three-components of the coordinate velocity of the photon. The null condition (\[Null\_Condition1\]) and geodesic equation (\[Geodetic\_Equation1\]) have equivalent physical content because (\[Null\_Condition1\]) is a first integral of (\[Geodetic\_Equation1\]). As mentioned above, the natural constant $c$ explicitly seen in both these equations (\[Geodetic\_Equation1\]) and (\[Null\_Condition1\]) means actually the speed of light, while the natural constant $c$ contained in the Christoffel symbols and metric tensor is related to the finite speed of gravity as stated already in the text below Eqs. (\[Introduction\_2\]) and (\[Introduction\_2a\]). Furthermore, it should be mentioned that the coordinate velocity of a light signal in the global system of curved space-time differs from the speed of light in flat space-time $\left|\dot{{\mbox{\boldmath$x$}}}\right| \neq c\,$; only in the local system of a free-falling observer both are equal.
### The initial value problem
The light signal is assumed to be emitted at the four-position of the light source, $\left(t_0,{\mbox{\boldmath$x$}}_0\right)$, as given in some global coordinate system $\left(t,{\mbox{\boldmath$x$}}\right)$. Then, a unique solution of the partial differential equation (\[Geodetic\_Equation1\]) is well-defined by the so-called initial-value problem (Cauchy problem), where the spatial position of the light source, ${\mbox{\boldmath$x$}}_0$, and the initial unit direction of the light ray, ${\mbox{\boldmath$\mu$}} = \dot{{\mbox{\boldmath$x$}}}\left(t_0\right) / \left|\dot{{\mbox{\boldmath$x$}}}\left(t_0\right)\right|$, are given. Usually, the initial value problem is often replaced by the so-called initial-boundary conditions [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991; @KK1992; @Kopeikin1997; @KSGE; @Zschocke1; @Zschocke2]: $$\begin{aligned}
{\mbox{\boldmath$x$}}_0 = {\mbox{\boldmath$x$}}\left(t\right)\bigg|_{t=t_0} \quad {\rm and} \quad {\mbox{\boldmath$\sigma$}} = \frac{\dot{{\mbox{\boldmath$x$}}}\left(t\right)}{c}\bigg|_{t = - \infty}\,,
\label{Initial_Boundary_Conditions}\end{aligned}$$
with ${\mbox{\boldmath$\sigma$}}$ being the unit-direction (${\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$\sigma$}} = 1$) of the light ray at past null infinity ${\cal J}^{-}$ (cf. notation in Section 34 in [@MTW] and Figure 34.2. in [@MTW]). The advantage for using initial-boundary conditions (\[Initial\_Boundary\_Conditions\]) rather than initial-value conditions when integrating the geodesic equation (\[Geodetic\_Equation1\]) is solely based on the integration constant which becomes simpler at past null infinity. One may easily find a unique relation between the tangent vectors ${\mbox{\boldmath$\sigma$}}$ and ${\mbox{\boldmath$\mu$}}$ (e.g. Section 3.2.3 in [@Brumberg1991]), so one verifies that there is a unique one-to-one correspondence between the initial-boundary problem (\[Initial\_Boundary\_Conditions\]) and the initial-value problem; more precisely, these statements are valid in case of a weak gravitational field and ordinary topology of space-time. According to (\[Initial\_Boundary\_Conditions\]), the solution for the light trajectory is a function of these initial-boundary conditions: ${\mbox{\boldmath$x$}}\left(t\right)= {\mbox{\boldmath$x$}}\left(t,{\mbox{\boldmath$x$}}_0,{\mbox{\boldmath$\sigma$}}\right)$.
### The boundary value problem
A unique solution of geodesic equation (\[Geodetic\_Equation1\]) can also be defined by the so-called boundary-value problem rather than the initial-boundary problem (\[Initial\_Boundary\_Conditions\]). In the boundary-value problem a light signal is supposed to be emitted at some initial space-time point $\left(t_0,{\mbox{\boldmath$x$}}_0\right)$ (source) which is received at another space-time point $\left(t_1,{\mbox{\boldmath$x$}}_1\right)$ (observer) [@Kopeikin_Efroimsky_Kaplan; @Klioner2003b; @Book_Clifford_Will; @Brumberg1991; @KK1992; @Klioner2003a]: $$\begin{aligned}
{\mbox{\boldmath$x$}}_0 = {\mbox{\boldmath$x$}}\left(t\right)\bigg|_{t=t_0} \quad {\rm and} \quad {\mbox{\boldmath$x$}}_1 = {\mbox{\boldmath$x$}}\left(t\right)\bigg|_{t=t_1}\,.
\label{Boundary_Value_Conditions}\end{aligned}$$
Accordingly, the solution of the light trajectory will be a function of these boundary conditions: ${\mbox{\boldmath$x$}}\left(t\right)= {\mbox{\boldmath$x$}}\left(t,{\mbox{\boldmath$x$}}_0,{\mbox{\boldmath$x$}}_1\right)$.
Because in reality any light source is located at some finite distance, the solution of the boundary-value problem is of decisive importance in practical astrometry [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991; @KK1992]. Accordingly, the primary aim of our investigation is to determine the solution of the boundary-value problem (\[Boundary\_Value\_Conditions\]) when the solution of the initial-boundary problem (\[Initial\_Boundary\_Conditions\]) is given.
The geodesic equation for light propagation in 2PN approximation
----------------------------------------------------------------
The metric enters the geodesic equation (\[Geodetic\_Equation1\]) in virtue of the Christoffel symbols (\[Christoffel\_Symbols\]). It is, however, impossible to determine the Solar System metric without taking recourse to an approximation scheme. Such an approximative approach is possible, because in the Solar System the gravitational fields are weak, $m_A/P_A \ll 1$ (Schwarzschild radius $m_A = G\,M_A/c^2$ with $M_A$ and $P_A$ being mass and equatorial radius of body $A$) and the motions of matter are slow as compared with the speed of light $v_A/c \ll 1$ (we have in mind that $v_A$ is just the orbital velocity of the body, but in general could also be rotational motion of extended bodies, convection currents inside the massive bodies, oscillations of the bodies, etc.). Accordingly, a series expansion in inverse powers of the natural constant $c$ is meaningful, $$\begin{aligned}
\fl \hspace{1.0cm} g_{\alpha \beta}\left(t,{\mbox{\boldmath$x$}}\right) = \eta_{\alpha \beta}
+ h^{(2)}_{\alpha\beta}\left(t,{\mbox{\boldmath$x$}}\right) + h^{(3)}_{\alpha\beta}\left(t,{\mbox{\boldmath$x$}}\right)
+ h^{(4)}_{\alpha\beta}\left(t,{\mbox{\boldmath$x$}}\right) + {\cal O} \left(c^{-5}\right),
\label{2PN_A}\end{aligned}$$
where $h^{(n)}_{\alpha\beta} \sim {\cal O} \left(c^{-n}\right)$ are tiny perturbations of the flat Minkowskian metric, that is $\left|h^{(n)}_{\alpha\beta}\right| \ll 1$ for any $\alpha,\beta$. Here, in line of the comments made above regarding the physical meaning of the natural constant $c$, we just notice that the post-Newtonian expansion of the metric tensor (\[2PN\_A\]) is of course an expansion with respect to the inverse power of the speed of gravity.
The series expansion (\[2PN\_A\]) includes all terms up to the fifth order and is called post-post-Newtonian (2PN) approximation of the metric tensor. The validity of the post-Newtonian expansion (\[2PN\_A\]) is restricted to the near-zone region of the Solar System where the retardations are small by definition [@Kopeikin_Efroimsky_Kaplan; @MTW; @Poisson_Lecture_Notes; @Poisson_Will; @Expansion_2PN]; see also the Fig. 7.7 in [@Kopeikin_Efroimsky_Kaplan] or Fig. 36.3 in [@MTW]. The near-zone of a gravitating system is defined as spatial region with the boundary $\left|{\mbox{\boldmath$x$}}\right| \ll \lambda_{\rm gr}\,$, where $\lambda_{\rm gr}$ is a characteristic wavelength of gravitational waves emitted by the system and the origin of spatial axes is assumed to be located at the center-of-mass of the gravitational system or somewhere nearby. For the Solar System one obtains about $\lambda_{\rm gr} \sim 10^{17}\,{\rm m}$ which is the lowest wavelength of gravitational radiation emitted by Jupiter during its revolution around the barycenter of the Solar System [@Kopeikin_Efroimsky_Kaplan; @Zschocke2; @MTW]. A more accurate statement is achieved by the fact that the term near-zone is intrinsically connected with orbital accelerations $a_A$ of the massive bodies $A=1,...,N$ which constitute gravitational system. In mathematical terms it requires $$\begin{aligned}
\frac{a_A\left(t\right)\,r_A\left(t\right)}{c^2} \ll \frac{v_A\left(t\right)}{c} \ll 1
\label{Near_Zone_1}\end{aligned}$$
for each massive body $A$; here $v_A\left(t\right)$ is the orbital velocity and $r_A\left(t\right) = \left| {\mbox{\boldmath$x$}} - {\mbox{\boldmath$x$}}_A\left(t\right)\right|$ is the spatial distance of some field point ${\mbox{\boldmath$x$}}$ from the massive body $A$ located at ${\mbox{\boldmath$x$}}_A\left(t\right)$. The condition (\[Near\_Zone\_1\]) has already been stated by by Eq. (B7) in [@Zschocke3] or Eq. (97) in [@Zschocke4] and follows from $\left|h^{(4)}_{\alpha\beta}\right| \ll \left|h^{(2)}_{\alpha\beta}\right| \ll 1$, where the metric coefficients for a system of $N$ moving monopoles are given by Eqs. (24) - (27) in [@Zschocke4] [^1]. Using the numerical values of the most massive Solar System bodies as given in Table \[Table1\] we find the spatial radius of the near-zone to be about $$\begin{aligned}
\left| {\mbox{\boldmath$x$}}\right| \le 10^{14}\,{\rm m}\,.
\label{Near_Zone_2}\end{aligned}$$
The results and considerations of our investigation are valid within this spatial region, which corresponds to about $4$ light-days.
By inserting the post-Newtonian expansion of the metric tensor (\[2PN\_A\]) into the geodesic equation (\[Geodetic\_Equation1\]) via the Christoffel symbols (\[Christoffel\_Symbols\]) one obtains the geodesic equation in the so-called post-post-Newtonian (2PN) approximation, which is given, for instance, in [@Zschocke4; @Zschocke3; @Bruegmann2005]. The formal solution of the geodesic equation in 2PN approximation reads [^2], $$\begin{aligned}
\fl \hspace{0.9cm} {\mbox{\boldmath$x$}}\left(t\right) = {\mbox{\boldmath$x$}}_0 + c \left(t - t_0\right) {\mbox{\boldmath$\sigma$}} + \Delta {\mbox{\boldmath$x$}}^{1 {\rm PN}}\left(t\right)
+ \Delta {\mbox{\boldmath$x$}}^{1.5 {\rm PN}}\left(t\right) + \Delta {\mbox{\boldmath$x$}}^{2 {\rm PN}}\left(t\right) + {\cal O} \left(c^{-5}\right).
\label{2PN_B}\end{aligned}$$
The first two terms on the r.h.s. in (\[2PN\_B\]) represent the unperturbed light ray (\[unperturbed\_lightray\_1\]), while the subsequent terms represent corrections to the unperturbed light ray. The physical meaning of the natural constant $c$ in the unperturbed light ray, ${\mbox{\boldmath$x$}}_0 + c \left(t - t_0\right) {\mbox{\boldmath$\sigma$}}$, is of course the speed of light in flat Minkowskian space-time; cf. comment below Eqs. (\[Geodetic\_Equation1\]) - (\[Null\_Condition1\]) regarding the geodesic equation and isotropic condition for light rays. It should be noticed that the post-Newtonian correction terms $\Delta {\mbox{\boldmath$x$}}^{n {\rm PN}}\left(t\right)$ originate from the post-Newtonian expansion of the metric tensor (\[2PN\_A\]), which is an expansion in inverses powers of $c$, meaning the speed of gravity. However, in order to compute these correction terms $\Delta {\mbox{\boldmath$x$}}^{n {\rm PN}}\left(t\right)$, the integration of geodesic equation proceeded along the unperturbed light ray [@Zschocke3; @Zschocke4], where the meaning of $c$ is the speed of light. Therefore, the correction terms $\Delta {\mbox{\boldmath$x$}}^{n {\rm PN}}\left(t\right)$ in (\[2PN\_B\]) contain the natural constant $c$ in two different meanings, namely the speed of light and the speed of gravity. One might believe that this kind of entanglement makes it impossible to separate the impact of the finite speed of gravity and the finite speed of light in these correction terms. This is, however, not true. The terms related to the characteristics of the gravity field and the terms related to the light characteristics can clearly be separated in the solution of the light-ray trajectory (\[2PN\_B\]); cf. comments below Eqs. (\[Four\_Vector\_sigma\]) - (\[Four\_Vector\_r\_A\]).
For an overview of the state-of-the-art in the theory of light propagation we refer to the text books [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991] and the articles [@Klioner2003b; @Zschocke1; @Zschocke2; @KK1992; @Kopeikin1997; @KS1999; @KSGE; @Klioner2003a; @KopeikinMashhoon2002]. According to these references, an impressive progress in the determination of the correction terms $\Delta {\mbox{\boldmath$x$}}^{1 {\rm PN}}\left(t\right)$ and $\Delta {\mbox{\boldmath$x$}}^{1.5 {\rm PN}}\left(t\right)$ has been made during recent decades.
On the other side, the knowledge of the correction terms $\Delta {\mbox{\boldmath$x$}}^{2 {\rm PN}}\left(t\right)$ is pretty much limited thus far. In fact, the problem of light propagation in 2PN approximation, that means the determination of the light trajectory (\[2PN\_B\]) as function of coordinate time, has only been considered for the following rather restricted situations [^3]:
1. 2PN light trajectory in the field of one monopole at rest [@Brumberg1991; @Brumberg1987] [^4],
2. 2PN light trajectory in the field of two point-like bodies in slow motion [@Bruegmann2005],
where [@Bruegmann2005] was not intended for light propagation in the Solar System.
It is, however, clear that for astrometry on the micro-arcsecond and sub-micro-arcsecond level it is indispensable to determine the light trajectory in the second post-Newtonian approximation for more realistic gravitational systems, especially where the motion of the bodies is taken into account [@Conference_Cambridge; @Deng_Xie; @Deng_2015; @Minazzoli2; @Xu_Wu; @Xu_Gong_Wu_Soffel_Klioner; @Minazzoli1; @2PN_Light_PropagationA; @Xie_Huang]. Already for micro-arcsecond astrometry it is necessary to account for the motion of the Solar System bodies, where it is sufficient to determine the light trajectory in the field of one monopole at rest, ${\mbox{\boldmath$x$}}_A = {\rm const}$, and then simply to insert the retarded position of the body, ${\mbox{\boldmath$x$}}_A = {\mbox{\boldmath$x$}}_A\left(s_1\right)$, where $s_1$ is the retarded instant of time as defined by Eq. (\[retarded\_time\_s\_1\]). But for the sub-micro-arcsecond astrometry such a simplified access is insufficient, because terms which are proportional to the orbital velocity of the body contribute on such level of precision in light deflection. In order to account for those terms in the 2PN solution of the light trajectory which are proportional to the orbital velocity of the body, one needs to consider the equation of motion for light signals propagating in the gravitational field of moving bodies. On these grounds, an analytical solution for the light trajectory in 2PN approximation in the gravitational field of one arbitrarily moving pointlike monopole has recently been determined in [@Zschocke3; @Zschocke4], where the so-called initial-value problem (\[Initial\_Boundary\_Conditions\]) has been solved:
1. 2PN light trajectory in field of one arbitrarily moving monopole [@Zschocke3; @Zschocke4].
Because in reality any light source is located at some finite distance, the consideration of the boundary-value problem (\[Boundary\_Value\_Conditions\]) is of fundamental importance for the unique interpretation of astrometric observations [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991; @KK1992]. Needless to say, that this fact becomes of particular importance for astrometry of Solar System objects, say for astrometric measurements in the near-zone of the Solar System, which will be the primary topic of this investigation.
The organization of the article is aligned as follows. In Section \[Section3\] the main results of the initial-boundary value problem of 2PN light propagation are summarized, which were recently obtained in [@Zschocke3; @Zschocke4]. Section \[Section4\] defines the boundary-value problem, and series expansions in the near-zone of the Solar System are considered. The three fundamental transformations of the boundary-value problem are derived in the Sections \[Section5\] and \[Section6\] and \[Section7\]. An estimation of the numerical magnitude of each individual term and the resulting simplified transformations are also given in these Sections. The impact of higher order terms beyond 2PN approximation is considered in Section \[Section\_3PN\]. The summary and outlook can be found in Section \[Section8\]. The notation, some relations, and details of the calculations are delegated to appendices.
The initial-boundary value problem in 2PN approximation {#Section3}
=======================================================
So as not to have to look up in the literature the main results of our articles [@Zschocke3; @Zschocke4], that is the solution in 2PN approximation for coordinate velocity and trajectory of a light signal propagating in the field of one moving monopole, will be summarized for subsequent considerations.
As formulated in the introductory section, a unique solution of (\[Geodetic\_Equation1\]) is well-defined by initial-boundary conditions, $$\begin{aligned}
{\mbox{\boldmath$x$}}_0 &=& {\mbox{\boldmath$x$}}\left(t\right)\bigg|_{t=t_0} \quad {\rm and} \quad {\mbox{\boldmath$\sigma$}} = \frac{\dot{{\mbox{\boldmath$x$}}}\left(t\right)}{c}\bigg|_{t = - \infty}\,,
\label{Introduction_6}\end{aligned}$$
with ${\mbox{\boldmath$x$}}_0$ being the position of the light source at the moment $t_0$ of emission of the light-signal and ${\mbox{\boldmath$\sigma$}}$ being the unit-direction (${\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$\sigma$}} = 1$) of the light ray at past null infinity.
![\[Diagram1\] A geometrical representation of light propagation through the gravitational field of one pointlike massive body $A$ moving along its worldline ${\mbox{\boldmath$x$}}_A\left(t\right)$; the diagram is not Minkowskian but a purely spatial picture, i.e. $\left(x^1,x^2,x^3\right)$ denote the three spatial axes of the BCRS. The three-vectors ${\mbox{\boldmath$r$}}_A\left(s\right)$, ${\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)$, and ${\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)$ are defined by Eqs. (\[vector\_B\]), (\[vector\_rA\_0\]), and (\[vector\_rA\_1\]), respectively; for ${\mbox{\boldmath$r$}}^{\rm N}_A\left(s\right)$ see footnote on p.18$\,$. The impact vectors ${\mbox{\boldmath$d$}}^{\rm N}\left(s\right)$ and ${\mbox{\boldmath$d$}}^{\rm N}\left(s_0\right)$ are given by Eqs. (\[Impact\_Vector\_Sigma\_s0\_Newtonian\]). The three-vectors ${\mbox{\boldmath$\sigma$}}$, ${\mbox{\boldmath$k$}}$, and ${\mbox{\boldmath$n$}}$ are shown, which are defined by the Eqs. (\[Introduction\_6\]), (\[Boundary\_3\]), and (\[Tangent\_Vector1\]), respectively. Their transformations among each other represent the fundamental aspects of the boundary value problem.](Diagram1)
The coordinate velocity of a light signal in 2PN approximation
--------------------------------------------------------------
The first integration of geodesic equation in 2PN approximation yields the coordinate velocity of a light signal and is given by (cf. Eq. (99) in [@Zschocke4]): $$\begin{aligned}
\fl \hspace{1.0cm} \frac{\dot{{\mbox{\boldmath$x$}}}\left(t\right)}{c} = {\mbox{\boldmath$\sigma$}}
+ m_A\,{\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$r$}}_A\left(s\right)\right)
+ m_A\,{\mbox{\boldmath$A$}}_2\left({\mbox{\boldmath$r$}}_A\left(s\right),{\mbox{\boldmath$v$}}_A\left(s\right)\right)
\nonumber\\
\fl \hspace{2.6cm} + \,m_A^2\,{\mbox{\boldmath$A$}}_3\left({\mbox{\boldmath$r$}}_A\left(s\right)\right)
+ m_A\,{\mbox{\boldmath$\epsilon$}}_1\left({\mbox{\boldmath$r$}}_A\left(s\right),{\mbox{\boldmath$v$}}_A\left(s\right)\right) + {\cal O}\left(c^{-5}\right),
\label{First_Integration}\end{aligned}$$
where the vectorial functions ${\mbox{\boldmath$A$}}_1$, ${\mbox{\boldmath$A$}}_2$, ${\mbox{\boldmath$A$}}_3$, and ${\mbox{\boldmath$\epsilon$}}_1$ are given in \[Appendix2\] by Eqs. (\[Vectorial\_Function\_A1\]) - (\[Vectorial\_Function\_A3\]) and Eq. (\[epsilon\_1\]), respectively. The argument ${\mbox{\boldmath$r$}}_A\left(s\right)$ in the vectorial functions in (\[First\_Integration\]) is [^5] $$\begin{aligned}
{\mbox{\boldmath$r$}}_A\left(s\right) &=& {\mbox{\boldmath$x$}}\left(t\right) - {\mbox{\boldmath$x$}}_A\left(s\right),
\label{vector_B}\end{aligned}$$
with ${\mbox{\boldmath$x$}}\left(t\right)$ being the exact spatial coordinate of the light signal at global coordinate time $t$, while ${\mbox{\boldmath$x$}}_A\left(s\right)$ is the spatial position of the body at retarded time $s$, which is defined by an implicit relation, $$\begin{aligned}
s = t - \frac{r_A\left(s\right)}{c}\,,
\label{retarded_time_s}\end{aligned}$$
where $r_A\left(s\right) = \left|{\mbox{\boldmath$r$}}_A\left(s\right)\right|$; here it should be noticed that the retardation (\[retarded\_time\_s\]) is due to the finite speed of propagation of gravity which equals the speed of light. The other argument ${\mbox{\boldmath$v$}}_A\left(s\right)$ in the vectorial functions in (\[First\_Integration\]) is the orbital velocity of the body at the retarded instant of time $s$. The retarded time (\[retarded\_time\_s\]) is a function of coordinate time and cannot be solved in closed form; only for the simple case of linear motion of the body a solution is possible as given by Eq. (3.14) in [@Kopeikin_CQG] or Eq. (9) in [@Zschocke_Soffel].
The trajectory of a light signal in 2PN approximation
-----------------------------------------------------
The second integration of geodesic equation in 2PN approximation yields the trajectory of a light signal and is given by (cf. Eq. (128) in [@Zschocke4]): $$\begin{aligned}
\fl \hspace{1.0cm} {\mbox{\boldmath$x$}}\left(t\right) = {\mbox{\boldmath$x$}}_0 + c \left(t - t_0\right) {\mbox{\boldmath$\sigma$}}
+ \,m_A\,\bigg(
{\mbox{\boldmath$B$}}_1\left({\mbox{\boldmath$r$}}_A\left(s\right)\right) - {\mbox{\boldmath$B$}}_1\left({\mbox{\boldmath$r$}}_A\left(s_0\right)\right)
\bigg)
\nonumber\\
\nonumber\\
\fl \hspace{2.7cm} + \,m_A\,\bigg({\mbox{\boldmath$B$}}^A_2\left({\mbox{\boldmath$r$}}_A\left(s\right),{\mbox{\boldmath$v$}}_A\left(s\right)\right)
- {\mbox{\boldmath$B$}}^A_2\left({\mbox{\boldmath$r$}}_A\left(s_0\right),{\mbox{\boldmath$v$}}_A\left(s\right)\right)\bigg)
\nonumber\\
\nonumber\\
\fl \hspace{2.7cm} + \,m_A\,\bigg({\mbox{\boldmath$B$}}^B_2\left({\mbox{\boldmath$r$}}_A\left(s\right),{\mbox{\boldmath$v$}}_A\left(s\right)\right)
- {\mbox{\boldmath$B$}}^B_2\left({\mbox{\boldmath$r$}}_A\left(s_0\right),{\mbox{\boldmath$v$}}_A\left(s_0\right)\right)\bigg)
\nonumber\\
\nonumber\\
\fl \hspace{2.7cm} + \,m_A^2\,\bigg({\mbox{\boldmath$B$}}_3\left({\mbox{\boldmath$r$}}_A\left(s\right)\right) - {\mbox{\boldmath$B$}}_3\left({\mbox{\boldmath$r$}}_A\left(s_0\right)\right) \bigg)
+ m_A\,{\mbox{\boldmath$\epsilon$}}_2\left(s,s_0\right) + ¬{\cal O}\left(c^{-5}\right),
\label{Second_Integration}\end{aligned}$$
where the vectorial functions ${\mbox{\boldmath$B$}}_1$, ${\mbox{\boldmath$B$}}^A_2$, ${\mbox{\boldmath$B$}}^B_2$, ${\mbox{\boldmath$B$}}_3$, and ${\mbox{\boldmath$\epsilon$}}_2$ are given in \[Appendix2\] by Eqs. (\[Vectorial\_Function\_C1\]) - (\[Vectorial\_Function\_C3\]) and Eqs. (\[epsilon\_3\]) - (\[epsilon\_3b\]), respectively. The argument ${\mbox{\boldmath$r$}}_A\left(s_0\right)$ reads $$\begin{aligned}
{\mbox{\boldmath$r$}}_A\left(s_0\right) = {\mbox{\boldmath$x$}}\left(t_0\right) - {\mbox{\boldmath$x$}}_A\left(s_0\right),
\label{vector_rA_00}\end{aligned}$$
with ${\mbox{\boldmath$x$}}\left(t_0\right)$ being the exact spatial coordinate of the light signal at the light source, while ${\mbox{\boldmath$x$}}_A\left(s_0\right)$ is the spatial position of the body at retarded time $s_0$, which reads $$\begin{aligned}
s_0 &=& t_0 - \frac{r_A\left(s_0\right)}{c}\,,
\label{retarded_time_s_00}\end{aligned}$$
where $r_A\left(s_0\right) = \left|{\mbox{\boldmath$r$}}_A\left(s_0\right)\right|$; let us notice here that the retarded time in (\[retarded\_time\_s\_00\]) is due to the finite speed of propagation of gravity which equals the speed of light.
The other argument ${\mbox{\boldmath$v$}}_A\left(s_0\right)$ in the vectorial functions in (\[Second\_Integration\]) is the orbital velocity of the body at the retarded instant of time $s_0$. As it has been emphasized in [@Zschocke4], it is important to realize that the velocity in the vectorial functions ${\mbox{\boldmath$B$}}^A_2$ in (\[Second\_Integration\]) is taken at the very same instant of retarded time $s$, which ensures the logarithm in (\[Vectorial\_Function\_C2\_A\]) in combination with (\[Second\_Integration\]) to be well-defined.
There seems to be a marginal difference between Eq. (\[Second\_Integration\]) and Eq. (128) in [@Zschocke4], namely the argument of the velocity term in the second line of both these equations are different. However, this difference is only apparent, because the relation (cf. Eq. (121) in [@Zschocke4]) $$\begin{aligned}
\frac{{\mbox{\boldmath$v$}}_A\left(s_0\right)}{c} = \frac{{\mbox{\boldmath$v$}}_A\left(s\right)}{c} + \frac{{\mbox{\boldmath$a$}}_A\left(s\right)}{c^2}\;c\left(s_0 - s\right)
+ {\cal O}\left(c^{-3}\right),
\label{Series_A}\end{aligned}$$
allows to replace ${\mbox{\boldmath$v$}}_A\left(s_0\right)$ by ${\mbox{\boldmath$v$}}_A\left(s\right)$. But according to this relation, such a replacement implies the occurrence of a term ${\mbox{\boldmath$a$}}_A\left(s\right) \left(s_0 - s\right)$ which is taken into account in the vectorial function ${\mbox{\boldmath$\epsilon$}}_2\left(s,s_0\right)$; cf. last term in (\[epsilon\_3b\]) and text below that equation. Here we also notice the following important relation (cf. Eq. (127) in [@Zschocke4]), $$\begin{aligned}
\fl c \left(s_0 - s\right) = r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right) - r_A\left(s_0\right)
+ {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}_A\left(s\right) + {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}_A\left(s_0\right),
\label{Series_B}\end{aligned}$$
which is valid up to terms of the order ${\cal O}\left(c^{-2}\right)$ and follows from (\[retarded\_time\_s\]) and (\[retarded\_time\_s\_00\]) in virtue of (\[Second\_Integration\]) with (\[vector\_B\]) and (\[vector\_rA\_00\]). It should be noticed that the solutions of coordinate velocity (\[First\_Integration\]) and trajectory (\[Second\_Integration\]) of a light signal as well as relation (\[Series\_B\]) are valid for any kind of configuration between source, body and observer.
Impact vectors in the initial value problem
-------------------------------------------
In the solution for the coordinate velocity (\[First\_Integration\]) and trajectory (\[Second\_Integration\]) of a light signal, the following expressions naturally appear, $$\begin{aligned}
{\mbox{\boldmath$d$}}_A\left(s\right) = {\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$r$}}_A\left(s\right) \times {\mbox{\boldmath$\sigma$}}\right),
\label{Impact_Vector_Sigma_s}
\\
{\mbox{\boldmath$d$}}_A\left(s_0\right) = {\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$r$}}_A\left(s_0\right) \times {\mbox{\boldmath$\sigma$}}\right),
\label{Impact_Vector_Sigma_s0}\end{aligned}$$
where the three-vectors ${\mbox{\boldmath$r$}}_A\left(s\right)$ and ${\mbox{\boldmath$r$}}_A\left(s_0\right)$ are defined by Eqs. (\[vector\_B\]) and (\[vector\_rA\_00\]). The three-vectors (\[Impact\_Vector\_Sigma\_s\]) and (\[Impact\_Vector\_Sigma\_s0\]) and their absolute values are called impact vectors and impact parameters [^6], respectively.
An important condition for the impact parameter $d_A\left(s\right)$ is imposed, which follows from the requirement that the light source should not be screened by the finite disk of the body, $$\begin{aligned}
d_A\left(s\right) \ge P_A \quad {\rm for} \quad {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right) \ge 0\,,
\label{Impact_Vector_Sigma_Constraint_1}\end{aligned}$$
cf. Section 4.2. in [@Article_Zschocke1] for the case of body at rest. If ${\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right) < 0$ then there is no constraint imposed for the impact parameter, $$\begin{aligned}
d_A\left(s\right) \ge 0 \quad {\rm for} \quad {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right) < 0\,.
\label{Impact_Vector_Sigma_Constraint_2}\end{aligned}$$
One may show that (\[Impact\_Vector\_Sigma\_Constraint\_1\]) implies $d_A\left(s_0\right) \ge P_A$ for ${\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right) \ge 0$, which is not an additional request but has the same meaning as (\[Impact\_Vector\_Sigma\_Constraint\_1\]). But because in the near-zone of the Solar System the impact parameter $d_A\left(s_0\right)$ is related to $d_A\left(s\right)$ via a series expansion, there is no need to impose additional constraints on $d_A\left(s_0\right)$. This issue will be considered in more detail within the boundary value problem.
The boundary value problem in 2PN approximation {#Section4}
===============================================
As formulated in the introductory section, a unique solution of (\[Geodetic\_Equation1\]) is also well-defined by boundary conditions, $$\begin{aligned}
{\mbox{\boldmath$x$}}_0 = {\mbox{\boldmath$x$}}\left(t\right)\bigg|_{t=t_0} \quad {\rm and} \quad {\mbox{\boldmath$x$}}_1 = {\mbox{\boldmath$x$}}\left(t\right)\bigg|_{t=t_1}\,,
\label{Boundary_1} \end{aligned}$$
where ${\mbox{\boldmath$x$}}_0$ is the point of emission of the light signal by the source and ${\mbox{\boldmath$x$}}_1$ is the point of reception of the light signal by the observer. The position of the observer ${\mbox{\boldmath$x$}}_1$ in the BCRS is known, while the position of the light source ${\mbox{\boldmath$x$}}_0$ has to be determined by a unique interpretation of astronomical observations, for it is the primary aim of astrometric data reduction.
In the theory of light propagation the unit-vector ${\mbox{\boldmath$k$}}$, which points from the light source towards the position of the observer, is of fundamental importance, $$\begin{aligned}
{\mbox{\boldmath$k$}} &=& \frac{{\mbox{\boldmath$R$}}}{R} \quad {\rm with} \quad {\mbox{\boldmath$R$}} = {\mbox{\boldmath$x$}}_1 - {\mbox{\boldmath$x$}}_0 \quad {\rm and} \quad R = \left|{\mbox{\boldmath$x$}}_1 - {\mbox{\boldmath$x$}}_0\right| \,.
\label{Boundary_3}\end{aligned}$$
A further important unit-vector is the normalized tangent along the light ray at the observer’s position, $$\begin{aligned}
{\mbox{\boldmath$n$}} = \frac{\dot{{\mbox{\boldmath$x$}}}\left(t_1\right)}{\left|\dot{{\mbox{\boldmath$x$}}}\left(t_1\right)\right|}\,.
\label{Tangent_Vector1} \end{aligned}$$
In Figure \[Diagram1\] these unit-vectors ${\mbox{\boldmath$n$}}$ and ${\mbox{\boldmath$k$}}$ are depicted which play the key role in the boundary value problem.
There are two specific cases for the retarded moment of time (\[retarded\_time\_s\]) which are of relevance in the boundary value problem:
\(i) The retarded instant of time $s_0$ with respect to the emission of the light signal at the four-coordinate of source $\left(c t_0, {\mbox{\boldmath$x$}}_0\right)$ (cf. Eq. (\[retarded\_time\_s\_00\])), $$\begin{aligned}
s_0 &=& t_0 - \frac{r_A^{\,0}\left(s_0\right)}{c} \quad {\rm with} \quad r_A^{\,0}\left(s_0\right) = \left|{\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|\,,
\label{retarded_time_s_0}\end{aligned}$$
where $$\begin{aligned}
{\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right) &=& {\mbox{\boldmath$x$}}_0 - {\mbox{\boldmath$x$}}_A\left(s_0\right),
\label{vector_rA_0}\end{aligned}$$
where the upper index $0$ refers to ${\mbox{\boldmath$x$}}_0$ and the argument $s_0$ refers to the body’s position ${\mbox{\boldmath$x$}}_A\left(s_0\right)$; here we notice again that the retarded time in (\[retarded\_time\_s\_0\]) is caused by the finite speed of propagation of gravity which equals the speed of light.
Actually, (\[vector\_rA\_0\]) coincides with (\[vector\_rA\_00\]) in view of ${\mbox{\boldmath$x$}}_0 = {\mbox{\boldmath$x$}}\left(t_0\right)$, but we will keep the notation (\[vector\_rA\_00\]) as is, in order not to change the notation for the initial-value problem as used in [@Zschocke4].
\(ii) The retarded instant of time with respect to the reception of the light signal at the four-coordinate of observer $\left(c t_1, {\mbox{\boldmath$x$}}_1\right)$, $$\begin{aligned}
s_1 &=& t_1 - \frac{r^{\,1}_A\left(s_1\right)}{c} \quad {\rm with} \quad r^{\,1}_A\left(s_1\right) = \left|{\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|\,,
\label{retarded_time_s_1}\end{aligned}$$
where $$\begin{aligned}
{\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) &=& {\mbox{\boldmath$x$}}_1 - {\mbox{\boldmath$x$}}_A\left(s_1\right),
\label{vector_rA_1}\end{aligned}$$
where the upper index $1$ refers to ${\mbox{\boldmath$x$}}_1$ and the argument $s_1$ refers to the body’s position ${\mbox{\boldmath$x$}}_A\left(s_1\right)$; let us recall that the retarded time in (\[retarded\_time\_s\_1\]) is due to the finite speed of propagation of gravity which equals the speed of light.
For the difference between these retarded instants of time the following relation holds $$\begin{aligned}
\fl c \left(s_0 - s_1\right) = \! \bigg(\!r_A^{\,1}\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)
- r^{\,0}_A\left(s_0\right) \! + \! {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\!\!\bigg)
\bigg(\!1 + \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\!\bigg)
\! + \! {\cal O}\left(c^{-2}\right),
\nonumber\\
\label{relation_k_1}\end{aligned}$$
which follows from (\[retarded\_time\_s\_0\]) and (\[retarded\_time\_s\_1\]) as well as (\[Second\_Integration\]) and (\[Boundary\_3\]); cf. Eq. (\[Series\_B\]) in combination with the fact that and ${\mbox{\boldmath$\sigma$}} = {\mbox{\boldmath$k$}} + {\cal O}\left(c^{-2}\right)$ and taking account of the below standing series expansion (\[series\_expansion\_body\_1\]). The relation (\[relation\_k\_1\]) is valid for any kind of configuration between source, body and observer.
Series expansion of the spatial position of the body
----------------------------------------------------
In the near-zone of the Solar System a series expansion of the spatial position of the body becomes meaningful. It is clear that the determination of $s_0$ requires the knowledge of the four-coordinate of the light source $\left(c t_0, {\mbox{\boldmath$x$}}_0\right)$, which initially is unknown but results from data reduction of astrometric observations. On the other side, the determination of $s_1$ requires the four-coordinate of the observer $\left(c t_1, {\mbox{\boldmath$x$}}_1\right)$ as well as the worldline of the body ${\mbox{\boldmath$x$}}_A\left(t\right)$, both of which are fundamental prerequisites for astrometric observations in the near-zone of the Solar System. Usually, the four-coordinates of the observer are provided by optical tracking of the spacecraft, while ${\mbox{\boldmath$x$}}_A\left(t\right)$ is provided by some Solar System ephemeris [@JPL]. Accordingly, we consider a series expansion of the body’s position around $s_1$, $$\begin{aligned}
\fl {\mbox{\boldmath$x$}}_A\left(s_0\right) = {\mbox{\boldmath$x$}}_A\left(s_1\right) + \frac{1}{1!} \,\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,c \left(s_0 - s_1\right)
+ \frac{1}{2!}\,\frac{{\mbox{\boldmath$a$}}_A\left(s_1\right)}{c^2}\,c^2 \left(s_0 - s_1\right)^2 + {\cal O}\left(c^{-3}\right),
\label{series_expansion_body_1}\end{aligned}$$
which relates the spatial position of the body at retarded time $s_1$ and at retarded time $s_0$, and where the expression for $c \left(s_0 - s_1\right)$ is given by Eq. (\[relation\_k\_1\]). The r.h.s. of (\[series\_expansion\_body\_1\]) still depends on $s_0$. So it turns out to be meaningful to introduce a further three-vector which is defined as follows, $$\begin{aligned}
{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right) &=& {\mbox{\boldmath$x$}}_0 - {\mbox{\boldmath$x$}}_A\left(s_1\right) \quad {\rm and} \quad
r^{\,0}_A\left(s_1\right) = \left| {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right) \right|,
\label{vector_rA_0_s1}\end{aligned}$$
where the upper index $0$ refers to ${\mbox{\boldmath$x$}}_0$ and the argument $s_1$ refers to the body’s position ${\mbox{\boldmath$x$}}_A\left(s_1\right)$. Using this three-vector one may show by iterative use of relation (\[series\_expansion\_body\_1\]) that the expression for $c \left(s_0 - s_1\right)$ as given by Eq. (\[relation\_k\_1\]) can also be expressed solely in terms of $s_1$ as follows, $$\begin{aligned}
\fl c \left(s_0 - s_1\right) = \! \bigg(\!r_A^{\,1}\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)
- r^{\,0}_A\left(s_1\right) + {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\!\!\bigg)
\bigg(\!1 + \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\!\bigg)
\! + \! {\cal O}\left(c^{-2}\right).
\nonumber\\
\label{relation_k_2}\end{aligned}$$
The series expansion (\[series\_expansion\_body\_1\]) is absolutely convergent in the near-zone of the Solar System where the time of light propagation is certainly less than the orbital period of any massive body orbiting around the barycenter of the Solar System. That means, according to the convergence criterion [@Mathematical_Methods], the following limit exists [^7] $$\begin{aligned}
\fl L = \lim_{n \rightarrow \infty}
\frac{\displaystyle \left|{\mbox{\boldmath$x$}}_A^{\left(n+1\right)}\left(s_1\right)\right|\,\frac{\left|s_0 - s_1\right|^{n+1}}{\left(n + 1\right)!}}
{\displaystyle \left|{\mbox{\boldmath$x$}}_A^{\left(n\right)}\left(s_1\right)\right|\,\frac{\left|s_0 - s_1\right|^{n}}{n!}}
< 1 \quad {\rm where} \quad
{\mbox{\boldmath$x$}}_A^{\left(n\right)}\left(s_1\right) = \frac{d^n {\mbox{\boldmath$x$}}_A\left(s\right)}{d\,s^n}\bigg|_{s=s_1}\,.
\label{Convergence_Criterion}\end{aligned}$$
Even though that terms proportional to the velocity of the body, ${\mbox{\boldmath$v$}}_A$, can be of the same magnitude or even much larger than the first term on the r.h.s. of the series expansion (\[series\_expansion\_body\_1\]), the series expansion converges so rapidly that just the first few terms up to order ${\cal O}\left(c^{-3}\right)$ were represented, while higher derivatives of the body’s position (jerk-term, snap-term, jounce-term, etc.) are not given explicitly. This fact can be seen by inserting the numerical parameters in Table \[Table1\] into the series expansion (\[series\_expansion\_body\_1\]). Finally, we notice that the expansion (\[series\_expansion\_body\_1\]) implies a series expansion of the spatial velocity of the body, $$\begin{aligned}
\frac{{\mbox{\boldmath$v$}}_A\left(s_0\right)}{c} = \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
+ \frac{{\mbox{\boldmath$a$}}_A\left(s_1\right)}{c^2}\,c \left(s_0 - s_1\right) + {\cal O}\left(c^{-3}\right),
\label{series_expansion_vA_2}\end{aligned}$$
where for $c \left(s_0 - s_1\right)$ one has to use relation (\[relation\_k\_2\]).
Impact vectors in the boundary value problem
--------------------------------------------
For the boundary value problem the relevant impact vectors are defined with respect to the unit vector ${\mbox{\boldmath$k$}}$ in Eq. (\[Boundary\_3\]). As we will see, the impact vector ${\mbox{\boldmath$d$}}^k_A$ at retarded time $s_0$ and $s_1$ will naturally appear in the solution of the boundary value problem, $$\begin{aligned}
{\mbox{\boldmath$d$}}^k_A\left(s_0\right) = {\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right) \times {\mbox{\boldmath$k$}}\right)
= {\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_0\right) \times {\mbox{\boldmath$k$}}\right)\,,
\label{Impact_Vector_k0}
\\
{\mbox{\boldmath$d$}}^k_A\left(s_1\right) = {\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) \times {\mbox{\boldmath$k$}}\right)
= {\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right) \times {\mbox{\boldmath$k$}}\right)\,,
\label{Impact_Vector_k1}\end{aligned}$$
where in the second expression on the r.h.s. in (\[Impact\_Vector\_k0\]) the three-vector $$\begin{aligned}
{\mbox{\boldmath$r$}}^{\,1}_A\left(s_0\right) = {\mbox{\boldmath$x$}}_1 - {\mbox{\boldmath$x$}}_A\left(s_0\right)
\label{vector_rA_1_s0}\end{aligned}$$
has been introduced. The first expression on the r.h.s. in (\[Impact\_Vector\_k0\]) and (\[Impact\_Vector\_k1\]) is regarded as the actual definition of the impact vector, while the second expression on the r.h.s. in (\[Impact\_Vector\_k0\]) and (\[Impact\_Vector\_k1\]) just establishes an equality. The notation impact vector for the three-vectors (\[Impact\_Vector\_k0\]) and (\[Impact\_Vector\_k1\]) becomes evident by their graphical representations as given by the Figures \[Diagram2\], \[Diagram3\] and \[Diagram4\]. For the same reason their absolute values, $$\begin{aligned}
d^k_A\left(s_0\right) = \left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right) \right|
= \left| {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_0\right) \right|\,,
\label{Impact_Parameter_k0}
\\
d^k_A\left(s_1\right) = \left| {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) \right|
= \left| {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right|\,,
\label{Impact_Parameter_k1}\end{aligned}$$
are called impact parameter. Like in Eq. (\[Impact\_Vector\_Sigma\_Constraint\_1\]), for the impact parameter at retarded time $s_1$ the following constraint is imposed, $$\begin{aligned}
d^k_A\left(s_1\right) \ge P_A \quad {\rm for} \quad {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) \ge 0\,,
\label{Impact_Vector_k_Constraint_1}\end{aligned}$$
which generalizes the constraint $d^k_A \ge P_A$ for light propagation in the field of bodies at rest (cf. Section 4.2 in [@Article_Zschocke1]) and just represents the fact that configurations where the light source can be seen by the observer in front of the finite sized body are excluded from the light propagation model. If ${\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) < 0$ then there is no constraint for the impact vector, $$\begin{aligned}
d^k_A\left(s_1\right) \ge 0 \quad {\rm for} \quad {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) < 0\,.
\label{Impact_Vector_k_Constraint_2}\end{aligned}$$
Actually, one may show that (\[Impact\_Vector\_k\_Constraint\_1\]) implies $d^k_A\left(s_0\right) \ge P_A$ if ${\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right) \ge 0$; such a configuration has been represented in Figure \[Diagram3\]. But there is no need for any constraint on the impact parameter $d^k_A\left(s_0\right)$, because this impact parameter is not independent of $d^k_A\left(s_1\right)$. This important issue will be considered in more detail in what follows.
As stated, the impact vectors (\[Impact\_Vector\_k0\]) and (\[Impact\_Vector\_k1\]) are not independent of each other but related via a series expansion. Such a relation is obtained by inserting (\[series\_expansion\_body\_1\]) into (\[vector\_rA\_1\_s0\]) and subsequently into the second term on the r.h.s. of (\[Impact\_Vector\_k0\]), which yields $$\begin{aligned}
\fl \hspace{1.75cm} {\mbox{\boldmath$d$}}^k_A\left(s_0\right) = {\mbox{\boldmath$d$}}^k_A\left(s_1\right)
- \frac{1}{1!}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)\,c \left(s_0 - s_1\right)
\nonumber\\
\nonumber\\
\fl \hspace{4.5cm} - \frac{1}{2!}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$a$}}_A\left(s_1\right)}{c^2} \times {\mbox{\boldmath$k$}} \right)\,c^2 \left(s_0 - s_1\right)^2
+ {\cal O}\left(c^{-3}\right),
\label{Impact_Vector_Relation1}\end{aligned}$$
where $c \left(s_0 - s_1\right)$ is given by Eq. (\[relation\_k\_2\]). For the absolute value we obtain from (\[Impact\_Vector\_Relation1\]) $$\begin{aligned}
\fl \left(d^k_A\left(s_0\right)\right)^2 \! = \left(d^k_A\left(s_1\right)\right)^2
- 2\,{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} c \left(s_0 - s_1\right)
- {\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot \frac{{\mbox{\boldmath$a$}}_A\left(s_1\right)}{c^2} c^2 \left(s_0 - s_1\right)^2
\nonumber\\
\nonumber\\
\fl \hspace{1.75cm} + \left|{\mbox{\boldmath$k$}} \times \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\right|^2\,c^2 \left(s_0 - s_1\right)^2
+ \, {\cal O}\left(c^{-3}\right),
\label{series_expansion_dA_2}\end{aligned}$$
where $\displaystyle {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$d$}}_A^k\left(s_1\right) = 0$ has been used. Whatever we need is a relation between the inverse of $d^k_A\left(s_0\right)$ and the inverse of $d^k_A\left(s_1\right)$. As mentioned above, the terms proportional to the velocity and acceleration of the body might become larger than the first term, hence a series expansion of the inverse of (\[series\_expansion\_dA\_2\]) is not necessarily possible in general. So we will have to use the exact identity, $$\begin{aligned}
\frac{1}{d^k_A\left(s_0\right)} = \frac{1}{d^k_A\left(s_1\right)}
+ \frac{\left(d^k_A\left(s_1\right)\right)^2 - \left(d^k_A\left(s_0\right)\right)^2}
{d^k_A\left(s_0\right)\,d^k_A\left(s_1\right)\,\left(d^k_A\left(s_0\right) + d^k_A\left(s_1\right)\right)} \,.
\label{Relation_d0_1}\end{aligned}$$
The latter is used in evaluating the following expansion of the inverse impact parameter, $$\begin{aligned}
\fl \frac{1}{d^k_A\left(s_0\right)} = \frac{1}{d^k_A\left(s_1\right)}
+ \frac{\displaystyle 2 \,{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot \frac{\displaystyle {\mbox{\boldmath$v$}}_A \left(s_1\right)}{c} c \left(s_0 - s_1\right)}
{d^k_A\left(s_0\right)\,d^k_A\left(s_1\right)\,\left(d^k_A\left(s_0\right) + d^k_A\left(s_1\right)\right)}
\nonumber\\
\nonumber\\
\fl \hspace{1.68cm}
+ \, \frac{\displaystyle {\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot \frac{\displaystyle {\mbox{\boldmath$a$}}_A \left(s_1\right)}{c^2} c^2 \left(s_0 - s_1\right)^2}
{d^k_A\left(s_0\right)\,d^k_A\left(s_1\right)\,\left(d^k_A\left(s_0\right) + d^k_A\left(s_1\right)\right)}
- \frac{\displaystyle \left|{\mbox{\boldmath$k$}} \times \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\right|^2 c^2 \left(s_0 - s_1\right)^2}
{d^k_A\left(s_0\right)\,d^k_A\left(s_1\right)\,\left(d^k_A\left(s_0\right) + d^k_A\left(s_1\right)\right)}
\nonumber\\
\nonumber\\
\fl \hspace{1.68cm} + \, {\cal O}\left(c^{-3}\right),
\label{Relation_d0_4}\end{aligned}$$
which is an incomplete series expansion because the r.h.s. still depends on $d^k_A\left(s_0\right)$.
A comment should be in order about these relations in (\[Relation\_d0\_1\]) and (\[Relation\_d0\_4\]). In contrast to $d^k_A\left(s_1\right)$, which must be larger than the equatorial radius $P_A$ of the massive body as long as ${\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) > 0$, there is no such kind of constraint for the impact parameter $d^k_A\left(s_0\right)$. In other words, the impact parameter $d^k_A\left(s_0\right)$ can become arbitrarily small and might even vanish, so that the limit $d^k_A\left(s_0\right) \rightarrow 0$ is quite possible; cf. the related comment below Eq. (B.12) in [@Zschocke4]. For such cases the relations (\[Relation\_d0\_1\]) and (\[Relation\_d0\_4\]) remain strictly valid, but the expressions on the l.h.s. and r.h.s. of these relations would become arbitrarily large. One has, however, to keep in mind that the inverse of the impact parameter $d^k_A\left(s_0\right)$ is only one piece of a more complex expression which, up to terms of the order ${\cal O}\left(c^{-5}\right)$, remains finite when inserting the r.h.s. of (\[Relation\_d0\_4\]), even in the limit $d^k_A\left(s_0\right) \rightarrow 0$. It is a remarkable feature of the 2PN solution that the constraint (\[Impact\_Vector\_k\_Constraint\_1\]) turns out to be sufficient to keep each term finite in each of the transformations of boundary value problem, regardless of how small the impact parameter $d^k_A\left(s_0\right)$ can be. But one has to bear in mind the fact that the impact vectors, the impact parameters, and the inverse of the impact parameters are not independent of each other, but related via Eqs. (\[Impact\_Vector\_Relation1\]), (\[series\_expansion\_dA\_2\]) and (\[Relation\_d0\_4\]).
Notation of four-vectors
------------------------
In what follows we will determine three fundamental transformations which comprise the boundary value problem, that means the transformations between ${\mbox{\boldmath$\sigma$}}$ in Eq. (\[Introduction\_6\]), ${\mbox{\boldmath$k$}}$ in Eq. (\[Boundary\_3\]), ${\mbox{\boldmath$n$}}$ in Eq. (\[Tangent\_Vector1\]), in their chain of reasoning. But before we proceed further, the following simplifying notation of four-dimensional vectors is introduced, as adopted from [@Kopeikin_Efroimsky_Kaplan; @KS1999; @KopeikinMashhoon2002], $$\begin{aligned}
\sigma^{\mu} = \left(1,{\mbox{\boldmath$\sigma$}}\right), \hspace{3.9cm} \eta_{\mu\nu}\,\sigma^{\mu} \sigma^{\nu} = 0 \,,
\label{Four_Vector_sigma}
\\
k^{\mu} = \left(1,{\mbox{\boldmath$k$}}\right), \hspace{4.0cm} \eta_{\mu\nu}\,k^{\mu} k^{\nu} = 0 \,,
\label{Four_Vector_k}
\\
\hspace{-0.5cm} r_A^{\mu}\left(s\right) = \left(r_A\left(s\right),{\mbox{\boldmath$r$}}_A\left(s\right)\right),
\hspace{1.47cm} \eta_{\mu\nu}\,r_A^{\mu}\left(s\right) r_A^{\nu}\left(s\right) = 0\,.
\label{Four_Vector_r_A}\end{aligned}$$
Each of these four-dimensional quantities, (\[Four\_Vector\_sigma\]) and (\[Four\_Vector\_k\]) and (\[Four\_Vector\_r\_A\]), is a null-vector with respect to the metric tensor $\eta_{\mu\nu}$ of the flat Minkowskian space-time. But one has to take care about their different meaning: the four-vectors (\[Four\_Vector\_sigma\]) and (\[Four\_Vector\_k\]) are, up to terms of the order ${\cal O}\left(c^{-2}\right)$, directed along the light ray which is a Bicharacteristic (\[Biharacteristics\_ED\]) of the covariant Maxwell equations in the curved space-time of the Solar System, while the four-vector (\[Four\_Vector\_r\_A\]) is directed along the Bicharacteristic (\[Biharacteristics\_GR\]) of the field equations of gravity; cf. comments made below Eq. (7.82) in [@Kopeikin_Efroimsky_Kaplan]. These facts allow formally to clearly separate the terms related to the characteristics of the gravity field from those terms related to the light characteristics; cf. text below Eq. (\[2PN\_B\]). But these remarks do not mean, that in concrete experiments the effects related to the speed of gravity can easily and clearly be separated from the effects related to the speed of light; cf. comments below Eqs. (\[Shapiro\_2\]). Furthermore, one should keep in mind that only (\[Four\_Vector\_sigma\]) is actually a physical four-vector, because it is defined in the asymptotic region of the Solar System which is Minkowskian, hence can be interpreted as a four-dimensional arrow pointing from one event to another. On the other side, the four-quantities (\[Four\_Vector\_k\]) and (\[Four\_Vector\_r\_A\]) are introduced as difference of two events in Riemannian space-time, hence they cannot be considered as physical four-vectors in the common sense, because in Riemannian space-time a physical four-vector is a (class of) directional derivative acting on some (arbitrary) scalar function; cf Sec. 9.2. in [@MTW]. Here, we consider four-quantities like (\[Four\_Vector\_sigma\]) - (\[Four\_Vector\_r\_A\]) as purely mathematical objects with whom it is allowed to apply usual vectorial operations; cf. text below Eq. (\[Relation\_alpha\]). In the solution of the light trajectory one encounters terms which, in the sense just described, are called four-scalars between the four-vectors $\sigma_{\mu} = \left(-1,{\mbox{\boldmath$\sigma$}}\right)$, $k_{\mu} = \left(-1,{\mbox{\boldmath$k$}}\right)$ and $r_A^{\mu}\left(s\right)$, given by [^8], $$\begin{aligned}
\sigma \cdot r_A\left(s\right) \equiv \sigma_{\mu}\,r_A^{\mu}\left(s\right)
= - \left(r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)\right),
\label{scalar_product_sigma}
\\
k \cdot r_A\left(s\right) \equiv k_{\mu}\,r_A^{\mu}\left(s\right) = - \left(r_A\left(s\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)\right).
\label{scalar_product}\end{aligned}$$
These four-vectors in (\[Four\_Vector\_sigma\]) and (\[Four\_Vector\_r\_A\]) and their scalar-product (\[scalar\_product\_sigma\]) do naturally appear as arguments of vectorial functions in the solution of the initial-boundary value problem for the light trajectory, while the four-vectors in (\[Four\_Vector\_k\]) and (\[Four\_Vector\_r\_A\]) and their scalar-product (\[scalar\_product\]) do naturally appear as arguments of vectorial functions in the solution of the boundary value problem for the light trajectory.
Here, we just have introduced the above standing notation in order to simplify the mathematical expressions in the boundary value problem of the theory of light propagation. In particular, for the two specific four-vectors, $$\begin{aligned}
r^{\,0\;\mu}_A \left(s_0\right) = \left(r^{\,0}_A \left(s_0\right), {\mbox{\boldmath$r$}}^{\,0}_A \left(s_0\right)\right) \quad {\rm where} \quad
r^{\,0}_A \left(s_0\right) = \left|{\mbox{\boldmath$r$}}^{\,0}_A \left(s_0\right)\right|,
\label{Four_Vector_1}
\\
r^{\,1\;\mu}_A \left(s_1\right) = \left(r^{\,1}_A\left(s_1\right), {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) \quad {\rm where} \quad
r^{\,1}_A \left(s_1\right) = \left|{\mbox{\boldmath$r$}}^{\,1}_A \left(s_1\right)\right|,
\label{Four_Vector_2}\end{aligned}$$
we obtain the following specific cases of four-scalar products $$\begin{aligned}
k \cdot r_A^{\,0}\left(s_0\right) = - \left(r_A^{\,0}\left(s_0\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right),
\label{scalar_product_1}
\\
k \cdot r^{\,1}_A\left(s_1\right) = - \left(r^{\,1}_A\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right),
\label{scalar_product_2}\end{aligned}$$
where the upper indices $0$ and $1$ refer to ${\mbox{\boldmath$x$}}_0$ and ${\mbox{\boldmath$x$}}_1$, respectively, as introduced in Eqs. (\[vector\_rA\_0\]) and (\[vector\_rA\_1\]), so they are of course not Lorentz indices. In line with this notation we also need to introduce the four-vector $$\begin{aligned}
r^{\,0\;\mu}_A \left(s_1\right) = \left(r^{\,0}_A \left(s_1\right), {\mbox{\boldmath$r$}}^{\,0}_A \left(s_1\right)\right) \quad {\rm where} \quad
r^{\,0}_A \left(s_1\right) = \left|{\mbox{\boldmath$r$}}^{\,0}_A \left(s_1\right)\right|,
\label{Four_Vector_3}\end{aligned}$$
and the four-scalar product $$\begin{aligned}
k \cdot r_A^{\,0}\left(s_1\right) = - \left(r_A^{\,0}\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right),
\label{scalar_product_3}\end{aligned}$$
where the three-vector ${\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)$ and its absolute value $r^{\,0}_A\left(s_1\right) = \left|{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right|$ were defined by (\[vector\_rA\_0\_s1\]).
Transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$ {#Section5}
===========================================================================
The most important relation in the formulation of the boundary value problem concerns the transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$, where the unit tangent vector ${\mbox{\boldmath$\sigma$}}$ of the light ray at past null infinity is defined by Eq. (\[Introduction\_6\]), while the unit vector ${\mbox{\boldmath$k$}}$ is defined by Eq. (\[Boundary\_3\]) and determines the unit direction from the light source towards the observer.
The implicit expression for the transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$
-----------------------------------------------------------------------------------------------------------
From (\[Second\_Integration\]) one finds the following formal expression, $$\begin{aligned}
\fl _{\rm N} & \hspace{-1.7cm} \biggr|& \hspace{-1.5cm} {{\mbox{\boldmath$\sigma$}}} = {{\mbox{\boldmath$k$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN}& \hspace{-1.7cm}\biggr|& \hspace{-1.5cm}
+ \frac{m_A}{R} \,
\Bigg({\mbox{\boldmath$k$}} \times \bigg[ {\mbox{\boldmath$k$}} \times
\left({\mbox{\boldmath$B$}}_1 \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}_1 \left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)\right) \bigg] \Bigg)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN}& \hspace{-1.7cm}\biggr|& \hspace{-1.5cm}
+ \frac{m_A}{R} \,
\Bigg({\mbox{\boldmath$k$}} \times \bigg[ {\mbox{\boldmath$k$}} \times
\left({\mbox{\boldmath$B$}}^A_2 \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right),{\mbox{\boldmath$v$}}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}^A_2\left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right),{\mbox{\boldmath$v$}}_A\left(s_1\right)\right)\right)
\bigg] \Bigg)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN}& \hspace{-1.7cm}\biggr|& \hspace{-1.5cm}
+ \frac{m_A}{R} \,
\Bigg({\mbox{\boldmath$k$}} \times \bigg[ {\mbox{\boldmath$k$}} \times
\left({\mbox{\boldmath$B$}}^B_2 \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right),{\mbox{\boldmath$v$}}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}^B_2\left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right),{\mbox{\boldmath$v$}}_A\left(s_0\right)\right)\right)
\bigg] \Bigg)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{-1.7cm}\biggr|& \hspace{-1.5cm}
+ \frac{m_A^2}{R} \,
\Bigg({\mbox{\boldmath$k$}} \times \bigg[ {\mbox{\boldmath$k$}} \times
\left({\mbox{\boldmath$B$}}_3\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}_3\left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)\right) \bigg] \Bigg)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{-1.7cm}\biggr|& \hspace{-1.5cm}
+ \frac{m_A^2}{R^2}\,\bigg[{\mbox{\boldmath$B$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}_1\left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)\bigg]
\times \bigg[{\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$B$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}_1\left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)\right)\bigg]
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{-1.7cm}\biggr|& \hspace{-1.5cm}
- \frac{3}{2}\,\frac{m_A^2}{R^2}\;{\mbox{\boldmath$k$}}\;
\bigg|{\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$B$}}_1 \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}_1 \left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)\right)\bigg|^2
+ \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_0\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN}& \hspace{-1.7cm}\biggr|& \hspace{-1.5cm}
+ {\cal O}\left(c^{-5}\right),
\label{Transformation_k_to_sigma_5}\end{aligned}$$
where relation (\[appendix\_E\_5\]) has been used in order to deduce (\[Transformation\_k\_to\_sigma\_5\]). From (\[Transformation\_k\_to\_sigma\_5\]) follows that ${\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$\sigma$}} = 1 + {\cal O}\left(c^{-5}\right)$ so that ${\mbox{\boldmath$\sigma$}}$ is still a unit vector up to terms beyond 2PN approximation. In the limit of body at rest the transformation (\[Transformation\_k\_to\_sigma\_5\]) agrees with Eq. (3.2.50) in [@Brumberg1991] and with Eq. (68) in [@Article_Zschocke1].
The meaning of the notation in the transformation (\[Transformation\_k\_to\_sigma\_5\]) and in each of the subsequent transformations is the following: 1PN terms are proportional to $m_A$, 1.5PN terms are proportional to $m_A\,v_A/c$, 2PN terms are proportional to either $m^2_A$ or $m_A\,v_A^2/c^2$.
The vectorial functions ${\mbox{\boldmath$B$}}_1\,,\dots\,,{\mbox{\boldmath$B$}}_3$ are given by Eqs. (\[Vectorial\_Function\_C1\]) - (\[Vectorial\_Function\_C3\]) in the \[Appendix2\], while $\hat{{\mbox{\boldmath$\epsilon$}}}_2$ is given by Eq. (\[epsilon2\]) in the \[Appendix\_epsilon\]. The expression for $R$ is given by Eq. (\[Boundary\_3\]). Furthermore, the three-vector ${\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)$ is given by (\[vector\_rA\_0\]), while the three-vector ${\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)$ is given by (\[vector\_rA\_1\]).
The vectorial functions ${\mbox{\boldmath$B$}}_1\,,\dots\,,{\mbox{\boldmath$B$}}_3$ as well as $\hat{{\mbox{\boldmath$\epsilon$}}}_2$ depend on ${\mbox{\boldmath$\sigma$}}$ rather than ${\mbox{\boldmath$k$}}$. Therefore, the expression (\[Transformation\_k\_to\_sigma\_5\]) represents, as it stands, an implicit form of the transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$. The explicit transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$ is arrived within the next section.
The explicit expression for the transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$
-----------------------------------------------------------------------------------------------------------
In the given approximation one may immediately replace ${\mbox{\boldmath$\sigma$}}$ by ${\mbox{\boldmath$k$}}$ in the 1.5PN and 2PN terms, because it would cause an error of the order ${\cal O}\left(c^{-5}\right)$ which is beyond 2PN approximation. That means, in the vectorial functions of the third until the seventh line in (\[Transformation\_k\_to\_sigma\_5\]) one may substitute ${\mbox{\boldmath$\sigma$}}$ by ${\mbox{\boldmath$k$}}$, while in the vectorial function in the second line in (\[Transformation\_k\_to\_sigma\_5\]) one needs to have the relation between ${\mbox{\boldmath$\sigma$}}$ and ${\mbox{\boldmath$k$}}$ in 1PN approximation as given by (\[appendix\_E\_10\]), which subsequently yields Eqs. (\[Relation\_Impact\_Vectors\_1\]) and (\[appendix\_E\_15\]). Using these relations one finally arrives at the following explicit expression for the transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$: $$\begin{aligned}
\fl _{\rm N} & \hspace{-0.15cm} \biggr|& {{\mbox{\boldmath$\sigma$}}} = {{\mbox{\boldmath$k$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_1}\biggr|&
- 2\,\frac{m_A}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_2}\biggr|&
+ 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_3}\biggr|&
- 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
+ 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_0\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_4}\biggr|&
+ 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_0\right)}{c}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl _{{\rm scaling}\;{\rm 2PN}}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_5} \biggr|&
- 2\,\frac{m_A^2}{R^2}{\mbox{\boldmath$k$}}
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right|^2
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_6}\biggr|&
- 2 \frac{m_A^2}{R^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)} + \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right|^2
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_7}\biggr|&
- 4\,\frac{m_A^2}{R}\,\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left(k \cdot r_A^{\,0}\left(s_0\right)\right)^2}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{-0.55cm} _{{\mbox{\boldmath$\rho$}}^A_8}\biggr|&
+ \frac{15}{4} \frac{m_A^2}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^3} \,
\left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)
\left(\arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|} + \frac{\pi}{2} \right)
\nonumber\\
\fl _{\rm 2PN}& \hspace{-0.55cm} _{{\mbox{\boldmath$\rho$}}^B_8}\biggr|&
- \frac{15}{4} \frac{m_A^2}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|^3} \,
\left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)
\left( \arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|} + \frac{\pi}{2}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_9}\biggr|&
- \frac{1}{4}\frac{m_A^2}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(r^{\,1}_A\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left(r^{\,0}_A\left(s_0\right)\right)^2}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{-0.15cm} \biggr|& + \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_0\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN}& \hspace{-0.15cm} \biggr|&
+ {\cal O}\left(c^{-5}\right),
\label{Transformation_k_to_sigma}\end{aligned}$$
where ${\mbox{\boldmath$\rho$}}_i = {\mbox{\boldmath$\rho$}}_i\left(s_1,s_0\right)$ with $i=1\,, \cdots \,,9$ that appear before the vertical lines are by definition equal to the expressions on the right of the vertical bars in each line, and the term $\hat{{\mbox{\boldmath$\epsilon$}}}_2$ is given by Eq. (\[Transformation\_k\_to\_sigma\_epsilon\]) in the \[Appendix\_epsilon\].
The transformation (\[Transformation\_k\_to\_sigma\]) allows the determination of ${\mbox{\boldmath$\sigma$}}$ for the given boundary conditions ${\mbox{\boldmath$x$}}_0$ and ${\mbox{\boldmath$x$}}_1$. In the limit of bodies at rest the relation (\[Transformation\_k\_to\_sigma\]) is in agreement with the expression as given by Eq. (3.2.52) in [@Brumberg1991] and Eq. (74) in [@Article_Zschocke1].
The term ${\mbox{\boldmath$\rho$}}_5$ in (\[Transformation\_k\_to\_sigma\]) is proportional to vector ${\mbox{\boldmath$k$}}$ and originates from the terms in the last two lines of (\[Transformation\_k\_to\_sigma\_5\]), where the vectorial relation ${\mbox{\boldmath$a$}} \times \left({\mbox{\boldmath$b$}} \times {\mbox{\boldmath$c$}}\right) = {\mbox{\boldmath$b$}} \left({\mbox{\boldmath$a$}} \cdot {\mbox{\boldmath$c$}} \right) - {\mbox{\boldmath$c$}} \left({\mbox{\boldmath$a$}} \cdot {\mbox{\boldmath$b$}} \right)$ has been used. In the transformation (\[Transformation\_k\_to\_sigma\]) a term proportional to vector ${\mbox{\boldmath$k$}}$ does not influence the angle $\delta\left({\mbox{\boldmath$\sigma$}},{\mbox{\boldmath$k$}}\right)$, which can be computed from the vector product ${\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$k$}}$. The only impact of that term proportional to vector ${\mbox{\boldmath$k$}}$ is to keep the vector ${\mbox{\boldmath$\sigma$}}$ to have unit length. Therefore, all terms proportional to vector ${\mbox{\boldmath$k$}}$ will be called [*scaling terms*]{}.
In anticipation of subsequent considerations, the notation [*enhanced 2PN terms*]{} in (\[Transformation\_k\_to\_sigma\]) for the 2PN terms ${\mbox{\boldmath$\rho$}}_6$ and ${\mbox{\boldmath$\rho$}}_7$ has been introduced whose meaning is as follows. The estimation of the upper limit of the sum of these terms is given by Eq. (\[angle\_2PN\_1\]), which recovers that their upper limit is proportional to the large factor $r^{\,1}_A\left(s_1\right)/P_A$ and are, therefore, called [*enhanced 2PN terms*]{} in (\[Transformation\_k\_to\_sigma\]) in order to distinguish them from standard 2PN terms in (\[Transformation\_k\_to\_sigma\]) which do not contain such a large factor. Originally, [*enhanced terms*]{} have been recovered for the case of 2PN light propagation in the field of one monopole at rest [@Article_Zschocke1; @Teyssandier; @AshbyBertotti2010]. In our detailed investigation in [@Article_Zschocke1] for light propagation in the gravitational field of one monopole at rest we have demonstrated that the mathematical origin of [*enhanced 2PN terms*]{} is solely caused by iterative procedure of the integration of the geodesic equation. The same conclusion is valid for the case of light propagation in the gravitational field of one monopole in motion. That means, solving iteratively the geodesic equation in 1PN approximation (i.e. the first four terms on the r.h.s. in the first line of Eq. (31) in [@Zschocke4] or Eq. (45) in [@Zschocke1] where the metric in 1PN approximation is given by the first two terms in Eq. (24) in [@Zschocke4]) then the first iteration contains terms proportional to $m_A$, the second iteration contains terms proportional to $m^2_A$, and so on. Using this iterative approach it is inevitable to encounter these so-called [*enhanced terms*]{} all of which contain that typical enhancing factor $r^{\,1}_A\left(s_1\right)/P_A$. It should also be noticed that the [*enhanced terms*]{} impose no limit on the distance between observer and light source, but impose only a constraint on the distance between observer and light-ray deflecting body. Here, we consider light deflection caused by Solar System bodies, where the distance between observer and massive body is limited by the near-zone of the Solar System as given by Eq. (\[Near\_Zone\_2\]). That fact elucidates the limitation of the post-Newtonian approach which is not applicable for the far-zone of the Solar System.
The simplified expression for the transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$
-------------------------------------------------------------------------------------------------------------
Two comments are in order about the transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$ as given by Eq. (\[Transformation\_k\_to\_sigma\]):
1\. The transformation (\[Transformation\_k\_to\_sigma\]) is of rather involved structure. In order to simplify the transformation one has to neglect all those terms whose magnitude it smaller than the envisaged accuracy of $1\,{\rm nas}$ in light deflection.
2\. The transformation (\[Transformation\_k\_to\_sigma\]) depends on the variables $m_A$, ${\mbox{\boldmath$x$}}_0$, ${\mbox{\boldmath$x$}}_1$, ${\mbox{\boldmath$x$}}_A\left(s_0\right)$ and ${\mbox{\boldmath$x$}}_A\left(s_1\right)$. As mentioned, while the four-coordinates of the observer $\left(c t_1,{\mbox{\boldmath$x$}}_1\right)$ are precisely known and the fundamental prerequisite of any astrometric measurement, the four-coordinates of the light source $\left(c t_0,{\mbox{\boldmath$x$}}_0\right)$ are not directly accessible but follow from data reduction of the astronomical observations. Stated differently, while the retarded instant of time $s_1$ as defined by (\[retarded\_time\_s\_1\]) is precisely known from the very beginning, the retarded instant of time $s_0$ as defined by Eq. (\[retarded\_time\_s\_0\]) is, first of all, an unknown parameter in the theory of light propagation.
In conclusion of these comments it becomes clear that practical astrometry necessitates a transformation (\[Transformation\_k\_to\_sigma\]) solely in terms of $s_1$ and where all those terms are neglected which contribute less than the given goal accuracy of $1\,{\rm nas}$ in light deflection. Such a transformation is obtained by means of a series expansion of each individual term in (\[Transformation\_k\_to\_sigma\]) around $s_1$, which reads $$\begin{aligned}
\fl \hspace{1.5cm} {\mbox{\boldmath$\rho$}}_i\left(s_1,s_0\right) = {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)
+ {\cal O}\left(c^{-5}\right) \quad {\rm for} \quad i = 1\,, \cdots \,, 4\,.
\label{series_expansion_rho}
\\
\fl \hspace{1.5cm} {\mbox{\boldmath$\rho$}}_i\left(s_1,s_0\right) = {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right)
\quad {\rm for} \quad i = 5\,, \cdots\,, 9\,.
\label{series_expansion_rho_2PN}
\\
\fl \hspace{1.5cm} \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_0\right) = \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right).
\label{series_expansion_epsilon_2} \end{aligned}$$
In \[Appendix\_EstimationA\] the results for the upper limits are presented, while the approach is described in \[Appendix\_Estimation1\] and a detailed example is given in \[Appendix\_Example\]. The results for the upper limits are given by Eqs. (\[Estimation\_rho1\]) and (\[Example\_rho1\_Delta\]), Eqs. (\[Term\_rho\_2\_2\]) and (\[Term\_rho\_2\_1\_E\]), Eqs. (\[Term\_rho\_3\_3\_A\]) and (\[Term\_rho\_3\_4\]), Eqs. (\[Estimation\_rho\_4\_1\]) and (\[Estimation\_rho\_4\_2\]), as well as Eqs. (\[Term\_rho\_5\_2\]), (\[Term\_rho\_6\_2\]), (\[Term\_rho\_7\_2\]), (\[Term\_rho\_8\_2\]), (\[Term\_rho\_9\_2\]), and (\[Appendix\_estimation\_epsilon\_3\]). Numerical values for the upper limits are given in Table \[Table2\]. These results can be summarized as follows: $$\begin{aligned}
\fl \hspace{1.5cm} \left|{\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)\right| \;\;\le 1\,{\rm nas}\;, \quad i = 3,5,8,9\,.
\label{series_expansion_rho_result_1}
\\
\nonumber\\
\fl \hspace{1.5cm} \left|\Delta {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)\right| \le 1\,{\rm nas}\;, \quad i = 1\,,2\,,3\,,4\,.
\label{series_expansion_rho_result_2}
\\
\nonumber\\
\fl \hspace{1.5cm} \left|\hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_1\right)\right| \;\;\; \le 1\,{\rm nas}\,.
\label{series_expansion_epsilon_2_result}\end{aligned}$$
Besides the fact that the absolute value of the [*scaling term*]{} ${\mbox{\boldmath$\rho$}}_5$ is less than $1\,{\rm nas}$, that term can be omitted anyway, because, as stated above already, it has no impact on the angle $\delta\left({\mbox{\boldmath$\sigma$}},{\mbox{\boldmath$k$}}\right)$ between ${\mbox{\boldmath$\sigma$}}$ and ${\mbox{\boldmath$k$}}$. For the absolute value of the total sum of all those neglected terms (\[series\_expansion\_rho\_result\_1\]) - (\[series\_expansion\_epsilon\_2\_result\]) which are not proportional to the three-vector ${\mbox{\boldmath$k$}}$, we get $$\begin{aligned}
\fl I_1 = \left|\sum\limits_{i=3,8,9} {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)
+ \sum\limits_{i=1,2,3,4} \Delta {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right) + \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_1\right)\right|
\nonumber\\
\nonumber\\
\fl \hspace{0.35cm} \le\, \frac{6\,m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c} + \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}
+ \frac{6\,m_A}{P_A}\,\frac{v_A^2\left(s_1\right)}{c^2} + \frac{6\,m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 18\,m_A\,\frac{a_A\left(s_1\right)}{c^2}\,.
\nonumber\\
\label{Sum_1}\end{aligned}$$
For the upper limits of the terms in (\[Sum\_1\]) we have used that $$\begin{aligned}
\fl \hspace{1.7cm} \left| {\mbox{\boldmath$\rho$}}_8 + {\mbox{\boldmath$\rho$}}_9 \right| \le \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}\,,
\label{Inequality_Sum_1_A}
\\
\nonumber\\
\fl \hspace{2.2cm} \left|\Delta {\mbox{\boldmath$\rho$}}_1 \right|
\le 6\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Inequality_Sum_1_B}
\\
\nonumber\\
\fl \left|\Delta {\mbox{\boldmath$\rho$}}_2 + \Delta {\mbox{\boldmath$\rho$}}_3 + \Delta {\mbox{\boldmath$\rho$}}_4 \right|
\le 6\,\frac{m_A}{r_A^{\,1}\left(s_1\right)}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 4\,\frac{m_A}{P_A}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 8\,m_A\,\frac{a_A\left(s_1\right)}{c^2}\,,
\label{Inequality_Sum_1_C}\end{aligned}$$
while $\left|{\mbox{\boldmath$\rho$}}_3\left(s_1,s_1\right)\right| = 0$ according to Eq. (\[Term\_rho\_3\_3\_A\]). The inequality (\[Inequality\_Sum\_1\_A\]) is not shown explicitly, but follows by using the approach as described in \[Appendix\_Estimation1\]. The inequality (\[Inequality\_Sum\_1\_B\]) has been shown in \[Appendix\_Example\] and is given by Eq. (\[Term\_rho\_10\_2\]). The inequality (\[Inequality\_Sum\_1\_C\]) follows from (\[Term\_rho\_2\_1\_E\]), (\[Term\_rho\_3\_4\]), and (\[Estimation\_rho\_4\_2\]), while the upper limit of $\left|\hat{{\mbox{\boldmath$\epsilon$}}}_2\right|$ is given by Eq. (\[Appendix\_estimation\_epsilon\_3\]). Using the numerical parameters as given by Table \[Table1\] one obtains $$\begin{aligned}
\fl \hspace{0.75cm} I_1 \le \; 1.0\,{\rm nas} \quad {\rm for}\;{\rm Sun}\;{\rm at}\,45^{\circ}\,\left({\rm solar}\;{\rm aspect}\;{\rm angle}\right),
\nonumber\\
\fl \hspace{1.15cm} \le \; 1.1\,{\rm nas} \quad {\rm for}\;{\rm Jupiter}\,,
\label{Total_Sum_1}\end{aligned}$$
and less than $0.38\,{\rm nas}$ for any other Solar System body. Accordingly, the terms (\[series\_expansion\_rho\_result\_1\]) - (\[series\_expansion\_epsilon\_2\_result\]) can be neglected for sub-$\,$ astrometry and even for astrometry on the level of $1.1\;{\rm nas}$ in light deflection. In this way one obtains the simplified transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$ fully in terms of $s_1$: $$\begin{aligned}
\fl _{\rm N} & \hspace{-0.15cm} \biggr|& {{\mbox{\boldmath$\sigma$}}} = {{\mbox{\boldmath$k$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_1}\biggr|&
- 2\,\frac{m_A}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_2}\biggr|&
+ 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_4}\biggr|&
+ 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_6}\biggr|&
- 2 \frac{m_A^2}{R^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)} + \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right)
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right|^2
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}}& \hspace{-0.5cm} _{{\mbox{\boldmath$\rho$}}_7}\biggr|&
- 4\,\frac{m_A^2}{R}\,\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r_A^{\,0}\left(s_1\right)\right)^2}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN}& \hspace{-0.15cm} \biggr|&
+ {\cal O}\left(c^{-5}\right),
\label{Simplified_Transformation_k_to_sigma}\end{aligned}$$
where ${\mbox{\boldmath$\rho$}}_i = {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)$ with $i=1,2,4,6,7$ that appear before the vertical lines are by definition equal to the expressions on the right of the vertical bars in each line. In the limit of monopole at rest this expression coincides with Eqs. (79) - (80) in [@Article_Zschocke1]. Using the approach and the results in the appendix one obtains for the upper limits of the 1PN, 1.5PN, and 2PN terms in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]): $$\begin{aligned}
\fl {\rm 1PN} \hspace{2.75cm}
\left|{\mbox{\boldmath$\rho$}}_1 \right| \le 4\,\frac{m_A}{P_A}\,,
\label{angle_1PN_1}
\\
\nonumber\\
\fl {\rm 1.5PN} \hspace{1.75cm}
\left|{\mbox{\boldmath$\rho$}}_2 + {\mbox{\boldmath$\rho$}}_4\right| \le 6\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{angle_15PN_1}
\\
\nonumber\\
\fl {\rm enhanced}\;{\rm 2PN} \hspace{0.5cm} \left|{\mbox{\boldmath$\rho$}}_6 + {\mbox{\boldmath$\rho$}}_7 \right| \le 16\,
\frac{m^2_A}{P^2_A}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,.
\label{angle_2PN_1}\end{aligned}$$
The reason of why there is a factor $6$ in (\[angle\_15PN\_1\]) rather than a factor $4$ is discussed in the text below Eq. (\[angle\_2PN\_3\]). In the limit of body at rest, the results (\[angle\_1PN\_1\]) and (\[angle\_2PN\_1\]) would coincide with Eqs. (76) and (77) in [@Article_Zschocke1], respectively.
The simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]) depends on the variables $m_A$, ${\mbox{\boldmath$x$}}_0$, ${\mbox{\boldmath$x$}}_1$, and ${\mbox{\boldmath$x$}}_A\left(s_1\right)$ and does not any longer depend on the retarded time $s_0$. The only unknown in (\[Simplified\_Transformation\_k\_to\_sigma\]) is the three-coordinate of the light source, ${\mbox{\boldmath$x$}}_0$, whose determination is the primary aim of data reduction of astronomical observations. For the neglected terms of the order ${\cal O}\left(c^{-5}\right)$ (2.5PN approximation) and of the order ${\cal O}\left(c^{-6}\right)$ (3PN approximation) we refer to Section \[Section\_3PN\], where some statements about their impact on light deflection are given.
Transformation from ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ {#Section6}
===========================================================================
Now we consider the transformation from ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$, where ${\mbox{\boldmath$\sigma$}}$ is the unit tangent vector along the light trajectory at past null infinity as defined by (\[Introduction\_6\]), while ${\mbox{\boldmath$n$}}$ is the unit tangent vector along the light trajectory at the observer’s position as defined by Eq. (\[Tangent\_Vector1\]).
The implicit expression for the transformation from ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$
-----------------------------------------------------------------------------------------------------------
By inserting (\[First\_Integration\]) into (\[Tangent\_Vector1\]) one obtains $$\begin{aligned}
\fl _{\rm N} & \hspace{-1.7cm} \biggr|& \hspace{-1.5cm} {{\mbox{\boldmath$n$}}} = {{\mbox{\boldmath$\sigma$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN} & \hspace{-1.7cm} \biggr|& \hspace{-1.5cm}
+ m_A\,{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) \times {\mbox{\boldmath$\sigma$}}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-1.7cm} \biggr|& \hspace{-1.5cm}
+ m_A\,
{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$A$}}_2\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right), {\mbox{\boldmath$v$}}_A\left(s_1\right)\right) \times {\mbox{\boldmath$\sigma$}}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-1.7cm} \biggr|& \hspace{-1.5cm}
+ m_A^2\,{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$A$}}_3\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) \times {\mbox{\boldmath$\sigma$}}\right)
- m_A^2\, {\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) \left({\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-1.7cm} \biggr|& \hspace{-1.5cm}
- \frac{1}{2}\,m_A^2\,{\mbox{\boldmath$\sigma$}}
\left({\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) \cdot {\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)\right)
+ \frac{3}{2}\,m_A^2\,{\mbox{\boldmath$\sigma$}} \left({\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)\right)^2
+ \hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN} & \hspace{-1.7cm} \biggr|& \hspace{-1.5cm}
+ {\cal O}\left(c^{-5}\right),
\label{Transformation_sigma_to_n_5} \end{aligned}$$
where the vectorial functions ${\mbox{\boldmath$A$}}_1$, ${\mbox{\boldmath$A$}}_2$, ${\mbox{\boldmath$A$}}_3$ are given by Eqs. (\[Vectorial\_Function\_A1\]) - (\[Vectorial\_Function\_A3\]) in the \[Appendix2\], while expression $\hat{{\mbox{\boldmath$\epsilon$}}}_1$ has been given by Eq. (\[epsilon1\]) in the \[Appendix\_epsilon\]. From (\[Transformation\_sigma\_to\_n\_5\]) follows that ${\mbox{\boldmath$n$}} \cdot {\mbox{\boldmath$n$}} = 1 + {\cal O}\left(c^{-5}\right)$ so that ${\mbox{\boldmath$n$}}$ is still a unit vector up to terms beyond 2PN approximation. The vectorial functions ${\mbox{\boldmath$A$}}_1\,,\,\dots\,,{\mbox{\boldmath$A$}}_3$ as well as $\hat{{\mbox{\boldmath$\epsilon$}}}_1$ depend on vector ${\mbox{\boldmath$\sigma$}}$. However, the aim is to achieve the transformation from ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ in terms of the boundary values (\[Boundary\_1\]), which implies to express these vectorial functions in terms of ${\mbox{\boldmath$k$}}$ rather than ${\mbox{\boldmath$\sigma$}}$. This will be done in the next section.
The explicit expression for the transformation from ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$
-----------------------------------------------------------------------------------------------------------
In the 1.5PN and 2PN terms one may immediately replace ${\mbox{\boldmath$\sigma$}}$ by ${\mbox{\boldmath$k$}}$, because such a replacement would cause an error of the order ${\cal O}\left(c^{-5}\right)$ which is beyond 2PN approximation. That means in the vectorial functions of the third until the fifth line in (\[Transformation\_sigma\_to\_n\_5\]) one may substitute ${\mbox{\boldmath$\sigma$}}$ by ${\mbox{\boldmath$k$}}$, while in the vectorial function in the second line in (\[Transformation\_sigma\_to\_n\_5\]) one has to use relation (\[appendix\_E\_10\]) and Eqs. (\[Relation\_Impact\_Vectors\_1\]) and (\[appendix\_E\_15\]). Using these relations one finally arrives at the following explicit expression for the transformation from ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$: $$\begin{aligned}
\fl _{\rm N} & \hspace{-0.1cm} \biggr|& \hspace{+0.05cm} {{\mbox{\boldmath$n$}}} = {{\mbox{\boldmath$\sigma$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_1}\biggr|&
+ 2\,m_A\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\nonumber\\
\nonumber\\
\fl _{{\rm scaling}\;{\rm 1.5PN}}& \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_2} \biggr|&
- 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,{\mbox{\boldmath$k$}}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_3} \biggr|&
- 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}}\cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_4} \biggr|&
+ 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
+ \frac{2\,m_A}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,{\mbox{\boldmath$d$}}^k_A\left(s_1\right)\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_5} \biggr|&
+ \frac{2\,m_A}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{{\rm scaling}\;{\rm 2PN}} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_6} \biggr|& - 2\,{\mbox{\boldmath$k$}}\,\frac{m_A^2}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\nonumber\\
\nonumber\\
\fl _{{\rm scaling}\;{\rm 2PN}} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_7} \biggr|&
+ 4\,\frac{{\mbox{\boldmath$k$}}}{R}\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_8} \biggr|&
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_9} \biggr|&
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)} \right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.55cm} _{{\mbox{\boldmath$\varphi$}}_{10}} \biggr|&
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)} \right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.55cm} _{{\mbox{\boldmath$\varphi$}}_{11}} \biggr|&
- 4\,\frac{m_A^2}{\left(r_A^{\,1}\left(s_1\right)\right)^2}\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.55cm} _{{\mbox{\boldmath$\varphi$}}_{12}} \biggr|&
- \frac{m_A^2}{2}\,{\mbox{\boldmath$d$}}_A^k\left(s_1\right)\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left(r^{\,1}_A\left(s_1\right)\right)^4}
- \frac{15}{4}\,\frac{m_A^2}{\left(r_A^{\,1}\left(s_1\right)\right)^2}\,{\mbox{\boldmath$d$}}^k_A\left(s_1\right)
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^2}
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.55cm} _{{\mbox{\boldmath$\varphi$}}_{13}} \biggr|& - \frac{15}{4}\,m_A^2\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^3}
\left(\arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|} + \frac{\pi}{2}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.025cm} \biggr|& + \hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN} & \hspace{-0.025cm} \biggr|& + {\cal O}\left(c^{-5}\right),
\label{Transformation_sigma_to_n}\end{aligned}$$
where ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1\right)$ for $i=1\,, \cdots \,,6,8,11,12,13$ and ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_0\right)$ for $i = 7,9,10$ that appear before the vertical lines are by definition equal to the expressions on the right of the vertical bars in each line, while the term $\hat{{\mbox{\boldmath$\epsilon$}}}_1$ is given by Eq. (\[Transformation\_sigma\_to\_n\_epsilon\]) in the \[Appendix\_epsilon\]. With the aid of the transformation (\[Transformation\_sigma\_to\_n\]) one may determine the difference between the vectors ${\mbox{\boldmath$n$}}$ and ${\mbox{\boldmath$\sigma$}}$ from the given boundary conditions ${\mbox{\boldmath$x$}}_0$ and ${\mbox{\boldmath$x$}}_1$. In the limit of bodies at rest the relation (\[Transformation\_sigma\_to\_n\]) is in agreement with the expression as given by Eq. (81) in [@Article_Zschocke1].
The 1.5PN [*scaling term*]{} ${\mbox{\boldmath$\varphi$}}_2$ in the third line of (\[Transformation\_sigma\_to\_n\]) originates from the term in the third line of (\[Transformation\_sigma\_to\_n\_5\]), where the vectorial relation ${\mbox{\boldmath$a$}} \times \left({\mbox{\boldmath$b$}} \times {\mbox{\boldmath$c$}}\right) = {\mbox{\boldmath$b$}} \left({\mbox{\boldmath$a$}} \cdot {\mbox{\boldmath$c$}} \right) - {\mbox{\boldmath$c$}} \left({\mbox{\boldmath$a$}} \cdot {\mbox{\boldmath$b$}} \right)$ has been used. The 2PN [*scaling term*]{} ${\mbox{\boldmath$\varphi$}}_6$ in the seventh line of (\[Transformation\_sigma\_to\_n\]) originates from the first term of the fifth line of (\[Transformation\_sigma\_to\_n\_5\]). The 2PN [*scaling term*]{} ${\mbox{\boldmath$\varphi$}}_7$ in the eighth line of (\[Transformation\_sigma\_to\_n\]) originates from the term in the second line of (\[Transformation\_sigma\_to\_n\_5\]), where relation (\[Relation\_Impact\_Vectors\_1\]) has to be used. In the transformation (\[Transformation\_sigma\_to\_n\]) a term proportional to vector ${\mbox{\boldmath$k$}}$ influences the angle $\delta\left({\mbox{\boldmath$\sigma$}},{\mbox{\boldmath$n$}}\right)$ between ${\mbox{\boldmath$\sigma$}}$ and ${\mbox{\boldmath$n$}}$ only beyond 2PN approximation, due to ${\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$k$}} = {\cal O}\left(c^{-2}\right)$. Hence, the only impact of these [*scaling terms*]{}, ${\mbox{\boldmath$\varphi$}}_2$, ${\mbox{\boldmath$\varphi$}}_6$, ${\mbox{\boldmath$\varphi$}}_7$, is to keep the vector ${\mbox{\boldmath$n$}}$ to have unit length.
Simplified expression for the transformation from ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$
---------------------------------------------------------------------------------------------------------
The transformation ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ as given by Eq. (\[Transformation\_sigma\_to\_n\]) contains many terms which contribute less than $1\,{\rm nas}$ in light deflection. Furthermore, as discussed above in the text before Eq. (\[series\_expansion\_body\_1\]) and in the second comment before Eq. (\[series\_expansion\_rho\]), while the four-coordinates of the observer $\left(c t_1,{\mbox{\boldmath$x$}}_1\right)$ are precisely known and the fundamental basis for any accurate astrometric measurement, the four-coordinates of the light source $\left(c t_0,{\mbox{\boldmath$x$}}_0\right)$ are, first of all, not available but the result of astrometric data reduction. That means, the retarded instant of time $s_1$ defined by (\[retarded\_time\_s\_1\]) is precisely known, while the retarded time $s_0$ defined by (\[retarded\_time\_s\_0\]) is, first of all, an unknown parameter in the theory of light propagation. Therefore, practical astrometry necessitates a transformation ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ as function of $s_1$ and which contains only those terms which are above the goal accuracy of $1\,{\rm nas}$. In (\[Transformation\_sigma\_to\_n\]) the terms depend only on $s_1$, except of ${\mbox{\boldmath$\varphi$}}_7$, ${\mbox{\boldmath$\varphi$}}_9$, and ${\mbox{\boldmath$\varphi$}}_{10}$. Hence, we consider only the series expansion of these three terms, which reads $$\begin{aligned}
\fl \hspace{1.5cm} {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_0\right) = {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_1\right)
+ {\cal O}\left(c^{-5}\right) \quad {\rm for} \quad i = 7,9,10\,.
\label{series_expansion_varphi}\end{aligned}$$
The upper limit of each individual term in the transformation (\[Transformation\_sigma\_to\_n\]) has been determined, given by (\[Appendix\_Estimation\_phi\_1\]), (\[Appendix\_Estimation\_phi\_2\]), (\[Appendix\_Estimation\_phi\_3\]), (\[Appendix\_Estimation\_phi\_4\]), (\[Appendix\_Estimation\_phi\_5\]), (\[Appendix\_Estimation\_phi\_6\]), (\[Appendix\_Estimation\_phi\_7\_2\]), (\[Appendix\_Estimation\_phi\_8\_2\]), (\[Appendix\_Estimation\_phi\_9\_2\]), (\[Appendix\_Estimation\_phi\_10\_2\]), (\[Appendix\_Estimation\_phi\_11\_2\]), (\[Appendix\_Estimation\_phi\_12\_2\]), (\[Appendix\_Estimation\_phi\_13\_2\]), and (\[Appendix\_estimation\_epsilon\_1\]). Numerical values are given in Table \[Table3\]. These results can be summarized as follows: $$\begin{aligned}
\fl \hspace{1.5cm} \left|{\mbox{\boldmath$\varphi$}}_i\left(s_1\right)\right| \;\;\le 1\,{\rm nas}\;, \quad i = 2,4,5,6,11,12,13\,.
\label{series_expansion_varphi_result_1}
\\
\nonumber\\
\fl \hspace{1.0cm} \left|{\mbox{\boldmath$\varphi$}}_i\left(s_1,s_1\right)\right| \;\;\le 1.3\,{\rm nas}\;, \quad i = 7\,.
\label{series_expansion_varphi_result_2}
\\
\nonumber\\
\fl \hspace{1.5cm} \left|\hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right)\right| \;\;\; \le 1\,{\rm nas}\,.
\label{series_expansion_epsilon_1_result}\end{aligned}$$
Besides the fact that the absolute value of the [*scaling terms*]{} ${\mbox{\boldmath$\varphi$}}_2$, ${\mbox{\boldmath$\varphi$}}_6$, ${\mbox{\boldmath$\varphi$}}_7$ is less than $1.3\,{\rm nas}$, these terms are irrelevant for the angle $\delta\left({\mbox{\boldmath$\sigma$}}, {\mbox{\boldmath$n$}}\right)$ between the vectors ${\mbox{\boldmath$\sigma$}}$ and , as stated above. Therefore, these scaling terms will be omitted in the simplified transformation. For the absolute value of the total sum of all those neglected terms (\[series\_expansion\_varphi\_result\_1\]) - (\[series\_expansion\_epsilon\_1\_result\]) which are not proportional to the three-vector ${\mbox{\boldmath$k$}}$, we get $$\begin{aligned}
\fl \hspace{0.75cm} I_2 = \left|\sum\limits_{i=4,5} {\mbox{\boldmath$\varphi$}}_i\left(s_1\right)
+ \sum\limits_{i=11,12,13} {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_1\right) + \hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right)\right|
\nonumber\\
\nonumber\\
\fl \hspace{1.15cm} \le\; 8\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c} + \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}
+ 18\,\frac{m_A}{P_A}\,\frac{v_A^2\left(s_1\right)}{c^2} + 8\,\frac{m_A}{r^{\,1}_A}\,\frac{v_A^2\left(s_1\right)}{c^2}\,.
\label{Sum_2}\end{aligned}$$
For the upper limits of the terms in (\[Sum\_2\]) we have used that $$\begin{aligned}
\fl \hspace{3.0cm} \left| {\mbox{\boldmath$\varphi$}}_4 + {\mbox{\boldmath$\varphi$}}_5 \right| \le 8\,\frac{m_A}{r_A^{\,1}\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Inequality_Sum_2_A}
\\
\nonumber\\
\fl \hspace{1.75cm} \left|{\mbox{\boldmath$\varphi$}}_{11} + {\mbox{\boldmath$\varphi$}}_{12} + {\mbox{\boldmath$\varphi$}}_{13} \right| \le \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}\,,
\label{Inequality_Sum_2_B}\end{aligned}$$
while the upper limit of $\left|\hat{{\mbox{\boldmath$\epsilon$}}}_1\right|$ is given by Eq. (\[Appendix\_estimation\_epsilon\_1\]). These inequalities, Eqs. (\[Inequality\_Sum\_2\_A\]) and (\[Inequality\_Sum\_2\_B\]), are not shown explicitly, but can be demonstrated with the aid of the approach as described in \[Appendix\_Estimation1\]. Using the numerical parameters as given by Table \[Table1\] one obtains for the absolute value of the total sum $$\begin{aligned}
\fl \hspace{0.75cm} I_2 \le \; 1.1\,{\rm nas} \quad {\rm for}\;{\rm Sun}\;{\rm at}\,45^{\circ}\,\left({\rm solar}\;{\rm aspect}\;{\rm angle}\right),
\nonumber\\
\fl \hspace{1.15cm} \le \; 1.2\,{\rm nas} \quad {\rm for}\;{\rm Jupiter}\,,
\label{Total_Sum_2}\end{aligned}$$
and less than $0.5\,{\rm nas}$ for any other Solar System body. Accordingly, the terms (\[series\_expansion\_varphi\_result\_1\]) - (\[series\_expansion\_epsilon\_1\_result\]) can be neglected for sub-$\,$ astrometry and even for astrometry on the level of $1.2\;{\rm nas}$ in light deflection. In this way one obtains the simplified transformation ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ fully in terms of $s_1$, which reads $$\begin{aligned}
\fl _{\rm N} & \hspace{-0.1cm} \biggr|& \hspace{+0.05cm} {{\mbox{\boldmath$n$}}} = {{\mbox{\boldmath$\sigma$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN}& \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_1}\biggr|&
+ 2\,m_A\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_3} \biggr|&
- 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}}\cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_8} \biggr|&
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.5cm} _{{\mbox{\boldmath$\varphi$}}_9} \biggr|&
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)} \right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.55cm} _{{\mbox{\boldmath$\varphi$}}_{10}} \biggr|&
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)} \right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN} & \hspace{-0.025cm} \biggr|& + {\cal O}\left(c^{-5}\right),
\label{Simplified_Transformation_sigma_to_n}\end{aligned}$$
where ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1\right)$ for $i=1,3,8$ and ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_1\right)$ for $i=9,10$ which appear before the vertical lines are by definition equal to the expressions on the right of the vertical bars in each line. In the limit of monopole at rest this expression coincides with Eqs. (85) - (86) in [@Article_Zschocke1]. By means of the approach and using the results in the appendix one obtains for the upper limits of the 1PN, 1.5PN, and 2PN terms in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]): $$\begin{aligned}
\fl {\rm 1PN} \hspace{3.9cm}
\left|{\mbox{\boldmath$\varphi$}}_1 \right| \le 4\,\frac{m_A}{P_A}\,,
\label{angle_1PN_2}
\\
\nonumber\\
\fl {\rm 1.5PN} \hspace{3.65cm}
\left|{\mbox{\boldmath$\varphi$}}_3 \right| \le 4\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{angle_15PN_2}
\\
\nonumber\\
\fl {\rm enhanced}\;{\rm 2PN} \hspace{0.5cm} \left|{\mbox{\boldmath$\varphi$}}_8 + {\mbox{\boldmath$\varphi$}}_9 + {\mbox{\boldmath$\varphi$}}_{10}\right| \le 16\,
\frac{m^2_A}{P^2_A}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,.
\label{angle_2PN_2}\end{aligned}$$
In the limit of body at rest, the results (\[angle\_1PN\_2\]) and (\[angle\_2PN\_2\]) would coincide with Eqs. (82) and (83) in [@Article_Zschocke1], respectively. The simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]) depends on the variables $m_A$, ${\mbox{\boldmath$x$}}_0$, ${\mbox{\boldmath$x$}}_1$, and ${\mbox{\boldmath$x$}}_A\left(s_1\right)$, but not anymore on the retarded time $s_0$. Thus, the only unknown in (\[Simplified\_Transformation\_sigma\_to\_n\]) is the three-coordinate of the light source, ${\mbox{\boldmath$x$}}_0$, whose determination is the fundamental aim of astrometric data reduction. Some statement about the neglected terms of the order ${\cal O}\left(c^{-5}\right)$ (2.5PN approximation) and of the order ${\cal O}\left(c^{-6}\right)$ (3PN approximation) are given in Section \[Section\_3PN\].
Transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$n$}}$ {#Section7}
======================================================================
The explicit expression for the transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$n$}}$
------------------------------------------------------------------------------------------------------
The actual aim of the boundary value problem is to establish a relation between the unit-vectors ${\mbox{\boldmath$k$}}$ and ${\mbox{\boldmath$n$}}$. From the transformations (\[Transformation\_k\_to\_sigma\]) and (\[Transformation\_sigma\_to\_n\]) we immediately obtain the transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$n$}}$: $$\begin{aligned}
\fl _{\rm N} & \hspace{0.1cm} \biggr|& \hspace{0.15cm} {{\mbox{\boldmath$n$}}} = {{\mbox{\boldmath$k$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN}& \hspace{-0.95cm} _{{\mbox{\boldmath$\rho$}}_1} + _{{\mbox{\boldmath$\varphi$}}_1} \biggr|& \hspace{0.0cm}
- 2\,\frac{m_A}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
+ 2\,m_A\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\rho$}}_2} \biggr|& \hspace{0.0cm}
+ 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\rho$}}_3} \biggr|& \hspace{0.0cm}
- 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
+ 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_0\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\rho$}}_4} \biggr|& \hspace{0.0cm}
+ 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_0\right)}{c}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\varphi$}}_3} \biggr|& \hspace{0.0cm}
- 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}}\cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\varphi$}}_4} \biggr|& \hspace{0.0cm}
+ 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
+ \frac{2\,m_A}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,{\mbox{\boldmath$d$}}^k_A\left(s_1\right)\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\varphi$}}_5} \biggr|& \hspace{0.0cm}
+ \frac{2\,m_A}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{{\rm scaling}\;{\rm 1.5PN}}& \hspace{-0.3cm} _{{\mbox{\boldmath$\varphi$}}_2} \biggr|& \hspace{0.0cm}
- 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,{\mbox{\boldmath$k$}}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{{\rm scaling}\;{\rm 2PN}} & \hspace{-1.00cm} _{{\mbox{\boldmath$\rho$}}_5} + _{{\mbox{\boldmath$\varphi$}}_6} \biggr|& \hspace{0.0cm}
- \frac{2\,m_A^2}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,{\mbox{\boldmath$k$}}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
- 2\,\frac{m_A^2}{R^2}{\mbox{\boldmath$k$}}
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right|^2
\nonumber\\
\nonumber\\
\fl _{{\rm scaling}\;{\rm 2PN}}& \hspace{-0.35cm} _{{\mbox{\boldmath$\varphi$}}_7} \biggr|& \hspace{0.0cm}
+ 4\,\frac{{\mbox{\boldmath$k$}}}{R}\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\rho$}}_6} \biggr|& \hspace{0.0cm}
- 2 \frac{m_A^2}{R^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
+ \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right|^2
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\rho$}}_7} \biggr|& \hspace{0.0cm}
- 4\,\frac{m_A^2}{R}\,\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left(k \cdot r_A^{\,0}\left(s_0\right)\right)^2}\right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\varphi$}}_8} \biggr|& \hspace{0.0cm}
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\varphi$}}_9} \biggr|& \hspace{0.0cm}
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)} \right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.35cm} _{{\mbox{\boldmath$\varphi$}}_{10}} \biggr|& \hspace{0.05cm}
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)} \right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.2cm} _{{\mbox{\boldmath$\rho$}}^A_{8}} \biggr|& \hspace{0.0cm}
+ \frac{15}{4} \frac{m_A^2}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^3} \,
\left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)
\left(\arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|} + \frac{\pi}{2} \right)
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.2cm} _{{\mbox{\boldmath$\rho$}}^B_{8}} \biggr|& \hspace{0.0cm}
- \frac{15}{4} \frac{m_A^2}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|^3} \,
\left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)
\left( \arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|} + \frac{\pi}{2} \right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{-0.95cm} _{{\mbox{\boldmath$\rho$}}_9} + _{{\mbox{\boldmath$\varphi$}}_{11}} \biggr|& \hspace{0.0cm}
- \frac{1}{4}\frac{m_A^2}{R}\left(\!\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(r_A^{\,1}\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left(r_A^{\,0}\left(s_0\right)\right)^2}\!\right)
- 4\,\frac{m_A^2}{\left(r_A^{\,1}\left(s_1\right)\right)^2}\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\varphi$}}_{12}} \biggr|& \hspace{0.0cm}
- \frac{m_A^2}{2}\,{\mbox{\boldmath$d$}}_A^k\left(s_1\right)\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left(r^{\,1}_A\left(s_1\right)\right)^4}
- \frac{15}{4}\,\frac{m_A^2}{\left(r_A^{\,1}\left(s_1\right)\right)^2}\,{\mbox{\boldmath$d$}}^k_A\left(s_1\right)
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^2}
\nonumber\\
\nonumber\\
\fl _{\rm 2PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\varphi$}}_{13}} \biggr|& \hspace{0.0cm}
- \frac{15}{4}\,m_A^2\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^3}
\left(\arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|} + \frac{\pi}{2} \right)
\nonumber\\
\nonumber\\
\fl _{\rm 2PN}& \hspace{0.25cm} \biggr|& \hspace{0.0cm}
+ \hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right) + \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_0\right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN} & \hspace{0.25cm} \biggr|& \hspace{0.0cm} + {\cal O}\left(c^{-5}\right),
\label{Transformation_k_to_n}\end{aligned}$$
where ${\mbox{\boldmath$\rho$}}_i = {\mbox{\boldmath$\rho$}}_i\left(s_1,s_0\right)$ with $i=1\,, \cdots \,,9$, and ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1\right)$ for $i=1\,, \cdots \,,6,8,11,12,13$, and ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_0\right)$ for $i = 7,9,10$ which appear before the vertical lines are by definition equal to the expressions on the right of the vertical bars in each line.
The transformation (\[Transformation\_k\_to\_n\]) allows one to determine the unit coordinate direction ${\mbox{\boldmath$n$}}$ at the observers position from the boundary values ${\mbox{\boldmath$x$}}_0$ and ${\mbox{\boldmath$x$}}_1$. In the limit of body at rest this expression coincides with Eq. (87) in [@Article_Zschocke1]. The terms $\hat{{\mbox{\boldmath$\epsilon$}}}_1$ and $\hat{{\mbox{\boldmath$\epsilon$}}}_2$ are given in the \[Appendix\_epsilon\] by Eqs. (\[Transformation\_sigma\_to\_n\_epsilon\]) and (\[Transformation\_k\_to\_sigma\_epsilon\]), respectively. In view of the remarkable algebraic structure in (\[Transformation\_k\_to\_n\]) it is evident how important the estimation of the upper limit of the individual terms is. Such an estimation allows for a considerably simpler structure of this transformation, which will be the topic of the next section.
Simplified expression for the transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$n$}}$
----------------------------------------------------------------------------------------------------
The simplified transformation from ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$n$}}$ follows from Eqs. (\[Simplified\_Transformation\_k\_to\_sigma\]) and (\[Simplified\_Transformation\_sigma\_to\_n\]), that means where all [*scaling terms*]{} are omitted and all terms are neglected whose individual contribution is less than $1\,{\rm nas}$ in light deflection for Sun at $45^{\circ}$ and all the other Solar System bodies. For the total sum of all those neglected terms which are not proportional to three-vector ${\mbox{\boldmath$k$}}$ one obtains $$\begin{aligned}
\fl I_3 = \bigg|\sum\limits_{i=3,8,9} {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)
+ \sum\limits_{i=1,2,3,4} \Delta {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)
+ \sum\limits_{i=4,5} {\mbox{\boldmath$\varphi$}}_i\left(s_1\right) + \sum\limits_{i=11,12,13} {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_1\right)
\nonumber\\
\nonumber\\
\fl \hspace{1.0cm} + \hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right) + \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_1\right) \bigg|
\nonumber\\
\nonumber\\
\fl \hspace{0.5cm} \le \frac{10\,m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c} + \frac{15}{4} \pi \frac{m_A^2}{P_A^2}
+ \frac{14\,m_A}{r_A^{\,1}\left(s_1\right)}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 24\,\frac{m_A}{P_A} \frac{v_A^2\left(s_1\right)}{c^2} + 18 \,m_A \,\frac{a_A\left(s_1\right)}{c^2}\,.
\nonumber\\
\label{Sum_3}\end{aligned}$$
For the upper limits of the terms in (\[Sum\_3\]) we have used that $$\begin{aligned}
\fl \left| {\mbox{\boldmath$\rho$}}_8 + {\mbox{\boldmath$\rho$}}_9 + {\mbox{\boldmath$\varphi$}}_{11} + {\mbox{\boldmath$\varphi$}}_{12} + {\mbox{\boldmath$\varphi$}}_{13} \right| \le \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}\,,
\label{Inequality_B}
\\
\nonumber\\
\fl \hspace{1.75cm} \left|\Delta {\mbox{\boldmath$\rho$}}_1 + {\mbox{\boldmath$\varphi$}}_4 + {\mbox{\boldmath$\varphi$}}_5\right|
\le 10\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Inequality_A}
\\
\nonumber\\
\fl \hspace{1.25cm} \left|\Delta {\mbox{\boldmath$\rho$}}_2 + \Delta {\mbox{\boldmath$\rho$}}_3 + \Delta {\mbox{\boldmath$\rho$}}_4 \right|
\le \frac{6\,m_A}{r_A^{\,1}\left(s_1\right)}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ \frac{4\,m_A}{P_A}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 8\,m_A\,\frac{a_A\left(s_1\right)}{c^2}\,,
\label{Inequality_C}\end{aligned}$$
while $\left|{\mbox{\boldmath$\rho$}}_3\left(s_1,s_1\right)\right| = 0$ according to Eq. (\[Term\_rho\_3\_3\_A\]). The inequality (\[Inequality\_B\]) is not shown explicitly, but its validity can be demonstrated by means of the approach described in \[Appendix\_Estimation1\]. The inequality (\[Inequality\_A\]) is shown in \[Appendix\_Inequality\_A\], while the inequality (\[Inequality\_C\]) follows from (\[Term\_rho\_2\_1\_E\]), (\[Term\_rho\_3\_4\]), and (\[Estimation\_rho\_4\_2\]). The upper limit of $\left|\hat{{\mbox{\boldmath$\epsilon$}}}_1\right|$ and $\left|\hat{{\mbox{\boldmath$\epsilon$}}}_2\right|$ are given by Eqs. (\[Appendix\_estimation\_epsilon\_1\]) and (\[Appendix\_estimation\_epsilon\_3\]), respectively. Using the numerical parameters as given by Table \[Table1\] one obtains $$\begin{aligned}
\fl \hspace{0.75cm} I_3 \le \; 1.3\,{\rm nas} \quad {\rm for}\;{\rm Sun}\;{\rm at}\,45^{\circ}\,\left({\rm solar}\;{\rm aspect}\;{\rm angle}\right),
\nonumber\\
\fl \hspace{1.15cm} \le \; 1.3\,{\rm nas} \quad {\rm for}\;{\rm Jupiter}\,,
\label{Total_Sum_3}\end{aligned}$$
and less than $0.64\,{\rm nas}$ for all the other Solar System bodies. Accordingly, by neglecting these terms in (\[Sum\_3\]) one obtains the simplified transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$n$}}$ fully in terms of $s_1$, which reads as follows: $$\begin{aligned}
\fl _{\rm N} & \hspace{0.1cm} \biggr|& \hspace{0.15cm} {{\mbox{\boldmath$n$}}} = {{\mbox{\boldmath$k$}}}
\nonumber\\
\nonumber\\
\fl _{\rm 1PN}& \hspace{-0.95cm} _{{\mbox{\boldmath$\rho$}}_1} + _{{\mbox{\boldmath$\varphi$}}_1} \biggr|& \hspace{0.0cm}
- 2\,\frac{m_A}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right)
+ 2\,m_A\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\rho$}}_2} \biggr|& \hspace{0.0cm}
+ 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\rho$}}_4} \biggr|& \hspace{0.0cm}
+ 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right)
\nonumber\\
\nonumber\\
\fl _{\rm 1.5PN} & \hspace{-0.3cm} _{{\mbox{\boldmath$\varphi$}}_3} \biggr|& \hspace{0.0cm}
- 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}}\cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\rho$}}_6} \biggr|& \hspace{0.0cm}
- 2 \frac{m_A^2}{R^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
+ \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right)
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right|^2
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\rho$}}_7} \biggr|& \hspace{0.0cm}
- 4\,\frac{m_A^2}{R}\,\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r_A^{\,0}\left(s_1\right)\right)^2}\right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\varphi$}}_8} \biggr|& \hspace{0.0cm}
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.25cm} _{{\mbox{\boldmath$\varphi$}}_9} \biggr|& \hspace{0.0cm}
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\left(\!\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)} \!\right)
\nonumber\\
\nonumber\\
\fl _{{\rm enhanced}\;{\rm 2PN}} & \hspace{-0.35cm} _{{\mbox{\boldmath$\varphi$}}_{10}} \biggr|& \hspace{0.05cm}
+ 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)} \right)
\nonumber\\
\nonumber\\
\fl _{\rm 2.5PN} & \hspace{0.25cm} \biggr|& \hspace{0.0cm} + {\cal O}\left(c^{-5}\right),
\label{Simplified_Transformation_k_to_n}\end{aligned}$$
where ${\mbox{\boldmath$\rho$}}_i = {\mbox{\boldmath$\rho$}}_i\left(s_1,s_1\right)$ for $i=1,2,4,6,7$, and ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1\right)$ for $i = 1,3,8$, and ${\mbox{\boldmath$\varphi$}}_i = {\mbox{\boldmath$\varphi$}}_i\left(s_1,s_1\right)$ for $i = 9,10$ which appear before the vertical lines are by definition equal to the expressions on the right of the vertical bars in each line. In the limit of body at rest this expression coincides with Eqs. (92) - (93) in [@Article_Zschocke1]. For the distance $R = \left|{\mbox{\boldmath$R$}}\right|$ one should implement the exact expression (\[Boundary\_3\]), because the approximative expression (\[vector\_R\_series\_6\]) is slightly more complicated and only in use for the estimations but not for astrometric data reduction. By means of the approach and results of the appendix, one obtains for the upper limits of the 1PN, 1.5PN, and 2PN terms of the simplified transformation (\[Simplified\_Transformation\_k\_to\_n\]): $$\begin{aligned}
\fl {\rm 1PN} \hspace{4.6cm}
\left|{\mbox{\boldmath$\rho$}}_1 + {\mbox{\boldmath$\varphi$}}_1\right| \le 4\,\frac{m_A}{P_A}\,,
\label{angle_1PN_3}
\\
\nonumber\\
\fl {\rm 1.5PN} \hspace{3.55cm}
\left|{\mbox{\boldmath$\rho$}}_2 + {\mbox{\boldmath$\rho$}}_4 + {\mbox{\boldmath$\varphi$}}_3 \right| \le 6\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{angle_15PN_3}
\\
\nonumber\\
\fl {\rm enhanced}\;{\rm 2PN} \hspace{0.5cm} \left|{\mbox{\boldmath$\rho$}}_6 + {\mbox{\boldmath$\rho$}}_7 + {\mbox{\boldmath$\varphi$}}_8 + {\mbox{\boldmath$\varphi$}}_9 + {\mbox{\boldmath$\varphi$}}_{10} \right|
\le 16\,\frac{m^2_A}{P^2_A}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,.
\label{angle_2PN_3}\end{aligned}$$
As outlined above, the 2PN term (\[angle\_2PN\_3\]) is a so-called [*enhanced term*]{} because of the factor $r^{\,1}_A\left(s_1\right)/P_A$.
A further comment is in order about the upper limit of the 1.5PN terms as given by (\[angle\_15PN\_3\]). In Eq. (179) in [@Zschocke2] the upper limit of the 1.5PN terms in light deflection was given by $\displaystyle \varphi_{\rm 1.5PN} \le 4\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}$ in agreement with the results in [@Klioner2003b; @KopeikinMakarov2007]. The marginal difference between the factor $6$ in Eq. (\[angle\_15PN\_3\]) and the factor $4$ in Eq. (179) in [@Zschocke2] originates from the logarithmic term ${\mbox{\boldmath$\rho$}}_2$ in the simplified transformation (\[Simplified\_Transformation\_k\_to\_n\]). That term has been estimated by Eq. (\[Term\_rho\_2\_2\]), according to which the term ${\mbox{\boldmath$\rho$}}_2$ would vanish in the limit of light sources at infinity. So the term ${\mbox{\boldmath$\rho$}}_2$ originates from the boundary value problem, which has not been on the scope of the investigations [@Klioner2003b; @Zschocke2; @KopeikinMakarov2007]. In particular, without the term ${\mbox{\boldmath$\rho$}}_2$ we would get the result as given by Eq. (179) in [@Zschocke2].
Let us summarize the variables on which the simplified transformations (\[Simplified\_Transformation\_k\_to\_sigma\]), (\[Simplified\_Transformation\_sigma\_to\_n\]), and (\[Simplified\_Transformation\_k\_to\_n\]) depend on, as there are: $m_A, {\mbox{\boldmath$x$}}_0, {\mbox{\boldmath$x$}}_1, {\mbox{\boldmath$x$}}_A\left(s_1\right)$. The values $m_A, {\mbox{\boldmath$x$}}_1, {\mbox{\boldmath$x$}}_A\left(s_1\right)$ are provided by some ephemerides and tracking of the orbit of the satellite (observer). Thus, the only unknown in these transformations remains the spatial position of the light source ${\mbox{\boldmath$x$}}_0$, which is the primary aim of astrometric data reduction.
Impact of higher order terms {#Section_3PN}
============================
The transformations (\[Transformation\_k\_to\_sigma\]), (\[Transformation\_sigma\_to\_n\]), (\[Transformation\_k\_to\_n\]) and their simplified versions (\[Simplified\_Transformation\_k\_to\_sigma\]), (\[Simplified\_Transformation\_sigma\_to\_n\]), (\[Simplified\_Transformation\_k\_to\_n\]) are valid up to terms of the order ${\cal O}\left(c^{-5}\right)$. So the question arises about the impact of these higher order terms. Are they relevant for nas-astrometry?
In order to address the problem we consider the light deflection angle $\varphi = \angle \left({\mbox{\boldmath$k$}}, {\mbox{\boldmath$n$}}\right)$. By including terms up to the order ${\cal O}\left(c^{-7}\right)$ the post-Newtonian expansion of the light deflection angle is $$\begin{aligned}
\fl \varphi = \arcsin \left| {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$n$}} \right| = \left| {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$n$}} \right|
+ \frac{1}{6}\,\left| {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$n$}} \right|^3 + {\cal O}\left(\left| {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$n$}} \right|^5\right)
\label{series_arcsin}
\\
\fl \hspace{0.3cm} = \varphi_{\rm 1PN} + \varphi_{\rm 1.5PN} + \varphi_{\rm 2PN}
+ \varphi_{\rm 2.5PN} + \varphi_{\rm 3PN} + \varphi_{\rm 3.5PN} + \varphi_{\rm 4PN} + \varphi_{\rm 4.5PN} + {\cal O}\left(c^{-10}\right),
\nonumber\\
\label{light_deflection_angle}\end{aligned}$$
where $\varphi_{\rm n PN} = {\cal O}\left(c^{- 2 n}\right)$. The first term on the r.h.s. of (\[series\_arcsin\]) contributes to any order, while the second term on the r.h.s. of (\[series\_arcsin\]) contributes to the order ${\cal O}\left(c^{-6}\right)$ and beyond. The 1PN, 1.5PN, and 2PN terms in (\[light\_deflection\_angle\]) can be obtained from the simplified transformation (\[Simplified\_Transformation\_k\_to\_n\]). One obtains $$\begin{aligned}
\varphi_{\rm 1PN} = \left| {\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$\rho$}}_1 + {\mbox{\boldmath$\varphi$}}_1\right)\right|
\le 4\,\frac{m_A}{P_A}\,,
\label{higher_order_terms_1PN}
\\
\nonumber\\
\hspace{-0.25cm} \varphi_{\rm 1.5PN} = \left| {\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$\rho$}}_2 + {\mbox{\boldmath$\rho$}}_4 + {\mbox{\boldmath$\varphi$}}_3\right)\right|
\le 6\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{higher_order_terms_15PN}
\\
\nonumber\\
\varphi_{\rm 2PN} = \left| {\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$\rho$}}_6 + {\mbox{\boldmath$\rho$}}_7 + {\mbox{\boldmath$\varphi$}}_8 + {\mbox{\boldmath$\varphi$}}_9 + {\mbox{\boldmath$\varphi$}}_{10} \right)\right|
\le 16\,\frac{m_A^2}{P_A^2}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A} \,,
\label{higher_order_terms_2PN}\end{aligned}$$
all of which are relevant on the nas-scale of accuracy. The next order beyond 2PN approximation would be 2.5PN terms. While they are out of the scope of the present investigation, a few comments can be stated already right now. Basically, there are three kind of 2.5PN terms, as there are $$\begin{aligned}
\varphi^A_{\rm 2.5PN} \sim \frac{m_A}{P_A}\,\frac{v^3_A\left(s_1\right)}{c^3} \ll 1\,{\rm nas}\;,
\label{higher_order_terms_25PN_A}
\\
\varphi^B_{\rm 2.5PN} \sim m_A\,\frac{v_A\left(s_1\right)}{c}\frac{a_A\left(s_1\right)}{c^2} \ll 1\,{\rm nas}\;,
\label{higher_order_terms_25PN_B}
\\
\varphi^C_{\rm 2.5PN} \sim \frac{m_A^2}{P_A^2}\,\frac{v_A\left(s_1\right)}{c}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,.
\label{higher_order_terms_25PN_C}\end{aligned}$$
The structure of the 2.5PN terms in (\[higher\_order\_terms\_25PN\_A\]) and (\[higher\_order\_terms\_25PN\_B\]) follows from a series expansion of the first post-Minkowskian (1PM) solution of a light signal propagating in the field of one arbitrarily moving monopole as found in [@KS1999]; for more explicit expressions of the coordinate velocity and light trajectory we refer to Eqs. (C.1) - (C.8) in [@Klioner2003a] or Eqs. (E.4) - (E.6) and (E.16) in [@Zschocke4]. So these terms in (\[higher\_order\_terms\_25PN\_A\]) and (\[higher\_order\_terms\_25PN\_B\]) have no enhancement factor and they are negligible even for highly precise measurements on the nas-scale of accuracy, also in case of some large numerical factor in front of these terms. But what about the 2.5PN terms in (\[higher\_order\_terms\_25PN\_C\])? They are connected with an enhancement factor $r^{\,1}_A\left(s_1\right)/P_A$ and might become large enough to be of relevance for nas-astrometry. As it stands, the term (\[higher\_order\_terms\_25PN\_C\]) is less than $1\,{\rm nas}$ for any Solar System body (even for grazing rays at the Sun), but certainly there will be some large numerical factor in front of this term. Then, the 2.5PN term (\[higher\_order\_terms\_25PN\_C\]) would be above the threshold of $1\,{\rm nas}$ for grazing rays at Jupiter. In order to determine more precisely the relevance of 2.5PN terms (\[higher\_order\_terms\_25PN\_C\]) for nas-astrometry, one should consider the 2PN light trajectory in the field of one monopole at rest, ${\mbox{\boldmath$x$}}_A = {\rm const}$, and then perform a Lorentz transformation in order to account for the translational motion of the body, which would yield all terms proportional to $m_A^2\,\left(v_A/c\right)^n$ with $n=1, 2, 3, ...$. Such an approach has already been developed in the first post-Minkowskian approximation [@Klioner2003a] and might be generalized for the case of 2.5PN light propagation in the field of one body in translational motion.
Let us now consider terms of the 3PN approximation, that means terms of the order ${\cal O}\left(c^{-6}\right)$ in (\[light\_deflection\_angle\]). Are they of relevance for nas-astrometry? A reliable answer can be found in the following manner. In [@Zschocke_Lense_Equation] a lens equation has been derived for the light deflection angle $\varphi$ in the field of one spherically symmetric body at rest, ${\mbox{\boldmath$x$}}_A = {\rm const}$, given by (cf. Eq. (15) in [@Zschocke_Lense_Equation] and shift of the origin of spatial axes by three-vector ${\mbox{\boldmath$x$}}_A$) $$\begin{aligned}
\fl \hspace{2.0cm} \varphi = \frac{1}{2} \left(\sqrt{\left(\frac{d^k_A}{r^{\,1}_A}\right)^2
+ 8\,\frac{m_A}{r^{\,1}_A}\, \frac{r^{\,0}_A\,r^{\,1}_A - {\mbox{\boldmath$r$}}^{\,0}_A \cdot {\mbox{\boldmath$r$}}^{\,1}_A}{R\,r^{\,1}_A} } - \frac{d^k_A}{r^{\,1}_A} \right)
+ {\cal O}\left(\frac{m_A^2}{P_A^2}\right),
\label{lens_equation}\end{aligned}$$
where ${\mbox{\boldmath$r$}}^{\,0}_A = {\mbox{\boldmath$x$}}_0 - {\mbox{\boldmath$x$}}_A$ and ${\mbox{\boldmath$r$}}^{\,1}_A = {\mbox{\boldmath$x$}}_1 - {\mbox{\boldmath$x$}}_A$ and the impact parameter $d^k_A = \left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\right| = \left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\right|$ is independent of time. The neglected terms of order ${\cal O}\left(m_A^2/P_A^2\right)$ has been shown to be less than $\displaystyle \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}$ which is less than $1\,{\rm nas}$ for Sun at $45^{\circ}$ (solar aspect angle adopted from the Gaia mission) and all the other Solar System bodies. The lens equation (\[lens\_equation\]) represents the total sum of all [*enhanced terms*]{}. Of course, for an observer located in the Solar System the lens effect (i.e. second image of the source caused by Solar System bodies) cannot be detected, and that is why the second solution with the lower sign in Eq. (15) in [@Zschocke_Lense_Equation] is omitted here for our considerations.
In the near-zone of the Solar System we have $m_A/d^k_A \ll 1$, which allows for a series expansion of the lens equation (\[lens\_equation\]) in terms of this small parameter. This possibility is utilized to get $$\begin{aligned}
\varphi = \varphi_{\rm 1PN} + \varphi_{\rm 2PN} + \varphi_{\rm 3PN} + {\cal O}\left(c^{-8}\right) + {\cal O}\left(\frac{m_A^2}{P_A^2}\right),
\label{series_expansion_lens_equation}\end{aligned}$$
which has already been given by Eq. (26) in [@Zschocke_Lense_Equation]; because the body is assumed to be at rest in (\[lens\_equation\]) there are no 1.5PN terms, 2.5PN terms and so on in the series expansion (\[series\_expansion\_lens\_equation\]). For the upper limits one obtains (cf. Eqs. (17) and (18) in [@Zschocke_Lense_Equation]), $$\begin{aligned}
\varphi_{\rm 1PN} \le 4\,\frac{m_A}{P_A}\,,
\label{series_expansion_lens_equation_1PN}
\\
\varphi_{\rm 2PN} \le 16\,\frac{m_A^2}{P_A^2}\,\frac{r^{\,1}_A}{P_A}\,.
\label{series_expansion_lens_equation_2PN}\end{aligned}$$
The above standing results in (\[higher\_order\_terms\_1PN\]) and (\[higher\_order\_terms\_2PN\]) coincide, in the limit of body at rest, with Eqs. (\[series\_expansion\_lens\_equation\_1PN\]) and (\[series\_expansion\_lens\_equation\_2PN\]). The 3PN term of light deflection for body at rest has already been considered in Eq. (27) in [@Zschocke_Lense_Equation] and reads: $$\begin{aligned}
\varphi_{\rm 3PN} \le 128\,\frac{m_A^3}{P_A^3}\,\left(\frac{r^{\,1}_A}{P_A}\right)^2\,.
\label{higher_order_terms_3PN}\end{aligned}$$
The same result has also been obtained within the Time Transfer Function approach in [@LinetTeyssandier2013_a; @LinetTeyssandier2013_b] (cf. Eq. (93) in [@LinetTeyssandier2013_a] or Eq. (21) in [@LinetTeyssandier2013_b]). One might believe that (\[higher\_order\_terms\_3PN\]) could also be concluded from the second term on the r.h.s. of (\[series\_arcsin\]), but this would be incomplete as long as the transformation ${\mbox{\boldmath$n$}}$ to ${\mbox{\boldmath$k$}}$ is only known in the 2PN approximation, because the first term on the r.h.s. of (\[series\_arcsin\]) contributes to any order. Inserting numerical parameter of Table \[Table1\] one obtains for grazing rays at Jupiter and Saturn about $\varphi_{\rm 3PN} = 32\,{\rm nas}$ and $\varphi_{\rm 3PN} = 7\,{\rm nas}$, respectively, in light deflection, while in the field of earth-like planets or Sun at $45^{\circ}$ they would contribute much less than $1\,{\rm nas}$; for grazing ray at the Sun the 3PN term (\[higher\_order\_terms\_3PN\]) amounts to be about $12 \cdot 10^3\,{\rm nas}$ in light deflection, as already noticed by Eq. (22) in [@LinetTeyssandier2013_b].
From these considerations it becomes certain, that [*enhanced terms*]{} in the third post-Newtonian (3PN) approximation have to be taken into account for astrometry on the nas-level of accuracy. But it is clear that such calculation would be a rather ambitious assignment of a task for moving bodies. Therefore, in order to get the light trajectory ${\mbox{\boldmath$x$}}\left(t\right)$ in the 3PN approximation for moving bodies, one should consider the much simpler case of 3PN light trajectory in the field of one monopole at rest, ${\mbox{\boldmath$x$}}_A = {\rm const}$, and then just take the retarded position of the massive body, ${\mbox{\boldmath$x$}}_A = {\mbox{\boldmath$x$}}_A\left(s_1\right)$, in order to account for the body’s motion.
Finally, from very similar considerations it becomes clear that 3.5PN terms and 4PN terms will not be of relevance for nas-astrometry. For instance, we would obtain $$\begin{aligned}
\varphi_{\rm 4PN} \le 1280\,\frac{m_A^4}{P_A^4}\,\left(\frac{r^{\,1}_A}{P_A}\right)^3\,,
\label{higher_order_terms_4PN}\end{aligned}$$
which is much less than $1\,{\rm nas}$ for Sun at $45^{\circ}$ and any other Solar System body; but we notice that for grazing ray at the Sun the 4PN term (\[higher\_order\_terms\_4PN\]) amounts to be about $50\,{\rm nas}$ in light deflection. So the strict statement is that the impact of [*enhanced terms*]{} becomes smaller and smaller the higher the post-Newtonian order is, and can be neglected from the 3.5PN order on, even for ultra-high precision of the nas-level of accuracy in astrometry, except for grazing rays at the Sun where the 4PN order has to be accounted for.
Summary {#Section8}
=======
Todays precision in angular measurements of celestial objects has reached a level of a few micro-arcseconds. In fact, the very recent Data Release 2 of the ESA astrometry mission Gaia contains precise positions, proper motions, and parallaxes for more than $1300$ million stars and provides astrometric data for parallaxes having uncertainties of only about $30$ $\;$ for bright stars with $V \!\! = \!\! 15\;{\rm mag}$ in stellar magnitude [@GAIA_DR2_1; @Gaia_Archive; @GAIA_DR2_2].
The impressive progress of the ESA cornerstone mission Gaia in astrometric precision has encouraged the astrometric science to further proceed in nearest future. Over the next coming years, the Gaia science community will embark on an intense series of workshops to develop the key science themes which will scope the requirements for a future astrometry mission. This will culminate in a detailed white paper which will be published to coincide with the first releases of Gaia data. Furthermore, among several astrometry missions proposed to ESA the M-5 mission proposals Gaia-NIR [@Gaia_NIR], Theia [@Theia], and NEAT [@NEAT1; @NEAT2; @NEAT3], are mentioned which in this order are designed for a highly precise measurement aiming at the $\;$ level, sub-$\mu{\rm as}$ level and even nas level of precision. Also feasibility studies of Earth-bounded telescopes are presently under consideration which aim at an accuracy of about $10\,{\rm nas}$ [@nas_telescopes].
Such ultra-highly precise accuracies on the sub--level presuppose corresponding advancements in the theory of light propagation in the Solar System. In particular, at such level of precision it is necessary to describe the propagation of a light signal in the gravitational field of $N$ Solar System bodies described by their full set of mass-multipoles $M_L^A$ and spin-multipoles $S_L^A$, allowing the bodies to have arbitrary shape, inner structure, oscillations and rotational motion. A remarkable and impressive progress has been achieved during recent years in determining the light trajectories in the gravitational field of bodies with higher multipoles, as there are:
$\bullet$ A general solution for the light-trajectory in the stationary gravitational field of a localized source at rest, ${\mbox{\boldmath$x$}}_A = {\rm const}$, with time-independent intrinsic multipoles, $M^A_L$ and $S^A_L$, has been determined in 1.5PN approximation in [@Kopeikin1997].
$\bullet$ Furthermore, the light-trajectory in the field of a localized source at rest with time-dependent intrinsic multipoles, $M^A_L\left(t\right)$ and $S^A_L\left(t\right)$, has been obtained in [@KopeikinKorobkovPolnarev2006; @KopeikinKorobkov2005] in 1PM approximation; see also [@KSGE]. Furthermore, the light trajectory in the field of an arbitrarily moving body with quadrupole-structure has been determined in [@KopeikinMakarov2007].
$\bullet$ In the investigation [@moving_axisymmetric_body] the light propagation in the field of an uniformly moving axisymmetric body has been determined in terms of the full mass-multipole structure of the body. Furthermore, an analytical formula for the time-delay caused by the gravitational field of a body in slow and uniform motion with arbitrary multipoles has been derived in [@Soffel_Han].
$\bullet$ A general solution for light trajectories in the field of arbitrarily moving bodies characterized by intrinsic multipoles has been determined in the 1PN approximation [@Zschocke1] where the moving bodies are endowed with time-dependent intrinsic mass-multipoles $M^A_L\left(t\right)$, as well as in the 1.5PN approximation [@Zschocke2] where the moving bodies are endowed with both time-dependent intrinsic mass-multipoles $M^A_L\left(t\right)$ and spin-multipoles $S^A_L\left(t\right)$.
Moreover, it is clear that astrometry on the sub-micro-arcsecond level necessitates to determine the light trajectory in the second post-Newtonian approximation [@Conference_Cambridge; @Deng_Xie; @Deng_2015; @Minazzoli2; @Xu_Wu; @Xu_Gong_Wu_Soffel_Klioner; @Minazzoli1; @2PN_Light_PropagationA; @Xie_Huang]. Thus far, the light trajectory in 2PN approximation has only been determined in the field of one monopole at rest [@Brumberg1991; @Brumberg1987], a result which has later been confirmed within several investigations [@Deng_Xie; @Deng_2015; @Minazzoli2; @Article_Zschocke1; @LLT2004; @TL2008; @Teyssandier; @HBL2014b; @AshbyBertotti2010; @Moving_Kerr_Black_Hole1]. Very recently, an analytical solution in 2PN approximation for the light trajectory in the field of one arbitrarily moving pointlike monopole has been obtained in [@Zschocke3; @Zschocke4]. That solution has solved the so-called initial-value problem (\[Initial\_Boundary\_Conditions\]). The initial value problem is just the first step in the theory of light propagation, while practical modeling of astronomical observations needs to solve the boundary value problem (\[Boundary\_Value\_Conditions\]), which is the primary topic of this investigation.
The solution of the boundary value problem (\[Boundary\_Value\_Conditions\]) comprises a set of altogether three transformations, which represent the first part of the main results of this investigation:
1. Transformation ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$\sigma$}}$ given by Eq. (\[Transformation\_k\_to\_sigma\]),
2. Transformation ${\mbox{\boldmath$\sigma$}} \rightarrow {\mbox{\boldmath$n$}}$ given by Eq. (\[Transformation\_sigma\_to\_n\]),
3. Transformation ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$n$}}$ given by Eq. (\[Transformation\_k\_to\_n\]).
These transformations are of rather involved structure which inherits two problems: (i) a highly effective algorithm in data reduction requires a simpler solution and (ii) a simplified solution reveals which terms are of relevance for a given goal accuracy in the sub-$\;$ domain. Therefore, in this investigation we have determined upper limits for each individual term in these transformations.
Furthermore, in meanwhile it has become a well-known fact that light propagation in second post-Newtonian approximation leads to the occurrence of so-called [*enhanced terms*]{}. The occurrence of [*enhanced terms*]{} have been recovered at the first time for the case of light propagation in the field of bodies at rest [@Article_Zschocke1; @Teyssandier; @AshbyBertotti2010]. Such [*enhanced terms*]{}, despite that they are of second post-Newtonian order, contain a large factor proportional to $r^{\,1}_A\left(s_1\right)/P_A \gg 1$, where $r^{\,1}_A\left(s_1\right)$ is the distance between body and observer and $P_A$ is the equatorial radius of the body. These [*enhanced 2PN terms*]{} are: ${\mbox{\boldmath$\rho$}}_6$ in (\[Term\_rho\_6\_1\]), ${\mbox{\boldmath$\rho$}}_7$ in (\[Term\_rho\_7\_1\]), ${\mbox{\boldmath$\varphi$}}_8$ in (\[Appendix\_Estimation\_phi\_8\_1\]), ${\mbox{\boldmath$\varphi$}}_9$ in (\[Appendix\_Estimation\_phi\_9\_1\]), and ${\mbox{\boldmath$\varphi$}}_{10}$ in (\[Appendix\_Estimation\_phi\_10\_1\]). The simplified transformations contain only those terms which are relevant for the given threshold in light deflection of at least $1.0\,{\rm nas}$, as there are: 1PN terms, 1.5PN terms and the just mentioned enhanced 2PN terms. These simplified transformations represent the second part of the main results of this investigation:
1. Simplified transformation ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$\sigma$}}$ given by Eq. (\[Simplified\_Transformation\_k\_to\_sigma\]),
2. Simplified transformation ${\mbox{\boldmath$\sigma$}} \rightarrow {\mbox{\boldmath$n$}}$ given by Eq. (\[Simplified\_Transformation\_sigma\_to\_n\]),
3. Simplified transformation ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$n$}}$ given by Eq. (\[Simplified\_Transformation\_k\_to\_n\]).
The simplified transformations ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$\sigma$}}$ and ${\mbox{\boldmath$\sigma$}} \rightarrow {\mbox{\boldmath$n$}}$ are valid with an accuracy of $1.0\,{\rm nas}$ and $1.2\,{\rm nas}$, respectively, while the simplified transformation ${\mbox{\boldmath$k$}} \rightarrow {\mbox{\boldmath$n$}}$ is valid with an accuracy of at least $1.3\,{\rm nas}$. These statements are valid for light deflection for Sun at $45^{\circ}$ (solar aspect angle adopted from the Gaia mission) and all the other Solar System bodies.
But one has to take care about the fact that higher order terms may also significantly contribute on the nas-level of accuracy. Therefore, the impact of possible 2.5PN and 3PN [*enhanced terms*]{} to order ${\cal O}\left(c^{-5}\right)$ and ${\cal O}\left(c^{-6}\right)$ has been considered. While it might be that 2.5PN terms are relevant, it has turned out that 3PN terms will certainly have an impact on the nas-scale of accuracy, namely about $32\,{\rm nas}$ for grazing rays at Jupiter and about $7\,{\rm nas}$ for grazing rays at Saturn. That means, in order to arrive at a light propagation model having an accuracy of $1\,{\rm nas}$ in angular determination, the 3PN solution for the light trajectory needs to be determined. For such a sophisticated calculation it would be sufficient to consider the case of one monopole at rest and then to take just the retarded position of the body at $s_1$ in order to account for the motion of the body. Furthermore, we have argued that [*enhanced terms*]{} in 3.5PN and 4PN approximation contribute certainly less than $1\,{\rm nas}$ for Sun at $45^{\circ}$ and all the other Solar System bodies, except for grazing rays at the Sun, where 4PN terms amount to be about $50\,{\rm nas}$ in light deflection.
The primary aim of our investigations is to develop a fully analytical model of light propagation in the gravitational field of the Solar System which allows for astrometry on the sub-$\;$ and even nas-level of accuracy. Before this aim is in reach, further aspects of the theory of light propagation are of decisive importance, for instance:
1. 1PN [@Zschocke1] and 1.5PN [@Zschocke2] light trajectory needs further to be investigated.
2. 2PN light trajectory in the field of $N$ moving monopoles.
3. 2PN effects of light propagation in the field of finite sized bodies at rest.
4. Enhanced terms in 2.5PN approximation in the field of one moving monopole.
5. Enhanced terms in 3PN approximation in the field of one monopole at rest.
6. Impact of the motion of source and observer.
Each of these and certainly further problems, for instance light propagation in the post-Minkowskian scheme (which allows for astrometry in the far-zone of the Solar System), need to be solved before light propagation models become feasible for astrometry on the sub--level or nas-level of accuracy.
Acknowledgment
==============
This work was funded by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) under grant number 263799048. Sincere gratitude is expressed to Prof. S.A. Klioner and Prof. M.H. Soffel for kind encouragement and enduring support. The author also wish to thank Dr. A.G. Butkevich, Prof. R. Schützhold, Priv.-Doz. Dr. G. Plunien, Prof. B. Kämpfer, and Prof. L.P. Csernai for inspiring discussions about general theory of relativity and astrometry during recent years.
Notation {#Appendix0}
========
Throughout the investigation the following notation is in use:
- $G$ is the Newtonian constant of gravitation
- $c$ is the vacuum speed of light in Minkowskian space-time
- $M_A$ is the rest mass of the body A
- $m_A = G\,M_A/c^2$ is the Schwarzschild radius of the body A
- $P_A$ denotes the equatorial radius of the body A
- $v_A$ denotes the orbital velocity of the body A
- $a_A$ denotes the orbital acceleration of the body A
- Theta-function: $\Theta\left(x\right) = 0$ for $x < 0$ and $\Theta\left(x\right) = 1$ for $x \ge 0$.
- Lower case Latin indices take values $1,2,3$
- $\delta_{ij} = \delta^{ij} = {\rm diag}\left(+1,+1,+1\right)$ is the Kronecker delta
- $\varepsilon_{ijk} = \varepsilon^{ijk}$ with $\varepsilon_{123} = + 1$ is the fully anti-symmetric Levi-Civita symbol
- Triplet of spatial coordinates (three-vectors) are in boldface: e.g. ${\mbox{\boldmath$a$}}$, ${\mbox{\boldmath$b$}}$, ${\mbox{\boldmath$k$}}$, ${\mbox{\boldmath$\sigma$}}$, ${\mbox{\boldmath$r$}}_A$
- Contravariant components of three-vectors: $a^{i} = \left(a^{\,1},a^2,a^3\right)$
- Absolute value of a three-vector: $a = |{\mbox{\boldmath$a$}}| = \sqrt{a^{\,1}\,a^{\,1}+a^2\,a^2+a^3\,a^3}$
- Scalar product of three-vectors: ${\mbox{\boldmath$a$}}\,\cdot\,{\mbox{\boldmath$b$}}=\delta_{ij}\,a^i\,b^j$
- Vector product of two three-vectors: $\left({\mbox{\boldmath$a$}}\times{\mbox{\boldmath$b$}}\right)^i=\varepsilon_{ijk}\,a^j\,b^k$
- Angle $\alpha$ between three-vectors ${\mbox{\boldmath$a$}}$ and ${\mbox{\boldmath$b$}}$ is determined by $\displaystyle \alpha = \arccos \frac{{\mbox{\boldmath$a$}} \cdot {\mbox{\boldmath$b$}}}{\left|{\mbox{\boldmath$a$}}\right|\,\left|{\mbox{\boldmath$b$}}\right|}$
- Lower case Greek indices take values 0,1,2,3
- $\eta_{\alpha\beta} = \eta^{\alpha \beta} = {\rm diag}\left(-1,+1,+1,+1\right)$ is the metric tensor of flat space-time
- $g_{\alpha\beta}$ and $g^{\alpha\beta}$ are the covariant and contravariant components of the metric tensor
- the signature of the metric tensor is adopted to be $\left(-,+,+,+\right)$
- Contravariant components of four-vectors: $a^{\mu} = \left(a^{\,0},a^{\,1},a^2,a^3\right)$
- Scalar product of four-vectors: $a \cdot b = \eta_{\mu \nu}\,a^{\mu}\,b^{\mu}$ in Minkowskian metric $\eta_{\mu\nu}$
- $f_{,\mu} = \partial_{\mu}\,f = \frac{\displaystyle \partial f}{\displaystyle \partial x^{\mu}}$ denotes partial derivative of $f$ with respect to $x^{\mu}$
- $A^{\alpha}_{\,;\,\mu} = A^{\alpha}_{\,,\,\mu} + \Gamma^{\alpha}_{\mu\nu}\,A^{\nu}$ is covariant derivative of first rank tensor.
- $B^{\alpha\beta}_{\;\;\;\;;\,\mu} = B^{\alpha\beta}_{\;\;\;\;,\,\mu} + \Gamma^{\alpha}_{\mu\nu}\,B^{\nu\beta} + \Gamma^{\beta}_{\mu\nu}\,B^{\alpha\nu}$ is covariant derivative of second rank tensor.
- Einstein’s convention is used, i.e. repeated indices are implicitly summed over
- $\displaystyle 1\,{\rm mas}\; ({\rm milli-arcsecond}) = \frac{\pi}{180 \times 60 \times 60}\,10^{-3}\,{\rm rad} \simeq 4.85 \times 10^{-9}\,{\rm rad}$
- $\displaystyle 1\,{\hbox{\rm $\mu$as}}\; ({\rm micro-arcsecond}) = \frac{\pi}{180 \times 60 \times 60}\,10^{-6}\,{\rm rad} \simeq 4.85 \times 10^{-12}\,{\rm rad}$
- $\displaystyle 1\,{\rm nas}\; ({\rm nano-arcsecond}) = \frac{\pi}{180 \times 60 \times 60}\,10^{-9}\,{\rm rad} \simeq 4.85 \times 10^{-15}\,{\rm rad}$
The vectorial functions for light propagation in 2PN approximation {#Appendix2}
==================================================================
The vectorial functions for the coordinate velocity of a light signal
---------------------------------------------------------------------
The vectorial functions ${\mbox{\boldmath$A$}}_1$, ${\mbox{\boldmath$A$}}_2$, ${\mbox{\boldmath$A$}}_3$, and ${\mbox{\boldmath$\epsilon$}}_1$ are given by $$\begin{aligned}
\fl {\mbox{\boldmath$A$}}_1\left({\mbox{\boldmath$x$}}\right) =
- 2\,\left(\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}
{x \left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right)}
+ \frac{{\mbox{\boldmath$\sigma$}}}{x} \right),
\label{Vectorial_Function_A1}
\\
\nonumber\\
\fl {\mbox{\boldmath$A$}}_2\left({\mbox{\boldmath$x$}},{\mbox{\boldmath$v$}}\right) = + \,2\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}
{x \left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right)}\,
\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$v$}}}{c}
+ \frac{4}{x}\,\frac{{\mbox{\boldmath$v$}}}{c}
+ 2\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}
{x^2}\,\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$v$}}}{c}
- 2\,\frac{{\mbox{\boldmath$\sigma$}}}{x^2}\,
\frac{{\mbox{\boldmath$x$}} \cdot {\mbox{\boldmath$v$}}}{c}
\nonumber\\
\nonumber\\
\fl \hspace{1.8cm} -\,2\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}
{x^2\,\left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right)}\,
\frac{\left({\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right) \right) \cdot {\mbox{\boldmath$v$}}}{c}\,,
\label{Vectorial_Function_A2}
\\
\nonumber\\
\fl {\mbox{\boldmath$A$}}_3\left({\mbox{\boldmath$x$}}\right) = - \frac{1}{2}\,\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}{x^4}\,{\mbox{\boldmath$x$}}
+ 8\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x^2\,\left(x - {\mbox{\boldmath$\sigma$}} \cdot{\mbox{\boldmath$x$}} \right)}\,
+ 4\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x\,\left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}} \right)^2}\,
- 4\,\frac{{\mbox{\boldmath$\sigma$}}}{x\,\left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}} \right)}
+ \frac{9}{2}\,\frac{{\mbox{\boldmath$\sigma$}}}{x^2}
\nonumber\\
\nonumber\\
\fl \hspace{1.6cm} -\,\frac{15}{4}\,\left({\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right)\,
\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x^2\,\left|{\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$x$}}\right|^2}
- \frac{15}{4}\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{\left|{\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$x$}}\right|^3}
\left(\arctan \frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}{\left|{\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$x$}}\right|} + \frac{\pi}{2}\right),
\label{Vectorial_Function_A3}\end{aligned}$$
and the vectorial function ${\mbox{\boldmath$\epsilon$}}_1$ is given as follows, $$\begin{aligned}
\fl {\mbox{\boldmath$\epsilon$}}_1\left({\mbox{\boldmath$x$}},{\mbox{\boldmath$v$}}\right) =
- \frac{v^2}{c^2}\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}\,\frac{1}{x}
- 2\,\left(\frac{{\mbox{\boldmath$v$}} \cdot {\mbox{\boldmath$x$}}}{c\,x}\right)^2\,
\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}\,\frac{1}{x}
\nonumber\\
\fl \hspace{0.35cm} -\, 2\,
\left(\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$v$}}}{c}\right)^2\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}\,\frac{1}{x}
+\, 4\,\left(\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$v$}}}{c}\right) \, \left(\frac{{\mbox{\boldmath$v$}} \cdot {\mbox{\boldmath$x$}}}{c\,x}\right) \,
\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}\,\frac{1}{x}
\nonumber\\
\fl \hspace{0.35cm} +\, 4\,\frac{{\mbox{\boldmath$v$}}}{c}\,\left(\frac{{\mbox{\boldmath$v$}} \cdot {\mbox{\boldmath$x$}}}{c\,x}\right) \,\frac{1}{x}
- 4\,\frac{{\mbox{\boldmath$v$}}}{c}\,\left(\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$v$}}}{c}\right)\,\frac{1}{x}
- \, \frac{v^2}{c^2}\,\frac{{\mbox{\boldmath$\sigma$}}}{x}
- 2\, \left(\frac{{\mbox{\boldmath$v$}} \cdot {\mbox{\boldmath$x$}}}{c\,x}\right)^2\,\frac{{\mbox{\boldmath$\sigma$}}}{x}
+ 2\, \left(\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$v$}}}{c}\right)^2\,\frac{{\mbox{\boldmath$\sigma$}}}{x}\,.
\label{epsilon_1}\end{aligned}$$
The vectorial functions for the trajectory of a light signal
------------------------------------------------------------
The vectorial functions for the second integration of geodesic equation (\[Second\_Integration\]) are given as follows: $$\begin{aligned}
\fl {\mbox{\boldmath$B$}}_1\left({\mbox{\boldmath$x$}}\right) = -\,2\, \frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}
+\,2\,{\mbox{\boldmath$\sigma$}}\,\ln \left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right),
\label{Vectorial_Function_C1}
\\
\nonumber\\
\fl {\mbox{\boldmath$B$}}^A_2\left({\mbox{\boldmath$x$}},{\mbox{\boldmath$v$}}\right) = - 2\,\frac{{\mbox{\boldmath$v$}}}{c}\,\ln \left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right),
\label{Vectorial_Function_C2_A}
\\
\nonumber\\
\fl {\mbox{\boldmath$B$}}^B_2\left({\mbox{\boldmath$x$}},{\mbox{\boldmath$v$}}\right) =
+ 2\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}\,\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$v$}}}{c}
+ 2\,\frac{{\mbox{\boldmath$v$}}}{c}\,,
\label{Vectorial_Function_C2_B}
\\
\nonumber\\
\fl {\mbox{\boldmath$B$}}_3\left({\mbox{\boldmath$x$}}\right) = + 4\,\frac{{\mbox{\boldmath$\sigma$}}}{x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}
+\,4\,\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}{\left(x - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right)^2}
+\,\frac{1}{4}\,\frac{{\mbox{\boldmath$x$}}}{x^2}
-\,\frac{15}{4}\,\frac{{\mbox{\boldmath$\sigma$}}}{\left|{\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$x$}}\right|} \,
\arctan \frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}{\left|{\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$x$}}\right|}
\nonumber\\
\nonumber\\
\fl \hspace{1.55cm} -\,\frac{15}{4}\,\left({\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}\right) \frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$x$}} \times {\mbox{\boldmath$\sigma$}}\right)}
{\left|{\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$x$}}\right|^3} \left(\arctan \frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$x$}}}{\left|{\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$x$}}\right|}
+ \frac{\pi}{2}\right).
\label{Vectorial_Function_C3}\end{aligned}$$
We notice that the second term in the vectorial function ${\mbox{\boldmath$B$}}^B_2$ would vanish in case of $N$ bodies; cf. relation (C.20) in [@Zschocke4]. The vectorial function ${\mbox{\boldmath$\epsilon$}}_2$ with well-defined logarithms is given as follows: $$\begin{aligned}
\fl {\mbox{\boldmath$\epsilon$}}_2\left(s,s_0\right) = {\mbox{\boldmath$\epsilon$}}^{\rm A}_2\left(s,s_0\right) + {\mbox{\boldmath$\epsilon$}}^{\rm B}_2\left(s,s_0\right),
\label{epsilon_3}
\\
\nonumber\\
\fl {\mbox{\boldmath$\epsilon$}}^{\rm A}_2\left(s,s_0\right) = - \frac{v_A^2\left(s\right)}{c^2}\,
\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$r$}}_A\left(s\right) \times {\mbox{\boldmath$\sigma$}}\right)}{r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)}
+ \frac{v_A^2\left(s_0\right)}{c^2}\,
\frac{{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$r$}}_A\left(s_0\right) \times {\mbox{\boldmath$\sigma$}}\right)}{r_A\left(s_0\right)
- {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl \hspace{1.9cm} + \frac{v_A^2\left(s_0\right)}{c^2}\, {\mbox{\boldmath$\sigma$}}\,
\ln \frac{r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)}{r_A\left(s_0\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right)}\,,
\label{epsilon_3a}
\\
\nonumber\\
\fl {\mbox{\boldmath$\epsilon$}}^{\rm B}_2\left(s,s_0\right) = + 2\,{\mbox{\boldmath$d$}}_A\left(s_0\right)\,\frac{{\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$a$}}_A\left(s_0\right)}{c^2}\,
\,\ln \frac{r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)}{r_A\left(s_0\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl \hspace{0.5cm} + 2\,\frac{{\mbox{\boldmath$a$}}_A\left(s_0\right)}{c^2}
\left[r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)
- r_A\left(s_0\right) + {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right) \right]
\nonumber\\
\nonumber\\
\fl \hspace{0.5cm} - 2\,\frac{{\mbox{\boldmath$a$}}_A\left(s_0\right)}{c^2}
\left(r_A\left(s_0\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right) \right)
\ln \frac{r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)}{r_A\left(s_0\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl \hspace{0.5cm} + 2\,\frac{{\mbox{\boldmath$a$}}_A\left(s\right)}{c^2}
\left(r_A\left(s_0\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right) - r_A\left(s\right)
+ {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right) \right)
\ln \frac{r_A\left(s\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)}{r_A\left(s_0\right) - {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$r$}}_A\left(s_0\right)}\,.
\nonumber\\
\label{epsilon_3b}\end{aligned}$$
As mentioned above (cf. text below Eq. (\[Series\_A\])) the last term in (\[epsilon\_3b\]) is caused by the replacement of ${\mbox{\boldmath$v$}}_A\left(s_0\right)$ in Eq. (128) in [@Zschocke4] by ${\mbox{\boldmath$v$}}_A\left(s\right)$ according to the series expansion (\[Series\_A\]) which, however, implies to account just for the last term in (\[epsilon\_3b\]). Of course, since ${\mbox{\boldmath$a$}}_A\left(s\right) = {\mbox{\boldmath$a$}}_A\left(s_0\right) + {\cal O}\left(c^{-1}\right)$, the last two terms in (\[epsilon\_3b\]) can be combined to simplify the expression (\[epsilon\_3b\]); cf. the vectorial function (\[Transformation\_k\_to\_sigma\_epsilon\]) where the last two terms in (\[epsilon\_3b\]) have been combined.
Some useful relations for the transformations {#Appendix5}
=============================================
First of all, we notice two important relations between ${\mbox{\boldmath$\sigma$}}$ and ${\mbox{\boldmath$k$}}$, namely $$\begin{aligned}
\fl \hspace{2.0cm} {\mbox{\boldmath$\sigma$}} = {\mbox{\boldmath$k$}} - 2\,\frac{m_A}{R}\,
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
+ {\cal O}\left(c^{-3}\right),
\label{appendix_E_10}\end{aligned}$$
which is just the term in the second line in (\[Transformation\_k\_to\_sigma\]), and $$\begin{aligned}
\fl \hspace{2.0cm} {\mbox{\boldmath$\sigma$}} \cdot {\mbox{\boldmath$k$}} = 1 - \frac{1}{2}\,\frac{m_A^2}{R^2}\,
\bigg|{\mbox{\boldmath$k$}} \times \left( {\mbox{\boldmath$B$}}_1 \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) - {\mbox{\boldmath$B$}}_1 \left({\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right) \right)\bigg|^2
+ {\cal O}\left(c^{-5}\right),
\label{appendix_E_5}\end{aligned}$$
which has already been given by Eq. (157) in [@Zschocke4] and which is needed in order to obtain the formal expression in (\[Transformation\_k\_to\_sigma\_5\]).
According to (\[appendix\_E\_10\]), up to the 1.5PN approximation there is no need to distinguish between the impact vectors (\[Impact\_Vector\_Sigma\_s\]), (\[Impact\_Vector\_Sigma\_s0\]) and (\[Impact\_Vector\_k0\]), (\[Impact\_Vector\_k1\]), simply because of ${\mbox{\boldmath$\sigma$}} = {\mbox{\boldmath$k$}} + {\cal O}\left(c^{-2}\right)$. However, beyond the 1.5PN approximation one has carefully to distinguish between these impact vectors. These impact vectors are related to each other, $$\begin{aligned}
\fl {\mbox{\boldmath$d$}}_A\left(s\right) = {\mbox{\boldmath$d$}}^k_A\left(s\right)
+ 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}}\,\left(
\frac{{\mbox{\boldmath$d$}}^k_A\left(s\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)} \right)
\nonumber\\
\nonumber\\
\fl \hspace{2.5cm} + 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}_A\left(s\right)\,\left(
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}_A^k\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
+ {\cal O}\left(c^{-3}\right),
\label{Relation_Impact_Vectors_1}\end{aligned}$$
which follows from (\[appendix\_E\_10\]). Actually, what we need is the relation between these impact vectors for the specific case of the retarded moment of emission $s_0$ and the retarded moment of reception $s_1$ of the light signal, which is easily obtained from (\[Relation\_Impact\_Vectors\_1\]) just by specifying either $s=s_0$ or $s=s_1$. Furthermore, we notice the following relation which follows from (\[appendix\_E\_10\]), $$\begin{aligned}
\fl \frac{1}{\sigma \cdot r_A\left(s\right)} \! = \! \frac{1}{k \cdot r_A\left(s\right)}
+ \frac{1}{R}\, \frac{2\,m_A}{\left(k \cdot r_A\left(s\right)\right)^2}
\!\left(\!\frac{{\mbox{\boldmath$d$}}^k_A\left(s\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\!\right)\!
+ {\cal O}\left(c^{-3}\right),
\nonumber\\
\label{appendix_E_15}\end{aligned}$$
from which one may deduce the expressions for the specific cases $s=s_0$ or $s=s_1$.
Parameters for massive Solar System bodies
==========================================
In order to quantify the numerical magnitude of the upper limits we will use the parameters of the most massive bodies of the Solar System as presented in Table \[Table1\].
[@cccccc]{} Object & $m_A\,[{\rm m}]$ & $P_A\,[{\rm 10^6\,m}]$ & $v_A/c$ & $a_A\,[{\rm 10^{-3}\,m/s^2}]$ & $r^{\,1}_A\left(s_1\right)\,[{\rm 10^{12}\,m}]$\
Sun & $1476$ & $ 696$ & $4.0 \cdot 10^{-8}$ & $ - $ & $0.149$\
Mercury & $ 0.245 \cdot 10^{-3}$ & $ 2.440$ & $15.8 \cdot 10^{-5}$ & $ 38.73 $ & $0.208$\
Venus & $ 3.615 \cdot 10^{-3} $& $ 6.052$ & $11.7 \cdot 10^{-5}$ & $ 11.34 $ & $0.258$\
Earth & $ 4.438 \cdot 10^{-3}$ & $ 6.378$ & $9.9 \cdot 10^{-5}$ & $ 5.93$ & $0.0015$\
Mars & $ 0.477 \cdot 10^{-3}$ & $ 3.396$ & $8.0 \cdot 10^{-5}$ & $ 2.55$ & $0.399$\
Jupiter & $1.410$ & $ 71.49$ & $4.4 \cdot 10^{-5}$ & $0.21$ & $ 0.898$\
Saturn & $ 0.422 $ & $ 60.27$ & $3.2 \cdot 10^{-5}$ & $ 0.06$ & $ 1.646$\
Uranus & $ 0.064 $ & $ 25.56$ & $2.3 \cdot 10^{-5}$ & $ 0.016$ & $ 3.142$\
Neptune & $ 0.076 $ & $ 24.76$ & $1.8 \cdot 10^{-5}$ & $ 0.0065$ & $4.638$\
\
The approach for the estimation of the upper limits {#Appendix_Estimation1}
===================================================
Preliminary remarks
-------------------
The transformations ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$ and ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ were given by Eqs. (\[Transformation\_k\_to\_sigma\]) and (\[Transformation\_sigma\_to\_n\]), respectively, and the transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$n$}}$ was given by Eq. (\[Transformation\_k\_to\_n\]). These formulae are of rather involved algebraic structure and it is necessary to simplify these expressions by estimations of the upper limit of each individual term which allows to neglect all those terms which contribute less than $1\,{\rm nas}$. The estimation of the terms for light propagation in the gravitational field of moving bodies is considerably more complicated than in case of bodies at rest as presented in our article [@Article_Zschocke1]. This fact is mainly caused by the circumstance that the impact vectors do not coincide for moving bodies, ${\mbox{\boldmath$d$}}_A^k\left(s_0\right) \neq {\mbox{\boldmath$d$}}_A^k\left(s_1\right)$, while in case of bodies at rest the impact vector ${\mbox{\boldmath$d$}}_A^k$ is constant. In the following the approach is described, while in a subsequent \[Appendix\_Example\] an example is considered in more detail.
![A geometrical illustration of a configuration of region A (Eq. (\[Config\_k1\])), where the massive body is located between the observer at ${\mbox{\boldmath$x$}}_1$ and the light source at ${\mbox{\boldmath$x$}}_0$, i.e. $\frac{\pi}{2} \le \alpha_0 \le \pi$ and $0 \le \alpha_1 \le \frac{\pi}{2}$.[]{data-label="Diagram2"}](Diagram2)
![A geometrical illustration of a configuration of region B (Eq. (\[Config\_k2\])), where the light source at ${\mbox{\boldmath$x$}}_0$ is located between the massive body and observer at ${\mbox{\boldmath$x$}}_1$, i.e. $0 \le \alpha_0 \le \frac{\pi}{2}$ and $0 \le \alpha_1 \le \frac{\pi}{2}$ and the condition $0 \le x \le 1$.[]{data-label="Diagram3"}](Diagram3)
![A geometrical illustration of a configuration of region C (Eq. (\[Config\_k3\])), where the observer at ${\mbox{\boldmath$x$}}_1$ is located between the massive body and light source at ${\mbox{\boldmath$x$}}_0$, i.e. $\frac{\pi}{2} \le \alpha_0 \le \pi$ and $\frac{\pi}{2} \le \alpha_1 \le \pi$ and the condition $x \ge 1$.[]{data-label="Diagram4"}](Diagram4)
The distance vector ${\mbox{\boldmath$R$}}$
-------------------------------------------
The following notation for the angles is introduced, $$\begin{aligned}
\fl \hspace{1.0cm} 0 \le \alpha_0 = \angle \left({\mbox{\boldmath$k$}},{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right) \le \pi \quad {\rm and} \quad
0 \le \alpha_1 = \angle \left({\mbox{\boldmath$k$}},{\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right) \le \pi\;.
\label{alpha0_alpha1}\end{aligned}$$
For the estimations it is reasonable to express the distance vector ${\mbox{\boldmath$R$}}$ in (\[Boundary\_3\]) in terms of ${\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)$ and ${\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)$ as follows, $$\begin{aligned}
{\mbox{\boldmath$R$}} = {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) - {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right),
\label{vector_R}\end{aligned}$$
where ${\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)$ and ${\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)$ are given by Eqs. (\[vector\_rA\_1\]) and (\[vector\_rA\_0\_s1\]), respectively. The three-vector ${\mbox{\boldmath$R$}}$ in Eq. (\[Boundary\_3\]) is time-independent. The vector ${\mbox{\boldmath$R$}}$ in Eq. (\[vector\_R\]) is identical to vector ${\mbox{\boldmath$R$}}$ in Eq. (\[Boundary\_3\]), hence also time-independent. The absolute value is $$\begin{aligned}
R = \sqrt{\left(r^{\,0}_A\left(s_1\right)\right)^2 + \left(r^{\,1}_A\left(s_1\right)\right)^2
- 2\,r_A^{\,0}\left(s_1\right) r^{\,1}_A\left(s_1\right)\,\cos \left(\alpha_0 - \alpha_1\right)}\;,
\label{vector_R_series_6}\end{aligned}$$
where $$\begin{aligned}
\angle \left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\,,\,{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right)
= \angle \left({\mbox{\boldmath$k$}}\,,\,{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right) - \angle \left({\mbox{\boldmath$k$}}\,,\,{\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)
\label{Relation_alpha} \end{aligned}$$
has been used; cf. Eq. (69) in [@Article_Zschocke1] for the same angular relation in case of body at rest and notice that $\alpha_0 \ge \alpha_1$ in any astrometric configuration. In order to show the validity of the angular relation (\[Relation\_alpha\]) one should keep in mind that the usual vector operations of Euclidean space can be applied to three-vectors like ${\mbox{\boldmath$k$}}$, ${\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)$, ${\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)$ [@Kopeikin_Efroimsky_Kaplan; @Brumberg1991; @Thorne; @Poisson_Lecture_Notes; @Poisson_Will]; e.g. text below Eq. (3.1.45) in [@Brumberg1991]. Accordingly, relation (\[Relation\_alpha\]) asserts the following, $$\begin{aligned}
\arccos \frac{{\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,1}_A\left(s_1\right)\,r^{\,0}_A\left(s_1\right)}
\! = \! \arccos \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}
- \arccos \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{r^{\,1}_A\left(s_1\right)}\,.
\label{Proof_Relation_alpha_1}\end{aligned}$$
Using $\arccos x - \arccos y = \arccos\left(x\,y + \sqrt{1 - x^2}\,\sqrt{1 - y^2} \right)$ (cf. Eq. (4.4.33) on p. 80 in [@Abramowitz_Stegun]) as well as Eqs. (\[Boundary\_3\]) and (\[vector\_R\]), one may demonstrate the validity of the angular relation (\[Relation\_alpha\]).
Furthermore, for the estimations it is convenient to introduce the ratio $$\begin{aligned}
x = \frac{r_A^{\,0}\left(s_1\right)}{r^{\,1}_A\left(s_1\right)} \quad {\rm with} \quad x \ge 0\,.
\label{ratio_x_1}\end{aligned}$$
Formally, the astrometric configurations allow all individual values for angles, that means $0 \le \alpha_0 \le \pi$ and $0 \le \alpha_1 \le \pi$ as already noticed in their definitions (\[alpha0\_alpha1\]). However, from (\[Impact\_Parameter\_k1\]) we get $r^{\,0}_A\left(s_1\right)\,\sin \alpha_0 = r^{\,1}_A\left(s_1\right)\,\sin \alpha_1$, hence as soon as the parameter $x$ in (\[ratio\_x\_1\]) is fixed, the combinations of these both angles are not arbitrary anymore, but restricted by the relation (note that $\sin \alpha_0 \ge 0$ as well as $\sin \alpha_1 \ge 0$) $$\begin{aligned}
x = \frac{\sin \alpha_1}{\sin \alpha_0}\,.
\label{ratio_x_2} \end{aligned}$$
So, relation (\[ratio\_x\_1\]) is the exact definition of parameter $x$, while (\[ratio\_x\_2\]) follows from (\[Impact\_Parameter\_k1\]). Finally, from (\[ratio\_x\_2\]) we deduce $$\fl \begin{array}[c]{l}
\displaystyle
\alpha_0 = \left \{ \begin{array}[c]{l}
\displaystyle
\pi - \arcsin\left(\frac{\sin \alpha_1}{x}\right)
\quad \mbox{for} \quad \frac{\pi}{2} \le \alpha_0 \le \pi \quad {\rm and} \quad x \ge 0 \\
\\
\displaystyle
\arcsin\left(\frac{\sin \alpha_1}{x}\right)
\quad \mbox{for} \quad 0 \le \alpha_0 \le \frac{\pi}{2} \quad {\rm and} \quad 1 \ge x \ge 0
\end{array} \right \} \;,
\end{array}
\label{ratio_x_3}$$
while the region $0 \le \alpha_0 \le \pi/2$ and $x > 1$ is not possible, because in such configurations the light signal would not be received by an observer; see also comment below Eqs. (\[Config\_k1\]) - (\[Config\_k3\]). The relations in (\[ratio\_x\_3\]) are needed if one evaluates the term $\cos \left(\alpha_0 - \alpha_1\right)$ in Eq. (\[vector\_R\_series\_6\]) for the distance $R$. In the estimations, the variables $x$ and $\alpha_1$ are considered as independent of each other, but restricted by the possible configurations as defined in Eqs. (\[Config\_k1\]) - (\[Config\_k3\]). That means, in region A and C, as defined below by Eqs. (\[Config\_k1\]) and (\[Config\_k3\]), one has to use the first line of (\[ratio\_x\_3\]), while in region B, as defined below by Eq. (\[Config\_k2\]), one has to use the relation in the second line of (\[ratio\_x\_3\]). We also notice that (\[ratio\_x\_2\]) implies $\displaystyle 0 \le \frac{\sin \alpha_1}{x} \le 1$, so that the relations in (\[ratio\_x\_3\]) are uniquely defined.
The possible configurations
---------------------------
The approach for the estimation of the upper limit of each individual term is the following. We separate all possible configurations into three angular areas, $$\begin{aligned}
{\rm A:}\; \frac{\pi}{2} \le \alpha_0 \le \pi \;,\; 0 \le \alpha_1 \le \frac{\pi}{2} \;: \quad
d^k_A\left(s_1\right) \ge P_A\;,
x \ge 0\,,
\label{Config_k1}
\\
{\rm B:}\;\; 0 \le \alpha_0 < \frac{\pi}{2} \;,\; 0 \le \alpha_1 \le \frac{\pi}{2} \;: \quad
d^k_A\left(s_1\right) \ge P_A\;,\; 1 \ge x \ge 0\,,
\label{Config_k2}
\\
{\rm C:}\; \frac{\pi}{2} \le \alpha_0 \le \pi \;,\; \frac{\pi}{2} < \alpha_1 \le \pi \;: \quad
d^k_A\left(s_1\right) \ge 0\;\; \;,\; x \ge 1\,.
\label{Config_k3}\end{aligned}$$
A graphical representation of a typical configuration belonging to region A, B, and C, is given by the Figures \[Diagram2\], \[Diagram3\], and \[Diagram4\], respectively. The constraints for the impact parameter were given by Eqs. (\[Impact\_Vector\_k\_Constraint\_1\]) and (\[Impact\_Vector\_k\_Constraint\_2\]), while the constraints $x \le 1$ in (\[Config\_k2\]) and $x \ge 1$ in (\[Config\_k3\]) are necessary because otherwise the light signal cannot be received by the observer.
The approach for the estimations {#Approach_Appendix}
--------------------------------
The determination of the upper limit of each individual term in the transformations ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$ in (\[Transformation\_k\_to\_sigma\]) and ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ in (\[Transformation\_sigma\_to\_n\]) proceeds as follows:
1. Series expansion of the individual expression in terms of $s_1$,
2. Inserting (\[vector\_R\_series\_6\]) for the absolute value $R$ of the distance vector,
3. Rewriting the expression in terms of the variables $x$ (\[ratio\_x\_1\]) and $\alpha_0, \alpha_1$ (\[alpha0\_alpha1\]),
4. Using relations (\[ratio\_x\_3\]) in line with the regions (\[Config\_k1\]) - (\[Config\_k3\]),
5. Estimation of the term for each possible region separately.
An example: the estimation of $\rho_1$ {#Appendix_Example}
======================================
The estimation of the upper limit of each individual term implies some algebraic effort. So an example is considered in some more detail, which comprehensively elucidates the basic steps about how the approach runs. Accordingly, we shall consider the determination of the upper limit of the term in the second line of (\[Transformation\_k\_to\_sigma\]), which reads $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_1\left(s_1,s_0\right) = - 2\,\frac{m_A}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right).
\label{Example_rho_1_1}\end{aligned}$$
In what follows an upper limit of this expression will be given by means of the approach as just described in the previous section.
Series expansion of ${\mbox{\boldmath$\rho$}}_1$
------------------------------------------------
For the impact vector ${\mbox{\boldmath$d$}}^k_A\left(s_0\right)$ in (\[Example\_rho\_1\_1\]) the series expansion (\[Impact\_Vector\_Relation1\]) is used. For the four-scaler product $k \cdot r_A^{\,0}\left(s_0\right) = - \left(r^{\,0}_A\left(s_0\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)$ in (\[Example\_rho\_1\_1\]) the series expansions $$\begin{aligned}
\fl {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right) = {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right) + \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\bigg(r_A^{\,0}\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)
- r^{\,1}_A\left(s_1\right) + {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\bigg)
\nonumber\\
\fl \hspace{1.5cm} + \,{\cal O}\left(c^{-2}\right),
\label{series_expansion_r0}
\\
\nonumber\\
\fl r^{\,0}_A\left(s_0\right) = r^{\,0}_A\left(s_1\right)
+ \frac{{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)} \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\bigg(r_A^{\,0}\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)
- r^{\,1}_A\left(s_1\right) + {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\bigg)
\nonumber\\
\fl \hspace{1.5cm} + \,{\cal O}\left(c^{-2}\right),
\label{series_expansion_r0_B}\end{aligned}$$
are applied, which follow by inserting the expansion (\[series\_expansion\_body\_1\]) into the definition (\[vector\_rA\_0\]) by keeping in mind relation (\[relation\_k\_2\]); recall that $r^{\,0}_A\left(s_0\right) = \left|{\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|$ and $r^{\,0}_A\left(s_1\right) = \left|{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right|$ in (\[series\_expansion\_r0\_B\]) are absolute values of three-vectors. Using these relations, one obtains the following expansion for the term ${\mbox{\boldmath$\rho$}}_1$ in (\[Example\_rho\_1\_1\]), $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_1\left(s_1,s_0\right) = {\mbox{\boldmath$\rho$}}_1\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}_1\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{series_expansion_rho_1}\end{aligned}$$
where $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_1\left(s_1,s_1\right) = - 2\,\frac{m_A}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right),
\label{series_expansion_rho_1_A}
\\
\nonumber\\
\fl \Delta {\mbox{\boldmath$\rho$}}_1\left(s_1,s_1\right) = + 2\,\frac{m_A}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\left({\mbox{\boldmath$k$}} - \frac{{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}\right)
\cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\;
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl \hspace{2.25cm} - 2\,\frac{m_A}{R}\; {\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl \hspace{2.25cm} + {\cal O}\left(\frac{v_A^2\left(s_1\right)}{c^2}\right)
+ {\cal O}\left(\frac{a_A\left(s_1\right)}{c^2}\right),
\label{series_expansion_rho_1_B}\end{aligned}$$
where the absolute value $R$ is still given by the exact expression (\[Boundary\_3\]). In what follows we show that $$\begin{aligned}
\fl \hspace{1.0cm} \rho_1\left(s_1,s_1\right) \le 4\,\frac{m_A}{P_A}\,,
\label{Estimation_rho1}
\\
\nonumber\\
\fl \hspace{0.75cm} \Delta \rho_1\left(s_1,s_1\right) \le 6\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}
+ {\cal O}\left(\frac{v_A^2\left(s_1\right)}{c^2}\right) + {\cal O}\left(\frac{a_A\left(s_1\right)}{c^2}\right),
\label{Example_rho1_Delta}\end{aligned}$$
for any kind of astrometric configuration. Because of $$\begin{aligned}
\Delta \rho_1\left(s_1,s_1\right) \le 1\,{\rm nas}\,,
\label{Example_rho_1_3}\end{aligned}$$
for all Solar System bodies, only the term (\[series\_expansion\_rho\_1\_A\]) is taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]).
First of all, we continue the exemplifying considerations with the expression (\[series\_expansion\_rho\_1\_A\]), while the estimation of the term (\[series\_expansion\_rho\_1\_B\]) proceeds in similar manner and is considered afterwards.
Estimation of (\[series\_expansion\_rho\_1\_A\])
------------------------------------------------
###
Using the relations $$\begin{aligned}
\fl \frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}
= - \frac{r^{\,1}_A\left(s_1\right) + {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left(d^k_A\left(s_1\right)\right)^2} \;\; {\rm and} \;\;
\frac{1}{k \cdot r^{\,0}_A\left(s_1\right)}
= - \frac{r^{\,0}_A\left(s_1\right) + {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{\left(d^k_A\left(s_1\right)\right)^2}\,,
\label{Example_rho_1_4}\end{aligned}$$
we get for the absolute value of (\[series\_expansion\_rho\_1\_A\]), $$\begin{aligned}
\fl \rho_1\left(s_1,s_1\right) = 2\,\frac{m_A}{d^k_A\left(s_1\right)}\,
\left|\frac{r^{\,1}_A\left(s_1\right) + {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)
- r^{\,0}_A\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{R}\right|\,.
\label{Example_rho_1_6}\end{aligned}$$
Now we insert for the absolute value $R$ the expression (\[vector\_R\_series\_6\]), and then we can rewrite (\[Example\_rho\_1\_6\]) in terms of the variables $x$ (\[ratio\_x\_1\]) as well as the angles $\alpha_0$, $\alpha_1$ (\[alpha0\_alpha1\]) as follows, $$\begin{aligned}
\fl \hspace{1.0cm} \rho_1\left(s_1,s_1\right) = 2\,\frac{m_A}{d^k_A\left(s_1\right)}\,
\left|\frac{1 + \cos \alpha_1 - x - x\,\cos \alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos \left(\alpha_0 - \alpha_1\right)}}\right|\,.
\label{Example_rho_1_7_A}\end{aligned}$$
Keeping in mind that in region A the first line of the angular relations (\[ratio\_x\_3\]) is valid, we find that (\[Example\_rho\_1\_7\_A\]) depends on two variables only, namely $x$ and $\alpha_1$. One may demonstrate with the aid of the computer algebra system [*Maple*]{} [@Maple] that $$\begin{aligned}
\fl f_1 = \left|\frac{1 + \cos \alpha_1 - x - x\,\cos \alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos \left(\alpha_0 - \alpha_1\right)}}\right|
\le 2\,,\quad {\rm for} \quad 0 \le \alpha_1 \le \frac{\pi}{2} \quad {\rm and} \quad x \ge 0\,.
\label{Term_rho_1_6_B}\end{aligned}$$
Inserting (\[Term\_rho\_1\_6\_B\]) into (\[Example\_rho\_1\_7\_A\]) yields for the upper limit $$\begin{aligned}
\rho_1\left(s_1,s_1\right) \le 4\,\frac{m_A}{d^k_A\left(s_1\right)}\,,
\label{Example_rho_1_7}\end{aligned}$$
which validates the upper limit (\[Estimation\_rho1\]) for region A, because $d^k_A\left(s_1\right) \ge P_A$ in region A.
###
The same steps as in the previous Section yield the same result as given by Eq. (\[Example\_rho\_1\_7\_A\]). Keeping in mind that in region B the second line of the angular relations (\[ratio\_x\_3\]) is valid, one may show that the inequality (\[Term\_rho\_1\_6\_B\]) is also valid for region B, $$\begin{aligned}
\fl f_1 = \left|\frac{1 + \cos \alpha_1 - x - x\,\cos \alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos \left(\alpha_0 - \alpha_1\right)}}\right|
\le 2 \quad {\rm for} \quad 0 \le \alpha_1 \le \frac{\pi}{2} \quad {\rm and} \quad 1 \ge x \ge 0\,.
\label{Term_rho_1_6_C}\end{aligned}$$
Hence, one obtains that (\[Example\_rho\_1\_7\]) is also valid in region B, $$\begin{aligned}
\rho_1\left(s_1,s_1\right) \le 4\,\frac{m_A}{d^k_A\left(s_1\right)}\,,
\label{Example_rho_1_8}\end{aligned}$$
which confirms the validity of the upper limit (\[Estimation\_rho1\]) for region B, because $d^k_A\left(s_1\right) \ge P_A$ in region B.
###
In this angular region the impact parameter $d^k_A\left(s_1\right)$ can be arbitrarily small. Therefore, an estimation for the expression in the first line of (\[series\_expansion\_rho\_1\]) is only meaningful if $d^k_A\left(s_1\right)$ does not appear in the denominator. But due to $r_A\left(s_1\right) \gg P_A$, we may get an upper limit where $r^{\,1}_A\left(s_1\right)$ appears in the denominator rather than $d^k_A\left(s_1\right)$. Hence, we reshape identically the expression in (\[Example\_rho\_1\_6\]), which is also valid for region C, as follows, $$\begin{aligned}
\fl \rho_1\left(s_1,s_1\right) = 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{r^{\,1}_A\left(s_1\right)}{d^k_A\left(s_1\right)}\,
\left|\frac{r^{\,1}_A\left(s_1\right) + {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)
- r^{\,0}_A\left(s_1\right) - {\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{R}\right|\,.
\nonumber\\
\label{Example_rho_1_9}\end{aligned}$$
Inserting the expression (\[vector\_R\_series\_6\]) for the distance $R$ and using the notation (\[alpha0\_alpha1\]) for the angles $\alpha_0,\alpha_1$, one obtains $$\begin{aligned}
\fl \rho_1\left(s_1,s_1\right) = 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\left|\frac{1}{\sin \alpha_1}\,
\frac{1 + \cos \alpha_1 - x - x\,\cos\alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos\left(\alpha_0 - \alpha_1\right)}} \right|\,.
\label{Example_rho_1_10}\end{aligned}$$
Keeping in mind that in region C the first line of the angular relations (\[ratio\_x\_3\]) is valid, relation (\[Example\_rho\_1\_10\]) depends on two variables only, namely $x$ and $\alpha_1$. Then, one may show that $$\begin{aligned}
\fl f_2 = \left|\frac{1}{\sin \alpha_1}\,
\frac{1 + \cos \alpha_1 - x - x\,\cos\alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos\left(\alpha_0 - \alpha_1\right)}} \right| \le 1
\quad {\rm for} \quad \frac{\pi}{2} \le \alpha_1 \le \pi \quad {\rm and} \quad x \ge 1\,,
\nonumber\\
\label{Example_rho_1_11}\end{aligned}$$
which can demonstrated with the aid of the computer algebra system [*Maple*]{} [@Maple]. Hence, by inserting (\[Example\_rho\_1\_11\]) into (\[Example\_rho\_1\_10\]) we get $$\begin{aligned}
\rho_1\left(s_1,s_1\right) \le \frac{2\,m_A}{r^{\,1}_A\left(s_1\right)}\,,
\label{Example_rho_1_12}\end{aligned}$$
which also confirms (\[Estimation\_rho1\]) because $r^{\,1}_A\left(s_1\right) \gg P_A$. The upper limits (\[Example\_rho\_1\_7\]), (\[Example\_rho\_1\_8\]), and (\[Example\_rho\_1\_12\]) confirm the estimation given by Eq. (\[Estimation\_rho1\]) for any astrometric configuration.
Estimation of (\[series\_expansion\_rho\_1\_B\])
------------------------------------------------
The expression (\[series\_expansion\_rho\_1\_B\]) is separated into two pieces, $$\begin{aligned}
\fl \Delta {\mbox{\boldmath$\rho$}}_1\left(s_1,s_1\right) = \Delta {\mbox{\boldmath$\rho$}}^A_1\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}^B_1\left(s_1,s_1\right)
+ {\cal O}\left(\frac{v_A^2\left(s_1\right)}{c^2}\right) + {\cal O}\left(\frac{a_A\left(s_1\right)}{c^2}\right),
\label{series_expansion_rho_1_B_0}\end{aligned}$$
where $$\begin{aligned}
\fl \Delta {\mbox{\boldmath$\rho$}}^A_1\left(s_1,s_1\right) = + 2\,\frac{m_A}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\left({\mbox{\boldmath$k$}} - \frac{{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}\right) \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\;
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,,
\nonumber\\
\label{series_expansion_rho_1_B_1}
\\
\fl \Delta {\mbox{\boldmath$\rho$}}^B_1\left(s_1,s_1\right) = - 2\,\frac{m_A}{R}\; {\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\;.
\label{series_expansion_rho_1_B_2}\end{aligned}$$
The estimation of these terms proceeds in the same way as the estimation of (\[series\_expansion\_rho\_1\_A\]). Inserting the expression (\[vector\_R\_series\_6\]) for the distance $R$ and using the notation (\[alpha0\_alpha1\]) for the angles $\alpha_0,\alpha_1$, one obtains $$\begin{aligned}
\fl \Delta {\mbox{\boldmath$\rho$}}^A_1\left(s_1,s_1\right) = 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,
\left|\frac{\sqrt{2}}{x^2}\,\frac{1 - \cos \alpha_1 - x + x\,\cos\alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos \left(\alpha_0 - \alpha_1\right)}}\,
\frac{\sin \alpha_1}{\left(1 - \cos \alpha_0\right)^{3/2}}\right|,
\nonumber\\
\label{Term_rho_10_1_A}
\\
\fl \Delta {\mbox{\boldmath$\rho$}}^B_1\left(s_1,s_1\right) = 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,
\left|\frac{1}{x}\,\frac{1 - \cos \alpha_1 - x + x\,\cos\alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos \left(\alpha_0 - \alpha_1\right)}}\,
\frac{1}{1 - \cos \alpha_0}\right|.
\label{Term_rho_11_1_A}\end{aligned}$$
where in (\[Term\_rho\_10\_1\_A\]) we have used $\left|{\mbox{\boldmath$k$}} - {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)/r^{\,0}_A\left(s_1\right)\right| = \sqrt{2\,\left(1 - \cos \alpha_0\right)}$. For each region A, B, and C one obtains the following inequality, $$\begin{aligned}
\fl \hspace{1.0cm} f_3 = \left|\frac{\sqrt{2}}{x^2}\,
\frac{1 - \cos \alpha_1 - x + x\,\cos\alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos \left(\alpha_0 - \alpha_1\right)}}\,
\frac{\sin \alpha_1}{\left(1 - \cos \alpha_0\right)^{3/2}}\right| \le 2\,,
\label{Term_rho_10_1_B}
\\
\nonumber\\
\fl \hspace{1.0cm} f_4 = \left|\frac{1}{x}\,
\frac{1 - \cos \alpha_1 - x + x\,\cos\alpha_0}{\sqrt{1 + x^2 - 2\,x\,\cos \left(\alpha_0 - \alpha_1\right)}}\,
\frac{1}{1 - \cos \alpha_0}\right| \le 1\,.
\label{Term_rho_11_1_B}\end{aligned}$$
Hence, inserting (\[Term\_rho\_10\_1\_B\]) into (\[Term\_rho\_10\_1\_A\]) and (\[Term\_rho\_11\_1\_B\]) into (\[Term\_rho\_11\_1\_A\]) yields for the upper limit of (\[series\_expansion\_rho\_1\_B\_0\]) $$\begin{aligned}
\Delta \rho_1\left(s_1,s_1\right) \le 6\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Term_rho_10_2}\end{aligned}$$
which is less than $1\,{\rm nas}$ for any Solar System body, as already stated in Eq. (\[Example\_rho\_1\_3\]).
The calculation of the remaining terms of order ${\cal O}\left(v_A^2/c^2\right)$ and ${\cal O}\left(a_A/c^2\right)$ in (\[series\_expansion\_rho\_1\_B\_0\]) involves a considerable algebraic effort which we believe cannot be of much interest. To present all these detailed calculations explicitly here would be disadvantageous to the clarity. So we are obliged to confine ourselves here by the statement that these terms turn out to be even smaller than the terms (\[series\_expansion\_rho\_1\_B\_1\]) and (\[series\_expansion\_rho\_1\_B\_2\]) and will, also for this reason, not be presented here in their explicit form.
Estimation of the terms in the transformation ${\mbox{\boldmath$k$}}$ to ${\mbox{\boldmath$\sigma$}}$ {#Appendix_EstimationA}
=====================================================================================================
Estimation of $\rho_2$ {#Estimation_rho2}
----------------------
The term in the third line of (\[Transformation\_k\_to\_sigma\]) is denoted as ${\mbox{\boldmath$\rho$}}_2$ and reads $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_2\left(s_1,s_0\right) = 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)}
\label{Term_rho_2_1_A}
\\
\nonumber\\
\hspace{1.5cm} = {\mbox{\boldmath$\rho$}}_2\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}_2\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Term_rho_2_1_B}\end{aligned}$$
where $$\begin{aligned}
\fl \hspace{0.25cm} {\mbox{\boldmath$\rho$}}_2\left(s_1,s_1\right) = 2\,\frac{m_A}{R}\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\,,
\label{Term_rho_2_1_C}
\\
\nonumber\\
\fl \Delta {\mbox{\boldmath$\rho$}}_2\left(s_1,s_1\right)
\!=\! 2 \frac{m_A}{R} {\mbox{\boldmath$k$}} \times \left(\!\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\!\right)\!\!
\left(\!{\mbox{\boldmath$k$}} - \frac{{\mbox{\boldmath$r$}}^0_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}\!\right) \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,.
\nonumber\\
\label{Term_rho_2_1_D}\end{aligned}$$
The upper limits of the absolute values are given by $$\begin{aligned}
\rho_2\left(s_1,s_1\right) \le 2\,\frac{m_A}{\sqrt{P_A\,r^{\,0}_A\left(s_1\right)}}\,\frac{v_A\left(s_1\right)}{c}
\le 2\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Term_rho_2_2}
\\
\nonumber\\
\Delta \rho_2\left(s_1,s_1\right) \le 4\,\frac{m_A}{P_A}\,\frac{v^2_A\left(s_1\right)}{c^2} \ll 1\,{\rm nas}\,.
\label{Term_rho_2_1_E}\end{aligned}$$
For the second estimation in (\[Term\_rho\_2\_2\]) we have taken into account that $r^{\,0}_A\left(s_1\right) \simeq P_A$ is quite possible, for instance by a moon orbiting around a planet. Hence, for all Solar System bodies only the term (\[Term\_rho\_2\_1\_C\]) is taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]).
Estimation of $\rho_3$
----------------------
The term in the fourth line of (\[Transformation\_k\_to\_sigma\]) is denoted as ${\mbox{\boldmath$\rho$}}_3$ and reads $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_3\left(s_1,s_0\right) = - 2\,\frac{m_A}{R}\,
\left( {\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} - \frac{{\mbox{\boldmath$v$}}_A\left(s_0\right)}{c}\right) \times {\mbox{\boldmath$k$}} \right)
\label{Term_rho_3_1}
\\
\nonumber\\
\hspace{1.6cm} = {\mbox{\boldmath$\rho$}}_3\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}_3\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Term_rho_3_2}\end{aligned}$$
where $$\begin{aligned}
\fl {\mbox{\boldmath$\rho$}}_3\left(s_1,s_1\right) = 0\,,
\label{Term_rho_3_3_A}
\\
\fl \Delta {\mbox{\boldmath$\rho$}}_3\left(s_1,s_1\right) = - 2\,m_A\,{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$a$}}_A\left(s_1\right)}{c^2} \times {\mbox{\boldmath$k$}}\right)
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{R}\,.
\label{Term_rho_3_3}\end{aligned}$$
The upper limit of the absolute value $\rho_3$ is given by $$\begin{aligned}
\Delta \rho_3\left(s_1,s_1\right) \le 4\,m_A\,\frac{a_A\left(s_1\right)}{c^2} \ll 1\,{\rm nas}\,,
\label{Term_rho_3_4}\end{aligned}$$
hence the term ${\mbox{\boldmath$\rho$}}_3$ is not taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]). As already mentioned above (see text below Eq. (\[Vectorial\_Function\_C3\])), the term (\[Term\_rho\_3\_1\]) actually vanishes in case of $N$ moving monopoles; cf. Eq. (C.20) in [@Zschocke4].
Estimation of $\rho_4$ {#Estimation_rho4}
----------------------
The term in the fifth line of (\[Transformation\_k\_to\_sigma\]) is denoted as ${\mbox{\boldmath$\rho$}}_4$ and reads $$\begin{aligned}
\fl \hspace{1.0cm} {\mbox{\boldmath$\rho$}}_4\left(s_1,s_0\right) = 2\,\frac{m_A}{R}\,\left(\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_0\right)}{c}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\label{Term_rho_4_1}
\\
\nonumber\\
\fl \hspace{2.6cm} = {\mbox{\boldmath$\rho$}}_4\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}_4\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Term_rho_4_2}\end{aligned}$$
where $$\begin{aligned}
\fl {\mbox{\boldmath$\rho$}}_4\left(s_1,s_1\right) = 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right),
\label{Term_rho_4_3}
\\
\nonumber\\
\fl \Delta {\mbox{\boldmath$\rho$}}_4\left(s_1,s_1\right) = + 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
{\mbox{\boldmath$k$}} \times \left(\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \times {\mbox{\boldmath$k$}}\right)
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl \hspace{0.75cm} - 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\left({\mbox{\boldmath$k$}} - \frac{{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}\right) \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\;
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl \hspace{0.75cm} - 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$a$}}_A\left(s_1\right)}{c^2}\,
{\mbox{\boldmath$d$}}^k_A\left(s_1\right)\,
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,.
\label{Term_rho_4_5}\end{aligned}$$
The upper limit of the absolute value $\rho_4$ is given by $$\begin{aligned}
\rho_4\left(s_1,s_1\right) \le 4\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Estimation_rho_4_1}
\\
\nonumber\\
\hspace{-0.25cm} \Delta \rho_4\left(s_1,s_1\right) \le 6\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 4\,m_A\,\frac{a_A\left(s_1\right)}{c^2} \ll 1\,{\rm nas}\,,
\label{Estimation_rho_4_2}\end{aligned}$$
for all Solar System bodies. Hence, only the term (\[Term\_rho\_4\_3\]) is taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]).
Estimation of $\rho_5$
----------------------
The term in the sixth line of (\[Transformation\_k\_to\_sigma\]) is denoted as ${\mbox{\boldmath$\rho$}}_5$ and reads $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_5\left(s_1,s_0\right) = - 2\,\frac{m_A^2}{R^2}\,{\mbox{\boldmath$k$}}\,
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right|^2
\label{Term_rho_5_1}
\\
\nonumber\\
\hspace{1.6cm} = {\mbox{\boldmath$\rho$}}_5\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Term_rho_5_1_A} \end{aligned}$$
where $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_5\left(s_1,s_1\right) = - 2\,\frac{m_A^2}{R^2}\,{\mbox{\boldmath$k$}}\,
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right|^2\,.
\label{Term_rho_5_1_B} \end{aligned}$$
The upper limit of the absolute value is given by $$\begin{aligned}
\rho_5\left(s_1,s_1\right) \le 8\,\frac{m_A^2}{P^2_A}\,,
\label{Term_rho_5_2}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies. Furthermore, (\[Term\_rho\_5\_1\]) is a [*scaling term*]{}.
Estimation of $\rho_6$
----------------------
The term in the seventh line of (\[Transformation\_k\_to\_sigma\]) is denoted as ${\mbox{\boldmath$\rho$}}_6$ and reads $$\begin{aligned}
\fl \hspace{0.25cm} {\mbox{\boldmath$\rho$}}_6\left(s_1,s_0\right) = - 2\,\frac{m_A^2}{R^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
+ \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right|^2
\label{Term_rho_6_1}
\\
\nonumber\\
\fl \hspace{1.85cm} = {\mbox{\boldmath$\rho$}}_6\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Term_rho_6_1_B}\end{aligned}$$
where $$\begin{aligned}
\fl {\mbox{\boldmath$\rho$}}_6\left(s_1,s_1\right) = - 2\,\frac{m_A^2}{R^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
+ \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right)
\left|\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right|^2\,.
\label{Term_rho_6_1_A}\end{aligned}$$
The upper limit of the absolute value is given by $$\begin{aligned}
\rho_6\left(s_1,s_1\right) \le 16\,\frac{m_A^2}{P_A^2}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,,
\label{Term_rho_6_2}\end{aligned}$$
which contains the large factor $r^{\,1}_A\left(s_1\right)/P_A$ and, therefore, is an [*enhanced term*]{}, hence (\[Term\_rho\_6\_1\_A\]) has necessarily to be taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]).
Estimation of $\rho_7$
----------------------
The term in the eighth line of (\[Transformation\_k\_to\_sigma\]) is denoted as ${\mbox{\boldmath$\rho$}}_7$ and reads $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_7\left(s_1,s_0\right) = - 4\,\frac{m_A^2}{R}\,\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left(k \cdot r_A^{\,0}\left(s_0\right)\right)^2}\right)
\label{Term_rho_7_1}
\\
\nonumber\\
\hspace{1.6cm} = {\mbox{\boldmath$\rho$}}_7\left(s_1,s_1\right) + {\cal O} \left(c^{-5}\right),
\label{Term_rho_7_1_A}\end{aligned}$$
where $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_7\left(s_1,s_1\right) = - 4\,\frac{m_A^2}{R}\,\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r_A^{\,0}\left(s_1\right)\right)^2}\right).
\label{Term_rho_7_1_B}\end{aligned}$$
The upper limit of the absolute value is given by $$\begin{aligned}
\rho_7\left(s_1,s_1\right) \le 16\,\frac{m_A^2}{P_A^2}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,,
\label{Term_rho_7_2}\end{aligned}$$
which is an [*enhanced term*]{} because of the large factor $r^{\,1}_A\left(s_1\right)/P_A$, hence (\[Term\_rho\_7\_1\_B\]) must be taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]).
Estimation of $\rho_8$
----------------------
The terms in the ninth and tenth line of (\[Transformation\_k\_to\_sigma\]) are combined to a term ${\mbox{\boldmath$\rho$}}_8$, $$\begin{aligned}
\fl {\mbox{\boldmath$\rho$}}_8\left(s_1,s_0\right) = {\mbox{\boldmath$\rho$}}^A_8\left(s_1\right) + {\mbox{\boldmath$\rho$}}^B_8\left(s_0\right),
\label{Term_rho_8_1}
\\
\nonumber\\
\fl {\mbox{\boldmath$\rho$}}^A_8\left(s_1\right) = + \frac{15}{4} \frac{m_A^2}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^3} \,
\left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)
\left(\arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|} + \frac{\pi}{2} \right),
\label{Term_rho_8_1_A}
\\
\nonumber\\
\fl {\mbox{\boldmath$\rho$}}^B_8\left(s_0\right) = - \frac{15}{4} \frac{m_A^2}{R}
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|^3} \,
\left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right)
\left( \arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_0\right)\right|}
+ \frac{\pi}{2} \right).
\label{Term_rho_8_1_B}\end{aligned}$$
The series expansion of ${\mbox{\boldmath$\rho$}}_8$ in (\[Term\_rho\_8\_1\]) reads $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_8\left(s_1,s_0\right) = {\mbox{\boldmath$\rho$}}_8\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Term_rho_8_1_C}\end{aligned}$$
where $$\begin{aligned}
\fl {\mbox{\boldmath$\rho$}}_8\left(s_1,s_1\right) = \frac{15}{4} \frac{m_A^2}{R}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^3}\,
\bigg[\left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right)
\left(\arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|} + \frac{\pi}{2} \right)
\nonumber\\
\fl \hspace{5.0cm} - \left({\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right)
\left( \arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)\right|}
+ \frac{\pi}{2} \right) \bigg].
\label{Term_rho_8_1_D}\end{aligned}$$
The upper limit of the absolute value is given by $$\begin{aligned}
\rho_8\left(s_1,s_1\right) \le \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}\,,
\label{Term_rho_8_2}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies and for the Sun at $45^{\circ}$. Hence (\[Term\_rho\_8\_1\_D\]) is not taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]).
Estimation of $\rho_9$
----------------------
The term in the eleventh line of (\[Transformation\_k\_to\_sigma\]) is denoted as ${\mbox{\boldmath$\rho$}}_9$ and reads $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_9\left(s_1,s_0\right) = - \frac{1}{4}\frac{m_A^2}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(r_A^{\,1}\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{\left(r_A^{\,0}\left(s_0\right)\right)^2}\right)
\label{Term_rho_9_1}
\\
\nonumber\\
\hspace{1.6cm} = {\mbox{\boldmath$\rho$}}_9\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Term_rho_9_1_A}\end{aligned}$$
where $$\begin{aligned}
{\mbox{\boldmath$\rho$}}_9\left(s_1,s_1\right) = - \frac{1}{4}\frac{m_A^2}{R}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(r_A^{\,1}\left(s_1\right)\right)^2}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(r_A^{\,0}\left(s_1\right)\right)^2}\right).
\label{Term_rho_9_1_B}\end{aligned}$$
The upper limit of the absolute value is $$\begin{aligned}
\rho_9\left(s_1,s_1\right) \le \frac{1}{4}\,\frac{m_A^2}{P_A^2}\,,
\label{Term_rho_9_2}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies and for the Sun at $45^{\circ}$. Hence (\[Term\_rho\_9\_1\]) is not taken into account in the simplified transformation (\[Simplified\_Transformation\_k\_to\_sigma\]).
Numerical values for the upper limits $\rho_1$, $\dots$ , $\rho_9$ are presented in Table \[Table2\].
[@ccccccccccc]{} & $\rho_1$ & $\rho_2$ & $\rho_3$ & $\rho_4$ & $\rho_5$ & $\rho_6$ &$\rho_7$& $\rho_8 \dots \rho_9$& $\hat{\epsilon}_2$\
Sun at $45^{\circ}$ & $1.2 \cdot 10^7$ & $-$ & $-$ & $-$ & $-$ & $0.95$ & $0.95$ & $-$ & $-$\
Mercury & $0.8 \cdot 10^5$ & $6.5$ & $-$ & $13.1$ & $-$ & $2.8$ & $2.8$ & $-$ & $-$\
Venus & $0.5 \cdot 10^6$ & $28.8$ & $-$ & $57.7$ & $-$ & $50.2$ & $50.2$ & $-$ & $-$\
Earth & $0.6 \cdot 10^6$ & $28.4$ & $-$ & $56.8$ & $-$ & $-$ & $-$ & $-$ & $-$\
Mars & $0.1 \cdot 10^6$ & $4.6$ & $-$ & $9.3$ & $-$ & $7.6$ & $7.6$ & $-$ & $-$\
Jupiter & $1.6 \cdot 10^7$ & $358.0$ & $-$ & $716.0$ & $-$ & $1.6 \cdot 10^4$ & $1.6 \cdot 10^4$ & $-$ & $-$\
Saturn & $0.6 \cdot 10^7$ & $92.4$ & $-$ & $184.9$ & $-$ & $4.4 \cdot 10^3$ & $4.4 \cdot 10^3$ & $-$ & $-$\
Uranus & $0.2 \cdot 10^7$ & $23.8$ & $-$ & $47.5$ & $-$ & $2.5 \cdot 10^3$ & $2.5 \cdot 10^3$ & $-$ & $-$\
Neptune & $0.2 \cdot 10^7$ & $22.8$ & $-$ & $45.6$ & $-$ & $5.8 \cdot 10^3$ & $5.8 \cdot 10^3$ & $-$ & $-$\
\
Estimation of the terms in the transformation ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ {#Appendix_EstimationB}
=====================================================================================================
The transformation ${\mbox{\boldmath$\sigma$}}$ to ${\mbox{\boldmath$n$}}$ is given by Eq. (\[Transformation\_sigma\_to\_n\]). In what follows an upper limit of each individual term of this transformation is given. The approach is the same as described and used in the previous sections. If the terms depend solely on the retarded $s_1$ then a series expansion is not necessary. The estimations are straightforward and they are just given.
Estimation of $\varphi_1$
-------------------------
The term in the second line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_1$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_1\left(s_1\right)
= 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\left|\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\right|,
\label{Appendix_Estimation_phi_1_A}
\\
\nonumber\\
\varphi_1\left(s_1\right) \le 4\,\frac{m_A}{P_A}\,,
\label{Appendix_Estimation_phi_1}\end{aligned}$$
which has to be taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_2$
-------------------------
The term in the third line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_2$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_2\left(s_1\right) = - 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,{\mbox{\boldmath$k$}}\,\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_2_A}
\\
\nonumber\\
\varphi_2\left(s_1\right) \le 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_2}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies. Furthermore, (\[Appendix\_Estimation\_phi\_2\_A\]) is a [*scaling term*]{}.
Estimation of $\varphi_3$
-------------------------
The term in the fourth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_3$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_3\left(s_1\right) = - 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}}\cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_3_A}
\\
\nonumber\\
\varphi_3\left(s_1\right) \le 4\,\frac{m_A}{P_A}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_3}\end{aligned}$$
which has to be taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_4$
-------------------------
The term in the fifth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_4$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_4\left(s_1\right) = 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
+ 2\,\frac{m_A}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,{\mbox{\boldmath$d$}}^k_A\left(s_1\right)\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_4_A}
\\
\nonumber\\
\varphi_4\left(s_1\right) \le 6\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_4} \end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies, hence (\[Appendix\_Estimation\_phi\_4\_A\]) is not taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_5$
-------------------------
The term in the sixth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_5$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_5\left(s_1\right)
= 2\,\frac{m_A}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_5_A}
\\
\nonumber\\
\varphi_5\left(s_1\right) \le 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Estimation_phi_5}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies, hence (\[Appendix\_Estimation\_phi\_5\_A\]) is not taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_6$
-------------------------
The term in the seventh line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_6$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_6\left(s_1\right) = - 2\,{\mbox{\boldmath$k$}}\,\frac{m_A^2}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}\,,
\label{Appendix_Estimation_phi_6_A}
\\
\nonumber\\
\varphi_6\left(s_1\right) \le 8\,\frac{m_A^2}{P_A^2}\,,
\label{Appendix_Estimation_phi_6}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies. Furthermore, (\[Appendix\_Estimation\_phi\_6\_A\]) is a [*scaling term*]{}.
Estimation of $\varphi_7$
-------------------------
The term in the eighth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_7$ and reads $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_7\left(s_1,s_0\right) = 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}}}{R}\, \left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)}\right)
\label{Appendix_Estimation_phi_7_1}
\\
\nonumber\\
\fl \hspace{1.6cm} = {\mbox{\boldmath$\varphi$}}_7\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right), \end{aligned}$$
where $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_7\left(s_1,s_1\right)
= 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}}}{R}\, \left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)}\right).
\label{Appendix_Estimation_phi_7_1_B}\end{aligned}$$
For the upper limit one finds $$\begin{aligned}
\varphi_7\left(s_1,s_1\right) \le 16\,\frac{m_A^2}{P_A^2}\,,
\label{Appendix_Estimation_phi_7_2}\end{aligned}$$
which is less than $1.3\,{\rm nas}$ for all Solar System bodies. Furthermore, (\[Appendix\_Estimation\_phi\_7\_1\]) is a [*scaling term*]{}.
Estimation of $\varphi_8$
-------------------------
The term in the ninth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_8$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_8\left(s_1\right)
= 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}\,,
\label{Appendix_Estimation_phi_8_1}
\\
\nonumber\\
\varphi_8\left(s_1\right) \le 16\,\frac{m_A^2}{P_A^2}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,,
\label{Appendix_Estimation_phi_8_2}\end{aligned}$$
which is an [*enhanced term*]{} because of the large factor $r^{\,1}_A\left(s_1\right)/P_A$, hence (\[Appendix\_Estimation\_phi\_8\_1\]) must necessarily to be taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_9$
-------------------------
The term in the tenth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_9$ and reads $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_9 \left(s_1,s_0\right) = 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)} \right)
\nonumber\\
\label{Appendix_Estimation_phi_9_1}
\\
\fl \hspace{1.6cm} = {\mbox{\boldmath$\varphi$}}_9 \left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Appendix_Estimation_phi_9_1_A}\end{aligned}$$
where $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_9\left(s_1,s_1\right) = 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{\left(k \cdot r^{\,1}_A\left(s_1\right)\right)^2}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)} \right).
\nonumber\\
\label{Appendix_Estimation_phi_9_1_B}\end{aligned}$$
For the upper limit one finds $$\begin{aligned}
\varphi_9\left(s_1,s_1\right) \le 32\,\frac{m_A^2}{P_A^2}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,,
\label{Appendix_Estimation_phi_9_2}\end{aligned}$$
which contains the large factor $r^{\,1}_A\left(s_1\right)/P_A$ hence is an [*enhanced term*]{}, so that (\[Appendix\_Estimation\_phi\_9\_1\_B\]) must be taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_{10}$
----------------------------
The term in the eleventh line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_{10}$ and reads $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_{10}\left(s_1,s_0\right) = 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)} \right)
\label{Appendix_Estimation_phi_10_1}
\\
\nonumber\\
\fl \hspace{1.75cm} = {\mbox{\boldmath$\varphi$}}_{10}\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right),
\label{Appendix_Estimation_phi_10_1_A}\end{aligned}$$
where $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_{10}\left(s_1,s_1\right) = 4\,\frac{m_A^2}{r^{\,1}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
- \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_1\right)} \right)\,.
\label{Appendix_Estimation_phi_10_1_B}\end{aligned}$$
For the upper limit one finds $$\begin{aligned}
\varphi_{10}\left(s_1,s_1\right) \le 16\,\frac{m_A^2}{P_A^2}\,\frac{r^{\,1}_A\left(s_1\right)}{P_A}\,,
\label{Appendix_Estimation_phi_10_2}\end{aligned}$$
which contains the large factor $r^{\,1}_A\left(s_1\right)/P_A$ hence is an [*enhanced term*]{}, so that (\[Appendix\_Estimation\_phi\_10\_1\_B\]) must be taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_{11}$
----------------------------
The term in the twelfth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_{11}$ and reads $$\begin{aligned}
{\mbox{\boldmath$\varphi$}}_{11}\left(s_1\right) = - 4\,\frac{m_A^2}{\left(r_A^{\,1}\left(s_1\right)\right)^2}\,
\frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,,
\label{Appendix_Estimation_phi_11_1}
\\
\nonumber\\
\varphi_{11}\left(s_1\right) \le 8\,\frac{m_A^2}{P_A\,r_A^{\,1}\left(s_1\right)}\,,
\label{Appendix_Estimation_phi_11_2}\end{aligned}$$
which contributes less than $1\,{\rm nas}$ for all Solar System bodies, hence (\[Appendix\_Estimation\_phi\_11\_1\]) is not taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_{12}$
----------------------------
The term in the thirteenth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_{12}$ and reads $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_{12}\left(s_1\right)
= - \frac{m_A^2}{2}\,{\mbox{\boldmath$d$}}_A^k\left(s_1\right)\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left(r^{\,1}_A\left(s_1\right)\right)^4}
- \frac{15}{4}\,\frac{m_A^2}{\left(r_A^{\,1}\left(s_1\right)\right)^2}\,{\mbox{\boldmath$d$}}^k_A\left(s_1\right)
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^2}\,,
\label{Appendix_Estimation_phi_12_1}
\\
\nonumber\\
\fl \varphi_{12}\left(s_1\right) \le \frac{1}{2}\,\frac{m_A^2}{\left(r^{\,1}_A\left(s_1\right)\right)^2}
+ \frac{15}{4}\,\frac{m_A^2}{P_A\,r^{\,1}_A\left(s_1\right)}\,,
\label{Appendix_Estimation_phi_12_2}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies, hence (\[Appendix\_Estimation\_phi\_12\_1\]) is not taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
Estimation of $\varphi_{13}$
----------------------------
The term in the fourteenth line of (\[Transformation\_sigma\_to\_n\]) is denoted as ${\mbox{\boldmath$\varphi$}}_{13}$ and reads $$\begin{aligned}
\fl {\mbox{\boldmath$\varphi$}}_{13}\left(s_1\right) = - \frac{15}{4}\,m_A^2\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|^3}
\left(\arctan \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)}{\left|{\mbox{\boldmath$k$}} \times {\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right)\right|} + \frac{\pi}{2} \right),
\label{Appendix_Estimation_phi_13_1}
\\
\nonumber\\
\fl \varphi_{13}\left(s_1\right) \le \frac{15}{4}\,\pi\,\frac{m_A^2}{P_A^2}\,,
\label{Appendix_Estimation_phi_13_2}\end{aligned}$$
which is less than $1\,{\rm nas}$ for all Solar System bodies, hence (\[Appendix\_Estimation\_phi\_13\_1\]) is not taken into account in the simplified transformation (\[Simplified\_Transformation\_sigma\_to\_n\]).
The numerical values for the upper limits are presented in Table \[Table3\].
[@cccccccccccc]{} & $\varphi_1$ & $\varphi_2$ & $\varphi_3$ & $\varphi_4 \dots \varphi_6$ & $\varphi_7$ & $\varphi_8$ & $\varphi_9$ & $\varphi_{10}$ & $\varphi_{11} \dots \varphi_{13}$ & $\hat{\epsilon}_1$\
Sun at $45^{\circ}$ & $1.2 \cdot 10^7$ & $-$ & $-$ & $-$ & $-$ & $-$ & $1.8$ & $-$ & $-$ & $-$\
Mercury & $\!0.8 \cdot 10^5\!$&$-$&$\!13.1\!$& $-$& $-$& $2.8$ & $\!5.7\!$ & $\!2.8\!$ & $-$ & $-$\
Venus & $\!0.5 \cdot 10^6\!$&$-$&$\!57.7\!$& $-$& $-$& $50.2$ & $\!100.4\!$ & $\!50.2\!$ & $-$ & $-$\
Earth & $\!0.6 \cdot 10^6\!$&$-$&$\!56.8\!$& $-$& $-$& $-$ & $-$ & $-$ & $-$ & $-$\
Mars & $\!0.1 \cdot 10^6\!$&$-$&$\!9.3\!$&$-$ &$-$& $7.7$ & $15.3$ & $7.7$ & $-$ & $-$\
Jupiter & $\!1.6 \cdot 10^7\!$&$-$&$\!716.0\!$&$-$&$1.3$ & $\!1.6 \cdot 10^4\!$ & $\!3.2 \cdot 10^4\!$ & $\!1.6 \cdot 10^4\!$ & $-$ & $-$\
Saturn & $\!0.6 \cdot 10^7\!$&$-$&$\!184.9\!$&$-$&$-$& $\!4.4 \cdot 10^3\!$ & $\!8.8 \cdot 10^3\!$ & $\!4.4 \cdot 10^3\!$ & $-$ & $-$\
Uranus & $\!0.2 \cdot 10^7\!$ & $-$ & $\!47.5\!$&$-$&$-$&$\!2.5 \cdot 10^3\!$ & $\!5.1 \cdot 10^3\!$ & $\!2.5 \cdot 10^3\!$ & $-$ & $-$\
Neptune & $\!0.2 \cdot 10^7\!$ & $-$ & $\!45.6\!$&$-$&$-$&$\!5.8 \cdot 10^3\!$ & $\!1.2 \cdot 10^4\!$ & $\!5.8 \cdot 10^3\!$ & $-$ & $-$\
\
The terms $\hat{{\mbox{\boldmath$\epsilon$}}}_1$ and $\hat{{\mbox{\boldmath$\epsilon$}}}_2$ {#Appendix_epsilon}
===========================================================================================
The upper limit of the term $\hat{{\mbox{\boldmath$\epsilon$}}}_1$
------------------------------------------------------------------
The expression of the term $\hat{{\mbox{\boldmath$\epsilon$}}}_1$ in (\[Transformation\_sigma\_to\_n\_5\]) and (\[Transformation\_sigma\_to\_n\]) reads $$\begin{aligned}
\hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right)
= m_A\,{\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$\epsilon$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right),{\mbox{\boldmath$v$}}_A\left(s_1\right)\right) \times {\mbox{\boldmath$\sigma$}}\right),
\label{epsilon1}\end{aligned}$$
where ${\mbox{\boldmath$\epsilon$}}_1$ is given by Eq. (\[epsilon\_1\]). The vectorial term $\hat{{\mbox{\boldmath$\epsilon$}}}_1$ is of order ${\cal O}\left(c^{-4}\right)$. Due to ${\mbox{\boldmath$\sigma$}} = {\mbox{\boldmath$k$}} + {\cal O}\left(c^{-2}\right)$ we may replace the unit-vector ${\mbox{\boldmath$\sigma$}}$ in (\[epsilon1\]) by the unit-vector ${\mbox{\boldmath$k$}}$, because such a replacement would cause an error of the order ${\cal O}\left(c^{-6}\right)$ which is beyond 2PN approximation. Hence, we get $$\begin{aligned}
\fl \hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right)
= m_A\,{\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$\epsilon$}}_1\left({\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right),{\mbox{\boldmath$v$}}_A\left(s_1\right)\right) \times {\mbox{\boldmath$k$}}\right) + {\cal O}\left(c^{-6}\right)
\nonumber\\
\nonumber\\
\fl \hspace{0.25cm} = + 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$v$}}_A\left(s_1\right) \times {\mbox{\boldmath$k$}}\right)}{c}
\frac{{\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{r^{\,1}_A\left(s_1\right)\,c}
\nonumber\\
\nonumber\\
\fl \hspace{0.5cm} - 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$v$}}_A\left(s_1\right) \times {\mbox{\boldmath$k$}}\right)}{c}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\nonumber\\
\fl \hspace{0.5cm} - \frac{m_A}{r^{\,1}_A\left(s_1\right)} \frac{{\mbox{\boldmath$d$}}_A^k\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
\left[\frac{v^2_A\left(s_1\right)}{c^2}
+ 2 \left(\frac{{\mbox{\boldmath$r$}}^{\,1}_A\left(s_1\right) \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{r^{\,1}_A\left(s_1\right)\,c}
+ \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\right)^2\right] + {\cal O}\left(c^{-6}\right).
\nonumber\\
\label{Transformation_sigma_to_n_epsilon}\end{aligned}$$
The upper limit of the absolute value of $\hat{\epsilon}_1 = \left|\hat{{\mbox{\boldmath$\epsilon$}}}_1\right|$ is given by $$\begin{aligned}
\hat{\epsilon}_1 = \left|\hat{{\mbox{\boldmath$\epsilon$}}}_1\left(s_1\right)\right| \le 8\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 18\,\frac{m_A}{P_A}\,\frac{v_A^2\left(s_1\right)}{c^2} + {\cal O}\left(c^{-6}\right).
\label{Appendix_estimation_epsilon_1}\end{aligned}$$
The upper limit of the term $\hat{{\mbox{\boldmath$\epsilon$}}}_2$
------------------------------------------------------------------
The expression of the term $\hat{{\mbox{\boldmath$\epsilon$}}}_2$ in (\[Transformation\_k\_to\_sigma\_5\]) and (\[Transformation\_k\_to\_sigma\]) reads as follows: $$\begin{aligned}
\hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_0\right)
= \frac{m_A}{R}\,\Bigg({\mbox{\boldmath$\sigma$}} \times \bigg[ {\mbox{\boldmath$\sigma$}} \times {\mbox{\boldmath$\epsilon$}}_2\left(s_1,s_0\right)\bigg]\Bigg),
\label{epsilon2}\end{aligned}$$
where ${\mbox{\boldmath$\epsilon$}}_2$ is given by Eq. (\[epsilon\_3\]). Because the vectorial term $\hat{{\mbox{\boldmath$\epsilon$}}}_2$ is of order ${\cal O}\left(c^{-4}\right)$, we ${\mbox{\boldmath$\sigma$}} = {\mbox{\boldmath$k$}} + {\cal O}\left(c^{-2}\right)$ we may replace the unit-vector ${\mbox{\boldmath$\sigma$}}$ in (\[epsilon2\]) by the unit-vector ${\mbox{\boldmath$k$}}$, because such a replacement would cause an error of the order ${\cal O}\left(c^{-6}\right)$ which is beyond 2PN approximation. So, we obtain $$\begin{aligned}
\fl \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_0\right)
= \frac{m_A}{R}\,\Bigg({\mbox{\boldmath$k$}} \times \bigg[ {\mbox{\boldmath$k$}} \times {\mbox{\boldmath$\epsilon$}}_2\left(s_1,s_0\right)\bigg]\Bigg) + {\cal O}\left(c^{-6}\right)
\nonumber\\
\nonumber\\
\fl \hspace{0.5cm} = - \frac{m_A}{R}\,\frac{v_A^2\left(s_1\right)}{c^2}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}
+ \frac{m_A}{R}\,\frac{v_A^2\left(s_0\right)}{c^2}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_0\right)}{k \cdot r_A^{\,0}\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl \hspace{0.75cm} - 2\,\frac{m_A}{R}\,{\mbox{\boldmath$d$}}^k_A\left(s_0\right) \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$a$}}_A\left(s_1\right)}{c^2}
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)}
\nonumber\\
\nonumber\\
\fl \hspace{0.75cm} + 2\,\frac{m_A}{R} \frac{{\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$a$}}_A\left(s_1\right) \times {\mbox{\boldmath$k$}}\right)}{c^2}
\left[k \cdot r^{\,1}_A\left(s_1\right) - k \cdot r_A^{\,0}\left(s_0\right)\right]
\nonumber\\
\nonumber\\
\fl \hspace{0.75cm} - 2\,\frac{m_A}{R}\,\frac{{\mbox{\boldmath$k$}} \times \left({\mbox{\boldmath$a$}}_A\left(s_1\right) \times {\mbox{\boldmath$k$}} \right)}{c^2}
\left(k \cdot r^{\,1}_A\left(s_1\right) \right)
\ln \frac{k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r_A^{\,0}\left(s_0\right)} + {\cal O}\left(c^{-6}\right),
\label{Transformation_k_to_sigma_epsilon}\end{aligned}$$
where all the acceleration terms carry the same argument because of ${\mbox{\boldmath$a$}}_A\left(s_0\right) = {\mbox{\boldmath$a$}}_A\left(s_1\right) + {\cal O}\left(c^{-1}\right)$. In this respect we recall that ${\mbox{\boldmath$d$}}_A^k\left(s_0\right) = {\mbox{\boldmath$d$}}_A^k\left(s_1\right) + {\cal O}\left(c^{-1}\right)$ and ${\mbox{\boldmath$v$}}_A\left(s_0\right) = {\mbox{\boldmath$v$}}_A\left(s_1\right) + {\cal O}\left(c^{-1}\right)$, hence also the impact vectors and velocities in (\[Transformation\_k\_to\_sigma\_epsilon\]) may actually be written such that they carry the same argument. Here we also notice that the origin of last term in (\[Transformation\_k\_to\_sigma\_epsilon\]) is just the combination of the last both terms in (\[epsilon\_3b\]). The series expansion of (\[Transformation\_k\_to\_sigma\_epsilon\]) around $s_1$ reads $$\begin{aligned}
\hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_0\right) = \hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_1\right) + {\cal O}\left(c^{-5}\right).
\label{Appendix_series_expansion_epsilon_2}\end{aligned}$$
For the upper limit of the absolute value of (\[Appendix\_series\_expansion\_epsilon\_2\]) one finds $$\begin{aligned}
\hat{\epsilon}_2 = \left|\hat{{\mbox{\boldmath$\epsilon$}}}_2\left(s_1,s_1\right)\right|
\le 2\,\frac{m_A}{P_A}\,\frac{v_A^2\left(s_1\right)}{c^2}
+ 10\,m_A\,\frac{a_A\left(s_1\right)}{c^2}\,.
\label{Appendix_estimation_epsilon_3}\end{aligned}$$
Proof of inequality (\[Inequality\_A\]) {#Appendix_Inequality_A}
=======================================
We will show the inequality (\[Inequality\_A\]), which reads $$\begin{aligned}
\fl \left|\Delta {\mbox{\boldmath$\rho$}}_1^A\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}_1^B\left(s_1,s_1\right)
+ {\mbox{\boldmath$\varphi$}}_4\left(s_1\right) + {\mbox{\boldmath$\varphi$}}_5\left(s_1\right)\right|
\le 10\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Inequality_A_5}\end{aligned}$$
where $\Delta {\mbox{\boldmath$\rho$}}_1^A$, $\Delta {\mbox{\boldmath$\rho$}}_1^B$, ${\mbox{\boldmath$\varphi$}}_4$, and ${\mbox{\boldmath$\varphi$}}_5$ are given by Eqs. (\[series\_expansion\_rho\_1\_B\_1\]), (\[series\_expansion\_rho\_1\_B\_2\]), (\[Appendix\_Estimation\_phi\_4\_A\]), and (\[Appendix\_Estimation\_phi\_5\_A\]). From (\[Impact\_Vector\_k1\]) follows the relation $$\begin{aligned}
\fl \hspace{1.5cm} \frac{{\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)} \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
= \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)} \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
+ \left(\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\right) \frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$r$}}^{\,0}_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}\,,
\label{Appendix_Inequality_A_10}\end{aligned}$$
which allows to rewrite the term $\Delta {\mbox{\boldmath$\rho$}}_1^A$ in (\[series\_expansion\_rho\_1\_B\_1\]) in the form $$\begin{aligned}
\fl \Delta {\mbox{\boldmath$\rho$}}_1^A\left(s_1,s_1\right) = - 2\,\frac{m_A}{R}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\right)
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\nonumber\\
\nonumber\\
\fl \hspace{2.35cm} - 2\,\frac{m_A}{R}\, \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}
\left(\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)} \cdot \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} \right)
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,.
\label{Appendix_Inequality_A_15}\end{aligned}$$
The term $\Delta {\mbox{\boldmath$\rho$}}_1^B$ in (\[series\_expansion\_rho\_1\_B\_2\]) is written as follows, $$\begin{aligned}
\fl \hspace{1.5cm} \Delta {\mbox{\boldmath$\rho$}}^B_1\left(s_1,s_1\right) = - 2\,\frac{m_A}{R}\, \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,,
\label{Appendix_Inequality_A_20}\end{aligned}$$
while the term proportional to three-vector ${\mbox{\boldmath$k$}}$ is omitted because it does not contribute to the light deflection. Using the expressions (\[Appendix\_Inequality\_A\_15\]) - (\[Appendix\_Inequality\_A\_20\]) for $\Delta {\mbox{\boldmath$\rho$}}^A_1$ and $\Delta {\mbox{\boldmath$\rho$}}^B_1$ as well as Eqs. (\[Appendix\_Estimation\_phi\_4\_A\]) and (\[Appendix\_Estimation\_phi\_5\_A\]) for ${\mbox{\boldmath$\varphi$}}_4$ and ${\mbox{\boldmath$\varphi$}}_5$ we get $$\begin{aligned}
\fl \left|\Delta {\mbox{\boldmath$\rho$}}_1^A\left(s_1,s_1\right) + \Delta {\mbox{\boldmath$\rho$}}_1^B\left(s_1,s_1\right)
+ {\mbox{\boldmath$\varphi$}}_4\left(s_1\right) + {\mbox{\boldmath$\varphi$}}_5\left(s_1\right)\right| = \left| {\mbox{\boldmath$T$}}_1 + {\mbox{\boldmath$T$}}_2 + {\mbox{\boldmath$T$}}_3 \right|,
\label{Appendix_Inequality_A_25}\end{aligned}$$
where the terms of same algebraic structure are grouped together, $$\begin{aligned}
\fl {\mbox{\boldmath$T$}}_1 = + 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
- 2\,\frac{m_A}{R}\, \frac{{\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,,
\label{Appendix_Inequality_A_30}
\\
\nonumber\\
\fl {\mbox{\boldmath$T$}}_2 = + \frac{2\,m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c} - 2\,\frac{m_A}{R}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{r^{\,0}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$k$}} \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,,
\nonumber\\
\label{Appendix_Inequality_A_35}
\\
\fl {\mbox{\boldmath$T$}}_3 = + 2\,\frac{m_A}{\left(r^{\,1}_A\left(s_1\right)\right)^2}\,\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,1}_A\left(s_1\right)}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}
\nonumber\\
\fl \hspace{0.9cm} - 2\,\frac{m_A}{r^{\,0}_A\left(s_1\right)}\, \frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,\frac{1}{R}\,
\frac{{\mbox{\boldmath$d$}}^k_A\left(s_1\right) \cdot {\mbox{\boldmath$v$}}_A\left(s_1\right)}{c}\,
\frac{k \cdot r^{\,0}_A\left(s_1\right) - k \cdot r^{\,1}_A\left(s_1\right)}{k \cdot r^{\,0}_A\left(s_1\right)}\,.
\label{Appendix_Inequality_A_40}\end{aligned}$$
Then, using the approach described above (items 2. - 5. in \[Approach\_Appendix\]) one may demonstrate that the upper limits are $$\begin{aligned}
\left|{\mbox{\boldmath$T$}}_1\right| \le 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Inequality_A_45}
\\
\left|{\mbox{\boldmath$T$}}_2\right| \le 2\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Inequality_A_50}
\\
\left|{\mbox{\boldmath$T$}}_3\right| \le 4\,\frac{m_A}{r^{\,1}_A\left(s_1\right)}\,\frac{v_A\left(s_1\right)}{c}\,,
\label{Appendix_Inequality_A_55}\end{aligned}$$
while their total sum confirms the asserted inequality (\[Appendix\_Inequality\_A\_5\]), that is (\[Inequality\_A\]).
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[^1]: Let us recall that the harmonic gauge condition (\[harmonic\_gauge\_condition\_1\]) still inherits a residual gauge freedom, so the harmonic coordinates actually refer to a class of reference systems. A unique choice of harmonic coordinates is provided by the Barycentric Celestial Reference System (BCRS) [@IAU_Resolution1], which defines the origin of spatial coordinates at the barycenter of the Solar System, a stipulation which removes the residual gauge freedom. The metric coefficients for a system of $N$ moving monopoles, which have been presented by Eqs. (24) - (27) in [@Zschocke4], are given in the BCRS, so they do not contain any gauge terms.
[^2]: The notation in Eq. (\[2PN\_B\]) has been adjusted to the standard notation commonly used in the literature [@Brumberg1991; @Zschocke1; @Zschocke2; @Zschocke3; @Zschocke4]. A reconcilable notation for the series expansions (\[2PN\_A\]) and (\[2PN\_B\]) can be achieved by noticing that $\Delta {\mbox{\boldmath$x$}}^{\left(2\right)} \equiv \Delta {\mbox{\boldmath$x$}}^{1 {\rm PN}}$ and $\Delta {\mbox{\boldmath$x$}}^{\left(3\right)} \equiv \Delta {\mbox{\boldmath$x$}}^{1.5 {\rm PN}}$ and $\Delta {\mbox{\boldmath$x$}}^{\left(4\right)} \equiv \Delta {\mbox{\boldmath$x$}}^{2 {\rm PN}}$.
[^3]: Let us notice here that the light deflection in 2PN approximation in the field of one monopole at rest has been determined a long time ago [@EpsteinShapiro; @FischbachFreeman; @RM1; @RM2; @RM3; @Cowling; @BodennerWill2003]. But a unique interpretation of astrometric observations requires the knowledge of the propagation of the light signal, i.e. the determination of the light trajectory as function of coordinate time (\[2PN\_B\]). We also notice the investigation in [@Moving_Kerr_Black_Hole2] where the problem of time delay in the field of one monopole in uniform motion has been considered, but this investigation was not aiming at astrometric measurements in the Solar System.
[^4]: The results of [@Brumberg1991; @Brumberg1987] were later confirmed in several related investigations [@Deng_Xie; @Deng_2015; @Minazzoli2; @Article_Zschocke1; @LLT2004; @TL2008; @Teyssandier; @HBL2014b; @AshbyBertotti2010; @Moving_Kerr_Black_Hole1].
[^5]: The approximative arguments in the vectorial functions in Eqs. (99) and (128) in [@Zschocke4] can be replaced by their exact value ${\mbox{\boldmath$r$}}_A\left(s\right)$, because such replacement causes an error of the order ${\cal O}\left(c^{-5}\right)$ which is beyond 2PN approximation.
[^6]: One may also define impact vectors with respect to the unperturbed light ray, $$\begin{aligned}
{\mbox{\boldmath$d$}}^{\rm N}_A\left(s\right) = {\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$r$}}^{\rm N}_A\left(s\right) \times {\mbox{\boldmath$\sigma$}}\right)
\quad {\rm and} \quad {\mbox{\boldmath$d$}}^{\rm N}_A\left(s_0\right) = {\mbox{\boldmath$\sigma$}} \times \left({\mbox{\boldmath$r$}}^{\rm N}_A\left(s_0\right) \times {\mbox{\boldmath$\sigma$}}\right),
\label{Impact_Vector_Sigma_s0_Newtonian}\end{aligned}$$
where ${\mbox{\boldmath$r$}}^{\rm N}_A\left(s\right) = {\mbox{\boldmath$x$}}_0 + c\,{\mbox{\boldmath$\sigma$}} \left(t-t_0\right) - {\mbox{\boldmath$x$}}_A\left(s\right)$ and ${\mbox{\boldmath$r$}}^{\rm N}_A\left(s_0\right) = {\mbox{\boldmath$x$}}_0 - {\mbox{\boldmath$x$}}_A\left(s_0\right) = {\mbox{\boldmath$r$}}_A\left(s_0\right)$. They are illustrated in Figure \[Diagram1\]. Due to ${\mbox{\boldmath$d$}}_A\left(s\right) = {\mbox{\boldmath$d$}}^{\rm N}_A\left(s\right) + {\cal O}\left(c^{-2}\right)$ and ${\mbox{\boldmath$d$}}_A\left(s_0\right) = {\mbox{\boldmath$d$}}^{\rm N}_A\left(s_0\right)$, the impact vector ${\mbox{\boldmath$d$}}_A\left(s\right)$ differs marginal from ${\mbox{\boldmath$d$}}^{\rm N}_A\left(s\right)$, while impact vector ${\mbox{\boldmath$d$}}_A\left(s_0\right)$ is even identical to ${\mbox{\boldmath$d$}}^{\rm N}_A\left(s_0\right)$. The graphical representation of ${\mbox{\boldmath$d$}}^{\rm N}_A\left(s\right)$ and ${\mbox{\boldmath$d$}}^{\rm N}_A\left(s_0\right)$ in Figure \[Diagram1\] makes it evident why these terms are called impact vectors.
[^7]: For instance, the worldline of a body in a two-dimensional circular orbit of radius $r$ is $\displaystyle {\mbox{\boldmath$x$}}_A\left(t\right) = \left(r\,\cos \omega t\;,\;r\,\sin \omega t \right)^{\rm T}$ where $\omega = 2\,\pi/T$ is the angular frequency with $T$ being the orbital period. One gets $\left|{\mbox{\boldmath$x$}}_A^{(n)}\right| = r\,\omega^n$, hence the limit $\displaystyle L = \lim_{n \rightarrow \infty} \frac{2\,\pi}{T}\,\frac{\left|s_0 - s_1\right|}{n + 1} = 0$.
[^8]: The notation $k^{\mu}$ is employed in [@Kopeikin_Efroimsky_Kaplan] (cf. text below Eq. (7.82) in [@Kopeikin_Efroimsky_Kaplan]) for what we call $\sigma^{\mu}$ (cf. Eq. (\[Four\_Vector\_sigma\])). Furthermore, our three-vector ${\mbox{\boldmath$k$}}$ in Eq. (\[Boundary\_3\]) coincides, up to a minus sign, with the three-vector ${\mbox{\boldmath$K$}}$ used in [@Kopeikin_Efroimsky_Kaplan; @KS1999; @KopeikinMashhoon2002] (cf. Eq. (7.66) in [@Kopeikin_Efroimsky_Kaplan] or Eq. (36) in [@KS1999] or Eq. (44) in [@KopeikinMashhoon2002]). It will certainly not cause any kind of confusion that $r_A\left(s\right) \equiv r_A^{\mu}\left(s\right)$ on the l.h.s. in (\[scalar\_product\]) denotes the four-vector, while $r_A\left(s\right) \equiv \left|{\mbox{\boldmath$r$}}_A\left(s\right)\right|$ on the r.h.s. in (\[scalar\_product\]) denotes the absolute value of the three-vector. Throughout the manuscript a single four-vector carries always a Lorentz-index. Only in four-scalar products the four-vectors do not carry a Lorentz index, but then there will always be a dot among these four-vectors. In three-scalar products there is also a dot among the three-vectors, but the three-vectors are always in bold. For the details of notation in use we refer to \[Appendix0\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Hohenberg-Kohn theorem of density functional theory (DFT) for the case of electrons interacting with an external magnetic field (that couples to spin only) is examined in more detail than previously. An unexpected generalization is obtained: in certain cases (which include half metallic ferromagnets and magnetic insulators) the ground state, and hence the spin density matrix, is invariant for some non-zero range of a shift in uniform magnetic field. In such cases the ground state energy is not a functional of the spin density matrix alone. The energy gap in an insulator or a half metal is shown to be a ground state property of the N-electron system in magnetic DFT.'
address: |
$^{\dag }$Institut für Festkörper- und Werkstofforschung Dresden e.V., Postfach 270016, D-01171 Dresden, Germany\
$^{\dag \dag }$Department of Physics, University of California, Davis, CA 95616\
author:
- 'H. Eschrig$^{\dag }$ and W. E. Pickett$^{\dag \dag}$'
title: Density Functional Theory of Magnetic Systems Revisited
---
The half metallic state of a ferromagnet has been receiving greatly increasing attention since its prediction from band theory[@HM] to be the ground state of important magnetic materials such as CrO$_2$,[@cro2] NiMnSb,[@nimnsb] and Sr$_2$FeMoO$_6$[@doubperov] and several other intermetallics and oxides, and its unusual physical properties.[@irkhin] Such systems have become very attractive for magnetoelectronic applications, where control of the spin degree of freedom is already leading to new devices.[@prinz] Materials thought to be half metals have been connected with the phenomena of colossal magnetoresistance (CMR),[@CMR] large tunneling MR,[@soulen] and large, low-field intergrain MR,[@IMR] and they would be optimal for applications of spin valve systems[@nimnsb] for non-volatile magnetic memory and for high density magnetic storage.
The half metallic state is, in a one-electron picture, a collinear magnetic state in which one spin direction is metallic while the other is gapped (‘insulating’). This state is half metallic in another sense: the absence of low energy spin flips leads to a vanishing magnetic susceptibility like an insulator, but its charge response (conductivity) is that of a metal. These properties combine to give a one-electron description of a spin-charge separated state. In fact, almost all understanding of half metals so far is based on the one-electron picture, which opens up questions such as (1) what is a half metal in many body context, and (2) are there other unusual possibilities in magnetic systems? One general characterization might be in terms of conductivity (charge response) and susceptibility (spin response) alluded to above: in an insulator both vanish, in a conventional (even ferromagnetic) metal both are non-zero, and in a half metal the conductivity is non-zero while the susceptibility vanishes. A clear many body formulation is however lacking.
Since density functional theory (DFT) is a rigorous many body theory for (chosen) ground state properties, we revisit the foundations of DFT with magnetic properties in mind. The first Hohenberg-Kohn (HK) theorem[@HK], which is the basis for the DFT of spin-independent particles, demonstrates the existence of a unique map $$n({\bf r}) \mapsto v({\bf r})~ {\rm mod~(constant)},
\label{eq1}$$ where $v$ is the external potential and $n$ is the ground state particle density. According to the second HK theorem, the ground state energy and density are obtained as the solution to a variational principle: $$E[v-{\mu}] = min_n \{F[n] + \int n ( v-{\mu}) d^3 r\},$$ with $\mu$ the chemical potential. Although the variational principle has been put on an independent, more general basis,[@LL] the uniqueness of the map (\[eq1\]) remains an important issue regarding the existence and uniqueness of the functional derivative $\delta F/\delta n = -(v-\mu)$. DFT, as extended by Kohn and Sham[@KS] and many others, forms the basis of our understanding of the electronic behavior of real materials. The theory has been extended to electrons with spin[@vBH; @IEG] and also applied heavily, however the HK theorem for interacting particles with spin has repeatedly been stated to be analagous to the HK theorem, although this was already questioned in [@vBH]. Zero susceptibility, however, would imply that the ground state spin density does not change when an external magnetic field is changed. In this paper we construct a more revealing generalization of the HK theorem, obtain explicitly the conditions that allow half metallicity, and demonstrate some unexpected consequences.
We consider the system in an external magnetic field ${\bf B}({\bf r})$ in the (commonly considered) non-relativistic limit, in which the field acts only on the electron spin and the dipolar interaction between spins is neglected. The potentials can be combined into 2$\times $2 spin matrix $$u_{ss^{\prime }}({\bf r})=v({\bf r})\delta _{ss^{\prime }}-\mu _{B}{\bf B}(%
{\bf r})\cdot {\vec \sigma }_{ss^{\prime }}.$$ The external field ${\bf B}$ may vary in magnitude and direction.
Realizing that two different scalar potentials cannot lead to the same ground state $\Psi$, the original derivation of HK[@HK] concluded that if $\Psi\mapsto v-\mu$ is unique then $n\mapsto
v-\mu$ is unique. Following the original derivation of HK[@HK], we begin by supposing that there are two different potentials $u, u^{\prime}$ which lead to the same ground state $\Psi$. We show that $\Psi\mapsto (v-\mu,\vec B)$ is [*not*]{} a unique mapping in general.
The many-body Hamiltonian of the system is $$\hat{H}=\hat{T}+\hat{W}+\hat{U},
\label{Hamil}$$ where $\hat{T}$ is the kinetic energy operator, $\hat{W}$ is the Coulomb interaction energy, and $\hat{U}$ is the interaction with the external potential. The fermionic many-particle Schrödinger equation is (in atomic units) $$\begin{aligned}
\Bigl[\sum_{i}^{N}\frac{-\nabla _{i}^{2}}{2} + \sum_{i<j}^{N}w({\bf r}%
_{i},{\bf r}_{j}) \Bigr] \Psi ({\bf r}_{1}\alpha_{1},..,{\bf r}_N{\alpha_N})
\\ \nonumber
+ \sum_{i}^{N}\sum_{\beta _{i}}u_{\alpha _{i},\beta _{i}}({\bf r}_{i})
\Psi ({\bf r}%
_{1}\alpha_{1},..,{\bf r}_i\beta_i,..,{\bf r}_{N}\alpha_{N})\\ \nonumber
=E\Psi ({\bf r}_{1}\alpha _{1},...,{\bf r}_{N} \alpha _{N}),
%\sum_{i}^{N}\sum_{\beta _{i}}[\frac{-\nabla _{i}^{2}}{2} \delta _{\alpha
%_{i},\beta _{i}}+u_{\alpha _{i},\beta _{i}}({\bf r}_{i})] \hfill \\ \nonumber
% \times \Psi ({\bf r}%
%_{1}\alpha_{1},..,{\bf r}_i\beta_i,..,{\bf r}_{N}\alpha_{N}) \\ \nonumber
% + \Bigl[\sum_{i<j}^{N}w({\bf r}%
%_{i},{\bf r}_{j}) \Bigr]
% \Psi ({\bf r}_{1}\alpha _{1},...,{\bf r}_{N}\alpha
%_{N}) \\ \nonumber
% =E\Psi ({\bf r}_{1}\alpha _{1},...,{\bf r}_{N} \alpha _{N}),\end{aligned}$$ where ${\bf r}_i, \alpha_i$ are the space and spin coordinates of the $i$-th electron; $w({\bf r},{\bf r}^{~\prime }) = e^2/|{\bf r}-{\bf r}^{\prime}|$ is the Coulomb interaction.
Assume there are two external potentials $u,
u^{\prime}$ with energies $E, E^{\prime}$ that have the same ground state wave function $\Psi ({\bf r}_{1}\alpha _{1},...,{\bf r}_{N}\alpha _{N})$. Subtracting the two many-particle Schrödinger equations leads to $$\begin{aligned}
\sum_{i=1}^{N} \sum_{\beta_i} \Delta u_{\alpha _{i},\beta _{i}}({\bf r}_{i})
\Psi ({\bf r}_{1}\alpha_{1},..,{\bf r}_i\beta_i,..,{\bf r}%
_{N}\alpha_{N})= \\ \nonumber
\Delta E ~\Psi ({\bf r}_{1}\alpha _{1},...,{\bf r}%
_{N}\alpha _{N}),\end{aligned}$$ where $\Delta u = u-u^{\prime}, \Delta E = E-E^{\prime}$. Now, we perform a unitary spin rotation $Q_{ss^{\prime }}({\bf r})$ at each point of space that diagonalizes the difference in potentials ([*i.e.*]{} rotates $\vec B$ to lie along the $\hat z$ direction: $$\{Q({\bf r})[\Delta u({\bf r})]Q^{\dag }({\bf r})\}_{ss^{\prime }} =
\Delta \tilde{u}_s({\bf r}) \delta _{ss^{\prime }}.
\label{diag}$$ The wavefunction is transformed according to $$\begin{aligned}
\prod_{i}^{N}Q_{\alpha _{i}\alpha _{i}^{\prime }}({\bf r}_{i})\Psi ({\bf r}%
_{1}\alpha _{1}^{\prime },...,{\bf r}_{N}\alpha _{N}^{\prime })
\equiv \tilde{%
\Psi}({\bf r}_{1}\alpha _{1},...,{\bf r}_{N}\alpha _{N}), \nonumber\end{aligned}$$ where $\prod_{i}^{N}Q_{\alpha _{i}\alpha _{i}^{\prime }}({\bf r}_{i})$ is the operator that rotates each of the ($\alpha _{i}$). Collecting these results gives $$\begin{aligned}
\sum_{i=1}^{N}\Delta \tilde{u}_{\alpha _{i}}({\bf r}_{i})
\tilde{\Psi}({\bf r}_{1}\alpha _{1},...,{\bf r}%
_{N}\alpha _{N})
=\Delta E ~\tilde{\Psi}({\bf r}_{1}\alpha _{1},...,{\bf r%
}_{N}\alpha _{N}). \nonumber\end{aligned}$$
$\tilde{\Psi}$ is some $\{{\bf r}_i\}$-dependent multi-component function of the $2^{N}$ possible spin configurations, at least one of which must be non-zero. Choose a non-zero component $%
\tilde \Psi _{c}$ and denote by $N_{\uparrow }$ the number of $\alpha _{i}=\uparrow
$ values in this component. Since $\tilde{\Psi}$ is antisymmetric (as was $%
\Psi $) with respect to permutations of (${\bf r}_{i}\alpha _{i}$) with ($%
{\bf r}_{j}\alpha _{j}$), we may renumber the particle indices in such a way that $\alpha _{1}=\alpha _{2}=...=\alpha _{N_{\uparrow }}$, $\alpha
_{N_{\uparrow }+1}=\alpha _{N_{\uparrow }+2}=...=\alpha _{N}.$ This ordering lets us write $$\begin{aligned}
\left\{ \sum_{i=1}^{N_{\uparrow}} \Delta \tilde{u}_{\uparrow }({\bf r}_{i})
+\sum_{i=N_{\uparrow}+1}^N \Delta \tilde{u}_{\downarrow }({\bf r}_{i})
\right\} \tilde{\Psi_c}({\bf r}%
_{1}\uparrow ,...,{\bf r}_{N}\downarrow ) \\ \nonumber
=\Delta E \tilde{\Psi_c}({\bf r}%
_{1}\uparrow ,...,{\bf r}_{N}\downarrow ).
\label{spinsplit}\end{aligned}$$ This equation must hold for all values of (${\bf r}_{1},...,{\bf r}_{N})$. (We suppose $u, u^{\prime}$ are analytic in ${\bf r}$ except possibly at isolated points, so that $\tilde{\Psi_c}$ is non-zero almost everywhere.) By varying only ${\bf r}_1$, and then separately varying only ${\bf r}_N$, we obtain $$%\left\{ \sum_{i=1}^{N_{\uparrow}}\Delta \tilde{u}_{\uparrow }({\bf r}_{i})
% +\sum_{i=N_{\uparrow}+1}^{N}\Delta \tilde{u}_{\downarrow }
% ({\bf r}_{i}) \right\} \equiv C
\Delta \tilde u_{\uparrow} = C_{\uparrow},~~
\Delta \tilde u_{\downarrow} = C_{\downarrow},$$ where $C_{\uparrow}, C_{\downarrow}$ are constants. The special cases $N_{\uparrow}=0$ or $N_{\uparrow}=N$ do not lead to new consequences. For further analysis, we consider separate cases.
[*Case A: impure spin states.*]{} Suppose that there are at least two components of $\tilde{\Psi}$ with different values of $N_{\uparrow }$ and hence $N_{\downarrow }=N-N_{\uparrow
}$. Then $$N_{\uparrow }C_{\uparrow }+(N-N_{\uparrow })C_{\downarrow }=\Delta E.$$ Since this holds for two different values of $N_{\uparrow }$, it follows that $C_{\uparrow }=C_{\downarrow } \equiv C,$ which leads to $\Delta \tilde u_{\uparrow}=\Delta \tilde u_{\downarrow}$ so $\Delta u_{\alpha,\beta}=C\delta_{\alpha,\beta}$. By the ground state energy minimum principle, this recovers the usual Hohenberg-Kohn result $$\begin{aligned}
n_{ss^{\prime}}=n^{\prime}_{ss^{\prime}} \rightarrow \left(
\begin{array}{c}
v({\bf r}) - v^{\prime}({\bf r}) \equiv C, \\
{\bf B}({\bf r}) - {\bf B}^{\prime}({\bf r}) \equiv 0,
\end{array}
\right) \label{conditions}\end{aligned}$$ implying a non-zero susceptibility.
[*Case B: pure spin states.*]{} Suppose now that all non-zero components of $\tilde{\Psi}$ have the same value of $N_{\uparrow }$ and $N_{\downarrow }$. These may be considered as “pure spin” states, eigenfunctions of the operator $\hat {S}^z$ = $\sum_i \sigma^z_{\alpha_i \beta_i}/2$ with eigenvalues $S_z=N_{\uparrow }-(N/2)=(N_{\uparrow }-
N_{\downarrow })/2$. Then $C_{\uparrow }$ and $C_{\downarrow }$ need not be equal and we can write $$\begin{aligned}
\Delta \tilde{u}& = & \left(
\begin{array}{cc}
C_{\uparrow } & 0 \\
0 & C_{\downarrow }
\end{array}
\right)
=\bar C {\bf 1} - \mu_B \bar{B}\sigma^{z},\end{aligned}$$ where $\bar{C} = (C_{\uparrow} + C_{\downarrow}$)/2 and $-\mu_B \bar{B} = (C_{\uparrow} - C_{\downarrow}$)/2.
Backtransforming according to the inverse of Eq. (\[diag\]) gives $$\Delta u_{\alpha \beta }({\bf r})=\bar C \delta _{\alpha
\beta }+\mu_B \bar{B}[Q^{\dag }({\bf r})\sigma _{z}Q({\bf r})]_{\alpha \beta }.$$ The last term on the right is position-dependent, non-diagonal, and non-vanishing in general. In this case the conditions for identical ground state wavefunctions are $$\Psi =\Psi ^{\prime }\rightarrow
\left(
\begin{array}{ccc}
v({\bf r})-v^{\prime }({\bf r}) & = & \bar{C} \hfill \\
{\bf B}({\bf r})-{\bf B}^{\prime }({\bf r}) & = & \bar{B} ~ \hat e({\bf r}).
\end{array}
\right) .
\label{condition3}$$ where $\hat e$ is the unit vector $\frac{1}{2}Tr\{\vec{\sigma}
Q^{\dag }({\bf r})\sigma _{z}Q({\bf r})\}$. The result (\[conditions\]) is modified accordingly. This result is a highly non-trivial generalization of the HK theorem: [*two magnetic fields whose difference is constant in magnitude, but possibly is non-unidirectional, may give rise to the same ground state.*]{}
Now we investigate the conditions on $Q$ for which $\tilde \Psi$ is an eigenstate of $\hat {S}^z$, [*i.e.*]{} $\tilde \Psi$ describes a collinear spin arrangement. Considering the Hamiltonian Eq. (\[Hamil\]), there must be an operator $$\hat U_o = \sum_{i=1}^N Q^*_{\alpha^{\prime}_i \alpha_i}
({\bf r}_i) \sigma^z_{\alpha^{\prime}_i \beta^{\prime}_i}
Q_{\beta^{\prime}_i \beta_i}({\bf r}_i)$$ that commutes with $\hat T + \hat U + \hat W$. We now specialize to the particular case where one of the external fields, $\bf B^{\prime}$, is zero. Since the interaction $\hat W$ is spin independent, $\hat U_o$ will commute with $H^{\prime}$ if and only if it commutes with $\hat T$. One can show that this necessitates that $Q$ be ${\bf r}$-independent, so that $\Psi$ itself is an eigenstate of $\hat {S}^z$, and hence is a collinear spin state. The second condition in Eq. (\[condition3\]) reduces to ${\bf B} - {\bf B^{\prime}} = B \hat z$: a turning on of a uniform magnetic field leaves the ground state invariant. Restated: in the subspace of collinear magnetizations, [*the ground state determines the magnetic field only up to some codirectional uniform field*]{}. A direct corollary is that there is no longer any ground state energy functional $E[n]$ of the density $n_{ss'}(\bf r)$ alone (see Eq. (\[eq18\]) below).
In the collinear situation, inserting $\Delta u = \mu_B B\sigma_z$ into Eq. (6) yields $$\Delta E(B) = -(N_\uparrow-N_\downarrow) \mu_B B,
\label{eq18}$$ which gives the well known dependence of energy vs. field for a system of fixed spin. Consider as a simple example a Be atom in a uniform magnetic field, with its ground state characterized as $1s^22s^2$ ($N=4$, $N_\uparrow=2$). The lowest excited $\hat S_z$-eigenstate is $1s^22s2p$ with $N_\uparrow=3$. Its excitation energy is that of a $2s\rightarrow 2p$ promotion. There is another excited state $1s2s^22p$ with the same $N_\uparrow$, but the much higher excitation energy of a $1s\rightarrow 2p$ core excitation. The energetically lowest $N_\uparrow=4$ state is $1s2s2p^2$ whose excitation energy is roughly the sum of the previous two. The situation is sketched in Fig. 1(a), where the lines with positive slopes correspond to states with all spins reversed.
Since states with $N_\uparrow=N/2\pm n$ are degenerate for $B=0$, Fig. 1(a) may be supplemented symmetrically to the vertical axis. Hence, for $|B| < B_0$ the groundstate is $1s^22s^2$ with energy $E_0$, for $B_0 < B < B_1$ the ground state is $1s^22s2p$ with energy $E_1-2\mu_B B$, and for $B\ge B_1$ the ground state is $1s2s2p^2$ with energy $E_3-4\mu_B B$. The ground state does not change with field except at certain isolated values.
In an extended system, say a non-magnetic insulator with gap $\Delta_g$, there is a continuum above $\Delta_g$ (one excited electron with reversed spin), another continuum above $2\Delta_g$ (two excited electrons) and so on, as illustrated in Fig. 1(b). In an extended system one would prefer to consider the intensive quantity $$\frac{\Delta E(B)}{N}
= -\mu_B B \left(\frac{N_\uparrow - N_\downarrow}{N}\right)$$ instead of $\Delta E$ itself. Then, one finds that for $\mu_B B <
\Delta_g$ the groundstate is independent of $B$, beyond which the state changes and $\Delta E/N$ veers off. Thus while the gap $\Delta_g$ is not a ground state property of the $N$ particle system in paramagnetic DFT (it involves the N$\pm$1 particle ground states), it is a ground state property in the presence of a uniform field.
For a stoichiometric half metal with moment per cell $\mu_B {\cal M}~({\cal M}$ an integer) the picture is related, except there is an overall bias – a slope of -$\mu_B {\cal M}$ in the energy per cell – and the positive and negative $B$ directions are not symmetric. The situation that is sketched in Fig. 1(c) has a gap $\Delta_v+\Delta_c$ for $\downarrow$ spin states, with no gap for $\uparrow$ spin. The chemical potential $\mu$ corresponds to the energy to remove an $\uparrow$ spin, and the quantities $\Delta_v = \mu_B B_v$, $\Delta_c = \mu_B |B_c|$ represent the energy, or field, required to flip a spin from $\downarrow$ to $\uparrow$, or vice versa. Note again that the interval of $B$ for which the state does not change, which is the gap in the $\downarrow$ spectrum, is a ground state property of the $N$ particle system in an external magnetic field.
It is useful to consider the form of constrained DFT in which $N_{\uparrow}$ and $N_{\downarrow}$ are specified, which leads to two associated chemical potentials $\mu_{\uparrow}$, $\mu_{\downarrow}$. Then as $N_s$ is changed to $N_s \pm 1$, $\mu_s$ may vary only to order 1/$N_s$ (metallic behavior) or it may jump discontinuously across a gap, just as is the case for insulators.[@1983] The half metal is defined as that situation in which one and only one of $\mu_s$ (we have choosen $\downarrow$) is discontinuous upon addition of one electron. For an insulator, there is a discontinuity in $\mu$ for both spins.
We now consider the KS eigenvalue spectrum. As long as the external field shifts the bands sufficiently little not to disturb the half metallicity ($B_c < B < B_v$), the ground state, and hence the charge density in each spin channel, remain unchanged. Using the same arguments as were applied to establish the discontinuity in $v_{xc}(N(\mu))$ for an insulator as $\mu$ crosses the gap (the kinetic energy is discontinuous across the gap)[@1983], one finds that there is a discontinuity in $v_{xc,\downarrow}(N_{\uparrow},N_{\downarrow})$ if the filling with $N_{\downarrow}$ moves $\mu_{\downarrow}$ across the gap.[@define]
The Kohn-Sham gap $\varepsilon_{g\downarrow}$ is smaller than the true (quasiparticle) gap $\Delta_g = \Delta_c + \Delta_v$. When the magnetic field is large enough that $\mu$ reaches the KS band minimum $\varepsilon_c(N_{\downarrow})$, the occupation of that channel becomes N$_{\downarrow}+\epsilon$ (with $\epsilon \rightarrow 0$). This is the point of the discontinuity, where the KS conduction eigenvalue (in fact, the entire $\downarrow$ spectrum) jumps upward. By comparison with Dyson’s equation, and the fact that the system’s ground state spin densities must be the same whether obtained from DFT or the quasiparticle Greens function, this jump must be such as to make $\varepsilon_{c\downarrow}(N_{\uparrow},%
N_{\downarrow}+\epsilon) \equiv$ $\Delta_c$, the quasiparticle conduction band edge, for $\epsilon \rightarrow$0.
It is apparent then that the KS gap in the insulating channel is not equal to the true gap in that channel, and that $\varepsilon_c(N_{\uparrow},N_{\downarrow}) - \mu$ is not the true spin flip energy (which is $\Delta_c$ - $\mu$). By our definition,[@define] as the reverse field is applied and $\mu$ is driven toward the valence band maximum $\varepsilon_{v\downarrow}$, there is no discontinuity, and the other spin flip energy – a true excitation energy – is given correctly by DFT. Needless to say, an approximation such as the local density approximation that interpolates across the discontinuity, will fail to predict both $\Delta_c$ and $\Delta_v$.
We now summarize. We have presented new, rigorous results for the Hohenberg-Kohn mapping in a magnetic field. We obtain conditions that characterize half metals: (1) two collinear systems in different uniform magnetic fields may have the same half metallic (or magnetic insulating) ground state; (2) exactly one of the chemical potentials $\mu_s$ is discontinuous upon particle addition to a half metal. We have pointed out other consequences, primary among them being that the ground state energy of a system is no longer a unique functional of the density $n_{ss'}$ when magnetic fields are allowed (although the ground state itself is), and that the gap in a half metal is a ground state property of the $N$ particle system. These results are only exact in the non-relativistic ($c \rightarrow \infty$) limit. For $c$ finite, half metallicity is an approximate notion due to orbital currents and orbital moments and spin-orbit coupling that mixes them, and the general theory[@vignale] probably restores the conventional theorems of DFT. Still, the notion of half metallicity will be an important model limit.
We acknowledge stimulating discussions with I. I. Mazin throughout the course of this work, and helpful interaction with W. Kohn during the early stages. This study was begun when the authors were at the Institute of Theoretical Physics at the University of California at Santa Barbara, which is supported by National Science Foundation Grant PHY-9407194. W.E.P. was supported by National Science Foundation Grant DMR-9802076.
R. A. de Groot [*et al.*]{}, Phys. Rev. Lett. [**50**]{}, 2024 (1983). A half metallic band structure was reported earlier by J. I. Horikawa [*et al.*]{}, J. Phys. C [**15**]{}, 2613 (1982).
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C. Hordequin [*et al.*]{}, J. Magn. Magn. Mat. [**183**]{}, 225 (1998), and references therein.
K.-I. Kobayashi [*et al.*]{}, Nature [**395**]{}, 677 (1998).
V. Yu. Irkhin and M. Katsnel’son, Usp. Fiz. Nauk [**164**]{}, 705 (1994) \[Sov. Phys. Usp. [**37**]{}, 659 (1994)\].
G. A. Prinz, Science [**282**]{}, 1660 (1998).
R. von Helmolt [*et al.*]{}, Phys. Rev. Lett. [**71**]{}, 2331 (1993); S. Jin [*et al*]{}., Science [**264**]{}, 413 (1994).
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M. Levy, Proc. Natl. Acad. Sci. [**76**]{}, 6062 (1979); E. H. Lieb, in [*Density Functional Methods in Physics*]{}, eds. R. M. Dreizler and J. da Providencia (Plenum, New York, 1985), p. 81.
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|
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"pile_set_name": "ArXiv"
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|
---
abstract: 'The concept of operator left residuation has been introduced by the authors in their previous paper [@CL1]. Modifications of so-called quantum structures, in particular orthomodular posets, like pseudo-orthomodular, pseudo-Boolean and Boolean posets are investigated here in order to show that they are operator left residuated or even operator residuated. In fact they satisfy more general sufficient conditions for operator residuation assumed for bounded posets equipped with a unary operation. It is shown that these conditions may be also necessary if a generalized version using subsets instead of single elements is considered. The above listed posets can serve as an algebraic semantics for the logic of quantum mechanics in a broad sense. Moreover, our approach shows connections to substructural logics via the considered residuation.'
author:
- Ivan Chajda and Helmut Länger
title: Left residuated operators induced by posets with a unary operation
---
[**AMS Subject Classification:**]{} 06A11, 06C15, 06E75, 03G12, 03G25
[**Keywords:**]{} operator residuation, operator left adjointness, Boolean poset, pseudo-Boolean poset, pseudo-orthomodular poset, generalized operator residuation
Introduction
============
It was shown by G. Birkhoff and J. von Neumann ([@BV]) and, independently, by K. Husimi (H) that orthomodular lattices can serve as an algebraic semantic of the logic of quantum mechanics. Namely, the class of event-state systems in quantum mechanics is usually identified with the set of projection operators on a Hilbert space $\mathbf H$ and this set is in a bijective correspondence with the set of all closed linear subspaces of $\mathbf H$. However, certain doubts concerning the relevance of this representation arose when it was shown that the class of orthomodular lattices arising in this way does not generate the variety of orthomodular lattices. In other words, there exist orthomodular lattices which do not represent a physical system in the logic of quantum mechanics. The reason is that some equational properties of event-state systems are not fairly reflected by the proposed mathematical abstraction. This was the reason for alternative approaches, see e.g. [@GLP16] and [@GLP]. In particular, an algebraic semantic for the logic of quantum mechanics was found among orthomodular posets and their modifications.
Orthomodular lattices have similar properties as algebraic structures used for substructural logics, see e.g. [@GJKO]. The authors proved in [@CL17a] and [@CL17b] that every orthomodular lattice can be converted into a so-called left residuated l-groupoid. They showed in [@CL2] that this result can be easily extended to a certain class of bounded lattices with a unary operation which, of course, contains the variety of orthomodular lattices. Hence, the natural question arises if this approach can be extended to ordered sets with a unary operation. For this purpose, so-called residuated operators were introduced in [@CL1] and several classes of ordered sets with a unary operation turned out to be operator residuated. A prominent role among these posets play the so-called pseudo-orthomodular posets which are a direct generalization of orthomodular lattices, but serve also as good candidates for an algebraic semantic of the logic of quantum mechanics. And again, there arises the natural question if the posets listed in [@CL1] really exhaust all possible cases.
The aim of the present paper is to provide several simple conditions under which a bounded poset with a unary operation can be organized into an operator left residuated poset. Similarly as it was done for lattices in [@CL2], we ask if these conditions are not only sufficient but also necessary. It is shown that if subsets instead of single elements are considered then these generalized conditions characterize the class of posets which can be converted into operator residuated ones.
Adjointness of operators
========================
Recall from [@BV] that an [*orthomodular lattice*]{} is a bounded lattice $(L,\vee,\wedge,{}',0,1)$ with a unary oparation $'$ which is a complementation and an antitone involution (see e.g.K) satisfying the [*orthomodular law*]{} $$x\leq y\text{ implies }x=y\wedge(x\vee y')$$ or, equivalently, $$x\leq y\text{ implies }y=x\vee(y\wedge x').$$ A [*left residuated lattice*]{} (or [*integral l-groupoid*]{} in the terminology of [@GJKO]) is a bounded lattice $(L,\vee,\wedge,\odot,\rightarrow,0,1)$ with two more binary operations $\odot$ and $\rightarrow$ satisfying $$\begin{aligned}
& x\odot1\approx1\odot x\approx x, \\
& x\odot y\leq z\text{ if and only if }x\leq y\rightarrow z\text{ (the so-called {\em left adjointness})}.\end{aligned}$$ We put $x':=x\rightarrow0$. If $\odot$ is, moreover, commutative then we call the previous property simply [*adjointness*]{}. If $\odot$ is associative and monotonous in every variable then it is called a [*t-norm*]{} (see [@GJKO]).
It was shown by the authors in [@CL17a] and [@CL17b] that taking $$\begin{aligned}
x\odot y & :=(x\vee y')\wedge y, \\
x\rightarrow y & :=(y\wedge x)\vee x'\end{aligned}$$ in an orthomodular lattice $\mathbf L=(L,\vee,\wedge,{}',0,1)$ yields a left residuated lattice $(L,\vee,\wedge,\odot,$ $\rightarrow,0,1)$ where $x'=x\rightarrow0$ coincides with the complementation in $\mathbf L$.
However, as shown in [@CL1], if $(P,\leq,{}',0,1)$ is an orthomodular poset (or even a Boolean poset) then such operations $\odot$ and $\rightarrow$ need not exist. In order to avoid these complications we study bounded ordered sets with a unary operation. We introduced in [@CL1] the following notion:
\[def1\] An [*operator left residuated poset*]{} is an ordered seventuple $\mathbf P=(P,\leq,{}',M,R,0,1)$ where $(P,\leq,{}',0,1)$ is a bounded poset with a unary operation and $M$ and $R$ are mappings from $P^2$ to $2^P$ satisfying the following conditions for all $x,y,z\in P$:
1. $M(x,1)\approx M(1,x)\approx L(x)$,
2. $M(x,y)\subseteq L(z)$ if and only if $L(x)\subseteq R(y,z)$,
3. $R(x,0)\approx L(x')$.
Condition [(ii)]{} is called [*operator left adjointness*]{}. If $M$ is [*commutative*]{} then [(ii)]{} is called [*operator adjointness*]{} only and $\mathbf P$ is called an [*operator residuated poset*]{}.
In [@CL1], the definition contains one more condition which, however, follows from (i) and (ii), see the following lemma.
\[lem5\] Every operator left residuated poset $(P,\leq,{}',M,R,0,1)$ satisfies the following condition for all $x,y\in P$: $$R(x,y)=P\text{ if and only if }x\leq y.$$
For $x,y\in P$ the following are equivalent: $$\begin{aligned}
R(x,y) & =P, \\
L(1) & \subseteq R(x,y), \\
M(1,x) & \subseteq L(y), \\
L(x) & \subseteq L(y), \\
x & \leq y.\end{aligned}$$
For posets $(P,\leq,{}')$ with a unary operation we define the following two conditions: $$\begin{aligned}
L(x) & \subseteq L(U(L(U(x,y'),y),y'))\text{ for all }x,y\in P,\label{equ1} \\
L(U(L(x,y),y'),y) & \subseteq L(x)\text{ for all }x,y\in P,\label{equ2}\end{aligned}$$ and the following two mappings from $P^2$ to $2^P$: $$\begin{aligned}
M(x,y) & :=L(U(x,y'),y)\text{ for all }x,y\in P,\label{equ3} \\
R(x,y) & :=L(U(L(y,x),x'))\text{ for all }x,y\in P.\label{equ4}\end{aligned}$$
\[lem1\] Let $(P,\leq,{}')$ be a poset with a unary operation and $M$ and $R$ defined by [(\[equ3\])]{} and [(\[equ4\])]{}, respectively. Then [(\[equ1\])]{} implies $M(x,y)\subseteq L(z)\Rightarrow L(x)\subseteq R(y,z)$ and [(\[equ2\])]{} implies $L(x)\subseteq R(y,z)\Rightarrow M(x,y)\subseteq L(z)$.
Assume $a,b,c\in P$. If (\[equ1\]) and $M(a,b)\subseteq L(c)$ then $$\begin{aligned}
L(a) & \subseteq L(U(L(U(a,b'),b),b'))=L(U(L(U(a,b'),b)\cap L(b),b'))= \\
& =L(U(M(a,b)\cap L(b),b'))\subseteq L(U(L(c)\cap L(b),b'))=L(U(L(c,b),b'))=R(b,c).\end{aligned}$$ If (\[equ2\]) and $L(a)\subseteq R(b,c)$ then $$\begin{aligned}
M(a,b) & =L(U(a,b'),b)=L(U(a)\cap U(b'),b)=L(U(L(a))\cap U(b'),b)\subseteq \\
& \subseteq L(U(R(b,c))\cap U(b'),b)=L(U(L(U(L(c,b),b')))\cap U(b'),b)= \\
& =L(U(L(c,b),b')\cap U(b'),b)=L(U(L(c,b),b'),b)\subseteq L(c).\end{aligned}$$
\[def2\] Recall [(]{}e.g. from [[@CR])]{} that a [*distributive poset*]{} is a poset $(P,\leq)$ satisfying one of the following equivalent identities: $$\begin{aligned}
L(U(x,y),z) & \approx L(U(L(x,z),L(y,z))), \\
U(L(x,y),z) & \approx U(L(U(x,z),U(y,z))).\end{aligned}$$ A [*poset with complementation*]{} is an ordered quintuple $\mathbf P=(P,\leq,{}',0,1)$ such that $(P,\leq,0,1)$ is a bounded poset and $'$ is a unary operation on $P$ satisfying the following conditions for all $x,y\in P$:
1. $L(x,x')\approx\{0\}$ and $U(x,x')\approx\{1\}$,
2. $x\leq y$ implies $y'\leq x'$,
3. $(x')'\approx x$.
As mentioned in the introduction, we introduce several kinds of posets with complementation which generalize orthomodular lattices.
The poset $\mathbf P$ with complementation is called a [*Boolean poset*]{} if $(P,\leq)$ is distributive. Of course, every Boolean algebra is a Boolean poset but there are interesting examples of Boolean posets which are not lattices, see e.g. [@CL2]. In every case, Boolean posets are orthomodular posets and hence they can be considered as quantum structures.
The poset $\mathbf P$ with complementation is called a [*pseudo-Boolean poset*]{} if it satisfies one of the following equivalent identities: $$\begin{aligned}
L(U(x,y),y') & \approx L(x,y'), \\
U(L(x,y),y') & \approx U(x,y').\end{aligned}$$ Pseudo-Boolean posets are certain generalization of Boolean ones but they are closely connected to the following posets.
The poset $\mathbf P$ with complementation is called a [*pseudo-orthomodular poset*]{} if it satisfies one of the following equivalent identities: $$\begin{aligned}
L(U(L(x,y),y'),y) & \approx L(x,y), \\
U(L(U(x,y),y'),y) & \approx U(x,y).\end{aligned}$$ It is evident that pseudo-orthomodular posets are generalizations of orthomodular lattices. Namely, if a poset $(P,\leq,{}',0,1)$ with complementation is a lattice satisfying these identities then $U(x,y)\approx U(x\vee y)$ and $L(x,y)\approx L(x\wedge y)$. Thus our equalities yield $$\begin{aligned}
L(((x\wedge y)\vee y')\wedge y) & \approx L((x\wedge y)\vee y',y)\approx L(U((x\wedge y)\vee y'),y)\approx L(U(x\wedge y,y'),y)\approx \\
& \approx L(U(L(x\wedge y),y'),y)\approx L(U(L(x,y),y'),y)\approx L(x,y)\approx L(x\wedge y)\end{aligned}$$ whence $$((x\wedge y)\vee y')\wedge y\approx x\wedge y$$ which is the orthomodular law. Hence, these posets can serve as an algebraic semantics of the logic of quantum mechanics. The advantage of this concept is that we need not assume $x\leq y$ as in the definition of orthomodular posets.
It is easy to see that every Boolean poset is pseudo-Boolean and every pseudo-Boolean poset is pseudo-orthomodular (cf. [@CL1]).
An example of a pseudo-orthomodular poset which is neither Boolean nor orthomodular is depicted in Fig. 1. It is not orthomodular because e.g. $b\leq c'$, but $b\vee c$ does not exist. It is evident that it is not Boolean because it is not distributive.
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\[th1\] Let $(P,\leq,{}',0,1)$ be a bounded poset with a unary operation satisfying both conditions [(\[equ1\])]{} and [(\[equ2\])]{} and the identity $1'\approx0$ and $M$ and $R$ defined by [(\[equ3\])]{} and [(\[equ4\])]{}, respectively. Then $(P,\leq,{}',M,R,0,1)$ is an operator left residuated poset.
1. $M(x,1)\approx L(U(x,1'),1)\approx L(U(x,1'))\approx L(U(x,0))\approx L(U(x))\approx L(x)$,\
$M(1,x)\approx L(U(1,x'),x)\approx L(1,x)\approx L(x)$,
2. follows from Lemma \[lem1\],
3. $R(x,0)\approx L(U(L(0,x),x'))\approx L(U(0,x'))\approx L(U(x'))\approx L(x')$.
We show that the posets mentioned above are among those assumed in Theorem \[th1\].
\[ex1\] Every pseudo-orthomodular poset satisfies both [(\[equ1\])]{} and [(\[equ2\])]{}. This can be seen as follows: If $(P,\leq,{}',0,1)$ is a pseudo-orthomodular poset and $a,b\in P$ then $$\begin{aligned}
L(a) & =L(U(a))\subseteq L(U(a,b'))=L(U(L(U(a,b'),b),b')), \\
L(U(L(a,b),b'),b) & =L(a,b)\subseteq L(a).\end{aligned}$$
For posets $(P,\leq,{}')$ with a unary operation we define the following two conditions: $$\begin{aligned}
L(x) & \subseteq L(U(L(x,y),y'))\text{ for all }x,y\in P,\label{equ7} \\
L(U(x,y'),y) & \subseteq L(x)\text{ for all }x,y\in P\label{equ8}\end{aligned}$$ and the following two mappings from $P^2$ to $2^P$: $$\begin{aligned}
M(x,y) & :=L(x,y)\text{ for all }x,y\in P,\label{equ9} \\
R(x,y) & :=L(U(y,x'))\text{ for all }x,y\in P.\label{equ10}\end{aligned}$$ Observe that $M$ is commutative.
\[lem3\] Let $(P,\leq,{}')$ be a poset with a unary operation and $M$ and $R$ defined by [(\[equ9\])]{} and [(\[equ10\])]{}, respectively. Then [(\[equ7\])]{} implies $M(x,y)\subseteq L(z)\Rightarrow L(x)\subseteq R(y,z)$ and [(\[equ8\])]{} implies $L(x)\subseteq R(y,z)\Rightarrow M(x,y)\subseteq L(z)$.
Assume $a,b,c\in P$. If (\[equ7\]) and $M(a,b)\subseteq L(c)$ then $$\begin{aligned}
L(a) & \subseteq L(U(L(a,b),b'))=L(U(L(a,b))\cap U(b'))=L(U(M(a,b))\cap U(b'))\subseteq \\
& \subseteq L(U(L(c))\cap U(b'))=L(U(c)\cap U(b'))=L(U(c,b'))=R(b,c).\end{aligned}$$ If (\[equ8\]) and $L(a)\subseteq R(b,c)$ then $$M(a,b)=L(a,b)=L(a)\cap L(b)\subseteq R(b,c)\cap L(b)=L(U(c,b'))\cap L(b)=L(U(c,b'),b)\subseteq L(c).$$
\[th2\] Let $(P,\leq,{}',0,1)$ be a bounded poset with a unary operation satisfying both conditions [(\[equ7\])]{} and [(\[equ8\])]{} and $M$ and $R$ defined by [(\[equ9\])]{} and [(\[equ10\])]{}, respectively. Then $(P,\leq,{}',M,R,0,1)$ is an operator residuated poset.
1. $M(x,1)\approx L(x,1)\approx L(x)$,\
$M(1,x)\approx L(1,x)\approx L(x)$,
2. follows from Lemma \[lem3\],
3. $R(x,0)\approx L(U(0,x'))\approx L(U(x'))\approx L(x')$ .
Since $M$ is commutative, $(P,\leq,{}',M,R,0,1)$ is an operator residuated poset.
Again, pseudo-Boolean and hence also Boolean posets are among those posets assumed in Theorem \[th2\], see the following example.
\[ex2\] Every pseudo-Boolean poset satisfies [(\[equ7\])]{} and [(\[equ8\])]{}. This can be seen as follows: If $(P,\leq,{}',0,1)$ is a pseudo-Boolean poset and $a,b\in P$ then $$\begin{aligned}
L(a) & =L(U(a))\subseteq L(U(a,b'))=L(U(L(a,b),b')), \\
L(U(a,b'),b) & =L(a,b)\subseteq L(a).\end{aligned}$$
Combining Theorems \[th1\] and \[th2\] and Examples \[ex1\] and \[ex2\] we conclude
\[cor1\] If $(P,\leq,{}',0,1)$ is a pseudo-Boolean poset and $M$ and $R$ are defined by [(\[equ9\])]{} and [(\[equ10\])]{}, respectively, then $(P,\leq,{}',M,R,0,1)$ is an operator residuated poset. If $(P,\leq,{}',0,1)$ is a pseudo-orthomodular poset satisfying the identity $1'\approx0$ and $M$ and $R$ are defined by [(\[equ3\])]{} and [(\[equ4\])]{}, respectively, then $(P,\leq,{}',M,R,0,1)$ is an operator left residuated poset.
It is well-known that in a residuated lattice each of the operations $\odot$ and $\rightarrow$ determines the other one. We can prove a similar result also for the posets listed above.
1. If $(P,\leq,{}',0,1)$ is a pseudo-orthomodular poset and $M$ and $R$ are defined by [(\[equ3\])]{} and [(\[equ4\])]{}, respectively, then $$\begin{aligned}
L((M(y',x))') & \approx R(x,y), \\
L((R(y,x'))') & \approx M(x,y). \end{aligned}$$
2. If $(P,\leq,{}',0,1)$ is a poset with complementation and $M$ and $R$ are defined by [(\[equ9\])]{} and [(\[equ10\])]{}, respectively, then $$\begin{aligned}
L((M(y',x))') & \approx R(x,y), \\
L((R(y,x'))') & \approx M(x,y). \end{aligned}$$
<!-- -->
1. $$\begin{aligned}
L((M(y',x))') & \approx L((L(U(y',x'),x))')\approx L(U(L(y,x),x'))\approx R(x,y), \\
L((R(y,x'))') & \approx L((L(U(L(x',y),y')))')\approx L(U(L(U(x,y'),y)))\approx L(U(x,y'),y)\approx \\
& \approx M(x,y),\end{aligned}$$
2. $$\begin{aligned}
L((M(y',x))') & \approx L((L(y',x))')\approx L(U(y,x'))\approx R(x,y), \\
L((R(y,x'))') & \approx L((L(U(x',y')))')\approx L(U(L(x,y)))\approx L(x,y)\approx M(x,y).\end{aligned}$$
A characterization of posets satisfying generalized operator residuation
========================================================================
The conditions (\[equ7\]),(\[equ8\]) as well as (\[equ9\]),(\[equ10\]) which are formulated for variables can be expressed also for subsets of $P$ in the following way:
For posets $(P,\leq,{}')$ with a unary operation we define $A':=\{x'\mid x\in A\}$ for all subsets $A$ of $P$. Moreover, we define the following two conditions: $$\begin{aligned}
L(A) & \subseteq L(U(L(A,B),B'))\text{ for all }A,B\subseteq P,\label{equ11} \\
L(U(A,B'),B) & \subseteq L(A)\text{ for all }A,B\subseteq P\label{equ12}\end{aligned}$$ and the following two binary operations on $2^P$: $$\begin{aligned}
M(A,B) & :=L(A,B)\text{ for all }A,B\subseteq P,\label{equ13} \\
R(A,B) & :=L(U(B,A'))\text{ for all }A,B\subseteq P.\label{equ14}\end{aligned}$$ In case $A=\{x\}$ and $B=\{y\}$ we will write simply $M(x,y)$ and $R(x,y)$ as previously. Of course, taking singletons in (\[equ11\]) and (\[equ12\]) instead of $A$ and $B$ yields (\[equ7\]) and (\[equ8\]), respectively. Hence, the new conditions and definitions include the previous ones as a particular case. Also our definition of operator adjointness can be extended to subsets of $P$ as follows: $$\begin{aligned}
& \text{for all }A,B,C\subseteq P, M(A,B)\subseteq L(C)\text{ implies }L(A)\subseteq R(B,C),\label{equ15} \\
& \text{for all }A,B,C\subseteq P, L(A)\subseteq R(B,C)\text{ implies }M(A,B)\subseteq L(C).\label{equ16}\end{aligned}$$ The ordered seventuple $\mathbf P=(P,\leq,{}',M,R,0,1)$ will be called a [*generalized operator left residuated poset*]{} if it satisfies (i) and (iii) of Definition \[def1\] as well as (\[equ15\]) and (\[equ16\]), i.e. if for all $A,B,C\subseteq P$, $$\begin{aligned}
& M(A,B)\subseteq L(C)\text{ is equivalent to }L(A)\subseteq R(B,C).\label{equ5}\end{aligned}$$ Condition (\[equ5\]) will be called [*generalized operator left adjointness*]{}. If $M$ is commutative then (\[equ5\]) is called [*generalized operator adjointness*]{} only and $\mathbf P$ is called a [*generalized operator residuated poset*]{}. It is evident that taking singletons instead of $A,B,C$ in generalized operator adjointness, we obtain condition (ii) from Definition \[def1\]. Now we are able to prove a result analogous to Lemma \[lem3\], but in a stronger version.
\[th3\] Let $(P,\leq,{}')$ be a poset with a unary operation and $M$ and $R$ defined by [(\[equ13\])]{} and [(\[equ14\])]{}, respectively. Then [(\[equ11\])]{} and [(\[equ15\])]{} are equivalent, and [(\[equ12\])]{} and [(\[equ16\])]{} are equivalent.
Assume $A,B,C\subseteq P$.\
(\[equ11\]) $\Rightarrow$ (\[equ15\]):\
If $M(A,B)\subseteq L(C)$ then $$\begin{aligned}
L(A) & \subseteq L(U(L(A,B),B'))=L(U(L(A,B))\cap U(B'))=L(U(M(A,B))\cap U(B'))\subseteq \\
& \subseteq L(U(L(C))\cap U(B'))\subseteq L(U(C)\cap U(B'))=L(U(C,B'))=R(B,C).\end{aligned}$$ (\[equ15\]) $\Rightarrow$ (\[equ11\]):\
Any of the following assertions implies the next one: $$\begin{aligned}
L(A,B) & \subseteq L(A,B), \\
M(A,B) & \subseteq L(A,B), \\
M(A,B) & \subseteq L(L(A,B)), \\
L(A) & \subseteq R(B,L(A,B)), \\
L(A) & \subseteq L(U(L(A,B),B')).\end{aligned}$$ (\[equ12\]) $\Rightarrow $(\[equ16\]):\
If $L(A)\subseteq R(B,C)$ then $$\begin{aligned}
M(A,B) & =L(A,B)=L(A)\cap L(B)\subseteq R(B,C)\cap L(B)=L(U(C,B'))\cap L(B)= \\
& =L(U(C,B'),B)\subseteq L(C).\end{aligned}$$ (\[equ16\]) $\Rightarrow$ (\[equ12\]):\
Any of the following assertions implies the next one: $$\begin{aligned}
L(U(A,B')) & \subseteq L(U(A,B')), \\
L(U(A,B')) & \subseteq R(B,A), \\
M(U(A,B'),B) & \subseteq L(A), \\
L(U(A,B'),B) & \subseteq L(A).\end{aligned}$$
By Theorem \[th3\] we obtain a stronger version of a result analogous to Theorem \[th2\].
\[cor2\] Let $(P,\leq,{}')$ be a poset with a unary operation and $M$ and $R$ defined by [(\[equ13\])]{} and [(\[equ14\])]{}, respectively. Then $(P,\leq,{}',M,R,0,1)$ is a generalized operator residuated poset if and only if it satisfies both conditions [(\[equ11\])]{} and [(\[equ12\])]{}.
[99]{} G. Birkhoff and J. von Neumann, The logic of quantum mechanics. Ann. of Math. [**37**]{} (1936), 823–843. I. Chajda and H. Länger, Residuation in orthomodular lattices. Topol. Algebra Appl. [**5**]{} (2017), 1–5. I. Chajda and H. Länger, Orthomodular lattices can be converted into left residuated l-groupoids. Miskolc Math. Notes [**18**]{} (2017), 685–689. I. Chajda and H. Länger, Residuated operators in complemented posets. Asian European J. Math. (to appear). I. Chajda and H. Länger, Left residuated lattices induced by lattices with a unary operation. Soft Computing (submitted). I. Chajda and J. Rachunek, Forbidden configurations for distributive and modular ordered sets. Order [**5**]{} (1989), 407–423. N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam 2017. ISBN 978-0-444-52141-5. R. Giuntini, A. Ledda and F. Paoli, A new view of effects in a Hilbert space, Studia Logica [**104**]{} (2016), 1145–1177. R. Giuntini, A. Ledda and F. Paoli, On some properties of PBZ\*-lattices. Preprint. H K. Husimi, Studies on the foundation of quantum mechanics. I. Proc. Phys.-Math. Soc. Japan [**19**]{} (1937), 766–789. K J. A. Kalman, Lattices with involution. Trans. Amer. Math. Soc. [**87**]{} (1958), 485–491.
Authors’ addresses:
Ivan Chajda\
Palacký University Olomouc\
Faculty of Science\
Department of Algebra and Geometry\
17. listopadu 12\
771 46 Olomouc\
Czech Republic\
[email protected]
Helmut Länger\
TU Wien\
Faculty of Mathematics and Geoinformation\
Institute of Discrete Mathematics and Geometry\
Wiedner Hauptstraße 8-10\
1040 Vienna\
Austria, and\
Palacký University Olomouc\
Faculty of Science\
Department of Algebra and Geometry\
17. listopadu 12\
771 46 Olomouc\
Czech Republic\
[email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Arkaprabha Sarangi & Isabelle Cherchneff'
date: 'Submitted 12 September 2014; Accepted 16 December 2014'
title: 'Condensation of dust in the ejecta of type II-P supernovae'
---
[We study the production of dust in Type II-P supernova ejecta by coupling the gas-phase chemistry to the dust nucleation and condensation phases. We consider two supernova progenitor masses with homogeneous and clumpy ejecta to assess the chemical type and quantity of dust that forms. Grain size distributions are derived for all dust components as a function of post-explosion time. ]{} [The chemistry of the gas phase and the simultaneous formation of dust clusters are described by a chemical network that includes all possible processes that are efficient at high gas temperatures and densities. The formation of key bimolecular species (e.g., CO, SiO) and dust clusters of silicates, alumina, silica, metal carbides, metal sulphides, pure metals, and amorphous carbon is considered. A set of stiff, coupled, ordinary, differential equations is solved for the gas conditions pertaining to supernova explosions. These master equations are coupled to a dust condensation formalism based on Brownian coagulation. ]{} [We find that Type II-P supernovae produce dust grains of various chemical compositions and size distributions as a function of post-explosion time. The grain size distributions gain in complexity with time, are slewed towards large grains, and differ from the usual Mathis, Rumpl, & Nordsieck power-law distribution characterising interstellar dust. Gas density enhancements in the form of ejecta clumps strongly affect the chemical composition of dust and the grain size distributions. Some dust type, such as forsterite and pure metallic grains, are highly dependent on clumpiness. Specifically, a clumpy ejecta produces large grains over 0.1 [$\mu$m]{}, and the final dust mass for the 19 [M$_{\odot}$]{} progenitor reaches 0.14 [M$_{\odot}$]{}. Clumps also favour the formation of specific molecules, such as CO$_2$, in the oxygen-rich zones. Conversely, the carbon and alumina dust masses are primarily controlled by the mass yields of alumina and carbon in the ejecta zones where the dust is produced. The supernova progenitor mass and the [$^{56}$Ni]{} mass also affect dust production. Our results highlight that dust synthesis in Type II-P supernovae is not a single and simple process, as often assumed. Several dust components form in the ejecta over time and the total dust mass gradually builds up over a time span around three to five years post-outburst. This gradual growth provides a possible explanation for the discrepancy between the small amounts of dust formed at early post-explosion times and the high dust masses derived from recent observations of supernova remnants. ]{}
Introduction
============
Ejecta zones Zone 1A Zone 1B Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Total
--------------------------------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- -------
Zone mass in [M$_{\odot}$]{} 0.11 0.302 1.68 0.141 0.486 0.774 0.358 3.85
Clump number 44 118 654 55 189 301 139 1500
$f_c$ 2.9(-2) 4.1(-3) 7.3(-2) 2.0(-2) 2.0(-2) 1.5(-2) 1.5(-2) –
$n_c$(day 100) in [cm$^{-3}$]{} 2.47(12) 2.83(13) 2.11(12) 8.60(12) 1.26(13) 4.24(13) 4.29(13) –
Important sources of cosmic dust include the explosion of supergiant stars as Type II supernovae (SNe). Dust formation was observed in SN1987A at infrared (IR) wavelengths a few hundred days after the explosion ([@luc89; @dan91; @wood93]), and this scenario has since been observed in several other Type II-P SNe (e.g., [@sug06; @ko09; @gal12; @sza13]). Analysis of the IR flux emitted by this warm dust has indicated fairly modest amounts of solid condensates in the ejecta, in the $10^{-5}-10^{-3}$ [M$_{\odot}$]{} range, while in the specific case of SN2003gd, a mass of 0.02 [M$_{\odot}$]{} at day 678 after outburst was inferred from Spitzer data, under the assumption that the ejecta was clumpy. However, Meikle et al. (2007) contest this high dust mass 660 days post-explosion by explaining that an IR echo from pre-existing circumstellar dust could contribute to the emission.
Recent observations of SN remnants (Thereafter SNRs) in the sub-millimetre (submm) with AKARI, Herschel and ALMA have brought evidence of large reservoirs of cool ejecta dust, specifically in the 330-year-old SNR Cas A, the 960-year-old Crab Nebula, and the young remnant of SN1987A. In Cas A, the dust mass estimated from AKARI and Herschel data ranges from 0.06 [M$_{\odot}$]{} ([@sib10]) to 0.075 [M$_{\odot}$]{} ([@bar10]). In the Crab Nebula, $0.02$ to $0.2$ [M$_{\odot}$]{} of dust were inferred from Herschel data ([@gom12; @tem13]). For SN1987A, very high dust masses in the range $0.4-0.8$ [M$_{\odot}$]{} were inferred from the submm fluxes measured with Herschel (Matsuura et al. 2012, 2014), and the mass was reduced to a lower limit of 0.2 [M$_{\odot}$]{} from the analysis of new ALMA data ([@ind14]). These observations clearly indicate a much higher mass of dust than that detected a few years after the outburst.
Whether these high dust masses are formed in the ejecta but remained undetected, or the dust continues to grow to high mass values in the remnant phase decades after the explosion is a highly debated issue. It has recently been proposed that dust cluster growth occured over a time span of a few years after the SN explosion ([@sar13], thereafter SC13). Furthermore, the growth of grains could not be ongoing after the nebular phase because the atomic and molecular accretion on the surface of dust grains could not proceed owing to a shortage of accreting species and the long time scale for accretion. These results contrast with a recent study by Wesson et al. (2015), who find the growth of dust grains continues from 1200 to 9200 days.
Indirect evidence of the formation and growth of dust grains in SNe is provided by the study of pre-solar grains from meteorites. Some of those bear the isotopic anomaly signatures characteristic of SNe, and include the presence of radiogenic $^{44}$ Ca, which stems from the decay of short-lived $^{44}$Ti, an isotope only produced in SNe ([@zin07]). Pre-solar grains of oxides, silicates, carbon, silicon carbide, silicon nitride, and silica formed in SN ejecta have been identified ([@zin07; @hop10; @haen13]). Typical lower limits for grain sizes are in the $0.1-1$ [$\mu$m]{} range, with some evidence of very large grains; e.g., one SiC grain with a radius of 35 [$\mu$m]{} has been identified ([@zin10]). The isotopic anomaly signatures of the pre-solar SN grains imply mixing in the ejecta, whereby the innermost and outermost zones might have been in contact during or after the explosion. These results indicate the dust formed in the SN ejecta can survive the SNR phase, be incorporated to the Interstellar Medium, and travel to the solar system.
Finally, the high masses of dust inferred from the observations of damped Ly$\alpha$ systems and quasars at high redshift ([@pei91; @pet94]) hint at a possible contribution of massive SNe, because massive stars evolve on short time scales that are compatible with the age of the Universe at high redshift ([@dwek11]). Massive SNe do form high masses of dust grains in their ejecta ([@noz03; @sch04; @cher10]), but these grains are heavily reprocessed in subsequent evolutionary phases, for example, in the remnant phase and the Interstellar Medium. When interstellar dust destruction is considered and a top-heavy initial mass function is assumed, a dust mass of $\sim 1$ [M$_{\odot}$]{} produced per SN is necessary to explain the high redshift dust ([@dwek07]). This value is higher than the dust masses derived from IR data of SNe and comparable, although higher, to the dust masses inferred from submm data of SNRs. It is thus paramount to shed light on the processes underpinning dust production in SNe, locally, and at high redshift.
Following our study on the molecule and cluster formation in Type II-P SNe of various progenitor masses (SC13), we present an exhaustive model of the dust production in SNe ejecta, which includes the coupling of the gas-phase and nucleation phase chemistry to the condensation of dust grains up to $\sim 5 $ years after the SN explosion. Size distributions for the various dust components are derived, and we explore the effect of a low $^{56}$Ni mass and ejecta clumpiness on the synthesis of molecules and dust grains. The derivation of grain chemical compositions, masses, and size distributions is important to model the dust fluxes in the IR and submm, and to assess the chances of survival of SN grains in the SN remnant phase. In § \[sec2\], we present the physical models used in the study, and in § \[sec3\], the chemical processes and the formalism underpinning the nucleation and the condensation phase. The results for homogeneous and clumpy ejecta are presented in § \[sec4\], and a discussion follows in § \[sec5\].
Physical model of the supernova ejecta {#sec2}
======================================
When a massive star explodes as a supernova, its helium core is crossed by the explosion blast wave that deposits energy to the gas. The reverse shock created at the base of the progenitor envelope propagates inward and triggers Rayleigh-Taylor instabilities and macroscopic mixing over time scales of a few days ([@jog10]). These instabilities result in the fragmentation of the He-core, but a chemical stratification persists over time. Radioactivity plays a crucial role in generating the light curve and impinges on the ejecta chemistry. The radioactive [$^{56}$Ni]{} produced in the explosion decays into [$^{56}$Co]{} on a time scale of a few days. In turn, [$^{56}$Co]{} decays into [$^{56}$Fe]{} with a half-life of $\sim$ 113 days. This decay sequence creates a flux of [$\gamma$-ray]{} photons that pervades the ejecta. The degrading of [$\gamma$-rays]{} to X-rays and ultraviolet (UV) photons occurs by Compton scattering and creates a population of fast Compton electrons in the ejecta. These fast electrons ionise the gas, and produce ions such as [Ar$^+$]{}, [Ne$^+$]{}, and [He$^+$]{}. These ions are detrimental to the survival of molecules in the ejecta gas ([@lepp90; @cher09], SC13).
We consider homogeneous, stratified ejecta, whose elemental compositions are given for a 15 [M$_{\odot}$]{} and 19 [M$_{\odot}$]{} stellar progenitors ([@rau02]). The ejecta consists of mass zones of specific chemical compositions, which are summarised in Table \[tab1\] of the Appendix A as a function of ejecta zoning. Each zone is microscopically mixed, and we assume no chemical leakage between the various zones. The ejecta gas temperature and density are derived from explosion models and vary with post-explosion time according to equations 1 and 2 in SC13 for both the 15 [M$_{\odot}$]{} and 19 [M$_{\odot}$]{} progenitors. For the gas density, a constant value of $1.1\times 10^{-11}$ g cm$^{-3}$ at day 100 is assumed in all He-core mass zones, hence the ejecta is homogeneous. Because each mass zone has a specific chemical composition, this constant gas density translates into different gas number densities at day 100 in the mass zones. The gas temperatures and number densities are listed in Table \[tab2\] of the Appendix A for both progenitor masses.
We also consider a non-homogeneous, clumpy ejecta for the 19 [M$_{\odot}$]{} stellar progenitor, which we choose as a surrogate to SN1987A and other massive SNe. We build up a simple model for a clumpy ejecta as follows: for each zone, we use the volume filling factor $f_c$ derived by Jerkstrand et al. (2011) in their modelling of the ultraviolet, optical, and near-IR emission lines observed in SN1987A. We assume a fiducial number of 1500 for the total number of clumps in the ejecta, in agreement with radiative transfer models of the IR spectral energy distribution of various SN ejecta (e.g., [@gal12]). Assuming the He-core mass for the 19 [M$_{\odot}$]{} stellar progenitor listed in Table \[tab2\], we derive a typical clump mass of $ \sim 2.6\times 10^{-3}$ [M$_{\odot}$]{} for all clumps, a value which agrees well with typical clump masses derived from 3-D explosion models (e.g., [@ham10]). The enhancement over the homogeneous gas density at day 100 is then given by $1/f_c$, and the gas number density $n_c$ in the clumps is estimated as $n_c=n_0 / f_c$, where $n_o$ is the gas number density for the homogeneous zone. All parameters are listed in Table \[tab3\] as a function of ejecta zones. The initial atomic yields are those of the 19 [M$_{\odot}$]{} homogeneous model given in Table \[tab1\], and the gas number density follows a time variation as in SC13. We will see in § \[clump\] that high amounts of CO form at early post-explosion time in the clumpy ejecta and will probably affect the cooling and temperature of the ejecta zones, but we do not consider molecular cooling at this stage of the modelling. Therefore, the gas temperature for the 19 [M$_{\odot}$]{} clumpy case is that given in Table \[tab2\].
Dust nucleation and condensation model {#sec3}
======================================
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
The nucleation of small dust clusters out of the gas phase in the ejecta of Type II-P SNe was studied by SC13. The formation of dimers of forsterite ([Mg$_2$SiO$_4$]{}) carbon, silicon carbide (SiC), pure metals that include silicon, iron, and magnesium, magnesia (MgO), iron sulphide (FeS), and iron oxide (FeO), has been modelled by using a chemical kinetic approach applied to the ejecta of SNe associated with stellar progenitors of various masses. In SC13, the formation of the molecule AlO was studied as an indicator of alumina formation, but no nucleation scheme for alumina clusters was implemented. As for carbon, the small carbon clusters were represented by the chain C$_{10}$.
In this paper, the growth of alumina, [Al$_2$O$_3$]{}, is described by the dimerisation of AlO, followed by the oxidation of the dimer through reactions with oxygen-bearing molecules (O$_2$, SO), which leads to the formation of the small cluster [Al$_2$O$_3$]{} ([@bis14]). We then consider the tetramer of [Al$_2$O$_3$]{}, the ([Al$_2$O$_3$]{})$_4$ molecule, to be the stable gas phase precursor of alumina dust. The growth of amorphous carbon is extended to carbon chains larger than C$_{10}$ and involves the synthesis of carbon chains through reactions with atomic carbon and C$_2$, the closure of these chains as rings when the number of carbon atoms N$_C \ge 10$, and the growth of rings, which is controlled by C$_2$ addition. The first stable cage structure that subsequently forms is C$_{28}$, and the cage keeps growing through C$_2$ addition to the carbon lattice without cage fragmentation ([@dun12]). The C$_2$ inclusion can proceed towards forming the fullerene cage, C$_{60}$, but the efficiency at synthesising fullerene depends on the gas number densities and the abundance of the growing agent, C$_2$. As in SC13, we consider the synthesis of pure metallic grains by forming the small tetramers (SiC)$_4$, (Si)$_4$, (Fe)$_4$, (FeS)$_4$, and the formation of the pentamer (SiO)$_5$ as a small cluster precursor of silica. However, the nucleation and condensation of silica is not considered in the present study because of the paucity of information available on these processes.
These molecular clusters act as dust seeds in the condensation phase, where they grow by coagulation to form large grains. Growth through accretion of atoms and molecules on the surface of dust grains may also occur if there is enough accreting species in the gas phase (i.e., Si, SiO, C, or C$_2$ depending on the ejecta zone) and available grain surfaces to accrete on. By the time small dust clusters have formed from the gas phase, the supply of growing gas-phase species has severely decreased in the dust-forming ejecta zones, as shown in SC13. Accretion can thus not proceed because of this shortage of accreting species, and we consider coagulation of small dust clusters the only dust growth mechanism before day 2000.
To describe dust coagulation, we use the formalism developed by Jacobson (2005) (thereafter J05), where the variation with time of the number density of a grain of specific volume $v$ is described by the integro-differential coagulation equation given by
$$\label{eq1}
\frac{dn_v (t)}{dt} = \frac{1}{2} \int\limits_{v_0}^v \beta_{v-v',v} n_{v-v'}n_{v'} dv' - n_{v} \int\limits_{v_0}^\infty \beta_{v,v'} n_{v'} dv'.$$
Here, $t$ is the time, $v'$ and $(v-v')$ are the volumes of the two coagulating particles, n$_v$ is the number density of grains with volume $v$, $\beta_{v, v'}'$ is the rate coefficient of coagulation between particles with volume $v$ and $v'$, and $v_0$ is the volume of the largest gas-phase cluster seed. The rate of coagulation $\beta_{v_i,v_j}$ for particles $i$ and $j$ is controlled by physical processes such as Brownian diffusion, convective Brownian motion enhancement, gravitational collection, turbulent inertial motion, and Van der Waal’s forces. Typical gas number densities in SN ejecta at 300 days post-explosion range between $10^9-10^{11}$ [cm$^{-3}$]{}. Therefore, the ejecta gas is characterised by a free-molecular regime, which is defined by $\lambda_p \gg a_i $, where $\lambda_p$ is the mean free-path of a particle in the gas and $a_i$ is the radius of the $i^{th}$particle. In a free-molecular regime, Brownian diffusion prevails in the coagulation process, whereas the other processes are relevant in case of larger particles and denser media. Brownian diffusion accounts for the scattering, collision, and coalescence of the grains through Brownian motion.
For the sake of simplicity, we rename the rate coefficient of coagulation $\beta_{v_i,v_j}$ as $\beta_{ij}$ between two grains $i$ and $j$ of radius $a_i$ and $a_j$, respectively. The rate is given by $$\label{eq2}
\beta_{ij} = \frac{4\pi(a_i + a_j)(D_{c,i} + D_{c,j})W_{i,j}}{\frac{(a_i + a_j)}{(a_i + a_j) + \sqrt{\delta_i^2 + \delta_j^2}} + \frac{4(D_{c,i} + D_{c,j})}{(a_i + a_j)\sqrt{v_{p,i}^2 + v_{p,j}^2}}}$$ where $D_{c,i}$ & $D_{c,j}$ are the diffusion coefficients for particle $i$ and $j$, respectively, $v_{p,m}$ is the mean thermal velocity for particle $m$, $\delta_i$ is the mean distance of particle $i$ from the centre of a sphere traveling a distance $\lambda_p$, and $W_{ij}$ is the enhancement factor due to the effect of Van der Waal’s dispersion forces (J05, [@sp06]). The formalism assumes the dust grains have a temperature similar to that of the local gas of the ejecta zone where the grains form at a specific epoch. In the free-molecular regime, we have $\sqrt{\delta_i^2 + \delta_j^2} \gg (a_i + a_j)$, and Equation \[eq2\] reduces to $$\label{eq3}
\beta_{ij} = K_{ij} \times W_{ij} = \pi(a_i + a_j)^2 \sqrt{v_{p,i}^2 + v_{p,j}^2} W_{ij},$$ where $K_{ij}$ is the Brownian diffusion term. The Van der Waal’s forces develop weak, local charge fluctuations and enhance the rate of coagulation for particles with size in the molecular range, leading to the enhancement factor $W_{ij}$. The interaction potential $V(r)$ between two particles separated by a distance $r$ is defined by the Hamaker’s theory and using London dispersion forces ([@st89; @al87]), and is given by
$$\begin{aligned}
\label{eq4}
V(r)& = & - {kT}\times \frac{A'}{12} \bigg(\frac{1}{(\frac{r}{a_i + a_j})^2-1} + \frac{1}{(\frac{r}{a_i + a_j})^2} \\ \nonumber
& & + 2\ln \bigg(1- \big(\frac{a_i + a_j}{r}\big)^{2} \bigg)\bigg) \\\end{aligned}$$
where $T$ is the gas temperature, and $A'$ is given by $$\label{eq5}
A' = \frac{A}{kT} \frac{4a_ia_j}{(a_i + a_j)^2}.$$ In the above equation, A is the Hamaker constant which varies according to the physical properties of individual dust species. The enhancement factor $W_{i,j}$ due to the Van der Waal’s dispersion forces is given by $$\label{eq6}
W_{ij} = \bigg(\frac{r_T}{a_i + a_j}\bigg)^2 e^{\frac{-V_{ij}(r_T)}{kT}}$$ where $r_T $ is the minimum separation between the two particles when the difference between attractive and repulsive potentials reaches a minimum, described by $\frac{\partial}{\partial r}(V(r)-2kT\ln r) = 0$ ([@st89]).
According to Equation \[eq3\], the coagulation rate, $\beta_{ij}$, is defined as the product of the Brownian diffusion term $K_{ij}$ and the enhancement factor due to coalescence $W_{ij}$. The Brownian diffusion term is controlled by the grain temperature, assumed to be equal to the gas temperature, while $W_{i,j}$ depends on the gas temperature. Both terms also depend on the ratio of the collider radii $f_a = a_i / a_j$, as shown in Figure \[fig0\], where the dependence of $K_{ij}$, $W_{ij}$, and $\beta_{ij}$ on collider and target sizes are shown. The diffusion term is large when $f_a \gg 1$ and reaches a minimum when $f_a \simeq1$. Conversely, the value taken by $W_{ij}$ is high for $f_a \rightarrow 1$, but is confined to a small range for all values of ($a_i$,$a_j$). Therefore, the rate of Brownian coagulation $\beta_{ij}$ is controlled by the variation of $K_{ij}$, as seen from Figure \[fig0\]. The coagulation rate is minimal when the two colliders are of the same size, and is high when $f_a$ is large. Therefore, the large grains present in the gas efficiently coagulate with the newly formed small grains, and this efficiency warrants the growth of large dust grains in the ejecta.
The evolution of the coagulation rate $\beta_{ij}$ is shown as a function of gas temperature in the ejecta and for specific pairs of collider sizes in Figure \[fig0\] (bottom-right panel). Coagulation becomes less efficient as the gas temperature decreases, and typical rate values for $\beta_{ij}$ are in the range $10^{-9}-10^{-4}$ [cm$^{-3}$]{} s$^{-1}$ for colliders and targets of all sizes and ejecta gas temperatures between 100 and 2000 days post-explosion.
A semi-implicit volume conserved model has been developed to solve Equation \[eq1\] (J05, [@sp06]). Dust grains are assumed to maintain a spherical morphology and compact structure. The grains of individual dust species are assigned to discrete bins, following a volume ratio distribution given by $v_n = \gamma^{n-1} v_0$, where $v_n$ is the volume of the $n^{th}$ bin, $\gamma$ is a constant defined as the ratio of the volumes of adjacent bins, and $v_0$ is the volume of the first bin, determined by the size of the gas-phase precursor. At each time step, the volume of the first bin, $v_0$ corresponds to the radius $a_0$ of the stable, largest clusters produced from chemical kinetics in the nucleation phase. Particles with volumes intermediate to any two consecutive bins are allocated following a volume fractionation formalism ([@mj94]). The various quantities are then calculated according to Equations \[eq3\] $-$ \[eq6\], Equation \[eq1\] is integrated, and the grain sizes of individual dust components are derived for each time step. Values for $a_0$ and the Hamaker coefficient $A$ are summarised in Table \[tab4\] for the dust types considered in this study.
Dust type $a_0$ ( Å) $A$ (10$^{-20}$ J)
----------------- ------------ --------------------
Forsterite 3.33 6.5
Alumina 3.45 15
Carbon 3.92 47
Pure Magnesium 2.29 30
Silicon Carbide 2.15 44
Pure Silicon 2.46 21
Pure Iron 2.81 30
Iron Sulphide 3.0 15
: Initial grain size $a_0$ equivalent to the size of the largest dust cluster formed in the gas phase from chemical kinetics and the Hamaker constant $A$ for different dust components. []{data-label="tab4"}
\
\
Results
=======
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
\[sec4\] The condensation phase is coupled to the nucleation phase to provide an exhaustive description of the synthesis of dust in Type II-P SN ejecta. We follow the gas phase chemistry by describing the formation of species (atoms, ions and molecules) and dust molecular clusters with a network of chemical reactions, as in SC13, and couple the chemistry to the simultaneous condensation of molecular clusters in dust grains from 100 to 2000 days after the SN explosion. Results on grain masses and size distributions for the various dust components are presented for a 15 [M$_{\odot}$]{} stellar progenitor, characterised by a homogeneous ejecta and a $^{56}$Ni mass equals to $0.075$ [M$_{\odot}$]{} (i.e., standard case). We further explore the effect of a low $^{56}$Ni mass (0.01 [M$_{\odot}$]{}) on the condensation of dust grains.. We also consider a 19 [M$_{\odot}$]{} stellar progenitor case as a surrogate of SN1987A and other massive SNe, for which we model the formation of dust in a homogeneous and a clumpy ejecta over a similar time span. The normalised size distributions for the 15 [M$_{\odot}$]{} standard case and the 19 [M$_{\odot}$]{} clumpy case are listed in Tables \[taba3\]$-$\[taba5\] in the Appendix A.
15 [M$_{\odot}$]{} progenitor: Standard Case {#sec1}
--------------------------------------------
The SN ejecta of our standard case follows the stratification given in Table \[tab1\], where the zones are as in SC13. The Si/S zone (zone 1A) is conducive to the formation of iron sulphide, pure silicon and iron clusters. The oxygen core of the ejecta includes zones 1B, 2 and 3. Oxygen-rich dust components such as silicates and alumina are synthesised in this region. An excess of magnesium in the ejecta also leads to the formation of pure magnesium clusters. Zones 4A and 4B, where most of CO molecules form, have little contribution to the total dust mass. Finally, the outermost zone of the helium core (zone 5) is characterised by a C/O ratio $>1$, and is conducive to the synthesis of carbon and silicon carbide dust.
The mass of the various dust components as a function of post-explosion time is illustrated in Figure \[fig1\], where the mass is summed over all zones and corresponds to grains with size larger than 10 Å. We consider that below this radius, the grains are molecular dust clusters. Silicates with forsterite stoichiometry ([Mg$_2$SiO$_4$]{}) starts forming mainly in zone 1B as early as 300 days post-explosion. Around day 600, the mass of forsterite is boosted by the formation of grains in zone 2. The formation of alumina in zones 2 and 3 adds up to the total dust mass in the oxygen core after day 700. Carbon grains form at late time ($t > 1000$ days) because the formation of carbon clusters is hampered by the presence of high quantities of [He$^+$]{} in zone 5 (SC13). Overall, the condensation follows the trends derived by SC13 for the nucleation of small dust clusters in the ejecta. Dust formation occurs by following several dust synthesis events at different times and in different zones of the ejecta, and results in a gradual build-up of the dust mass over a time span of a few years after the explosion. The total mass of dust at day 400 is $\sim1 \times 10^{-5}$ [M$_{\odot}$]{} and gradually grow to reach 0.035 [M$_{\odot}$]{} four years after the outburst. The general trend in dust production indicates a high efficiency of condensation of the gas-phase precursors that amounts to $\sim$ 99 %.
To explore how dust grains are distributed over size, we define for a specific dust type $i$ the grain size distribution $f_i(a)$ as
$$\label{eq8}
N_{tot, i}(a) = \sum_{a} f_i(a)\times \Delta{a},$$
where $N_{tot,i}(a)$ is the total number of grains with radius $a$ summed over all zones. The quantity $N_{tot, i}(a)$ is calculated from the number density $n_i(a)$ of grains with radius $a$ and by assuming spherical symmetry for the various ejecta zones. The size distribution $f_i(a)$ has thus the units of $N_{tot,i}(a)$ [ Å]{}$^{-1}$. The size distributions for the various dust types as a function of post-explosion time are presented in Figure \[fig2\].
![Dust mass distributions (in [M$_{\odot}$]{}) as a function of grain radius for the various dust types formed in the ejecta at 600 days (top), 900 days (middle), and 2000 days (bottom) after explosion.[]{data-label="fig3"}](f31 "fig:"){width="\columnwidth"} ![Dust mass distributions (in [M$_{\odot}$]{}) as a function of grain radius for the various dust types formed in the ejecta at 600 days (top), 900 days (middle), and 2000 days (bottom) after explosion.[]{data-label="fig3"}](f32 "fig:"){width="\columnwidth"} ![Dust mass distributions (in [M$_{\odot}$]{}) as a function of grain radius for the various dust types formed in the ejecta at 600 days (top), 900 days (middle), and 2000 days (bottom) after explosion.[]{data-label="fig3"}](f33 "fig:"){width="\columnwidth"}
Dust type $x_d$ $a_{peak}$ ( Å)
----------------- --------- --------- ----------------- --------- --------- --------- --------- ------ --------- --
500 700 900 1100 1200 1500 2000
Forsterite 1.1(-5) 4.2(-3) 5.3(-3) 5.5(-3) 5.5(-3) 5.6(-3) 5.6(-3) 15.8 64; 168
Alumina - 6.1(-6) 7.4(-3) 7.6(-3) 7.6(-3) 7.7(-3) 7.7(-3) 21.8 96
Carbon - - - 2.0(-2) 2.0(-2) 2.0(-2) 2.1(-2) 59.7 134
Pure Magnesium - 2.5(-6) 1.9(-4) 3.2(-4) 3.6(-4) 4.4(-4) 5.0(-4) 1.4 29
Pure Silicon - - 1.5(-4) 2.7(-4) 3.1(-4) 3.8(-4) 4.4(-4) 1.2 0.69
Pure Iron - - 4.1(-5) 9.1(-5) 9.4(-5) 1.2(-4) 1.4(-4) 0.4 36
Silicon Carbide - - - - 6.1(-6) 2.6(-5) 3.4(-5) 0.1 9
Iron Sulphide 1.7(-6) 1.8(-6) 1.8(-6) 1.8(-6) 1.8(-6) 1.8(-6) 1.8(-6) 0.05 21
Total 1.3(-5) 4.3(-3) 0.013 0.034 0.034 0.035 0.035 100
Small grains of silicate dust are produced at day 300 in the inner zone 1B. The average grain size remains small ($\sim 50$ Å) at day 500 as well as the mass produced, because zone 1B has a small mass compared to zone 2. The forsterite dimers efficiently starts forming larger grains at $\sim$ day 700 from zone 2 where the gas temperature is $\sim$ 1000 K, and the peak of the distribution curve shifts from 50 Å to 100 Å. The decrease in the number of small grains around 20 Å is attributed to the formation of larger grains. After 1000 days, the gas density is quite low and no new nucleation seeds (small clusters) form. The concentration of large grains ($a >200$ Å) is also low, and their growth is stopped. However small grains still participate in the coagulation process by replenishing the grain population with size 10 Å $<a<$ 100 Å, albeit with a lower efficiency. When coagulation is completed at day 2000, the silicate grain size distribution has a peak of 170 Å for the largest grain population, and most of the large grains come from zone 2.
Alumina is the second dust component to form in the O-rich core at $\sim$ day 700 after explosion. The alumina grains undergo very fast and efficient condensation to form large grains peaking at $\sim$ 60 Å at day 900, when the gas temperature is in the range $600-800$ K. As shown in Figure 13 of SC13, most atomic aluminium present in the O-rich core gets locked up in molecules and dust clusters, hence no new [Al$_2$O$_3$]{} dimers are formed after day 900 and a growth trend similar to that of forsterite applies to alumina. The alumina dust growth results in the production of large grains peaking around 100 Å at day 2000.
The formation of carbon dust in the outermost ejecta zone is strongly affected by the presence of [He$^+$]{}. The synthesis of stable C$_{28}$ cages occurs as late as 1050 days after outburst once the abundance of [He$^+$]{} ions has decreased to negligible values (SC13). Then the carbon chains, rings and cages form simultaneously with high abundances along with the efficient condensation of the cage C$_{28}$ in carbon grains. This results in a sudden high concentration of large carbon grains with peak radius of $\sim$ 100 Å at day 1200. The rate of coagulation is directly proportional to the thermal velocities and hence the gas temperatures, as seen from Equation \[eq3\]. Because of the low gas temperatures ($\sim$ 300 K) and densities at $t > 1200$ days, the condensation process becomes less efficient. However, owing to a high abundance of carbon cages in the gas phase, the condensation of grains proceeds even at these low gas temperatures. The carbon grain size distribution does not vary after day 1500, and shows a peak for large grains at $a \sim$ 150 Å.
Apart from the previous three prevalent dust components, iron sulphide, pure metal, and silicon carbide grains also condense in the ejecta. Iron sulphide, FeS, forms at day 300 in the innermost zone, zone 1A, and reaches its final mass and size distribution at day 500, with a peak of the distribution curve that lies at 25 Å. Pure silicon and iron grains start forming at day 700 in zone 1A, where most of the silicon is locked in the molecule SiS and in atomic form (SC13). Iron is mainly in atomic form in this zone but some pure iron clusters form after day 700. The size distribution of Si grains is the most extended of all pure metal dust grains, and peaks at $\sim$ 70 Å, although the overall grain population remains small because of the modest amounts of pure Si clusters formed in the gas phase. A similar scenario applies to pure iron grains, for which the distribution peaks at 40 Å at day 2000. The pure magnesium grains are synthesised from the Mg$_4$ clusters formed in the O-rich zones at day 600. The size distribution reaches its final shape at day 1200, with a peak at $\sim 30$ Å, and a small population of grains with sizes over 120 Å. Silicon carbide forms in the He/C-rich zone 5 after day 1100, and captures the available silicon and the carbon left over from the condensation of carbon dust. The low abundance of SiC clusters combined to the low gas temperature and densities at the epoch of its formation result in small masses of SiC grains characterised by small sizes in the range $5-11$ Å. The various dust masses versus post-explosion time and the radius $a_{peak}$ where the size distributions peak are summarised in Table \[tab5\].
In Figure \[fig2\] (bottom-right panel), the total dust size distribution is shown at day 900 and 2000, along with a dust size distribution for a similar dust mass following a power-law of exponent -3.5, and the Mathis-Rumpl-Nordsieck power law distribution (exponent -3.5) for grains with radius $a$ in the range $50-2500$ Å, characteristic of interstellar dust ([@mat77]). The total size distribution is the total number of grains at a particular size $a$, calculated by summing the $N_{tot,i} (a)$ for all dust types $i$ over a small size interval centred on $a$. The size distributions of the dust produced by SNe do not follow a power law, for both single dust types and the total dust distribution. Our size distributions have less small grains with radius $a < 30$ Å, and are skewed towards large grains in the size range $40-1000$ Å, compared with the a power law distribution of any exponent. Furthermore, our total number of grains are roughly one order of magnitude larger compared to the MRN profile around 200 Å. This result clearly indicates that the estimated dust mass that survives sputtering by shocks in the remnant phase may be incorrect if a MRN size distribution is assumed for the unshocked dust grains present in the SNR phase.
The dust mass distributions as a function of grain size are shown in Figure \[fig3\] at 600, 900, and 2000 days after explosion, and are derived as follows: for a specific chemical dust type $i$ and epoch, the mass fraction $x_{m,i}(a)$ of grains with radius $a$ is given by $$\label{eq9}
x_{m,i}(a) = \frac{n_i(a) \times v_n(a)}{\sum_{a} n_i(a) \times v_n(a)}$$ where $v_n(a)$ is the volume of the $n^{th}$ bin that corresponds to radius $a$, and $n_i(a)$ is the number density of grains with radius $a$. The total mass of dust $m_i(a)$ with radius $a$ is then given by $$\label{eq10}
m_i(a) = x_{m,i}(a)\times M_{tot,i} = N_{tot,i}(a) \times {v_n(a) \times \rho_{d,i}}$$ where $M_{tot,i}$ is the total mass of condensed dust of type $i$ in the ejecta, and $\rho_{d,i}$ the density of the dust material.
We see that the dust mass quickly grows with time and mainly resides in a population of large grains with sizes that range from 60 Å to 700 Å. Once coagulation has stopped, most of the dust mass is distributed in populations of carbon, forsterite, and alumina grains, while pure metals, FeS, and SiC grains do not significantly contribute to the total dust mass. For the standard case with 15 [M$_{\odot}$]{} progenitor, carbon dust dominates with a final mass amounting to $\sim 0.02$ [M$_{\odot}$]{}, while the alumina and forsterite masses are $0.008$ [M$_{\odot}$]{}, and $0.006$ [M$_{\odot}$]{}, respectively, as summarised in Table \[tab5\].
More generally, about 99% of the dust clusters formed in the gas phase efficiently condense to form dust, and only a small fraction remains in the gas phase. Coagulation is thus very efficient as soon as dust molecular clusters form in the ejecta gas. However, we see from Figure \[fig1\] that the dust size distributions are strongly time-dependent. The overall dust formation process in the ejecta results in a gradual build-up of various dust grain populations in the different ejecta zones. Therefore, the dust observed at IR wavelengths at post-explosion day 500 differs in type, mass, and size distribution, from the dust observed at submm wavelengths. This late dust is the outcome of a complete series of nucleation and coagulation events over a time span of $\sim$ 5 years after outburst.
15 [M$_{\odot}$]{} progenitor with [$^{56}$Ni]{} = 0.01 [M$_{\odot}$]{} {#lowni}
-----------------------------------------------------------------------
![Results for the homogeneous SN ejecta with 15 [M$_{\odot}$]{} progenitor and [$^{56}$Ni]{}$= 0.01$ [M$_{\odot}$]{}. Top: The total dust mass versus post-explosion time; Middle: The dust size distributions for forsterite grains in zone 1B and 2 at day 600 and 2000; Bottom: The dust size distributions at day 2000.[]{data-label="fig4"}](f41 "fig:"){width="\columnwidth"} ![Results for the homogeneous SN ejecta with 15 [M$_{\odot}$]{} progenitor and [$^{56}$Ni]{}$= 0.01$ [M$_{\odot}$]{}. Top: The total dust mass versus post-explosion time; Middle: The dust size distributions for forsterite grains in zone 1B and 2 at day 600 and 2000; Bottom: The dust size distributions at day 2000.[]{data-label="fig4"}](f42 "fig:"){width="\columnwidth"} ![Results for the homogeneous SN ejecta with 15 [M$_{\odot}$]{} progenitor and [$^{56}$Ni]{}$= 0.01$ [M$_{\odot}$]{}. Top: The total dust mass versus post-explosion time; Middle: The dust size distributions for forsterite grains in zone 1B and 2 at day 600 and 2000; Bottom: The dust size distributions at day 2000.[]{data-label="fig4"}](f43 "fig:"){width="\columnwidth"}
Several SN explosions have stellar progenitors in the mass range $8-15$ [M$_{\odot}$]{} and produce [$^{56}$Ni]{} mass between 0.01 and 0.02 [M$_{\odot}$]{}. The affect of [$^{56}$Ni]{} on the gas phase chemistry has been assessed and discussed in SC13. We extend our analysis to study the effect of a small [$^{56}$Ni]{} mass on the dust condensation process and the resulting dust masses and size distributions. The less efficient deposition of [$\gamma$-rays]{} energy due to a smaller [$^{56}$Ni]{} mass will decrease the ejecta gas temperature. However, we do not consider any change in the ejecta physical conditions caused by decreasing the [$^{56}$Ni]{} mass. The physical conditions and initial chemical compositions for each ejecta zones are then given in Table \[tab1\] and correspond to the standard case, but the [$^{56}$Ni]{} mass has been changed to $0.01$ [M$_{\odot}$]{}. The effect on the nucleation of dust clusters is that described in SC13. However, a low [$^{56}$Ni]{} also affects the dust mass and size distribution, as shown in Figure \[fig4\], where the mass of dust summed over all ejecta zones is displayed, along with the forsterite grain size distributions versus formation zones and post-explosion time, and the size distributions for the various dust components.
Forsterite, alumina and carbon dust are produced in zones 1B, 2 and 5, respectively, and these specific zones are characterised by high abundances of noble gases, which include argon, neon, and helium. The [$\gamma$-ray]{} flux induced by the decay of [$^{56}$Ni]{} is lower owing to the lower [$^{56}$Ni]{} mass. Therefore, the abundances of [Ar$^+$]{}, [Ne$^+$]{}, and [He$^+$]{} ions produced by the degrading of [$\gamma$-rays]{} are lower and dust clusters form early, starting at $\sim$ day 250 (SC13). The coagulation of dust clusters occurs at high gas density from a larger mass of clusters than in the standard case. This efficient coagulation ensues the production of large grains. Inspection of Figure \[fig4\] (middle panel) shows that for forsterite, the formation of large grains is already completed in zones 1B and 2 at day 600, with almost no variation of these large grain populations at later time. The peaks in the size distribution are from grains produced in zone 2, and correspond to 130 Å and 550 Å. For alumina, the distribution peaks around 120 Å and for carbon, the peak is located at 160 Å. Pure silicon and iron dust have similar distributions than for the standard case because both dust types form in zone 1A, which is deprived of inert gas elements.
The final dust mass produced in this ejecta is 0.055 [M$_{\odot}$]{}, which represents an increment of 57 % compared to the value for the standard case. Therefore, for an equal progenitor mass, the ejecta characterised by a low mass of produced [$^{56}$Ni]{} forms a larger quantity of dust grains as early as day 300 and with larger radii than in the case of a high [$^{56}$Ni]{} mass SN counterpart.
[l c c c]{}
Dust type & Mass ( [M$_{\odot}$]{}) & $x_d$ &$a_{peak}$ (Å)\
Forsterite & 2.5(-2) & 45.5&131; 543\
Carbon & 2.1(-2) &38.2 &162\
Alumina & 7.7(-3) & 14.0&117\
Pure Silicon & 4.4(-4) &0.80 &69\
Pure Magnesium & 4.3(-4) &0.78 &53\
Pure Iron & 1.4(-4) & 0.25&36\
Silicon Carbide & 8.8(-5) & 0.16&16\
Iron Sulphide & 2.8(-6) & 0.005& 26\
0.00
Total & 0.055 & 100&\
![Top: dust mass as a function of post-explosion time and dust types for the 19 [M$_{\odot}$]{} progenitor with homogeneous ejecta; Bottom: dust size distributions as function of dust type at day 2000. []{data-label="fig5"}](f51 "fig:"){width="\columnwidth"} ![Top: dust mass as a function of post-explosion time and dust types for the 19 [M$_{\odot}$]{} progenitor with homogeneous ejecta; Bottom: dust size distributions as function of dust type at day 2000. []{data-label="fig5"}](f52 "fig:"){width="\columnwidth"}
19 [M$_{\odot}$]{} model: homogenous and clumpy ejecta {#19ms}
------------------------------------------------------
For the SN originating from the explosion of a 19 [M$_{\odot}$]{} stellar progenitor, we consider two cases: 1) a homogeneous, stratified ejecta, and 2) a clumpy, stratified ejecta whose model is given in Table \[tab3\]. For the homogeneous case, we consider an initial gas density of $4.4 \times 10^{-12}$ g [cm$^{-3}$]{} for all mass zones at day 100. This value is a factor 2.5 smaller than that considered in SC13, and was arbitrary chosen to offer a better match between the modelled SiO masses for the clumpy case and those derived from observations - see § \[clump\]. As explained in § \[sec2\], the resulting gas number density per zone in the clumpy case is corrected by the volume filling factor derived by Jerkstrand et al. (2011).
### Homogeneous
Results on the dust mass and the grain sizes distributions for the various dust types synthesised in the ejecta are presented in Figure \[fig5\]. The final mass of dust formed stems from various dust production events in ejecta zones at different time, similar to what happened in our 15 [M$_{\odot}$]{} progenitor case. However, the slighlty lower number densities for each ejecta zone at day 100 lead to a delayed formation of forsterite clusters in zones 1B and 2, as seen in Figure \[fig5\], compared to the results presented in Figure 10 of SC13 for a similar progenitor mass. Iron sulphide, pure iron and silicon dust first condense in zone 1A, followed by forsterite in zone 1B, forsterite, alumina and pure magnesium in zones $2 -4$, and finally, carbon and silicon carbide dust in zone 5. The condensation of dust clusters is efficient as 96 % of the clusters initially formed in the gas phase condense into dust grains. The final dust mass at day 2000 amounts to 0.032 [M$_{\odot}$]{}, and details on the various dust populations, their masses and peak radii are given in Table \[tab7\].
The homogeneous ejecta forms medium-size grains, with distribution peak sizes for alumina, and forsterite ranging between $30-70$ Å, while the synthesis of carbon dust results in a population of small grains, that peaks around 20 Å. The largest grains correspond to the forsterite grains formed in zone 1B before day 600, as these grains had more time to grow than the forsterite grain population produced in zone 2. The distribution resembles that for the 15 [M$_{\odot}$]{} progenitor, except for the smaller contribution from carbon dust to the total dust budget, as seen from Table \[tab7\].
### Clumpy {#clump}
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
[l c c c c c c c r c]{}\
Dust type & Zone 1A & Zone 1B & Zone 2 & Zone 3 & Zone 4 & Zone 5 & Total & $x_d$ & $a_{peak}$ ( Å)\
Forsterite & - & 2.3(-3)& 5.1(-3)& 9.1(-5)& 1.0(-4) & - & 7.6(-3) & 23.8 & 35, 63\
Alumina & - & - & 1.7(-2) & 3.9(-6) & 6.5(-5) & - & 1.7(-2) & 53.0 & 44\
Carbon & -& - & - & - & - &7.5(-3) & 7.5(-3) & 23.4 & 16\
Pure Magnesium & -& - & 1.7(-5) & 2.1(-7) & 5.4(-7)& - & 1.8(-5) & 0.06 & 8\
Pure Silicon & 6.2(-5)& - & - & -&- & - & 6.2(-5) & 0.19 & 18\
Pure Iron & 3.0(-5)& - & - & -&- & - & 3.0(-5) & 0.09 & 14\
Silicon Carbide & -& - & - & - & - & 3.7(-6) & 4.1(-6) & 0.01 & 7\
Iron Sulphide & 9.4(-8) & - & - & -&- & - & 9.4(-8) & 3(-4) & 8\
Total & 9.2(-5) & 2.3(-3) & 2.2(-2) & 9.5(-5) & 1.7(-4) & 7.5(-3)& 0.032 & 100 &\
\
Dust type & Zone 1A & Zone 1B & Zone 2 & Zone 3 & Zone 4 & Zone 5 & Total & $x_d$ & $a_{peak}$ ( Å)\
Forsterite & - & 2.4(-3) & 4.8(-2) & 5.4(-4) & 1.6(-3) & 3.0(-4) & 5.3(-2) & 38.4 & 77, 661, 3170\
Alumina & - & 4.0(-4) & 1.7(-2) & 1.3(-5) & 6.3(-5) & - & 1.8(-2) & 13.0 & 211, 562\
Carbon & -& - & - & - & - &7.3(-3) & 7.3(-3) & 5.3 & 435\
Pure Magnesium & -& - & 2.1(-2) & 1.2(-3) & 4.0(-3)& - & 2.6(-2) & 18.8 & 252\
Pure Silicon & 1.7(-2)& - & - & -&- & - & 1.7(-2) & 12.3 & 1066\
Pure Iron & 1.7(-2)& - & - & -&- & - & 1.7(-2) &12.3& 1003\
Silicon Carbide & -& - & - & - & - & 1.7(-5) & 1.7(-5) & 0.01& 29\
Iron Sulphide & 1.1(-4) & - & - & -&- & - & 1.1(-4) & 0.08 & 57, 334\
Total & 3.4(-2) & 2.8(-3) & 8.6(-2) & 1.8(-3) & 5.6(-3) & 7.6(-3)& 0.138 & 100 &\
Results for the clumpy ejecta are presented in Figure \[fig6\], where the SiO mass and the dust mass variation versus post-explosion time, the grains size distributions, and the grain mass distributions are shown.
SC13 showed that SiO was a direct tracer of silicate dust condensation in the ejecta of supernovae. Here the decrease in SiO mass with time indeed agrees well with observational data for SNe with high progenitor masses, e.g., SN2004et (although the mass for this SN is still debated and may be lower than 20 [M$_{\odot}$]{}, e.g., Jerkstrand et al. 2012), and SN1987A. Compared to the homogeneous ejecta, forsterite formation in Zone 1B occurs at early time and higher gas density. Furthermore, Zone 1B has the lowest filling factor among all the zones, which causes zone 1B to be the densest zone in the oxygen-rich core. This leads to the formation of a population of large grains, which peaks at $\sim 0.4$ [$\mu$m]{} at day 2000, as seen in the dust size distributions (bottom-left panel). Forsterite also efficiently forms in zone 2 around day 400, and increases the total forsterite mass. This formation event is reflected in the size distribution by a population of large grains peaking at 660 Å.
A similar scenario applies to alumina which forms essentially in zone 1B at $\sim$ day 450 and zone 2 at $\sim$ day 700. The alumina size distribution thus shows a peak around $\sim$ 240 Å, which corresponds to the grains formed in zone 2, and a tail of large grains produced in zone 1B and with a size over 0.1 [$\mu$m]{}. The formation of three populations of grains of pure silicon, pure iron, and iron sulphide pertains to zone 1A. Despite their late formation at day 650, pure silicon and Fe grain condensation is boosted compared to the homogeneous case because of the higher gas density in zone 1A. This leads to grain populations that peak around 0.12 [$\mu$m]{} for both pure silicon and iron dust. This peak size is almost two orders of magnitude greater than in the homogeneous case. Grains of FeS condense at day 200, and thus grow over time to relatively large sizes ($\sim$ 400 Å). However the final FeS dust mass remains low owing to a modest mass of FeS clusters that form in zone 1A.
Finally, the outermost zone 5 forms carbon and SiC grains. The carbon grains form at day 870, a much earlier epoch than for the homogeneous case, where carbon forms at day 1350. This results in a size distribution dominated by larger grain sizes peaking at 520 Å, while the carbon size distribution peaks at 70 Å for the homogeneous ejecta.
Dust condensation is strongly sensitive to the ejecta number density. A clumpy ejecta leads to the synthesis of all types of dust at early epochs, and increases the final dust mass, compared to the homogeneous case. The dust components can be further categorised in two groups. Clusters of alumina and amorphous carbon undergo a fast and efficient formation, but their final mass is controlled by the availability of atomic Al and C, respectively, in the production region. Therefore, the final mass of these dust populations is not significantly sensitive to any change in ejecta parameters, as seen from Table \[tab7\], for the typical ejecta gas parameters derived from SN explosion models. Conversely, the silicate and pure metal dust mass is strongly dependent of the gas conditions. Silicate clusters are formed through a complex silicon-based chemistry, whose efficiency strongly depends on the gas-phase parameters. The mass of silicate grains thus varies by a factor of 10 to 20 from the homogeneous to the clumpy case. As for pure metallic grains, the clusters essentially form in the gas phase through trimolecular association reactions that include the bath gas as a collider. These processes become very effective as the gas number density is raised in the clumpy ejecta. The metallic grain masses thus increase by $\sim$ 3 orders of magnitude compared to the homogeneous case. The mass distribution as a function of grain size is shown in Figure \[fig6\] (bottom-right panel), and the masses for the various dust components are summarised in Table \[tab7\], along with the $a_{peak}$ radii that characterise each grain components. The distribution is skewed towards large grain sizes comprised between 100 Å and 1 [$\mu$m]{}, which contain most of the mass of forsterite, pure iron and silicon, alumina, and carbon grains. As seen from Table \[tab7\], the alumina and carbon mass are similar to those for the homogeneous case as their final masses are controlled by the initial abundances of Al in the ejecta and the C/O ratio of zone 5, respectively. Of interest is the strong contribution of $\sim 0.1$ [$\mu$m]{} grains of pure iron and silicon to the total dust mass, while these grains have a minor contribution to the size and mass distributions for the homogeneous case. The clumpy ejecta of the 19 [M$_{\odot}$]{} progenitor forms a total dust mass of $0.14$ [M$_{\odot}$]{}, which is greater than the dust mass derived for the homogeneous case by a factor of four.
![Mass of CO$_2$ formed in the clumpy ejecta of the 19 [M$_{\odot}$]{} progenitor as a function of post-explosion time and ejecta zone.[]{data-label="fig7"}](f71){width="\columnwidth"}
The high gas densities of the clumpy model enhance the efficiency of the gas-phase chemistry and boost the formation of specific molecules in the ejecta. This is the case for CO$_2$, as shown in Figure \[fig7\], where the mass as a function of post-explosion time and zoning is shown. The molecule forms at day 500 in zone 3, and grows in mass from a second formation event in zone 2 starting at day 800. At day 2000, the total CO$_2$ mass reaches $ \sim 1 \times 10^{-2}$ [M$_{\odot}$]{}. The molecules N$_2$ and NO also form with a mass of $7.5 \times 10^{-3}$ [M$_{\odot}$]{} and $1.1 \times 10^{-5}$ [M$_{\odot}$]{}, respectively, at day 2000. In the homogeneous case, CO$_2$ forms with a low mass of $3\times 10^{-4}$ [M$_{\odot}$]{}. A large reservoir of CO molecules is produced in the clumpy case, where the CO mass reaches $\sim 0.06$ [M$_{\odot}$]{} $0.1$ [M$_{\odot}$]{}, and $0.3$ [M$_{\odot}$]{} at day 2000 in zones 3, 2, and 4, respectively. A smaller mass of CO ($6\times 10^{-3}$ [M$_{\odot}$]{}) is produced in the carbon-rich zone 5, where carbon dust condenses. Thus, CO cannot be considered as a carbon dust tracer in SN ejecta, as already pointed out by SC13. The large CO reservoir in zones 2 and 3 provides an effective formation channel for CO$_2$ via CO recombination. Carbon dioxide is thus a tracer of clumpiness in the O-rich core of the 19 [M$_{\odot}$]{} ejecta.
{width="19cm"}
### Elemental depletion
We calculate the depletion of elements in molecules, dust clusters and grains to assess the effect of ejecta clumps on the depletion fraction. Results for both homogeneous and clumpy SN ejecta with the 19 [M$_{\odot}$]{} stellar progenitor are shown in Figure \[fig8\] for the various elements of interest.
In the homogeneous case, the fraction of elements staying in atomic form is high for oxygen and sulphur, and almost 100 % for both magnesium and iron. Carbon is essentially depleted in CO for both homogeneous and clumpy ejecta, with a few % going to amorphous carbon. The oxygen depletion changes drastically in the clumpy ejecta with $\sim 70$ % of oxygen in the form of O$_2$, and a larger depletion in SO and forsterite. While more than 96 % of magnesium is atomic for the homogeneous ejecta, almost 50 % of magnesium is trapped in forsterite and pure Mg when the ejecta is clumpy. For both cases, aluminium is heavily depleted in alumina, with a fraction exceeding 93 %, and the rest left in atomic form. Sulphur is not depleted in metal sulphides for both cases as the amount of FeS formed in both ejecta is small (see Table \[tab7\]). Sulphur is depleted in the molecules SO and SiS, and the depletion becomes total for the clumpy case. Finally, clumpiness has a strong effect on iron. In the homogeneous ejecta, almost all iron is in atomic form since the mass of formed FeS is very small. However, in the clumpy case, 56 % of iron is in large grains of pure iron, as seen in Figure \[fig6\].
For both cases, we see that a high fraction of Si atoms is trapped in clusters which enter the formation process of silica, SiO$_2$. These clusters labelled, Si$_x$O$_y$, will not all be included in the final silica mass, as their growth process is controlled by the amount of available SiO. As mentioned in § \[sec3\], we have not studied the nucleation and condensation of silica in this study, but an assessment of the silica mass based on the SiO mass at day 300 or the assumption that all Si$_x$O$_y$ clusters turn into silica leads to values ranging between $10^{-5}$ [M$_{\odot}$]{} and $10^{-2}$ [M$_{\odot}$]{}, respectively, at day 2000.
More generally, we see that a clumpy ejecta depletes almost all elements in molecules, dust clusters and grains, except for magnesium and iron, which retain some high mass fraction in atomic form. However, in contrast with the homogeneous ejecta, clumpiness also favours the depletion of $\sim$ 46 % of the magnesium mass in pure magnesium and silicate grains, whereas $\sim$ 56 % of the iron mass is depleted in pure iron grains.
Summary and discussion {#sec5}
======================
We have presented an exhaustive model of dust synthesis in the homogeneous and clumpy ejecta of Type II-P SNe, where the gas-phase chemistry, including the formation of dust clusters (i.e., nucleation phase), is coupled to the coagulation and coalescence of these clusters into dust grains (i.e., condensation phase). Our findings are summarised below.
- [As soon as dust clusters form from the gas phase, the coalescence and coagulation of these clusters is very efficient at growing dust grains with an appreciable size for all models. The effective condensation of grains confirms the crucial role of the nucleation phase as a bottleneck to dust production, as already highlighted by SC13. The nucleation of clusters indeed controls the chemical type and the final mass of the dust produced in SN ejecta.]{}
- [All SN ejecta form three prevalent populations of dust: silicates, carbon and alumina. Homogeneous ejecta form medium-size grains with size distributions peaking between $50$ Å and $100$ Å for these three dust components. However, the prevalence of a dust type depends on the elemental composition of the stellar progenitor before explosion, hence the progenitor mass. A low-mass progenitor predominantly forms carbon dust while a high-mass progenitor forms silicates and alumina. ]{}
- [The grain size distributions for all cases are different from a power-law distribution or the MRN power-law size distribution representative of interstellar dust. Specifically, all our derived size distributions are skewed towards large grains, with several distribution peaks that correspond to dust condensation events at different epochs and positions in the ejecta zones.]{}
- [A clumpy ejecta and/or a low [$^{56}$Ni]{} mass favour the formation of dust clusters and grains at early post-explosion time in the various zones. These grains have thus more time to grow in the nebular phase, and lead to the formation of large grains with size over $0.1$ [$\mu$m]{}. Furthermore, clumps boost the formation of pure metallic grains, including iron and silicon dust in the innermost He-core zones, and magnesium grains in the oxygen-rich zones. These metallic grains have individual masses comparable to that of alumina, and when summed over the three dust populations, their total mass is comparable to that of silicates. Iron sulphide grains represent a minor dust component for all ejecta cases. ]{}
{width="19cm"}
The modelling of the IR emission flux due to warm dust in Type II SN ejecta usually involves the use of a MRN power-law size distribution with index $\alpha \sim -3.5$ for dust grains. Two types of dust are assumed, either carbon or silicates, or a mixture of both, and dust masses are derived (e.g., [@sug06; @erc07; @ko09]). As our results illustrate, the final size distributions obtained for dust grains that condense in homogenous and clumpy ejecta do not follow such a power-law distribution. Furthermore, the dust chemical composition is much more complex than those usually assumed, is extremely time-dependent, and include several dust components with grain size distributions that reflect the He-core mass position and time at which dust forms. So fitting the dust contribution to the spectral energy distribution at different times requires the use of different dust chemical compositions and size distributions. This differs strongly from assuming a simple MRN distribution for one or two dust components.
Our results show that the synthesis of dust highly depends on two parameters, the mass of [$^{56}$Ni]{} produced in the explosion and the ejecta gas density. A low [$^{56}$Ni]{} mass favours the synthesis of dust at early post-explosion time in the various ejecta zones predominantly forming silicates, metal oxide, and carbon. For example, in SN 2003gd, which has a low-mass progenitor and a low [$^{56}$Ni]{} mass, a mid-IR excess, along with asymmetric blue-shifted emission lines and an increase in optical extinction, was observed as evidence for dust formation in the ejecta, as early as day 250 ([@hen05; @sug06]).
The dependence on gas density is even stronger and is well illustrated by our clumpy ejecta case. A clumpy ejecta favours the dust formation in all ejecta zones at early times, and result in several population of large grains a few years after explosion. Indeed, dust forms in the dense ejecta zones, and the early-formed grains have then time to grow to fairly large grains in the ejecta. A large fraction of these large grains will survive the non-thermal sputtering induced by the reverse shock, sheltered in the dense ejecta clumps, and the thermal sputtering in the hot, inter-clump medium once the clumps are disrupted during the SNR phase (Biscaro & Cherchneff, in preparation). Some graphite and silicate pre-solar grains found in meteorites have a SN origin ([@zin07; @hop10]). According to the present results, they may be identified as the largest grains of silicates and carbon that form in the dense ejecta clumps of Type II-P SNe. However, our results cannot explain the elemental mixing between ejecta zones as derived from the isotopic ratio analysis of these pre-solar grains, because our models use stratified ejecta with no leakage between zones. Clues on elemental mixing from different ejecta zones are provided by 3-D simulations of SN explosion from which the chemical composition of clumps are derived a few hours after outburst (e.g., [@ham10]), and further studies will consider these outputs as initial conditions for the physico-chemical model of clumpy SN ejecta.
Most important is the fact that grains respond differently to gas density enhancement. Silicate dust production is extremely dependent on gas density because the nucleation phase of this type of grains is characterised by complex chemical pathways, which are density-dependent. The nucleation phase thus controls the final amount of silicate dust mass that forms in the SN explosion, as shown by SC13. Other tracers of density increase in the ejecta of Type II-P SNe are pure metal dust, such as silicon, magnesium, and iron grains, and new molecules that only form with high abundance and mass in the clumpy ejecta, i.e., CO$_2$.
Conversely, the production of alumina and carbon is not too responsive to density increase and is limited by the availability of atomic carbon and aluminium in the ejecta zones where these specific dust grains form. For example, according to Figure \[fig8\] and for all SN progenitor masses, atomic carbon is essentially depleted in CO molecules, which primarily form in the oxygen-rich zone labelled 4, and in carbon dust produced in the outermost, C-rich zone 5. The efficiency at forming carbon dust depends on the C/O ratio of this ejecta zone. For the 15 [M$_{\odot}$]{} progenitor, the C/O ratio of zone 5 is high ($\sim 21$), and thus 78 % of the carbon mass yield gets into carbon dust while $\sim$ 20 % stay in the form of carbon chains in the gas phase. For the 19 [M$_{\odot}$]{} progenitor, the C/O is small ($\sim 4$), and only 58 % of the initial carbon mass yield gets into grains for both homogeneous and clumpy ejecta. The final mass of carbon dust formed in Type II-P SN ejecta is thus limited by the carbon mass yield of the outermost, carbon-rich, ejecta zone, and not by the total carbon yield of the ejected material, as it is often assumed (e.g., Matsuura et al. 2011, 2014). Although carbon is one important component of SN dust $\sim$ 5 years post-outburst, silicates, alumina, and pure metals are also important dust components, especially when ejecta clumpiness is taken into account. Actually, a satisfactory fit of the Herschel data on SN1987A requires a population of large ($a=0.5$ [$\mu$m]{}) iron grains ([@mat11]). Hence, the analysis of the submm flux emitted by cool, thermal dust in SN remnants must consider these various dust components as a whole to properly assess the mass of dust formed in the ejecta.
We conclude that the synthesis of dust in Type II-P is a multi-parameter-dependent process. To illustrate this point, our total modelled dust masses for various SN progenitors, the dust masses derived from mid-IR observation of several Type II-P SNe, and the dust masses assessed from submm data of SN remnants are plotted in Figure \[fig9\]. The dust masses derived before day 1000 from mid-IR data span a high value range comprised between $10^{-6}$ [M$_{\odot}$]{} and $10^{-2}$ [M$_{\odot}$]{}. This large spread in dust mass reflects the difference in SN progenitor mass, the mass of [$^{56}$Ni]{}, the likely presence of ejecta clumps, and the chemical composition of the dust that forms. Our modelled dust mass values for various progenitors, [$^{56}$Ni]{} mass, and clumpy/non clumpy ejecta models well reproduce this large spread, and point to the fact that dust synthesis in SNe depends on several parameters, and cannot be described by assuming a simple dust composition, size distribution, and dust temperature. As already pointed out by SC13, our results indicate a gradual increase of dust production and growth over a time span of $\sim 3-5$ years after explosion, to reach dust masses in the range $0.03-0.2$ [M$_{\odot}$]{}. This range agrees well with the latest submm dust masses derived from Herschel and ALMA data for SN1987A and other SN remnants. The trend of a gradual growth of dust grains over $\sim 3-5$ years is in contrast with recent radiative transfer study of SN1987A by Wesson et al. (2015), who conclude the major growth of dust grains has taken place between day 1200 and day 9200, and that the grains are very large ($a\sim 3-5$ [$\mu$m]{}). They argue that growth by atom accretion on the grain surface is not effective enough to form such large grains and propose coagulation instead. However, coagulation is usually characterised by larger time scales than accretion and is effective at high gas densities, as shown in this study. Therefore, the mechanism triggering dust growth at very late post-explosion time and low gas density has yet to be identified. The dust mass values derived in the present study are still well below the 1 [M$_{\odot}$]{} required to account for the observed dust mass in J1148+5251 at z=6.4. However primeval SNe are expected to be more massive than their local counterparts considered in this study ([@hir14]), and may produce larger dust masses ([@noz03; @sch04; @cher10]).
The presolar grains of SN origin found in meteorites may have formed in massive and dense Type II-P SNe akin to our clumpy 19 [M$_{\odot}$]{} surrogate model for SN1987A, as these large dust grains may survive the remnant phase sheltered in their dense, clumpy cradle. Out of the twelve Type II-P SNe plotted in Figure \[fig9\], eight SNe have masses of warm dust detected before 1100 days that agree with our homogeneous SN models, and for which rather small grains are synthesised. These grains may not survive the remnant phase. This study points to the fact that while some dense Type II-P SNe may contribute to providing dust to the Interstellar Medium and the solar system, many Type II-P SNe will probably be modest contributors to the dust budget of local galaxies.
The authors thank the anonymous referee for useful comments that helped improving the manuscript, John Plane for providing the initial version of the condensation code, and Rubina Kotak and Stefan Bromley for stimulating discussions. A.S. acknowledges support from the Swiss National Science Foundation through the subside 20GN21?128950 linked to the CoDustMas network of the European Science Foundation Eurogenesis programme.
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[l l l l l l l l l l l l l l]{} & & & & & & & && & && &\
\
1A & 35.5& 5.9(-2) & 0 & 1.5(-7) & 3.3(-6) & 0 & 1.4(-5) & 2.0(-5) & 3.2(-2) & 2.0(-2) & 4.0(-3) & 1.7(-2) & 2.8(-4)\
1B & 20.9& 2.1(-3) & 0 & 6.9(-6) & 4.4(-2) & 1.0(-5) & 3.9(-4) & 5.0(-5) & 3.1(-2) & 1.3(-2) & 7.4(-4) & 1.3(-4) & 1.3(-7)\
2 & 17.2 & 5.5(-3) & 0 & 9.3(-4) & 0.23 & 1.5(-2) & 1.6(-2) & 2.1(-3) & 2.1(-2) & 2.5(-3) & 4.1(-5) & 2.3(-5) & 0\
3 & 17.1 & 1.6(-2) & 0 & 2.8(-3) & 0.24 & 7.8(-2) & 1.8(-2) & 1.9(-3) & 1.8(-3) & 6.8(-5) & 1.7(-5) & 2.3(-5) & 0\
4A &15.0 & 0.37 & 6.1(-6) & 4.0(-2) & 0.15 & 3.0(-3) & 1.6(-4) & 2.3(-4) & 7.1(-5) & 3.5(-5) & 9.6(-6) & 2.2(-5) & 0\
4B & 10.7 & 0.74 & 3.1(-2) & 6.2(-2) & 0.11 & 1.4(-2) & 7.1(-4) & 1.9(-5) & 1.1(-4) & 4.4(-5) & 8.6(-6) & 4.0(-5) & 0\
5 & 4.1 & 21.3 & 0.71 & 2.7(-2) & 1.7(-3) & 1.2(-3) & 3.9(-4) & 5.3(-5) & 4.8(-4) & 2.9(-4) & 1.2(-5) & 8.4(-4) & 0\
6 & 4.1 & 1.2 & 0.34 & 9.1(-5) & 9.6(-5) & 5.5(-4) & 1.8(-4) & 2.4(-5) & 2.3(-4) & 1.4(-4) & 5.3(-6) & 4.1(-4) & 0\
& 1.1& 0.13& 0.78 &0.11 & 3.6(-2) & 4.4(-3) & 8.7(-2) & 3.6(-2) &4.8(-3) & 1.8(-2)& 2.8(-4)\
\
1A & 35.3 & 0.16 & 1.4(-6) & 8.1(-8) & 6.9(-7) & 0 & 1.7(-5) & 2.5(-5) & 3.8(-2) & 2.3(-2) & 4.5(-3) & 2.5(-2) & 3.3(-4)\
1B & 22.5& 1.3(-3) & 0 & 1.2(-4) & 0.12 & 1.1(-4) & 8.8(-4) & 2.2(-4) & 9.9(-2) & 5.6(-2) & 1.5(-2) & 3.1(-3) & 6.2(-6)\
2 & 16.9 & 6.5(-2) & 0 & 5.9(-2) & 1.2 & 0.3 & 8.4 (-2) & 9.1(-3) & 1.5(-2) & 1.3(-3) & 1.2(-4) & 7.5(-4) & 0\
3 & 15.1 & 0.4 & 0 & 2.9(-2) & 0.11 & 2.8(-3) & 2.0(-3) & 1.6(-5) & 1.1(-4) & 3.2(-5) & 8.9(-6) & 7.8(-5) & 0\
4 & 10.3 & 0.6 & 7.7(-2) & 0.13 & 0.26 & 1.3(-2) & 5.4(-3) & 5.3(-5) & 3.8(-4) & 1.4(-4) & 3.5(-5) & 3.8(-4) & 0\
5 & 4.1 & 3.9 & 0.74 & 1.3(-2) & 4.3(-3) & 1.1(-2) & 5.0(-4) & 6.0(-5) & 5.5(-4) & 3.2(-4) & 7.0(-5) & 9.8(-4) & 0\
6 & 4.1 & 1.8 & 0.35 & 1.7(-4) & 9.2(-5) & 5.7(-4) & 2.3(-4) & 3.2(-5) & 2.6(-4) & 1.5(-4) & 3.3(-5) & 4.6(-4) & 0\
& 1.2& 0.23 &1.69 &0.33 &9.3(-2) &9.5(-3) &0.15 & 8.1(-2)&2.0(-2) & 3.7(-2)& 3.4(-4)\
\
[ c l l l l l l l l l l l l l l ]{}\
Zones & & & & & & &\
Day & T & n$_{gas}$ & T & n$_{gas}$ & T & n$_{gas}$ & T & n$_{gas}$& T & n$_{gas}$ & T & n$_{gas}$ & T & n$_{gas}$\
100 & 12000 & 1.8(11) & 11600 & 3.1(11) & 10400 & 3.7(11) & 8779 & 3.8(11) & 7980 & 4.3(11) & 7580 & 6.1(11) & 6490 & 1.6(12)\
300 & 3006 & 6.7(9) & 2906 & 1.1(10) & 2605 & 1.4(10) & 2199 & 1.4(10) & 1998 & 1.6(10) & 1899 & 2.3(10) & 1626 & 5.9(10)\
600 & 1255 & 8.3(8) & 1213 & 1.4(9) & 1088 & 1.7(9) & 918 & 1.8(9) & 835 & 2.0(9) & 793 & 2.8(9) & 679 & 7.4(9)\
900 & 753 & 2.5(8) & 728 & 4.3(8) & 653 & 5.1(8)& 551 & 5.2(8) & 501 & 5.9(8) & 476 & 8.4(8) & 407 & 2.2(9)\
1200 & 524 & 1.0(8) & 507 & 1.8(8) & 454 & 2.1(8)& 383 & 2.2(8) & 349 & 2.5(8) & 331 & 3.5(8) & 283 & 9.3(8)\
1500 & 396 & 5.3(7) & 382 & 9.2(7) & 343 & 1.1(8)& 289 & 1.1(8) & 263 & 1.3(8) & 250 & 1.8(8) & 214 & 4.7(8)\
2000 & 275 & 2.3(7) & 266 & 3.9(7) & 239 & 4.6(7) & 201 & 4.8(7) & 183 & 5.4(7) & 174 & 7.6(7) & 149 & 2.0(8)\
\
Zones & & & & & & &\
Day& T & n$_{gas}$ & T & n$_{gas}$ & T & n$_{gas}$ & T & n$_{gas}$& T & n$_{gas}$ & T & n$_{gas}$ & T & n$_{gas}$\
100 & 12400 & 7.2(10) & 12000 & 1.2(11) & 9980 & 1.5(11) & 7190 & 1.7(11) & 6390 & 2.5(11) & 6000 & 6.3(11) & 5900 & 6.4(11)\
300 & 3106 & 2.7(9) & 3006 & 4.3(9) & 2500 & 5.7(9) & 1801 & 6.4(9) & 1601 & 9.3(9) & 1503 & 2.3(10) & 1478 & 2.4(10)\
600 & 1297 & 3.3(8) & 1255 & 5.4(8) & 1044 & 7.1(8) & 752 & 8.0(8) & 668 & 1.2(9) & 628 & 2.9(9) & 617 & 3.0(9)\
900 & 778 & 9.8(7) & 753 & 1.6(8) & 626 & 2.1(8)& 451 & 2.4(8) & 401 & 3.5(8) & 377 & 8.7(8) &370 & 8.8(8)\
1200 & 542 & 4.1(7) & 524 & 6.7(7) & 436 & 8.9(7)& 314 & 1.0(8) & 279 & 1.5(8) & 262 & 3.7(8) & 258 & 3.7(8)\
1500 & 409 & 2.1(7) & 396 & 3.4(7) & 329 & 4.6(7) & 237 & 5.1(7) & 211 & 7.5(7) & 198 & 1.9(8) & 195 & 1.9(8)\
2000 & 285 & 9.0(6) & 275 & 1.5(7) & 229 & 1.9(7) & 165 & 2.2(7) & 147 & 3.2(7) & 138 & 7.9(7) & 135 & 8.0(7)\
\
$a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$
--------- ----------- --------- ----------- --------- ----------- --------- -----------
3.33 5.76(-01) 3.45 6.20(-01) 3.93 2.43(-02) 2.46 6.23(-01)
4.05 1.29(-01) 4.19 1.38(-01) 4.78 3.35(-05) 2.99 1.42(-01)
4.93 8.55(-02) 5.10 8.84(-02) 5.81 1.01(-05) 3.63 9.13(-02)
6.00 5.44(-02) 6.20 5.10(-02) 7.07 5.48(-07) 4.42 5.21(-02)
7.30 4.14(-02) 7.55 3.26(-02) 8.60 5.55(-08) 5.38 3.17(-02)
8.88 3.51(-02) 9.18 2.25(-02) 10.46 3.41(-09) 6.54 1.98(-02)
10.80 2.95(-02) 11.17 1.63(-02) 12.72 8.30(-10) 7.96 1.27(-02)
13.14 2.14(-02) 13.59 1.13(-02) 15.48 6.63(-08) 9.68 8.24(-03)
15.98 1.22(-02) 16.53 6.91(-03) 18.83 2.33(-06) 11.77 5.41(-03)
19.44 6.00(-03) 20.10 3.81(-03) 22.90 3.78(-05) 14.32 3.56(-03)
23.64 3.05(-03) 24.46 2.13(-03) 27.86 3.33(-04) 17.42 2.33(-03)
28.76 1.64(-03) 29.75 1.21(-03) 33.89 1.82(-03) 21.19 1.50(-03)
34.99 9.89(-04) 36.19 6.49(-04) 41.23 6.83(-03) 25.78 9.49(-04)
42.56 8.15(-04) 44.02 4.16(-04) 50.15 1.92(-02) 31.36 6.45(-04)
51.77 7.75(-04) 53.55 4.80(-04) 61.00 4.30(-02) 38.14 5.82(-04)
62.97 6.36(-04) 65.14 6.95(-04) 74.21 8.09(-02) 46.40 6.97(-04)
76.60 4.07(-04) 79.24 9.08(-04) 90.27 1.31(-01) 56.44 8.65(-04)
93.18 2.49(-04) 96.39 9.78(-04) 109.81 1.81(-01) 68.66 9.38(-04)
113.35 2.12(-04) 117.25 8.16(-04) 133.57 2.04(-01) 83.52 7.99(-04)
137.88 2.15(-04) 142.62 4.98(-04) 162.48 1.73(-01) 101.59 4.76(-04)
167.73 1.94(-04) 173.49 2.16(-04) 197.65 9.75(-02) 123.58 1.77(-04)
204.03 1.36(-04) 211.04 6.33(-05) 240.43 3.17(-02) 150.33 3.73(-05)
248.19 6.58(-05) 256.72 1.15(-05) 292.47 5.29(-03) 182.87 4.27(-06)
301.91 1.92(-05) 312.29 1.21(-06) 355.77 4.27(-04) 222.45 2.75(-07)
367.26 3.13(-06) 379.88 7.28(-08) 432.77 1.71(-05) 270.60 1.10(-08)
446.74 2.77(-07) 462.10 2.68(-09) 526.44 3.76(-07) 329.16 3.06(-10)
543.44 1.41(-08) 562.11 6.30(-11) 640.39 5.03(-09) 400.41 6.04(-12)
661.06 4.41(-10) 683.78 9.99(-13) 778.99 4.36(-11) 487.07 8.34(-14)
804.14 9.02(-12) 831.78 1.09(-14) 947.60 2.54(-13) 592.49 8.21(-16)
978.19 1.25(-13) 1011.80 8.47(-17) 1152.70 1.02(-15) 720.73 5.85(-18)
1189.90 1.21(-15) 1230.80 4.69(-19) 1402.20 2.87(-18) 876.73 3.05(-20)
1447.50 8.32(-18) 1497.20 1.89(-21) 1705.70 5.76(-21) 1066.50 1.17(-22)
1760.70 4.10(-20) 1821.30 5.56(-24) 2074.90 8.36(-24) 1297.30 3.36(-25)
2141.80 1.48(-22) 2215.40 1.21(-26) 2523.90 8.85(-27) 1578.10 7.20(-28)
$a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$
--------- ----------- --------- ----------- --------- ----------- --------- -----------
2.28 6.05(-01) 2.81 6.11(-01) 3.03 5.45(-01) 2.70 9.45(-03)
2.78 1.42(-01) 3.42 1.42(-01) 3.68 1.73(-01) 3.29 1.51(-02)
3.38 9.31(-02) 4.16 9.21(-02) 4.48 1.23(-01) 4.00 3.73(-02)
4.11 5.49(-02) 5.06 5.33(-02) 5.45 7.00(-02) 4.87 7.27(-02)
5.00 3.46(-02) 6.15 3.28(-02) 6.63 3.35(-02) 5.92 1.22(-01)
6.08 2.21(-02) 7.49 2.08(-02) 8.06 1.27(-02) 7.20 1.75(-01)
7.40 1.42(-02) 9.11 1.34(-02) 9.81 5.12(-03) 8.76 2.08(-01)
9.00 9.44(-03) 11.08 8.63(-03) 11.93 4.18(-03) 10.66 1.90(-01)
10.95 6.42(-03) 13.48 5.52(-03) 14.52 5.42(-03) 12.96 1.18(-01)
13.31 4.38(-03) 16.39 3.50(-03) 17.66 7.02(-03) 15.77 4.39(-02)
16.20 3.00(-03) 19.94 2.49(-03) 21.48 7.91(-03) 19.18 8.56(-03)
19.70 2.26(-03) 24.26 2.41(-03) 26.13 7.04(-03) 23.33 8.11(-04)
23.97 2.10(-03) 29.51 2.85(-03) 31.79 4.43(-03) 28.38 3.76(-05)
29.15 2.22(-03) 35.89 3.22(-03) 38.67 1.75(-03) 34.53 9.31(-07)
35.46 2.16(-03) 43.66 2.98(-03) 47.03 3.88(-04) 42.00 1.38(-08)
43.14 1.66(-03) 53.11 2.01(-03) 57.21 4.62(-05) 51.09 1.30(-10)
52.47 8.78(-04) 64.61 8.71(-04) 69.60 3.01(-06) 62.15 8.22(-13)
63.83 2.90(-04) 78.59 2.18(-04) 84.66 1.16(-07) 75.60 3.55(-15)
77.65 5.54(-05) 95.60 2.95(-05) 102.99 2.85(-09) 91.96 1.07(-17)
94.45 5.93(-06) 116.30 2.17(-06) 125.28 4.74(-11) 111.87 2.31(-20)
114.89 3.67(-07) 141.47 9.24(-08) 152.39 5.50(-13) 136.08 3.57(-23)
139.76 1.40(-08) 172.09 2.47(-09) 185.37 4.56(-15) 165.53 4.01(-26)
170.01 3.48(-10) 209.33 4.38(-11) 225.50 2.76(-17) 201.36 3.30(-29)
206.81 5.92(-12) 254.64 5.34(-13) 274.30 1.26(-19) 244.95 2.01(-32)
251.57 7.04(-14) 309.75 4.60(-15) 333.67 4.46(-22) 297.96 9.09(-36)
306.02 5.97(-16) 376.80 2.85(-17) 405.89 1.24(-24) 362.45 2.78(-39)
372.26 3.68(-18) 458.35 1.31(-19) 493.75 2.77(-27) 440.90 0.00(00)
$a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$
---------- ----------- ---------- ----------- ---------- ----------- ---------- -----------
3.33 6.55(-01) 3.45 6.55(-01) 3.93 8.85(-02) 2.46 7.16(-01)
4.05 1.41(-01) 4.19 1.41(-01) 4.78 6.04(-06) 2.99 1.40(-01)
4.93 8.77(-02) 5.10 8.82(-02) 5.81 7.10(-07) 3.63 7.86(-02)
6.00 4.73(-02) 6.20 4.80(-02) 7.07 7.41(-11) 4.42 3.60(-02)
7.30 2.69(-02) 7.55 2.74(-02) 8.60 2.14(-13) 5.38 1.66(-02)
8.88 1.55(-02) 9.18 1.58(-02) 10.46 6.12(-14) 6.54 7.36(-03)
10.80 9.00(-03) 11.17 9.33(-03) 12.72 4.28(-14) 7.96 3.15(-03)
13.14 5.32(-03) 13.59 5.62(-03) 15.48 2.96(-14) 9.68 1.33(-03)
15.98 3.22(-03) 16.53 3.46(-03) 18.83 1.30(-14) 11.77 5.67(-04)
19.44 2.03(-03) 20.10 2.14(-03) 22.90 3.02(-15) 14.32 2.42(-04)
23.64 1.35(-03) 24.46 1.27(-03) 27.86 1.54(-14) 17.42 9.81(-05)
28.76 9.56(-04) 29.75 6.61(-04) 33.89 1.25(-11) 21.19 3.70(-05)
34.99 7.31(-04) 36.19 2.83(-04) 41.23 2.72(-09) 25.78 1.35(-05)
42.56 6.02(-04) 44.02 1.20(-04) 50.15 1.89(-07) 31.36 5.01(-06)
51.77 5.26(-04) 53.55 8.15(-05) 61.00 5.26(-06) 38.14 1.84(-06)
62.97 4.78(-04) 65.14 7.93(-05) 74.21 7.08(-05) 46.40 6.63(-07)
76.60 4.41(-04) 79.24 7.83(-05) 90.27 5.37(-04) 56.44 2.38(-07)
93.18 4.03(-04) 96.39 8.57(-05) 109.81 2.61(-03) 68.66 9.02(-08)
113.35 3.44(-04) 117.25 1.22(-04) 133.57 8.91(-03) 83.52 3.85(-08)
137.88 2.46(-04) 142.62 1.89(-04) 162.48 2.33(-02) 101.59 2.11(-08)
167.73 1.33(-04) 173.49 2.69(-04) 197.65 4.93(-02) 123.58 1.81(-08)
204.03 5.18(-05) 211.04 3.23(-04) 240.43 8.84(-02) 150.33 2.64(-08)
248.19 1.83(-05) 256.72 3.04(-04) 292.47 1.37(-01) 182.87 5.71(-08)
301.91 1.13(-05) 312.29 2.03(-04) 355.77 1.79(-01) 222.45 1.42(-07)
367.26 1.18(-05) 379.88 8.49(-05) 432.77 1.88(-01) 270.60 3.38(-07)
446.74 1.39(-05) 462.10 2.02(-05) 526.44 1.43(-01) 329.16 7.81(-07)
543.44 1.62(-05) 562.11 2.83(-06) 640.39 6.97(-02) 400.41 1.74(-06)
661.06 1.72(-05) 683.78 5.14(-07) 778.99 1.88(-02) 487.07 3.52(-06)
804.14 1.51(-05) 831.78 2.58(-07) 947.60 2.55(-03) 592.49 6.18(-06)
978.19 9.95(-06) 1011.80 1.18(-07) 1152.70 1.66(-04) 720.73 9.49(-06)
1189.90 4.43(-06) 1230.80 3.21(-08) 1402.20 5.53(-06) 876.73 1.26(-05)
1447.50 1.26(-06) 1497.20 4.80(-09) 1705.70 1.05(-07) 1066.50 1.40(-05)
1760.70 2.44(-07) 1821.30 3.93(-10) 2074.90 1.23(-09) 1297.30 1.20(-05)
2141.80 4.04(-08) 2215.40 1.86(-11) 2523.90 9.56(-12) 1578.10 7.12(-06)
2605.40 9.76(-09) 2695.00 5.54(-13) 3070.20 5.03(-14) 1919.70 2.59(-06)
3169.30 6.31(-09) 3278.30 1.10(-14) 3734.70 1.84(-16) 2335.20 5.23(-07)
3855.30 5.74(-09) 3987.80 1.50(-16) 4543.10 4.75(-19) 2840.60 5.60(-08)
4689.70 4.14(-09) 4850.90 1.45(-18) 5526.40 8.78(-22) 3455.40 3.28(-09)
5704.80 1.99(-09) 5900.90 1.00(-20) 6722.50 1.18(-24) 4203.30 1.15(-10)
6939.50 5.79(-10) 7178.00 5.08(-23) 8177.60 1.15(-27) 5113.10 2.61(-12)
8441.50 9.74(-11) 8731.70 1.90(-25) 9947.50 8.33(-31) 6219.80 4.07(-14)
10269.00 9.56(-12) 10622.00 5.31(-28) 12101.00 4.47(-34) 7566.00 4.50(-16)
12491.00 5.71(-13) 12920.00 1.12(-30) 14720.00 1.78(-37) 9203.60 3.61(-18)
15195.00 2.19(-14) 15717.00 1.80(-33) 17906.00 0.00(00) 11196.00 2.13(-20)
18484.00 5.61(-16) 19119.00 2.25(-36) 21781.00 0.00(00) 13619.00 9.39(-23)
22484.00 9.95(-18) 23257.00 2.20(-39) 26495.00 0.00(00) 16566.00 3.12(-25)
$a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$ $a$ (Å) $f(a)$
--------- ----------- ---------- ----------- ---------- ----------- ---------- -----------
2.28 6.37(-01) 2.81 6.99(-01) 3.03 6.56(-01) 2.70 2.00(-03)
2.78 1.41(-01) 3.42 1.38(-01) 3.68 1.40(-01) 3.29 6.88(-07)
3.38 8.95(-02) 4.16 7.97(-02) 4.48 8.63(-02) 4.00 2.31(-06)
4.11 5.00(-02) 5.06 3.88(-02) 5.45 4.61(-02) 4.87 3.60(-05)
5.00 2.99(-02) 6.15 2.01(-02) 6.63 2.61(-02) 5.92 3.23(-04)
6.08 1.84(-02) 7.49 1.09(-02) 8.06 1.50(-02) 7.20 1.79(-03)
7.40 1.15(-02) 9.11 6.16(-03) 9.81 8.67(-03) 8.76 6.78(-03)
9.00 7.35(-03) 11.08 3.43(-03) 11.93 5.06(-03) 10.66 1.92(-02)
10.95 4.75(-03) 13.48 1.81(-03) 14.52 3.00(-03) 12.96 4.33(-02)
13.31 3.12(-03) 16.39 9.07(-04) 17.66 1.85(-03) 15.77 8.17(-02)
16.20 2.08(-03) 19.94 4.49(-04) 21.48 1.25(-03) 19.18 1.32(-01)
19.70 1.39(-03) 24.26 2.22(-04) 26.13 1.04(-03) 23.33 1.84(-01)
23.97 9.42(-04) 29.51 1.07(-04) 31.79 1.19(-03) 28.38 2.09(-01)
29.15 6.46(-04) 35.89 4.92(-05) 38.67 1.60(-03) 34.53 1.79(-01)
35.46 4.48(-04) 43.66 2.12(-05) 47.03 2.01(-03) 42.00 1.02(-01)
43.14 3.12(-04) 53.11 8.75(-06) 57.21 2.09(-03) 51.09 3.35(-02)
52.47 2.15(-04) 64.61 3.56(-06) 69.60 1.63(-03) 62.15 5.68(-03)
63.83 1.43(-04) 78.59 1.46(-06) 84.66 8.50(-04) 75.60 4.65(-04)
77.65 9.02(-05) 95.60 6.24(-07) 102.99 2.65(-04) 91.96 1.89(-05)
94.45 5.78(-05) 116.30 3.06(-07) 125.28 4.84(-05) 111.87 4.20(-07)
114.89 4.81(-05) 141.47 2.10(-07) 152.39 1.14(-05) 136.08 5.66(-09)
139.76 5.74(-05) 172.09 2.34(-07) 185.37 1.36(-05) 165.53 4.95(-11)
170.01 7.58(-05) 209.33 3.43(-07) 225.50 2.15(-05) 201.36 2.90(-13)
206.81 9.11(-05) 254.64 6.16(-07) 274.30 3.02(-05) 244.95 1.17(-15)
251.57 9.20(-05) 309.75 1.37(-06) 333.67 3.57(-05) 297.96 3.32(-18)
306.02 7.19(-05) 376.80 3.07(-06) 405.89 3.36(-05) 362.45 6.73(-21)
372.26 3.99(-05) 458.35 6.05(-06) 493.75 2.26(-05) 440.90 9.82(-24)
452.83 1.53(-05) 557.56 1.03(-05) 600.61 9.68(-06) 536.33 1.05(-26)
550.84 5.11(-06) 678.24 1.52(-05) 730.61 2.40(-06) 652.42 8.18(-30)
670.06 2.07(-06) 825.04 1.91(-05) 888.74 3.28(-07) 793.62 4.74(-33)
815.09 7.70(-07) 1003.60 1.97(-05) 1081.10 2.50(-08) 965.40 2.05(-36)
991.51 1.79(-07) 1220.80 1.51(-05) 1315.10 1.13(-09) 1174.30 0.00(00)
1206.10 2.26(-08) 1485.10 7.66(-06) 1599.70 3.22(-11) 1428.50 0.00(00)
1467.20 1.54(-09) 1806.50 2.30(-06) 1946.00 6.12(-13) 1737.70 0.00(00)
1784.70 6.00(-11) 2197.50 3.76(-07) 2367.20 8.10(-15) 2113.80 0.00(00)
2171.00 1.46(-12) 2673.10 3.27(-08) 2879.50 7.73(-17) 2571.30 0.00(00)
2640.90 2.39(-14) 3251.70 1.60(-09) 3502.80 5.51(-19) 3127.90 0.00(00)
3212.50 2.71(-16) 3955.50 4.78(-11) 4260.90 3.01(-21) 3804.90 0.00(00)
3907.80 2.18(-18) 4811.60 9.45(-13) 5183.10 1.30(-23) 4628.40 0.00(00)
4753.60 1.28(-20) 5853.00 1.29(-14) 6305.00 4.45(-26) 5630.20 0.00(00)
5782.50 5.48(-23) 7119.90 1.25(-16) 7669.60 1.21(-28) 6848.80 0.00(00)
7034.10 1.75(-25) 8660.90 8.81(-19) 9329.70 2.59(-31) 8331.10 0.00(00)
8556.50 4.20(-28) 10535.00 4.56(-21) 11349.00 4.30(-34) 10134.00 0.00(00)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Classification problem is a significant topic in machine learning which aims to teach machines how to group together data by particular criteria. In this paper, a framework for ensemble learning (EL) method based on group decision making (GDM) has been proposed to resolve this issue. In this framework, base learners can be considered as decision makers, different categories can be seen as alternatives, classification results obtained by diverse base learners can be considered as performance ratings, and the precision, recall and accuracy which can reflect the performances of the classification methods can be employed to identify the weights of decision makers in GDM. Moreover, considering that the precision and recall defined in binary classification problem can not be used directly in multi-classification problem, the One vs Rest (OvR) has been proposed to obtain the precision and recall of the base learner for each category. The experimental results demonstrate that the proposed EL method based on GDM has higher accuracy than other 6 current popular classification methods in most instances, which verifies the effectiveness of the proposed method.'
author:
- 'Jingyi He, Xiaojun Zhou, Rundong Zhang, Chunhua Yang'
title: An ensemble learning framework based on group decision making
---
Introduction
============
Classification is a significant field of data mining. A classifier is constructed by a tagged training set. It is used to judge the class label for the testing set which is unlabeled. There are many machine learning methods that can be used to solve classification problems. However, there is no such a machine learning method which can perform better than others in classification overall. To solve this problem, ensemble learning (EL) method could be adopted, which is commonly regarded as the machine learning interpretation for the wisdom of the crowd [@1999Liu]. Learners which compose an ensemble are usually called base learners, and their results are generally combined by majority voting. The main premise of EL method is that by combining multiple models, the errors of a base learner will likely be compensated by base learners, and as a result, the overall prediction performance of the ensemble would be better than that of a base learner [@2015Yi]. Many studies have demonstrated that EL method could be used to solve classification problems. A new classification method based on EL method for real-time detection of seizure activity in epilepsy patients was discussed by Hosseini et al. [@2018Hosseini]. Mesquita et al. [@2017Mesquita] proposed ensemble methods for classification and regression using minimal learning machine. In recent years, class imbalance becomes a pressing issue, and some researches have shown that EL method can clearly enhance the classification performance in this respect [@2018Bi; @2019Chakraborty]. It is common for us to make a choice that taking various suggestions into consideration. The process of making a choice with taking many suggestions from our friends or experts into account is group decision making (GDM) [@1999Li]. GDM is a common occurrence of every aspect in our daily life. For example, shopping online considering our friends’ experience. GDM is a process of finding an optimal alternative that has the highest degree of satisfaction from a set of alternatives considering various advisors’ suggestions. And current literatures mainly address the combination of GDM with other discipline domains [@2015Ali; @2005Ye]. Ure$\tilde{n}$a et al. [@2019Kou] analyzed the trust and reputation in distributed networked scenarios in decision making to reach consensus. Dong et al. [@2019Dong] proposed a novel framework that hybridizes both the process of making closer opinions realized by consensus reaching processes and the evolving relationships among experts based on social network analysis.
As mentioned above, EL method is a comprehensive term for methods that combine multiple inducers to make a decision, typically in supervised machine learning tasks. The combining strategy in EL method is especially important, but EL method combining with GDM has not been studied fully yet. In this paper, a framework based on EL method and GDM has been proposed to settle classification problems. Experiments are carried out with 5 public data sets and 6 different base learners to compare their performance. Extensive experiments have verified that the EL method combining with GDM is better than most of other machine learning algorithms. It also demonstrates that the framework combining EL method based on GDM is effective.\
The remainder of this paper is organized as follows. In Section \[section2\], a new EL method called EL based on GDM will be proposed. In Section \[section3\], the experimental settings and the performance of the proposed method comparing with other methods will be presented. Finally, the conclusion and future work will be drawn in Section \[section4\].
Ensemble learning based on group decision making {#section2}
================================================
In this section, the framework for EL method based on GDM will be presented. First, how classical EL methods work and their differences will be introduced. Second, the GDM model will be illustrated. Finally, the framework for EL method based on GDM will be given in detail.
Classical ensemble learning methods {#section2.1}
-----------------------------------
Ensemble methods can be divided into two main frameworks: one is the dependent; the other is the independent [@2018O]. In the former, the output of each base learner works on the model of the next base learner. In the latter, each base learner works independently from other base learners [@2010Liu]. A brief review of two popular ensemble methods of both frameworks has been performed in sections below.
**Random Forest(RF)** is a typical representative of the independent framework. It consists of many independent decision trees, each tree is generated by the different training sets. In RF, a training set is produced by sampling with replacement [@2016M]. In order to produce a random tree which is still sufficiently accurate, RF has one difference from the classic decision tree. Before splitting each decision tree, the best feature should be selected from a feature set which is generated by random selection. The decision trees’ combination strategy for RF is majority vote.
**AdaBoost** is another widely used independent framework. The main idea of AdaBoost is that the next base learner is focused on the instances which are incorrectly classified by last base learner. This procedure is implemented by modifying the weights of instances [@2019Jiang]. The initial weights of all instances are the same. After each iteration, the weights of the correct instances will be reduced. Instead, the weights of the error instances will be increased. Besides, weights are also allocated to the base learners based on their overall predictive performance. Same as RF, weighted voting is adopted. In this case, each base learner has a different emphasis, and combining all of the base learners can allow the model to focus on all instances.\
In general, 5 differences in generation of base learners could be summarized as follows:
- Sample selection: Random Forest takes sampling with replacement, but AdaBoost uses the same training set for each iteration.
- Weights: Random Forest has the same sample weights and base learner weights, but AdaBoost has the different sample weights and base learner weights for each iteration.
- Parallel computing: Each tree in a Random Forest can be computed in parallel, but base learners in AdaBoost can only serial generated.
- Error: Random Forest focuses on reducing variance, but AdaBoost tends to reducing bias.
- Diversity: The diversities of base learners are ensured by different training sets in Random Forest, but AdaBoost is done by changing the weights of the instances.
It is obvious that the classical EL methods change the samples to produce different models. In this paper, 6 different machine learning methods could be used to achieve this goal.
Group decision making methods
-----------------------------
Considering the GDM problem of ranking $m$ alternatives, represented as $A_{1},A_{2},\ldots,A_{m}$ based on the descending order from ‘best’ to ‘worst’ [@1999Li]. Forming a committee of $K$ decision makers, denoted as $E_{1},E_{2},\ldots,E_{K}$ to identify $n$ decision criteria, called $C_{1},C_{2},\ldots,C_{n}$. Each decision maker can evaluate every alternatives and its corresponding criteria individually, then the performance ratings of every alternatives and its corresponding criteria, called $x_{ij}^{k}(i=1,2,\ldots,n;j=1,2,\ldots,m;k=1,2,\ldots,K)$ will be obtained. Also, the important weight of each criterion and each decision maker can be denoted as $w_{1},w_{2},\ldots,w_{n}$ and $W_{1},W_{2},\ldots,W_{K}$ with respect to some overall objectives, respectively [@2019Kou; @2019Dong; @2019Liu]. Performance ratings and importance weights assigned by the decision makers could be crisp numbers ranging from $0$ to $1$.\
The performance rating $x_{ij}^{k}$ assigned to alternative $A_{i}$ by decision maker $E_{k}$ for criterion $C_{j}$ can be used to measure how well $A_{i}$ satisfies $C_{j}$ for decision maker $E_{k}$ [@2000Chen]. Then the decision making data can be collected to form the matrices $X_{k}$:
$$X_{k}=\left[
\begin{array}{ccccc}
x_{11}^{k} & \cdots & x_{1j}^{k} & \cdots & x_{1m}^{k}\\
\vdots & \ddots & \vdots & \ddots & \vdots\\
x_{i1}^{k} & \cdots & x_{ij}^{k} & \cdots & x_{im}^{k}\\
\vdots & \ddots & \vdots & \ddots & \vdots\\
x_{n1}^{k} & \cdots & x_{nj}^{k} & \cdots & x_{nm}^{k}\\
\end{array}
\right]
\label{matrices}$$
The decision making method can be used to aggregate these matrices with importance weights to make the decision making process more conveniently [@2013Wan]. In this work, GDM is used for information fusion of the base learners in EL method.
A framework for ensemble learning based group decision making
-------------------------------------------------------------
In fact, multiple different classification methods can obtain absolutely different classification results even for solving the same classification problem. It is hard to find a overall best classification method. In order to get the best classification result obtained by combining multiple classification methods under the classification problem, a framework for EL method based on GDM can be employed. In EL method, each classification method called base learner. In this framework, base learners can be considered as decision makers, different categories can be seen as alternatives, and different classification results obtained by diverse base learners will be described as performance ratings. So, for a multi-classification problem which has $m$ categories, the decision making data can be collected to form the matrices $X_{k}$. The matrices $X_{k}$ can be written as Eq. (\[matrices\]). When $K$ base learners are used in EL method and they evaluate each category in n dimensions, the importance weights of each criterion can be represented as $w=[w_{1},w_{2},\ldots,w_{n}]$ and the importance of the weight of each decision maker can be represented as $W=[W_{1},W_{2},\ldots,W_{K}]$. The precision, recall and accuracy can reflect the performance of the classification method. In this study, a combination of these indexes is used, rather than a single index, to measure the performance of classification methods. The precision, recall and accuracy are respectively given from Eq. (\[precision\]) to Eq. (\[accuracy\]).
{width="12cm"}\
[p[2.5cm]{}< p[2cm]{}< p[1cm]{}< p[2cm]{}< ]{} Truth & & Prediction &\
(ll)[2-4]{} &Positive Class&& Negative Class\
Positive Class & $TP$&& $FN$\
Negative Class & $FP$&& $TN$\
\[tab\_dataset\]
$$P=\frac {TP}{TP+FP}
\label{precision}$$
$$R=\frac {TP}{TP+FN}
\label{recall}$$
$$A=\frac {TP+TN}{TP+FP+TN+FN}
\label{accuracy}$$
where $P,R,A$ are the precision, recall and accuracy. A class could be identified as positive class and the rest of classes belong to negative class.\
The precision is the number of correct positive results divided by the number of positive results predicted by the classifier. The recall is the number of correct positive results divided by the number of all relevant samples (all samples that should have been identified as positive). The accuracy is the ratio of number of correct predictions to the total number of input samples. The precision and recall are a pair of contradictory. For example, if all the instances are predicated to be positive class, the recall must be equal to one, but the precision will be very low. If there are K classification methods, the precision, recall and accuracy can be described as $P=[P_{1},P_{2},...,P_{K}],R=[R_{1},R_{2},...,R_{K}],A=[A_{1},A_{2},...,A_{K}]$, respectively. The P, R, A of each base learner can be identified as the components of P, R, A. If the dataset has a total of $m$ categories, the components of $P,R,A$ are defined as $P_{k}={[p_{1},p_{2},...,p_{m}]}^{T}$, $R_{k}={[r_{1},r_{2},...,r_{m}]}^{T}$, $A_{k}={[a_{1},a_{2},...,a_{m}]}^{T}(k=1,2,...,K)$, where $p_{i},r_{i},a_{i}$ represents the precision, recall and accuracy of base learner $k$ for category $i$. Based on the above description, the weight of each decision maker is defined as follows: $$W_{k}=P_{k}+R_{k}+A_{k}\quad(k=1,2,...,K)
\label{Wk}$$ From the Eq. (\[Wk\] ), it is obviously that with larger indicators come larger weights. Finally, classification results should be calculated as: $$H(x)=\arg\max\sum_{k=1}^{K}X_{k}W_{k}(x)$$ where $H(x)$ can be regarded as the alternative $x$ with the highest score.\
The framework for EL method based on GDM is illustrated in Fig. \[fig\_framework\]. As can be seen from this framework, a dataset is divided into training set and testing set. A training set is used to train models and the performance indexes can be obtained. Each model is a base learner. GDM has been introduced to fuse the information of these base learners. Decision makers give each alternative evaluating scores which are called performance ratings. The decision matrixes can be formed according to these performance ratings. The weights of decision makers are related to its performance indexes. In this framework, GDM is used to determine which alternative is the most suitable one.
Experimental Results {#section3}
====================
In this section, several examples are illustrated to show the effectiveness of our proposed method.
[p[1.8cm]{}< p[1.8cm]{}< p[1.8cm]{}< p[1.8cm]{}< p[1.8cm]{}< p[1.8cm]{}< p[4.5cm]{}<]{} DataSets & Instances & Classes & Attributes & Train & Test & Size of classes\
CMC & 1473 & 3 & 9 & 1179 & 294 & 629 ,333,511\
Glass & 214 & 6 & 9 & 172 & 42 & 29 ,76,70,17,13,19\
Seeds & 210 & 3 & 7 & 168 & 42 & 70 ,70,70\
Sonar & 208 & 2 & 60 & 167 & 41 & 97 ,111\
Wine & 178 & 3 & 13 & 143 & 35 & 59 ,71,48\
\[tab\_dataset\]
In this paper, experiments are carried out with 5 real-world data sets which are all available at the UCI Machine Learning Repository (<http://archive.ics.uci.edu>). Details about these datasets can be found in Table \[tab\_dataset\]. The table lists the number of instances, classes, features and instances of per class for each dataset. Specific tasks for the datasets are listed as follows:
(i) The Contraceptive Method Choice Dataset(CMC): The problem is to predict the current contraceptive method choice of a woman based on her demographic and socio-economic characteristics.
(ii) The Glass Identification Dataset(Glass): The purpose of this is determining whether the glass is a type of “float” glass or not.
(iii) The Seeds Dataset(Seeds): The instances belonging to three different varieties of wheat, the goal is figuring out what kind of wheat each instance is.
(iv) The Connectionist Bench Dataset(Sonar): The dataset contains signals obtained from a variety of different aspect angles. Whether the sample is a mine or a rock is judged by this information.
(v) The Wine Dataset(Wine): These data are the results of a chemical analysis of wines. Which kind of wine the instances belong to should be predicted.
First, 80$\%$ of each data set have been randomly selected as the training set and the rest as the testing set. By this way, each dataset is divided into two subsets. In this paper, 6 common classification methods are employed to settle the same classification problem. That is to say, we have 6 base learners which are Support Vector Machine(SVM) [@2018Yu], Back-Propagation(BP) Neural Network [@2019Han], Logistic Regression(LR) [@2019Hoang], k-Nearest Neighbor(KNN) [@2016Aburomman], Random Forest(RF), Extreme Learning Machine(ELM) [@2014Samat]. These classification methods have been widely used to solve so many kinds of different classification problems. These classification methods have been employed to ensure the diversity of base learners in this paper. They will be briefly introduced as follows:
(1) SVM: It plots each instance as a point in a $n$-dimensional space (where $n$ is the number of features), and the value of each feature is the value of a specific coordinate. SVM is trying to find a hyperplane that divides the dataset into two categories. It needs some treatments for multi-classification problems.
(2) BP: A neural network is an algorithm that endeavors to recognize underlying relationships in a set of data through a process that mimics the way the human brain operates. BP Neural Network is an algorithm widely used in the training of feedforward neural networks for supervised learning.
(3) LR: The core of the regression method is to find the most appropriate parameters for the function $f(x)$, so that the value of the function and the sample are closed. LR fits a function whose value reflects the probability of sample belonging to its class in probability.
(4) KNN: It is a simple algorithm that stores all available cases and categorizes new cases of its $K $ neighbors by majority votes. For instance, a good way to understand a stranger maybe get information from his friends.
(5) RF: For more details, see the Section \[section2.1\].
(6) ELM: ELM is regarded as an improvement on feedforward neural network and back propagation algorithm. The characteristic of ELM is that the weight of hidden layer nodes is randomly or artificially given. Also, the weight does not need to be updated and the learning process only calculates the output weight.
For each base learner, each training set can be used to train a model which can be used to classify the testing set. The model’s ability to categorize testing sets determines its weight. Taking CMC dataset as a motivating example, a training set is used to train 6 base learners, called BP, ELM, LR, SVM, FR, KNN. These base learners can be seen as 6 decision makers and 3 categories can be represented as 3 alternatives. According to the GDM, each decision maker assigns a score for every alternative. So 6 decision matrices can be shown as follows: $$X_{k}=[x_{11}^{k} , x_{12}^{k} , x_{13}^{k}]\\$$ where $X_{k}$ is the decision matrix of decision maker $k$.\
In order to obtain the performance indexes of the decision maker, the precision, recall and accuracy are calculated. In view of three categories, the components of $P,R,A$ are $P_{k}={[p_{1},p_{2},p_{3}]}^{T},R_{k}={[r_{1},r_{2},r_{3}]}^{T},A_{k}={[a_{1},a_{2},a_{3}]}^{T}$, respectively. Their concrete subvalues are given in the Table \[tab\_example\].
[p[1.5cm]{}< p[1.5cm]{}< p[1.4cm]{}< p[1.4cm]{}< p[1.4cm]{}<]{} Base & & & Classes &\
(ll)[3-5]{} learners & Indicators & First & Second & Third\
& Precision & 84.40 & 50.00 & 71.19\
SVM & Recall & 92.00 & 68.00 & 84.00\
& Accuracy & 51.19 & 51.19 & 51.19\
& Precision & 89.19 & 65.71 & 81.54\
BP & Recall & 90.83 & 69.70 & 84.13\
& Accuracy &59.30 & 90.83 & 59.30\
& Precision & 83.33 & 44.44 & 65.52\
LR & Recall & 93.45 & 69.56 & 84.44\
& Accuracy & 52.20 & 52.20 & 52.20\
& Precision & 75.73 & 46.81 & 60.94\
KNN & Recall & 88.64 & 68.75 & 79.59\
& Accuracy & 47.12 & 47.12 & 47.12\
& Precision & 86.13 & 52.50 & 73.24\
RF & Recall & 95.93 & 80.77 & 91.23\
& Accuracy & 64.75 & 64.75 & 64.75\
& Precision & 80.19 & 57.14 & 36.36\
ELM & Recall & 64.39 & 37.33 & 20.34\
& Accuracy & 42.37 & 42.37 & 42.37\
\[tab\_example\]
At last, classification results are calculated as Eq. (\[W\],\[H\]). $$W_{k}=P_{k}+R_{k}+A_{k}\quad(k=1,2,...,6)
\label{W}$$ $$H(x)=\arg\max\sum_{k=1}^{6}X_{k}W_{k}(x)
\label{H}$$ where $W_{k}$ is the weight of decision maker $k$.\
Subsequent results are listed in Table \[Accuracy\] and Fig. \[fig\_result\]. The performance of the base learners are compared with that of EL method based on GDM proposed in this paper. In Table \[Accuracy\], the highest accuracy has been noted for each dataset. As one can notice, considering the recently proposed EL method based on GDM, our method achieved higher classification rates in most of the data sets, and it is never the worst of all the base learners. It is very easy to find that our proposed method outperforms other algorithms in solving this classification problems from Fig. \[fig\_result\].
[p[1.925cm]{}< p[1.925cm]{}< p[1.925cm]{}< p[1.925cm]{}< p[1.925cm]{}< p[1.925cm]{}< p[1.925cm]{}< p[1.925cm]{}<]{} & BP & ELM & LR & SVM & FR & KNN & Ensemble\
CMC & 58.3 & 49.49 & 52.2 & 51.19 & **64.75** & 52.2 & 61.6\
Glass & 58.14 & 34.88 & **97.67** & **97.67** & 76.74 & 79.07 & **97.67**\
Seeds & 90.48 & 45.24 & 85.71 & 90.48 & 73.81 & **92.86** & **92.86**\
Sonar & 79.57 & 64.27 & **85.71** & 61.90 & 76.19 & **85.71** & 82.50\
Wine & **92.3** & 88.46 & **92.3** & **92.3** & 61.54 & 76.90 & **92.3**\
The numbers in boldface indicate the highest accuracy among all methods
\[Accuracy\]
![Comparison of experimental results.[]{data-label="fig_result"}](fig_result.pdf "fig:"){width="8.5cm"}\
Conclusion and Future Work {#section4}
==========================
Multiple different classification methods can obtain absolutely different classification results even for solving the same classification problem. A framework for EL method based on GDM has been proposed to obtain the consistent result. In EL method, each classification method is called base learner. In this paper, base learners can be considered as decision makers, different categories can be seen as alternatives, different classification results obtained by diverse base learners will be described as performance ratings, and the precision, recall and accuracy which can reflect the performance of the classification method can be employed to identify the weights of decision makers in GDM. 6 current popular machine learning methods have been employed to settle the same classification problem. 5 classical classification datasets have been adopted to conduct the experiment. The experimental results demonstrate that our proposed EL method based on GDM has the highest accuracy than other 6 current popular classification methods, which verifies that our proposed method is effective. In general, the strategy and method this paper proposed are efficient in dealing with the classification problem. In the future, there are three more perspectives can be further studied. The first is that more criteria of alternatives can be considered in the framework. The second is to utilize improved GDM method in this framework to settle the classification problem. The third is adopting the EL method based on GDM to solve real-world problems in industrial process.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present VLBA observations of the ground-state hydroxyl masers in W3(OH) at 0.02 [km s$^{-1}$]{} spectral resolution. Over 250 masers are detected, including 56 Zeeman pairs. Lineshapes are predominantly Gaussian or combinations of several Gaussians, with normalized deviations typically of the same magnitude as in masers in other species. Typical FWHM maser linewidths are 0.15 to 0.38 [km s$^{-1}$]{} and are larger in the 1665 MHz transition than in the other three ground-state transitions. The satellite-line 1612 and 1720 MHz masers show no evidence of $\sigma^{\pm2,3}$ components. The spatial positions of most masers are seen to vary across the line profile, with many spots showing clear, organized positional gradients. Equivalent line-of-sight velocity gradients in the plane of the sky typically range from 0.01 to 1 [km s$^{-1}$]{} AU$^{-1}$ (i.e., positional gradients of 1 to 100 AU ([km s$^{-1}$]{})$^{-1}$). Small velocity gradients in the 1667 MHz transition support theoretical predictions that 1667 MHz masers appear in regions with small velocity shifts along the amplification length. Deconvolved maser spot sizes appear to be larger in the line wings but do not support a spherical maser geometry.'
author:
- 'Vincent L. Fish, Walter F. Brisken, & Loránt O. Sjouwerman'
title: 'A Very High Spectral Resolution Study of Ground-State OH Masers in W3(OH)'
---
Introduction
============
In the presence of a magnetic field, the degeneracy of magnetic sublevels of a molecule is broken due to the Zeeman effect. Zeeman splitting of the hydroxyl radical (OH) is often used to infer magnetic field strengths, both in masers [e.g., @davies66] and in thermal gas [e.g., @turner70]. For the main-line, $F$-conserving transitions, the line splits into one $\pi$ component at the systemic velocity and two $\sigma$ components ($\sigma^+$ and $\sigma^-$) shifted in opposite senses with respect to the systemic velocity. For transitions in which $\Delta F = \pm 1$, such as the 1612 MHz ($F = 1
{\ensuremath{\rightarrow}}2$) and 1720 MHz ($F = 2 {\ensuremath{\rightarrow}}1$) transitions of OH, the splitting is more complicated (Figure \[fig-split\]). These ground-state satellite lines split into six $\sigma$ components ($\sigma^{\pm 1,2,3}$) and three $\pi$ components ($\pi^0, \pi^\pm$), with component intensities in local thermodynamic equilibrium (LTE) being strongest for the innermost $\sigma^{\pm 1}$ components (Figure \[fig-intensities\]). Excited-state satellite lines split into a larger number of components; for instance, the 6016 and 6049 MHz lines each split into 15 different lines in the presence of a magnetic field [@davies74].
With the exception of a single marginal Zeeman triplet at the $F$-conserving 1665 MHz transition in W75 N [@hutawarakorn02; @fish06], a full Zeeman pattern has never been observed in interstellar OH masers. In most sources, no clear $\pi$ components are seen at all. In the $F$-nonconserving satellite lines, theoretical considerations of cross-relaxation among magnetic sublevels due to trapped infrared radiation predict that even the $\sigma^{\pm 2}$ and $\sigma^{\pm 3}$ components should not be observable (@goldreich73b as well as the discussion in @lo75). Single-dish observations of the 1612 MHz OH transition in Orion A are suggestive of the presence of $\sigma^{\pm
2}$ and $\sigma^{\pm 3}$ components [@chaisson75; @hansen82] but are not conclusive, since it is not clear that all spectral features come from the same spatial region. Nevertheless, the possibility that $\sigma^{\pm 2}$ and $\sigma^{\pm 3}$ components may exist in OH masers presents practical difficulties for observers of satellite-line transitions, as noted by @fish03 and @hoffman05a. Conversion of the velocity difference of $\sigma$ components in a Zeeman pair to a magnetic field strength is dependent upon the Zeeman splitting coefficient, which is different depending on which $\sigma$ components are seen. Traditionally it has been assumed that only the $\sigma^{\pm 1}$ components are seen, for which a Zeeman splitting of 0.654 kHz mG$^{-1}$ is appropriate at 1612 and 1720 MHz. But it is possible that several $\sigma$ components overlap for small Zeeman splittings, in which case the Zeeman splitting coefficient appropriate for conversion to a magnetic field strength may be a weighted average of the splitting coefficients of the $\sigma$ components. Indeed, comparison of magnetic fields obtained from Zeeman splitting of the 1720 MHz transition are often a factor of 1.5 to 2 higher than those obtained in the same spatial region at 1665 or 1667 MHz [@fish03; @caswell04], although @gray92 do note an instance in W3(OH) in which the splitting between two 1720 MHz features of opposite polarization appears consistent with their interpretation as $\sigma^{\pm1}$ components with a splitting coefficient (between $\sigma$ components) of 0.12 [km s$^{-1}$]{} mG$^{-1}$. It has heretofore been unclear whether the @fish03 and @caswell04 results indicate that 1720 MHz masers prefer higher densities (which are correlated with magnetic field strength) or that multiple $\sigma$ components from the same Zeeman group are blended together. It is interesting to note that the Zeeman splitting coefficient between a blend of all $\sigma^+$ and $\sigma^-$ components in their LTE ratio of intensities is exactly twice the splitting coefficient of the $\sigma^{+1}$ and $\sigma^{-1}$ components alone.
High spectral-resolution observations of OH masers are also important in order to determine the maser lineshapes. Theoretical models suggest that maser lineshapes may be sensitive to the degree of saturation and to the amount of velocity redistribution of molecules along the maser amplification path [@goldreich74; @field94; @elitzur98]. While other masers have been observed at high spectral resolution, such as 12 GHz CH$_3$OH masers [@moscadelli03] and 22 GHz H$_2$O masers [@vlemmings05], OH masers have never been observed with both the high spectral resolution required to determine the shape of the line wings and the high angular resolution required to ensure that spatially-separated maser spots are not blended together in the beam. The lack of such observations may be due to instrumental limitations. Since the velocity extent of OH maser emission in massive star-forming regions is typically several tens of [km s$^{-1}$]{}, an appropriately wide bandwidth is usually selected in order to observe all maser spots simultaneously. Because the number of spectral channels allowed by the correlator is generally limited (e.g., 1024 channels per baseband channel for the Socorro correlator, or only 128 channels in full-polarization mode), ground-state masers are usually observed at 1 kHz (0.18 [km s$^{-1}$]{}) resolution to within a factor of two.
It is in these interests that we have undertaken observations of OH masers in two high-mass star-forming regions at very high spectral resolution. Orion KL was chosen in order to examine whether the features observed by @hansen82 consist of a single Zeeman group or several spatially-unrelated maser features. W3(OH) was chosen because it is a frequently studied massive star-forming region with a well understood magnetic field structure [@bloemhof92] and has several bright maser features at 1612 and 1720 MHz [@masheder94; @argon00; @wright04b].
Observations
============
The National Radio Astronomy Observatory[^1]’s Very Long Baseline Array (VLBA) was used to observe the ground-state OH masers in two massive star-forming regions: W3(OH) and Orion KL. Data were collected starting at approximately 09 00 UT on 2005 September 20 using all 10 antennas. Approximately 2.3 hours of on-source observing time was devoted to W3(OH) and 1.0 hours to Orion KL. DA193 was also observed as a bandpass calibrator.
All four ground-state transitions (1612.23101, 1665.40184, 1667.35903, and 1720.52998 MHz) were observed in dual circular polarization. A bandwidth of 62.5 kHz was divided into 512 spectral channels with 122 Hz channel spacing (0.02 [km s$^{-1}$]{} velocity spacing). The usable equivalent velocity bandwidth of about 10 [km s$^{-1}$]{} was centered at $-44$ [km s$^{-1}$]{} LSR for W3(OH) and $+10$ [km s$^{-1}$]{} for Orion KL. Many OH maser spots fall outside this velocity range in Orion KL, but the bandwidth was centered appropriately to include the 1612 MHz maser feature at $+8$ [km s$^{-1}$]{} for which @hansen82 claimed detection of $\sigma^{\pm2}$ components. The data were sampled in 4-level (2-bit) mode. A correlator averaging time of 4 seconds was used. Due to the large oversampling required to record 62.5 kHz of bandwidth, 122 Hz is the highest spectral resolution available to the Socorro correlator[^2]. Four of the stations (Brewster, North Liberty, Owens Valley, and St. Croix) used the original VLBA tape-based recording system, while the other six used the newer Mark 5 disk-based recording system.
The data were reduced using the NRAO Astronomical Image Processing System (AIPS). Left circular polarization (LCP) data from the North Liberty station were unusable due to anomalously low amplifier gain. Data from the Hancock station were discarded due to strong radio frequency interference (RFI). Weaker RFI contaminated some data from other stations as well. An auto-correlation bandpass was applied to the data (for further details see §\[gradient\]). Each of the four maser transition frequencies was self-calibrated and imaged separately. Additionally, the left and right circular polarizations were self-calibrated and imaged separately from each other in the 1665 and 1667 MHz transitions, due to RFI that affected each polarization differently. No polarization calibration was applied, as the VLBA polarization leakage is small enough for our scientific purposes. Image cubes were created in both circular polarizations. Each velocity channel was searched for maser emission, and one or more elliptical Gaussians were fitted to detected features using the fitting routines of the AIPS task `JMFIT`.
Results
=======
The detected spots are listed in Table \[tab-spots\] and shown in Figure \[fig-map\]. Symbols are plotted at the locations of peak emission both in LCP and in RCP (right circular polarization). Alignment of maser maps at different frequencies was accomplished by comparison with the data of @wright04b. We estimate that resulting relative positional errors between frequencies is $\sim 10$ mas, due partly to errors in estimating spot positions in the two epochs and partly to the inherent motion of the maser spots. Maser proper motions in W3(OH) are about 3 to 5 [km s$^{-1}$]{} [@bloemhof92], which corresponds to 3 to 5 mas in the 9-year baseline between the @wright04a [@wright04b] observations and the present data. Zeeman pairs are identified in Table \[tab-zeeman\].
Our map is qualitatively similar to the @wright04b map. We recover the vast majority of maser spots in their data. Omissions may be explained by the difference in sensitivity in the observations. The @wright04a [@wright04b] observations spent nearly a factor of 5 more time on source with a factor of 4 coarser velocity resolution. Additionally, data from the Hancock and (frequently) North Liberty VLBA stations were not usable in the present observations, further reducing our sensitivity.
We were unable to detect any satellite-line maser emission in Orion KL. Our pointing center was chosen to be 4 to the northeast of the group marked “Center” in the map of @johnston89, which is the probable location of the Stokes V spectrum interpreted as a 4-component Zeeman pattern centered at velocity 8.0 [km s$^{-1}$]{} by @hansen82. At this pointing center, peak amplitude loss to time-average smearing is small (5% at 7). Bandwidth smearing is negligible with our extremely narrow spectral channels. The nondetection is therefore likely due to a decrease in the flux density of the 8 [km s$^{-1}$]{} 1612 MHz features in the 20 years since the Johnston et al. observations. (Note that their measured flux densities are lower than those of Hansen seven years prior.) Because our primary goal in these observations was to address the issue of satellite-line Zeeman splitting and the Orion KL main-line data were of inferior quality to the W3(OH) data, the Orion KL data were not further analyzed.
Lineshapes and Gaussian Components {#lineshapes}
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Even at this high spectral resolution, the OH maser spectral profiles can usually be fitted well with one or a small number of Gaussian components in the spectral domain. The top panels in Figure \[fig-gaussians\] show selected maser spots with single-Gaussian fits. For spots weaker than $\sim 1$ Jybeam$^{-1}$ (about half the spots), a single Gaussian component usually fits the spectral profile due to signal-to-noise limitations. For spots with a larger signal-to-noise ratio, more complicated spectral profiles are seen as well. Some maser profiles appear skewed or have asymmetric tails. For the most part, these can be fairly well fitted by two or three Gaussian components, as shown in the middle and bottom panels of Figure \[fig-gaussians\]. In some instances, two or more maser lines at different velocities may appear at approximately the same spatial location. When multiple distinct peaks are present in the spectral domain at the same spatial location, we identify each peak as maser line for purposes of inclusion in Table \[tab-spots\]. It is observationally cleanest to identify these as separate features, although theoretical models indicate that under certain conditions the spectrum of a single masing spot could be multiply peaked [e.g., @nedoluha88; @field94].
The normalized deviation ($\delta$) of a lineshape from a Gaussian shape can be defined by $$\delta = \frac{\int\left[I(v) - a_1 \exp(-v^2/a_2)\right]^2\,dv}
{I_p^2 \, \Delta v},$$ where $I(v)$ is the intensity distribution as a function of velocity, $a_1$ and $a_2$ are parameters from the best Gaussian fit, $I_p$ is the peak intensity, and $\Delta v$ is the FWHM of the distribution [@watson02]. Figure \[fig-delta\] shows the distribution of $\delta$ (calculated over the entire range of channels in which the maser spot is detected) and the FWHM as a function of $I_p$ for masers with a peak intensity greater than 1 Jy beam$^{-1}$. Excluded from consideration are masers with multiple peaks or extended asymmetric tails (e.g., with profiles as in the bottom of Figure \[fig-gaussians\]) as well as maser features for which spatial blending with a second maser feature within the beam prohibits accurate determination of the spectral profiles of the two features individually.
Derived values of $\delta$ range from 0 to $6 \times 10^{-3}$, consistent with theoretical predictions by @watson03. The distribution is suggestive of a fall-off of $\delta$ for large values of $I_p$, but with only five data points having $I_p > 30$ Jy beam$^{-1}$, this result is not statistically significant. @watson03 predict that $\delta$ attains a maximal value when the stimulated emission rate, $R$, is a few times the pump loss rate, $\Gamma$, and then decreases with increasing $R/\Gamma$ (i.e., as the maser becomes increasingly saturated)[^3]. Their model also predicts that when $R/\Gamma$ is large enough for $\delta$ to decrease noticeably, the line profile should rebroaden to nearly the thermal Doppler linewidth. In our observations, the FWHM linewidth of maser spots has a range of 0.15 to 0.38 [km s$^{-1}$]{} and is independent of the peak intensity of the maser spot. This suggests that even the brightest masers in a typical star-forming region (which are clearly saturated) may not be sufficiently saturated to exhibit rebroadening.
The FWHM linewidth does appear to be a function of maser transition. As shown in Figure \[fig-delta\], 1665 MHz masers are on average broader than their other ground-state counterparts. The mean and sample standard deviation of the FWHM are $0.244 \pm 0.058$ [km s$^{-1}$]{} for the 1665 MHz transition and $0.192 \pm 0.025$ [km s$^{-1}$]{} for the other three transitions combined. Sixteen of the 28 1665 MHz masers brighter than 1 Jy beam$^{-1}$ are broader than 0.227 [km s$^{-1}$]{}, the linewidth of the broadest maser in any of the other three transitions. The spatial distribution of masers meeting the brightness and shape criteria for the above analysis is shown in Figure \[fig-mapfwhm\]. As can be seen by comparison with Figure \[fig-map\], nearly all of the broad 1665 MHz maser spots appear in regions where no ground-state OH masers are found except in the 1665 MHz transition. The only broad 1665 MHz masers found in proximity to masers of other ground-state transitions are in the cluster of masers near the origin. This region is notable for the existence of highly-excited OH [@baudry93; @baudry98] and is presumed to mark the location of an O-type star exciting the region. As such, it is likely that the physical conditions change in this region over a shorter linear scale than in other regions of W3(OH).
Satellite-Line Splitting {#satellite}
------------------------
There is no evidence of the presence of multiple $\sigma$ components in the single-polarization spectra of the satellite-line (1612 and 1720 MHz) transitions. We find four Zeeman pairs at 1720 MHz and seven at 1612 MHz. Figure \[fig-1720\] shows the LCP and RCP spectra of the brightest 1720 MHz Zeeman pair. The top panels show the best three-component fit to each polarization. The residuals suggest that a fourth, weak component may be required to fit the high-velocity tail of RCP emission. The magnetic field strength derived from applying the splitting coefficient appropriate for $\sigma^{\pm1}$ components to the velocities of the peak channels of emission in spots 190 and 192 is $+6.6$ mG.
The velocities of the two strongest, narrow Gaussian components in each circular polarization in the top panels of Figure \[fig-1720\] are consistent with a 2:1 splitting ratio centered at approximately $-45.26$ [km s$^{-1}$]{} to within the errors in determining the center velocities of the Gaussians. The only lines in a symmetric, incomplete Zeeman pattern with this ratio are the $\sigma^{\pm1}$ and $\pi^\pm$ components. Nevertheless, we reject the possibility that the two brightest Gaussians correspond to $\pi$ components for several reasons. First, they are only seen in one circular polarization, while $\pi$ components should be 100% linearly polarized (although see @fish06 for a discussion of the possibility of $\pi$ components with nonzero circular polarization fractions). Second, the $\pi^0$ component is missing from this pattern, although theory predicts that it should be stronger than $\pi^\pm$ components. Third, no other $\pi$ components are seen in the ground-state masers in W3(OH) [@garciabarreto88]. Linear polarization is rare in W3(OH); all masers are more circularly polarized than linearly polarized.
The bottom panels show the best fit constraining the center velocities of each component to be in the ratio expected from the Zeeman pattern of multiple $\sigma$ components in each polarization, as shown in Figure \[fig-intensities\]. It is not the case that the two Gaussian components closest to the systemic velocity are brightest. This argues against interpretation of the spectra as $\sigma^{\pm
1,2,3}$ components in their LTE ratios. It is more probable that the same factors that produce asymmetric, non-Gaussian lineshapes in the main-line transitions (as in the middle and bottom panels of Figures \[fig-gaussians\]) also produce non-Gaussian asymmetries in the $\sigma^{\pm 1}$ components at 1720 MHz. Indeed, at higher angular resolution, @masheder94 note that features 190 and 192 are each actually a cluster of several maser spots. This is consistent with the increasing intensities in each polarization toward higher velocity, suggesting that our observed features may be the result of blending of (at least) three nearby Zeeman pairs with a regular shift in velocity but approximately the same magnetic field (coincidentally also $+6.6$ mG if interpreted as $\sigma^{\pm 1}$ components). This magnetic field value is consistent with the two nearby Zeeman pairs to the northeast: $+6.4$ mG from spots 212 and 218 at 1665 MHz and $+6.8$ mG from spots 208 and 209 at 1720 MHz. (Note that this latter Zeeman pair, shown in the middle panel of Figure \[fig-satsplit\], is unquestionably comprised solely of $\sigma^{\pm1}$ components, since there is only one feature in each circular polarization and interpretation of these features as $\sigma^{\pm2}$ components would imply a magnetic field strength of $+2.2$ mG, a value too small to be consistent with the 1665 MHz magnetic field or any other magnetic field strength in the cluster of maser spots near the origin.)
Figure \[fig-satsplit\] shows the LCP and RCP spectra of the other two 1720 MHz Zeeman pairs and one 1612 MHz Zeeman pair in W3(OH). The multiple peaks in the single-polarization spectra of the top panel are again due to blending of two adjacent maser spots. It is clear that these are not due to spatially-shifted $\sigma^{\pm2,3}$ components from a single Zeeman pattern, since the spectra are not symmetric by reflection across a single, systemic velocity. We interpret the spectra as two Zeeman pairs, each with a different magnetic field strength. Asymmetric amplification of the various peaks in LCP and RCP may also be partly due to the large velocity range spanned — greater than 1.4 [km s$^{-1}$]{} from the low-velocity peak in LCP to the high-velocity peak in RCP. Since this is more than twice the turbulent velocity dispersion of a maser cluster in W3(OH) [@reid80], it would be expected that the emission from multiple maser spots in this velocity range might be amplified by different amounts.
The middle panel in Figure \[fig-satsplit\] shows a 1720 MHz Zeeman pair that is well fitted by a single Gaussian component in each polarization. The bottom shows a 1612 MHz Zeeman pair. It is clear from the velocities of the fit components that the lines are not produced from multiple $\sigma$ components of a single Zeeman pattern. Emission from other masers in the 1612 MHz transition is qualitatively similar to these Zeeman pairs. The image cubes of 1612 and 1720 MHz emission were searched thoroughly at the locations of the detected masers for indications of weak emission at other velocities. No emission was detected to within the limits of our noise except as listed in Table \[tab-spots\].
Positional Gradients {#gradient}
--------------------
In general, the position of a maser spot is seen to vary across the linewidth [e.g., @moscadelli03; @hoffman03]. Figure \[fig-gradients\] shows the maser position as a function of LSR velocity for a sample of maser spots. The position of the center of the best-fitting elliptical Gaussian usually varies linearly as a function of velocity. In some instances the position may trace out a curving structure rather than a straight line, but all maser spots display organization in their position as a function of frequency.
Table \[tab-spots\] includes the velocity gradient and position angle (degrees east of north) in the direction of increasing velocity for each maser spot. The velocity gradients were determined algorithmically. On both sides of the peak, the nearest channel with emission below half of the peak brightness was identified. The velocity difference between these two channels was divided by the difference in positions. For a Gaussian spectral profile this corresponds to dividing the FWHM by the difference of the positions across the FWHM, but it is algorithmically implementable for any emission spectrum, including spectra with multiple peaks, as in the middle and bottom panels of Figure \[fig-gaussians\]. For consistency with @moscadelli03, we report the velocity gradients in units of [km s$^{-1}$]{} mas$^{-1}$ rather than the positional gradient in mas ([km s$^{-1}$]{})$^{-1}$. A large positional gradient corresponds to a small velocity gradient, and vice versa.
These gradients appear to be real, not an artifact due to residual calibration or bandpass phase errors. Comparison of selected bright maser spots in different regions of W3(OH) indicate that positional gradients determined from applying the auto-correlation (real) bandpass are consistent with those determined from applying the cross-correlations (complex) bandpass to within measurement errors. Since the auto-correlation bandpass has a higher signal-to-noise ratio and the phases of the cross-correlation bandpass are constant with frequency over the region of interest, the auto-correlation bandpass was applied. In addition, combinations of plots of the Right Ascension or Declination positional gradients versus Right Ascension offset, Declination offset, or LSR velocity are all consistent with a random scatter about zero (as with Figure 6 in @moscadelli03), both for individual transitions and polarizations as well as for the ensemble of all maser spots with detected positional gradients as listed in Table \[tab-spots\]. However, the two-dimensional distribution does show larger velocity gradients (smaller positional gradients) near the origin (Figure \[fig-gradientmap\]). Note that the origin is *not* near the location of the reference spots for self-calibration except at 1720 MHz and the LCP polarization at 1665 MHz, nor is it near the pointing and correlation center [taken from @argon00], which is at $(\Delta\alpha,\Delta\delta) \approx (-884,+311)$ mas.
It is probable that some gradients are the result of two maser spots within a beamwidth that blend together spectrally. One clear instance of this is shown in the top panel of Figure \[fig-satsplit\]. Only one feature is detected in each circular polarization in each spectral channel. Yet it is clear from the spectra that there are at least two distinct maser spots in each polarization. The weaker peak is to the northwest of the strong peak (panel g of Figure \[fig-gradients\]). The centroid of the fitted Gaussian is effectively a weighted average of the two positions at velocities intermediate to the two peak velocities. This effect is more prominent in RCP due to the smaller velocity offset between the two peaks. Nevertheless, there is a real positional gradient associated with each of the maser spots as well, as is clearest in the uncontaminated blue wing of the bright features.
Velocity gradients of the RCP and LCP components of a Zeeman pair are generally aligned. Figure \[fig-zeemangrads\] shows the distribution of position angle differences between the velocity gradients of the RCP and LCP components of Zeeman pairs. These position angle differences are also shown for “echoes,” i.e., spectral features detected in the opposite circular polarization and same location and line-of-sight velocity as another strong, partially linearly polarized spectral feature due to the fact that both circular feeds of a telescope are sensitive to linear polarization. These detections are not a result of telescope polarization leakage; in most cases, the brightness of the weaker polarization feature is more than 25% of that of the stronger polarization feature, while polarization leakage of the VLBA feeds is only 2 to 3% [@wrobel05]. Since an echo is a second, weaker detection of a single maser spot, both a maser spot and its echo would be expected to have essentially the same gradient. We find this to be the case; for the 9 maser spots for which a gradient can be determined algorithmically both for itself and its echo, all have gradient polarization angle differences less than 40. Of the 23 Zeeman pairs for which gradients can be obtained for both components, the RCP and LCP components are aligned to within better than 45 for 18 of them. Larger deviations for the other pairs can usually be attributed to spatial blending with nearby maser spots. The alignment of RCP and LCP gradients is especially pronounced in the bright Zeeman pairs at 1720 MHz. Their spectra and positions are shown in Figures \[fig-1720\], \[fig-satsplit\], and \[fig-gradients\]. In each case, there is a clear, linear positional gradient that is similar for both components of the Zeeman pair.
The magnitude of the velocity gradient of a maser spot does not display a clear correlation with its peak brightness, as shown in Figure \[fig-gradflux\]. However, there does appear to be an absence of 1667 MHz maser spots with large velocity gradients. (That is, 1667 MHz masers appear to have large *positional* gradients as a function of line-of-sight velocity.) This is consistent with observations by @ramachandran06. It is unclear whether the line-of-sight velocity gradient projected onto the plane of the sky necessarily allows inference of the line-of-sight velocity gradient along the amplification path. Large velocity gradients along the amplification length may destroy the velocity coherence required for significant amplification, so the population of detectable maser spots may have an inherent bias in favor of areas where the projection of the velocity gradient along the line of sight is small. But in §\[transitions\] we present further evidence that the velocity gradient along the amplification path is indeed small in 1667 MHz masers.
There does not appear to exist a correlation between the orientation of the gradient of a maser spot and its proper motion vector. From the list of 1665 MHz maser spots for which @bloemhof92 were able to measure a proper motion, approximately three dozen spots with measurable positional gradients were recovered in our observations. Since the Bloemhof et al. data were not phase referenced, multiple reference frames consisting of their proper motions with an added constant vector were compared against our positional gradient vectors. No clear correlations were found. Proper motion maps of the OH masers in W3(OH) display a clear large-scale pattern of motions [@bloemhof92; @wright04a], while the map of gradients shows no such large-scale organization, with the possible exception of the cluster near the origin (Figure \[fig-gradientmap\]), where velocity gradients are large (i.e., positional gradients are small). If there is a connection between observed maser velocity gradients and material motions, it is probable that it is the turbulent motions that dominate, not the large-scale organized motions.
Likewise, the gradients do not correlate with linear polarization fraction (which is zero for most maser spots) or polarization position angle, as determined from @garciabarreto88. The magnetic field direction can theoretically be derived from the linear polarization fraction and position angle [e.g., @goldreich73a], although empirical data suggest that recovery of the full, three-dimensional orientation of the magnetic field may not actually be possible at OH maser sites [@fish06].
Deconvolved Sizes and Maser Geometry {#deconv}
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The apparent size of a maser may be a function of frequency offset from line center, due to saturation effects dependent on the maser geometry. For example, @elitzur90 calculates that the size of a spherical maser should increase exponentially with $|\nu -
\nu_0|/\Delta\nu_\mathrm{D}$, where $\nu_0$ is the line center frequency and $\Delta\nu_\mathrm{D}$ is the Doppler linewidth. This effect can be large; Elitzur calculates that the apparent spot size at half the Doppler width may be twice that at line center (peak flux).
Figure \[fig-deconv\] shows deconvolved spot sizes as a function of velocity offset from the channel of peak emission for 20 selected maser spots. Displayed masers were selected under the criteria that they have a peak flux density of at least 7 Jy and not have obvious spatial blending with other maser emission. Minimum nominal deconvolved spot sizes typically range from 3 to 6 mas, consistent with results obtained for 1665 MHz masers by @garciabarreto88, although the apparent sizes of 1720 MHz masers are much bigger than the $\leq 1.2$ mas upper limit obtained by @masheder94 (see §\[satellite\] for discussion of probable spatial blending in spot numbers 190 and 192). In general, maser spot sizes appear to increase toward the line wings, although the degree to which the spot size increases with frequency offset from center (or indeed whether it does at all) is different with each maser spot. In some spots, the spot size is a complicated function of frequency. It is possible that some maser spots display additional structure on scales smaller than the beam size, which could cause the spot size to be overestimated over part or all of the line profile. In any case, the variation of spot size over the observable line profile is sufficiently small and variable to preclude accurate determination of the functional form of the apparent spot size as a function of frequency (and therefore geometry).
The velocity offset at which the maser spot size doubles is generally greater than 0.5 [km s$^{-1}$]{}. For a spherical maser, this implies a Doppler width greater than 1.0 [km s$^{-1}$]{}, based on the @elitzur90 model, which would require a kinetic temperature in excess of 400 K. This value is more than a factor of two higher than the inferred effective temperature of the ambient radiation field [@walmsley86]. It is probable that the geometry of the OH masers in W3(OH) is not spherical. Other theoretical considerations lead @goldreich72 to conclude that a filamentary geometry is more typical of astrophysical masers. Alternatively, several different (spherical) clumps may overlap along the line of sight to produce a detectable maser.
Discussion
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$\sigma$ Components in Satellite-Line Transitions {#sigmas}
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We find no evidence of the presence of $\sigma^{\pm2,3}$ components in the 1612 and 1720 MHz satellite lines of OH. Some of the spectral profiles in the 1720 MHz transition appear to consist of several Gaussian components (Figures \[fig-1720\] and \[fig-satsplit\]), but the velocities and intensities of these components are not consistent with what is expected by theory (Figure \[fig-intensities\]). The nondetection of $\sigma^{\pm2,3}$ components lends support to the prediction that cross-relaxation across magnetic sublevels will favor amplification of the $\sigma^{\pm1}$ components over the other $\sigma$ components [@goldreich73b]. For the brightest 1720 MHz maser, our nondetection of accompanying maser components requires that $S_{\sigma^{\pm1}}/S_{\sigma^{\pm2}} >$ a few hundred. Future observations of sources with stronger satellite-line OH maser emission, such as G43.165$-$0.028 [@argon00] and G331.512$-$0.103 [@caswell99], could improve on this by more than a factor of 10.
Could $\sigma^{\pm2}$ components ever be observed in a maser source? The number of gain lengths for a $\sigma^{\pm1}$ component will be twice that of a $\sigma^{\pm2}$ component over the same physical region of space. For unsaturated amplification, the intensity depends exponentially on the number of gain lengths, effectively prohibiting detection of the $\sigma^{\pm2}$ components. (Since the number of gain lengths for the $\sigma^{\pm1}$ components is $\gtrsim 20$ [@goldreich73b], the $\sigma^{\pm2}$ components would be weaker by a factor of $\gtrsim e^{10}$). If the $\sigma^{\pm1}$ components are highly saturated, it is possible that the $\sigma^{\pm2}$ components would be detectable, provided that the populations of the magnetic sublevels are not redistributed by radiative transitions connecting these levels with the far infrared. It should be noted that two different radiative effects are likely operating in satellite-line masers. First, cross-relaxation of the magnetic sublevel populations due to radiative transitions connecting these levels with the far infrared will favor the $\sigma^{\pm1}$ components [@goldreich73b]. This effect can operate over frequency differences much greater than the linewidth of a single maser component. Second, velocity redistribution inherent in a three-dimensional geometry causes maser lines to remain narrow even during saturation, preventing rebroadening to the Doppler width [@field94]. Velocity redistribution causes flux from the linewings to move toward the line center, effectively changing the frequency on the order of a maser linewidth. Cross-relaxation of populations of the magnetic sublevels will be unavoidable in regions of strong infrared radiation. Thus, it is probable that $\sigma^{\pm2}$ components will not be detectable in massive star-forming regions.
It is also likely that 1720 MHz supernova remnant masers will not display evidence of $\sigma^{\pm2}$ components. Supernova remnant OH masers are collisionally pumped [@elitzur76; @frail94], so it may be possible to avoid far infrared cross-relaxation among magnetic sublevels. However, the Zeeman splitting is usually less than the maser linewidth [e.g., @hoffman05a; @hoffman05b], resulting in blending of multiple maser components into a single maser line. Velocity redistribution would likely destroy the signature of $\sigma^{\pm2,3}$ components, if emission in these modes is produced.
As mentioned in §\[introduction\], if $\sigma^{\pm2,3}$ components are not blended with $\sigma^{\pm1}$ components at 1720 MHz, there is observational evidence that the magnetic field, and hence the density, at sites of 1720 MHz maser emission in massive star-forming regions may be higher than at sites of 1665 and 1667 MHz maser emission [@fish03; @caswell04]. Our data indicate that $\sigma^{\pm2,3}$ components, if they exist, are so weak as to have no effect on the observed emission. Thus, the value of the magnetic field obtained from assuming a Zeeman splitting coefficient appropriate for pure $\sigma^{\pm 1}$ components is reliable. Using this coefficient, the three brightest 1720 MHz Zeeman pairs in W3(OH) are consistent with the magnetic field strengths derived from nearby main-line Zeeman pairs. Models of @pavlakis96 suggest that 1720 MHz maser activity may be favored at densities near or just above those for which 1665 MHz maser activity occurs. Thus, in an ensemble of OH maser sources, it would be expected that the magnetic fields derived from 1720 MHz Zeeman splitting would be skewed higher than those obtained at 1665 MHz (consistent with the findings of @fish03 and @caswell04) , although the magnetic field strengths derived at 1665 MHz and 1720 MHz would be similar in some of those sources (consistent with this work).
Comparison of Maser Transitions {#transitions}
-------------------------------
Our results for the properties of hydroxyl masers at high spectral resolution are remarkably similar to those found in a similar study of 12.2 GHz methanol masers in W3(OH) [@moscadelli03]. FWHM line widths of single-Gaussian fits range from 0.15 to 0.38 [km s$^{-1}$]{} in OH, as compared with the range 0.14 to 0.32 [km s$^{-1}$]{} in CH$_3$OH. Normalized deviations from a Gaussian shape are several $\times 10^{-3}$ for both the OH and CH$_3$OH masers. Gradients in the spot position as a function of velocity are observed in both species, with similar amplitudes. The OH masers in our sample have velocity gradients as a function of position (i.e., the inverse of a positional gradient as a function of line-of-sight velocity) of 0.01 to 1 [km s$^{-1}$]{} AU$^{-1}$ (with one outlier at 5 [km s$^{-1}$]{} AU$^{-1}$), as compared to 0.02 to 0.30 [km s$^{-1}$]{} AU$^{-1}$ in a smaller sample of CH$_3$OH masers [@moscadelli03]. The similar observational characteristics of OH and CH$_3$OH masers are not surprising given that these molecules form in the same environment [@hartquist95], are both excited under similar conditions [@cragg02], and appear in close proximity [@etoka05].
Since the ground-state transitions of OH have large Zeeman splitting coefficients, an apparent velocity gradient could be the result of a magnetic field gradient. Indeed, in the cluster of maser spots located near the origin in Figure \[fig-map\], the line-of-sight velocity gradients as a function of position are large, and the magnetic field strengths are large and change significantly on a small spatial scale [see Table \[tab-zeeman\] of the present work as well as Figure 13 of @wright04b]. Likewise, the velocity gradients are small in the cluster of masers near $(\Delta\alpha,\Delta\delta) =
(-800,-700)$ mas, where the magnetic field strengths are small and the gradient of the magnetic field as a function of position is small. But the observed velocity gradients cannot be entirely due to magnetic field gradients, since they are also observed in methanol masers [@moscadelli03], in which Zeeman splitting is negligible.
In the absence of velocity redistribution between velocity subgroups in the masing region, the linewidth of a saturated maser will in general increase as the amplification (and hence, intensity) of the maser increases [@goldreich74], although maser linewidths remain narrow even during saturated amplification when trapped infrared radiation is included in the theory. The lack of single-Gaussian lineshapes with FWHM greater than 0.4 [km s$^{-1}$]{} combined with the absence of a correlation between FWHM linewidth and maser flux density suggests that line rebroadening does not occur, even for the brightest OH masers. @field94 suggest that velocity redistribution is important at 1665 MHz, which would produce narrow, single-peaked maser lines, as observed. In extreme cases, increasing amplification may cause the line center to go into absorption, resulting in two very narrow maser lines at different velocities [@gray91; @field94]. The addition of a velocity gradient over the amplification length can also produce very strong, leptokurtic intensity profiles. However, large velocity gradients in the presence of complete velocity redistribution can also produce multiply-peaked spectral profiles, which we do not observe. While spectral profiles do sometimes exhibit more than one peak (as in Figure \[fig-gaussians\]), it is neither the case that the individual peaks in the spectrum are abnormally narrow nor that the overall spectrum resembles a single broad Gaussian whose center is strongly absorbed. It is more probable that these spectra are indicative of two or more spatially distinct maser spots blended within a beamwidth. VLBI studies of other sources find that it is common for several distinct maser spots to be found within several milliarcseconds of each other [e.g., @slysh01; @fish05].
We find that the linewidths of 1665 MHz masers are greater than the other ground-state OH masers. While 1665 MHz masers are usually the brightest OH masers in a source, the lack of a correlation between the linewidth and maser intensity indicates that saturated rebroadening is not the cause of the larger linewidths at 1665 MHz. One possible explanation may involve the large Zeeman splitting coefficient at 1665 MHz. A magnetic field gradient of 0.34 mG is sufficient to shift the center velocity of a 1665 MHz maser by the 0.2 [km s$^{-1}$]{} FWHM typical in other transitions; necessary magnetic fields for similar shifts are 0.56 mG at 1667 MHz and over 1.6 mG in the satellite-line transitions. Observations of a larger sample of interstellar maser sources suggest that the magnetic field strength typically varies by a few tenths to a full milligauss in a typical cluster (projected dimension of several $\times 10^{15}$ cm) of maser spots [@fish06]. Since the amplification length (along the line of sight) is likely a factor of a few smaller than the clustering scale, it is reasonable that the magnetic field strength may change by a few tenths of a milligauss over the amplification length. If so, and if velocity redistribution is not total, it is possible that the resulting spectral profile would be broader. Under these assumptions, it would be expected that broader 1665 MHz masers would appear in regions where the gradient of the magnetic field is large. The central cluster does contain several broad 1665 MHz maser spots, and it is clear that the magnetic field strength varies significantly over a small spatial scale in this region. However, the other broad OH masers in W3(OH) appear in regions where the magnetic field strength is sampled (in this study and in @wright04a [@wright04b]) by only one or a few Zeeman pairs, so it is difficult to obtain an estimate of the gradient of the magnetic field in these locations. Indeed, since regions of large magnetic field gradients may not favor amplification of both $\sigma$-components in a Zeeman pair [@cook66], it is possible that the magnetic field gradients in these regions are large.
A related possiblity is that the broad 1665 MHz masers sample a region of parameter space in which amplification of only the 1665 MHz masers is favored. Due to Zeeman splitting, a large magnetic field gradient might be expected to act akin to a large velocity gradient, although rigorous theoretical examination of the effect of magnetic field gradients in maser sites is lacking. With the exception of the central cluster of maser spots, in which physical conditions likely change substantially over a small spatial scale, all other broad 1665 MHz masers appear in regions where the only ground-state OH masers found are 1665 MHz masers. Models by @pavlakis96 indicate that for radiatively-pumped OH masers, amplification of 1667 MHz decreases significantly as the velocity gradient over the amplification path increases from 1 [km s$^{-1}$]{} to 2 [km s$^{-1}$]{}, while 1665 MHz maser amplification remains relatively unaffected. This is in excellent qualitative agreement with @gray92, who find that amplification of the 1667 MHz transition falls off with increasing velocity shift. In our observations, no broad 1665 MHz maser is found in the vicinity of 1667 MHz masers; in fact, 1667 MHz masers are the only ground-state transition absent from the highly active central cluster of masers. These facts fit well with the observation that 1667 MHz masers have small line-of-sight velocity gradients in the plane of the sky (see Figure \[fig-gradflux\]), suggesting that the gradient of the line-of-sight velocity along the amplification path may be small as well.
It should be noted that the central cluster of masers also includes a 4765 MHz maser [@gray01], for which inversion requires a small velocity gradient [@pavlakis96]. However, this maser is near the southern edge of the cluster [@etoka05], where the magnetic field gradient is small. It may be the case that even in clusters with large velocity gradients, subregions exist in which the velocity gradient along the line of sight is small. In any case, it is not yet established whether gradients in the centroid of a maser spot as a function of line-of-sight velocity also provide information as to the line-of-sight velocity distribution along the amplification path of a maser spot. If maser amplification is only favored for a narrow range of velocity gradients along the amplification path, an unavoidable observational bias will exist. But velocity redistribution may weaken the correlation between velocity gradients and maser gain. Further theoretical and observational work may be required to resolve these issues.
Conclusions
===========
We have observed over 250 ground-state OH maser spots at very high spectral resolution. Spectral profiles are generally well fit by one or a small number of Gaussian components. The data hint that deviations from Gaussianity may diminish for bright ($> 30$ Jy) masers, but our sample size of bright masers is too small to be conclusive. Maser FWHM linewidths range from 0.15 to 0.38 [km s$^{-1}$]{}, with 1665 MHz masers generally having broader profiles than other ground-state masers.
Consistent with theoretical predictions [@goldreich73b], we do not see $\sigma^{\pm 2,3}$ components in the 1612 and 1720 MHz satellite-line transitions. When satellite-line Zeeman pairs are seen, the magnetic fields are most consistent with values derived from main-line transitions if the splitting appropriate to $\sigma^{\pm 1}$ components is assumed.
Velocity gradients are common in OH masers. In W3(OH), 1667 MHz masers are seen to have large positional gradients (i.e., the position in the plane of the sky changes rapidly as a function of LSR velocity), corresponding to small velocity gradients. This is consistent with predictions by @pavlakis96, who find that small velocity gradients are required for significant amplification at 1667 MHz.
Maser spot sizes appear to be larger in the line wings than at line center. The increase of deconvolved spot size with frequency offset from center is small enough to argue against a spherical maser geometry [@elitzur90]. However, data of higher sensitivity and spatial resolution are required to conclusively argue for or against specific maser geometries.
We thank J. Romney, C. Walker, & L. Foley for their assistance in identifying allowed modes of operation of the Socorro correlator.
[*Facility: VLBA*]{}
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[^1]: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
[^2]: The minimum sample rate is 2.0 Msamples s$^{-1}$. With an oversampling factor of 16, the correlator playback interface cannot accumulate a 2048-bit Nyquist-sampled FFT segment before internal buffers are cleared automatically.
[^3]: In the saturated regime, the maser intensity increases as a polynomial function of $R$ depending on the geometry [@goldreich72], and thus $\log(I_p)
\propto \log(R/\Gamma)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginzburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfaces.'
author:
- Vassily Gorbounov
- Fyodor Malikov
title: '**Vertex Algebras and the Landau-Ginzburg/Calabi-Yau Correspondence**'
---
[^1]
[*To Borya Feigin on his 50th birthday*]{}
**Introduction**
Introduced in \[MSV\] for any smooth (algebraic, analytic, etc.) manifold $X$ there is a sheaf of vertex algebras $\Omega^{ch}_{X}$. For example, the vertex algebra of global sections over $\BC^{N}$, $\Omega^{ch}_{\BC^{N}}(\BC^{N})$ or simply $\Omega^{ch}(\BC^{N})$, is well known as “$bc-\beta\gamma$-system”; it is an apparently unsophisticated object. Despite various important contributions \[B, BD, BL, KV1\], however, very little is known about the cohomology vertex algebra $H^{*}(X,\Omega^{ch}_{X})$ even if $X$ is toric – except perhaps the case of $\BP^{2n}$ where at least the character of the space of global sections, $H^{0}(\BP^{2n},\Omega^{ch}_{\BP^{2n}})$, has been computed: it was shown to be equal to the elliptic genus of $\BP^{2n}$ in \[MS\].
Let $$\fF=\{f=0\}\subset\BP^{N-1}.
\eqno{(1)}$$ be a Calabi-Yau hypersurface. The present paper is devoted to an interplay between $\Omega^{ch}(\BC^{N})$ and closely related algebras on the one hand, and $H^{*}(\fF,\Omega^{ch}_{\fF})$ on the other. Let us now formulate the main result.
[*Preparations.*]{} Set $$\Lambda=\BZ^{N}\oplus(\BZ^{N})^{*}.$$ Associated to this lattice in the standard manner there are the lattice vertex algebra $V_{\Lambda}$ and the fermionic vertex algebra ( $bc$-system), $F_{\Lambda}$, which is none other than the vacuum representation of the infinite-dimensional Clifford algebra $Cl(\BC\otimes_{\BZ^{N}}\Lambda)$. In the important paper \[B\] Borisov introduces the vertex algebra $\BB_{\Lambda}=V_{\Lambda}\otimes F_{\Lambda}$ and the vertex algebra embedding $$\Omega^{ch}(\BC^{N})\hookrightarrow \BB_{\Lambda}.$$ Define the $\BZ_{N}$-action $$\BZ_{N}\times \BC^{N}\rightarrow \BC^{N},\;
(m,\vec{x})\mapsto \exp{(2\pi i\frac{m}{N})}\vec{x}.$$ There arises the vertex subalgebra of $\BZ_{N}$-invariants $$(\Omega^{ch}(\BC^{N}))^{\BZ_{N}}\subset \Omega^{ch}(\BC^{N}).$$ Let us now warp the lattice $\Lambda$: denote by $\BZ^{N}_{orb}$ the sublattice of $\BZ^{N}$ consisting $(m_{0},m_{1},..., m_{N-1})$ such that $\sum_{j}m_{j}$ is divisible by $N$ and define $$\Lambda_{orb}= \BZ^{N}_{orb}\oplus (\BZ^{N}_{orb})^{*}.$$ Just as above, there arises the vertex algebra $\BB_{\Lambda_{orb}}$. Note that $\BB_{\Lambda}$ and $\BB_{\Lambda_{orb}}$ have a non-empty intersection, which contains $(\Omega^{ch}(\BC^{N}))^{\BZ_{N}}$; thus $$(\Omega^{ch}(\BC^{N}))^{\BZ_{N}}\hookrightarrow \BB_{\Lambda_{orb}}.$$ Now we extend $(\Omega^{ch}(\BC^{N}))^{\BZ_{N}}$ inside $\BB_{\Lambda_{orb}}$ to a bi-differential vertex algebra.
Let $\{X_{i}\}\subset\BZ^{N}$, $\{X_{i}^{*}\}\subset(\BZ^{N})^{*}$ be the standard dual bases. Associated to them inside $V_{\Lambda}$ there are fields, such as $X_{i}(z)$, $X_{i}^{*}(z)$, $e^{X_{i}^{*}}(z)$. Let the corresponding tilded letters denote their superpartners inside $F_{\Lambda}$, e.g., $\tilde{X}_{i}(z)$, $\tilde{X}_{i}^{*}(z)$.
Denote $$X^{*}_{orb}=\frac{1}{N}(X^{*}_{0}+X^{*}_{1}+\cdots X_{N-1}^{*})
\in (\BZ^{N}_{orb})^{*}.$$ Form $$\widetilde{\text{LG}}=\bigoplus_{n=0}^{\infty}
\widetilde{\text{LG}}^{(n)},\;
\widetilde{\text{LG}}^{(n)}=(\Omega^{ch}(\BC^{N}))^{\BZ_{N}}e^{nX^{*}_{orb}}.$$ This is clearly a $\BZ_{+}$-graded subalgebra of $\BB_{\Lambda_{orb}}$ (but not of $\BB_{\Lambda}$).
Now define two operators $$D_{orb}=
(\sum_{j=0}^{N-1}e^{X^{*}_{orb}}(z)\tilde{X}^{*}_{j}(z))_{(0)},
d_{LG}=df(z)_{(0)}\in \text{End}(\widetilde{\text{LG}}),$$ where $f$ is the polynomial appearing in (1) and $df(z)$ is computed by using the definition $$d(x_{0}^{m_{0}}x_{1}^{m_{1}}\cdots x_{N-1}^{m_{N-1}})(z)=
e^{\sum_{j}m_{j}X_{j}}(z)\sum_{j}m_{j}\tilde{X}_{j}(z).$$
These are commuting, square zero derivations of $\widetilde{\text{LG}}$; thus we have obtained the bi-differential vertex algebra $(\widetilde{\text{LG}}; D_{orb}, d_{LG})$. Note that it is filtered by the bi-differential vertex ideals $$\widetilde{\text{LG}}^{\geq n}=\bigoplus_{m=n}^{\infty}
\widetilde{\text{LG}}^{(m)}.$$ Hence there arises the projective system of bi-differential vertex algebras $$\widetilde{\text{LG}}^{< n}=\widetilde{\text{LG}}/
\widetilde{\text{LG}}^{\geq n}.$$
[**Theorem 1.**]{} (cf. Theorems 4.7, 4.9) [*There is a spectral sequence* ]{} $$(E^{*,*}_{*}, d_{*})\Rightarrow H^{*}(\fF,\Omega^{ch}_{\fF})$$ [*that satisfies:*]{}
\(i) $$(E^{*,i-*}_{1}, d_{1})\sim
(H^{i}_{D_{orb}}(\widetilde{\text{LG}}^{< N}), d_{LG});$$ [*(ii) at the conformal weight zero component this spectral system degenerates in the 2nd term so that*]{} $$H^{*}(\fF,\Lambda^{*}\CT_{\fF}) \iso
H^{*}(\fF,\Omega^{ch}_{\fF})_{0}
\sim (E^{*,*}_{2})_{0}=H_{d_{LG}}(\widetilde{\text{LG}}^{< N})_{0},$$ [*where $\Lambda^{*}\CT_{\fF}$ is the algebra of polyvector fields over $\fF$. Further,*]{} $$H_{d_{LG}}(\widetilde{\text{LG}}^{(i)})_{0}=
\left\{\aligned
\BC &\text{ if } 1\leq i\leq N-1,\\
M_{f}^{\BZ_{N}} &\text{ if } i=0,
\endaligned\right.
\eqno{(2)}$$ [*where $M_{f}^{\BZ_{N}}=(\BC[x_{0},...,x_{N-1}]/<df>)^{\BZ_{N}}$ is the $\BZ_{N}$-invariant part of the Milnor ring. $\qed$*]{}
[*Remarks.*]{}
\(i) The sign $\sim$ in item (i) means that rather than being isomorphic the complexes are filtered, and the corresponding graded complexes are naturally isomorphic. This is not too serious a complication; in fact, $\sim$ is a genuine isomorphism if $i<N-1$, and there is a 1-step filtration if $i=N-1$.
\(ii) The reader familiar with previous work might expect $ H^{*}(\fF,\Omega^{*}_{\fF}) \iso
H^{*}(\fF,\Omega^{ch}_{\fF})_{0}$ instead of the first isomorphism in the item (ii) of the theorem. We have indeed changed the conformal grading and find this important; so much so that in the main body of the text (see especially 2.3.3) we change the terminology and notation: we write $\Lambda^{ch}\CT_{\fF}$ for $\Omega^{ch}_{\fF}$ with the changed grading and call it the [*the algebra of chiral polyvector fields.*]{} $\qed$
Now we would like to make two points. First, let us demonstrate how this result works.
[*Application: an elliptic genus formula.*]{} Let $\text{Ell}_{\fF}(\tau,s)$ be the 2-variable elliptic genus of $\fF$ as defined, for example, in \[BL\] or \[KYY\].
Introduce $$E(\tau,s)=\prod_{n=0}^{\infty}\frac{\left(1-e^{2\pi
i\left(\left(n+1\right)\tau+\left(1-1/N\right)s\right)}\right)^{N}\left(1-e^{2\pi
i\left(n\tau+\left(-1+1/N\right)s\right)}\right)^{N}}{\left(1-e^{2\pi
i\left(\left(n+1\right)\tau+s/N\right)}\right)^{N}\left(1-e^{2\pi
i\left(n\tau-s/N\right)}\right)^{N}}.$$ It follows easily from Theorem 1 (see 4.11- 4.12) that $$\Ell_{\fF}(\tau,s)= \frac{1}{N}\sum_{l=0}^{N-1}\sum_{j=0}^{N-1}
e^{\pi
i\left(N-2\right)\left\{-js+\left(j^{2}-j\right)\tau+j^{2}\right\}}E(\tau,s-j\tau-l).
\eqno{(3)}$$ The structure of this formula is rather clear: the infinite product $E(\tau,s)$ reflects the polynomial nature of the space $\Omega^{ch}(\BC^{N})$ of which it is indeed the Euler character, the summation with respect to $l$ extracts the $\BZ_{N}$-invariants, and the summation w.r.t. $j$ reminds of the summation over “twisted sectors” because the change of variable $s\mapsto s-j\tau$ is reminiscent of the spectral flow. All of this smacks of an orbifold, and indeed formula (3) was proposed in \[KYY\] as the elliptic genus of the Landau-Ginzburg orbifold. Furthermore, it was shown in \[KYY\] that the specialization of the r.h.s. of (3) to $\tau=i\infty$ or $s=0$ gives Vafa’s orbifold formulas for the Poincare polynomial and the Euler characteristic of the Fermat hypersurface respectively. These formulas have been proved and further discussed in \[OR, R\]. Of course, both easily follow from (3).
This brings about the 2nd point we would like to make.
[*Landau-Ginzburg orbifold interpretation.*]{} The Landau-Ginzburg model is associated to an affine manifold and a function over it known as superpotential. In the case where the manifold is $\BC^{N}$ and the function is a homogeneous polynomial $f$ with a unique singularity at 0, Witten’s discovery \[W2\] can perhaps be formulated in the language accessible to us as follows:
*1) There is an action of the $N=2$ superconformal algebra on $\Omega^{ch}(\BC^{N})$ such that it commutes with the differential $df(z)_{(0)}$.*
2\) The cohomology vertex algebra $H_{df(z)_{(0)}}(\Omega^{ch}(\BC^{N}))$ with thus defined action of the $N=2$ superconformal algebra is the chiral algebra attached to the Landau-Ginzburg model with superpotential $f$.
We find it convenient not to pass to the cohomology but to declare the Landau-Ginzburg model to be the differential vertex algebra $(\Omega^{ch}(\BC^{N}), df(z)_{(0)})$ with the above fixed $N=2$ superconformal algebra action.
We shall argue in 5.1.5 that alternatively one can think of$(\Omega^{ch}(\BC^{N}), df(z)_{(0)})$ as the “right definition” of the chiral de Rham complex $\Omega^{ch}_{\text{Spec}M_{f}}$ over the spectrum of the Milnor ring. $\qed$
Next consider the space $\Omega^{ch}(\BC^{N})e^{nX^{*}_{orb}}$. It does not belong to $\BB_{\Lambda_{orb}}$ but carries a canonical structure of a [*twisted*]{} $\Omega^{ch}(\BC^{N})$-module, and this is synonymous to being a twisted sector. Therefore, taking the direct sum of these, $0\leq n\leq N-1$, and then extracting $\BZ_{N}$-invariants corresponds accurately with what physicists call the Landau-Ginzburg orbifold, see e.g. \[V\]. In the notation we have adopted, the formula describing the outcome of this process is $(\widetilde{\text{LG}}^{< N}; d_{LG})$; see sect. 5 for details. Thus item (i) of Theorem 1 can be interpreted as follows:
[*there is a spectral sequence abutting to $ H^{*}(\fF,\Omega^{ch}_{\fF})$ whose 1st term is isomorphic to the $D_{orb}$-cohomology of the Landau-Ginzburg orbifold $H_{D_{orb}}(\widetilde{\text{LG}}^{< N})$.*]{}
One can say that the Landau-Ginzburg orbifold $(\widetilde{\text{LG}}^{< N}; d_{LG})$ approximates the vertex algebra $ H^{*}(\fF,\Omega^{ch}_{\fF})$. This approximation is consistent with the $N=2$ supersymmetry. Indeed, on the one hand, the $N=2$ superconformal algebra action on the Landau-Ginzburg model commutes with $D_{orb}$ and all higher differentials and thus descends to an action on $ H^{*}(\fF,\Omega^{ch}_{\fF})$. On the other hand, since $\fF$ is Calabi-Yau, $ H^{*}(\fF,\Omega^{ch}_{\fF})$ carries a canonical $N=2$ superconformal algebra action \[MSV\]. A direct computation (Lemmas 4.10.1, 5.1.1, 5.2.14) shows that
[*both the $N=2$ superconformal algebra actions coincide.*]{}
Furthermore, not only one of the spaces involved in assertion (i) of Theorem 1, but both assertions (i, ii) themselves are reminiscent of some of the important developments in string theory. In order to explain this we shall have to pluck courage and discuss a little more of physics.
It seems that except for the torus case, see e.g. a mathematical exposition in \[KO\], the space of states of the model describing the string propagation on a manifold is unknown even as a vector space to say nothing about its algebraic, vertex or otherwise, structure. One striking result towards understanding what this might be is Gepner’s model proposed in \[G\]. Gepner’s paper, a combination of guesswork and computational [*tour de force*]{}, is not an easy reading. More conceptual approach emerged soon afterwards, e.g. \[V,VW\], proclaiming that the Landau-Ginzburg orbifold is equivalent (in this or that sense) to the string theory on Calabi-Yau hypersurfaces in weighted projective spaces. (So far as we can tell, apart from both the theories carrying an $N=2$ superconformal algebra action with the same central charge and integral $U(1)$-charges, most of the supporting evidence amounted to the isomorphism of chiral rings – exactly as in Theorem 1 (ii).) This activity seems to have been crowned by Witten’s paper \[W1\] where it is asserted, and we cite, “that rather than Landau-Ginzburg being “equivalent” to Calabi-Yau, they are two different phases of the same system.”
Now, if one is allowed to think of a phase transition as a family where at certain values of the parameter something happens, then it seems that spectral sequences might be relevant. For example, the spectral sequence of Theorem 1 comes from a double complex equipped with two differentials. Let us denote them for the purposes of introduction by $d_{+}$ and $d_{-}$. The vertex algebra $H^{*}(\fF,\Omega^{ch}_{\fF})$ is the cohomology of the total differential $d_{+}+d_{-}$, and $H_{D_{orb}}(\widetilde{\text{LG}}^{< N})$ arises as the $d_{-}$-cohomology. Introduce a parameter, $t$, and form the differential $td_{+}+d_{-}$. This defines a family of complexes over a line such that at $t=0$ the cohomology is $H_{D_{orb}}(\widetilde{\text{LG}}^{< N})$, and elsewhere it is $H^{*}(\fF,\Omega^{ch}_{\fF})$.
Furthermore, the geometric background as explained in sect. 4 of \[W1\] is very similar to that we use in the proof of Theorem 1.
[*Sketch of proof.*]{} The proof is based on the computation of two spectral sequences, both due to \[B\]. The first allows, in a sense, to replace $\fF$ with the canonical line bundle, $\CL^{*}$, over $\BP^{N-1}$. The 1st term of this sequence is $H^{*}(\CL^{*},\Omega^{ch}_{\CL^{*}})$ and the corresponding differential $d_{1}$ has the meaning of the chiral Koszul differential [^2]. Most of Theorem 1 is about identification of the complex $(H^{*}(\CL^{*},\Omega^{ch}_{\CL^{*}}), d_{1})$.
We identify this space in the following way. Along with $\CL^{*}$ consider $\CL^{*}-0$ obtained by deleting the zero section. The Čech complex that computes the desired $H^{*}(\CL^{*},\Omega^{ch}_{\CL^{*}})$ naturally embeds into the analogous Čech complex over $\CL^{*}-0$. Write this down schematically as $$\check{C}(\CL^{*})\hookrightarrow \check{C}(\CL^{*}-0).$$ Proposed in \[B\] there is a vertex algebra resolution that allows to extend the latter embedding to a resolution of complexes: $$\check{C}(\CL^{*})\hookrightarrow \check{C}(\CL^{*}-0)
\rightarrow \check{C}(\CL^{*}-0)^{(1)}
\rightarrow \check{C}(\CL^{*}-0)^{(2)}\rightarrow\cdots.$$ This resolution serves the same purpose as the Cousin resolution but has a different flavor: its terms $\check{C}(\CL^{*}-0)^{(j)}$ are identified with each other as vector spaces, but are spectral flow transforms of each other as vertex modules. The latter property is responsible for the occurrence of the orbifold twisted sectors.
There arises then the bi-complex $$0\rightarrow \check{C}(\CL^{*}-0)
\rightarrow \check{C}(\CL^{*}-0)^{(1)}
\rightarrow \check{C}(\CL^{*}-0)^{(2)}\rightarrow\cdots.
\eqno{(4)}$$ whose total cohomology is the desired $H^{*}(\CL^{*},\Omega^{ch}_{\CL^{*}})$. Finally, and this is the geometry bit reminiscent of \[W1\], there is an isomorphism $$(\BC^{N}-0)/\BZ_{N}\iso \CL^{*}-0.$$ Pulling bi-complex (4) back onto $(\BC^{N}-0)/\BZ_{N}$ and further writing it down in terms of the coordinates on the universal covering space $\BC^{N}-0$ allows to compute all the terms of the corresponding spectral sequence and thus identify $(H^{*}(\CL^{*},\Omega^{ch}_{\CL^{*}}), d_{1})$ with $(H_{D_{orb}}(\widetilde{\text{LG}}^{< N}), d_{LG})$ as asserted in Theorem (i). (Note that these two pull-backs are made possible by the [*naturality*]{} property of $\Omega^{ch}_{X}$, see 2.1.) $\qed$
The “conformal weight zero component of this argument” gives a self-contained computation of the cohomology algebra of polyvector fields $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ along with explicit “vertex” formulas for the cocycles representing the cohomology classes, see 4.13. Classically, the computation uses $H^{*}(\fF,\BC)$ obtained in \[Gr\], the Serre duality, and variations of the Hodge structure \[Don, Theorem 2.2\] – and even then powers of the class representing hyperplane sections require special, although simple, treatment. In our approach, it is the other way around: the fact that the Milnor ring is realized inside $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ is almost obvious, even its Koszul resolution arises naturally at the 1st term of the spectral sequence. On the other hand, the Serre duals of powers of the hyperplane section are “interesting” because they are produced by the twisted sectors as indicated in Theorem 1 (ii), 1st line in (2). We restore the multiplicative structure of $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ by vertex algebra methods twice: first, in 4.13 in the context of the proof of Theorem 1; second, in 5.2.19 by way of testing the vertex algebra structure on the space $H_{d_{LG}}(\widetilde{\text{LG}}^{< N})$ we propose in Theorem 5.2.18 in the case of the “diagonal” $f$. We believe that this vertex algebra structure is exactly Witten’s chiral algebra of the Landau-Ginzburg orbifold. Note that, in general, orbifoldizing multiplicative structures is a problem. The vertex algebra structure of Theorem 5.2.18 owes its existence to a remarkable periodicity property of the category of unitary modules over the $N=2$ superconformal algebra \[FS\].
[**Acknowledgements.**]{} The indebtedness of this work to Borisov’s constructions \[B\] should be clear to any reader. We gratefully acknowledge many illuminating conversations with V.Batyrev, A.Gerasimov, A.Givental, C.Hertling, M.Kapranov, R.Kaufmann, Yu.I.Manin, A.Semikhatov, A.Vaintrob. Special thanks go to V.Schechtman who participated at the early stages of this research and was the first enthusiast of the algebra of chiral polyvector fields. This work was started in 2002 when we were visiting the Max-Planck-Institut für Mathematik in Bonn and finished a year later at the same place. We are grateful to the institute for excellent working conditions.
[**1. Vertex algebras** ]{}
This section is only a collection of well-known facts and examples that will be needed in the sequel and the reader may want to consult either \[K\] or \[FB-Z\] for more detail. We would like to single out sect. 1.12 on the spectral flow, the notion that seems to be not too popular in mathematics literature but is essential for understanding orbifolds and will reappear several times in sect. 2.3.5, 3.10, 4.6, 5.2.15. Our treatment of the spectral flow is greatly influenced by \[LVW\].
[**1.1.**]{} A vector space $V$ is called a supervector space if it is $\BZ_{2}$-graded, that is, $V=V^{(0)}\oplus V^{(1)}$. We define the parity $\pr(a)$ of $a\in V$ so that $\pr(a)=\epsilon$ if and only if $a\in V^{(\epsilon)}$. If $V$ and $W$ are supervector spaces, then $V\otimes W$ is also with $\pr(a\otimes
b)=\pr(a)+\pr(b)$, and so is $\text{Hom}_{\BC}(V,W)$.
Given a supervector space $V$, let $\text{Field}(V)$ be the subspace of $\text{End}(V)[[z,z^{\pm 1}]]$ consisting of such formal series $x(z)=\sum_{n\in\BZ}x_{(n)}z^{-n-1}$ that for any $a\in V$ $$x_{(n)}a=0\text{ if } n>>0.
\eqno{(1.1.1)}$$
[**1.2. Definition.**]{} A vertex algebra is a supervector space $V$ with a distinguished element $\b1\in V$ called [*vacuum*]{} and a parity preserving map $Y(.,z):\; V\rightarrow \text{Field}(V)$, $Y(a,z)=\sum_{n\in\BZ}a_{(n)}z^{-n-1}$, such that the following axioms hold:
\(i) [*vacuum*]{}: $$Y(\b1,z)=Id_{V},\; a_{(-1)}\b1=a;
\eqno{(1.2.1)}$$ (ii) [*Borcherds identity:*]{} for any $a,b\in V$ and any rational function $F(z,w)$ in $z$, $w$ with poles only at $z=0$, $w=0$, $z-w=0$ $$\aligned
&\text{Res}_{z-w}Y(Y(a,z-w)b,w)i_{w,z-w}F(z,w)\\
&=
\text{Res}_{z}\left(Y(a,z)Y(b,w)i_{z,w}F(z,w)-
(-1)^{\pr(a)\pr(b)}Y(b,w)Y(a,z)i_{w,z}F(z,w)\right).
\endaligned
\eqno{(1.2.2)}$$ $\qed$
In the latter formula the standard notation is used: $\text{Res}_{t}$ means the coefficient of $t^{-1}$ in the indicated formal Laurent expansion; $i_{\bullet,\bullet}$ specifies exactly which Laurent expansion is to be used, e.g. $i_{z,w}$ stands for the expansion in the domain $|w|<|z|$, $i_{w,z-w}$ for that in the domain $|z-w|<|w|$, etc.
One thinks of $a_{(n)}$ as the “ n-th multiplication by $a$”, so there arises a family of multiplications $$_{(n)}:\; V\otimes V\rightarrow V.
\eqno{(1.2.3)}$$
[**1.3. W- and cohomology vertex algebras.**]{}
If $F=1$, then (1.2.2) gives $$(a_{(0)}b)_{(n)}=[a_{(0)},b_{(n)}],\; n\in\BZ.
\eqno{(1.3.1)}$$ In other words, for any $a\in V$, $a_{(0)}\in\text{End}V$ is a derivation of all the multiplications. Hence, $$\text{Ker}a_{(0)}\subset V
\eqno{(1.3.2)}$$ is a vertex subalgebra known as a $W$-algebra.
Furthermore, suppose $a\in V$ is odd and $a_{(0)}a=0$. Then (1.3.1) implies that $a_{(0)}^{2}=0$ and $\text{Im}a_{(0)}\subset\text{Ker}a_{(0)}$ is an ideal. Therefore, the cohomology $$H_{a_{(0)}}(V) \eqd \text{Ker}a_{(0)}/\text{Im}a_{(0)}
\eqno{(1.3.3)}$$ carries a canonical vertex algebra structure. The vertex algebras to be used in this text will mostly be either W- or cohomology vertex algebras.
[**1.4. Chiral rings.**]{} Suppose that $V$ is graded so that $$V=\oplus_{n=0}^{\infty} V_{n}\text{ and }
V_{n\;(r)}V_{m}\subset V_{n+m-r-1}.
\eqno{(1.4.1)}$$ The grading satisfying this condition will be called [*conformal*]{}.
One can show that if (1.4.1) is valid, then $$_{(-1)}: V_{0}\otimes V_{0}\rightarrow V_{0}$$ is associative and supercommutative. In the context of the unitary $N=2$ supersymmetry, supercommutative associative algebras attached to graded vertex algebras in this way are often called [*chiral rings*]{} \[LVW\] – not to be confused with chiral algebras although these rings are indeed algebras. We shall take the liberty to call these rings chiral in any case.
[**1.5. Remarks.**]{}
\(i) Let $\delta(z-w)=\sum_{n\in\BZ}z^{n}w^{-n-1}$. It follows from (1.2.2) that $[Y(a,z), Y(b,w)]$ is local, that is, equals a linear combination of the delta-function derivatives, $\partial_{w}^{n}\delta(z-w)$, over fields in $w$.
\(ii) A vertex algebra $V$ is said to be generated by a collection of fields $Y(a_{\alpha},z)$, $\{a_{\alpha}\}\subset V$ if $V$ is the linear span of (non-commutative) monomials in $(a_{\alpha})_{(j)}$ applied to vacuum $\b1\in V$. The important [*reconstruction theorem*]{}, e.g. \[K, Theorem 4.5\], says, and we are omitting some details, that this can be reversed: if there is a collection of mutually local fields $v_{\alpha}(z)$ which generate $V$ from a fixed vector, then $V$ carries a unique vertex algebra structure such that $v_{\alpha}(z)=Y(v_{\alpha},z)$ for some $v_{\alpha}\in V$. Because of this we will allow ourselves in our list of well-known examples, which we are about to begin, to fix only a space $V$ and a collection of mutually local fields that generate this space. Typically, we shall have a Lie algebra, a collection of fields with values in this algebra, and a representation of this algebra such that nilpotency condition (1.1.1) is satisfied by the fields.
\(iii) It should be clear what a vertex algebra homomorphism is. As in (ii), speaking of homomorphisms we shall often specify only images of generating fields.
Now to some basic excamples.
[**1.6. $bc$-system.**]{} Let $Cl$ be the Lie superalgebra with basis $b_{(i)}, c_{(i)}$, $i\in\BZ$ (all odd), and $C$ (even) and commutation relations $$[b_{(i)},c_{(j)}]=\delta_{i,-j-1}C,\; [C,c_{i}]=[C,b_{i}]=0,\; i,j\in\BZ.
\eqno{(1.6.1)}$$ Let $$F=\text{Ind}_{Cl}^{Cl_{+}}\BC,
\eqno{(1.6.2)}$$ where $Cl_{+}$ is the Lie subalgebra spanned by $x_{(i)}$, $C$, $i\geq 0$, and $\BC$ is an $Cl_{+}$-module where $x_{(i)}$’s act by 0 and $C$ as multiplication by 1. In terms of fields $x(z)\sum_{i\in\BZ}x_{(i)}z^{-i-1}$, $x=b$ or $c$, (1.6.1) becomes: $$[b(z),c(z)]=\delta(z-w).
\eqno{(1.6.3)}$$ Hence the vertex algebra structure on $F$, see 1.5.
A little more generally, to any purely odd $\BC$-vector $W$ with a non-degenerate symmetric form $(.,.)$ one can attach the Lie superalgebra $Cl(W)=W\otimes\BC[t,t^{-1}]\oplus\BC C$ with the following bracket: if $x(z)=\sum_{i\in\BZ}(x\otimes t^{i})z^{-i-1}$, then $$[x(z),y(z)]=\delta(z-w)(x,y)C,\; [x(z),C]=0.
\eqno{(1.6.4)}$$ The corresponding vertex algebra is $$F_{W}=\text{Ind}_{Cl(W)_{+}}^{Cl(W)}\BC,
\eqno{(1.6.5)}$$ where $Cl(W)_{+}=W\otimes \BC[t]\oplus\BC C$, $W\otimes \BC[t]$ operates on $\BC$ by 0, and $C$ by 1.
[**1.7. $\beta\gamma$- and $bc-\beta\gamma$-system.**]{} The $\beta\gamma$-system is obtained by the “parity change” functor applied to the beginning of 1.6: the even Lie algebra $\fa$ is spanned by $\beta_{(i)}$, $\gamma_{(i)}$, $C$, the bracket is $$[\beta(z),\gamma(w)]=-[\gamma(z),\beta(w)]=\delta(z-w)C,\; [C, x(z)]=0.
\eqno{(1.7.1)}$$ The vertex algebra, $B$, is defined in the same way as $F$, see (1.6.2).
If $V$ and $W$ are vertex algebras, then $V\otimes W$ carries the standard vertex algebra structure. Denote $$FB=F\otimes B.
\eqno{(1.7.2)}$$ This algebra and its modifications are local models for the chiral de Rham complex.
[**1.8. Heisenberg algebra.**]{} Let now $\fh$ be a purely even vector space with a non-degenerate symmetric form $(.,.)$. There arises then the Heisenberg Lie algebra $$\hat{\fh}=\fh\otimes\BC[t,t^{-1}]\oplus\BC\cdot C$$ with bracket defined by letting fields be $x(z)=\sum_{i\in\BZ}(x\otimes t^{i})z^{-i-1}$ and then setting $$[a(z),b(w)]=(a,b)\partial_{w}\delta(z-w),\; [C,a(z)]=0.
\eqno{(1.8.1)}$$ The vertex algebra attached to this Lie algebra is $$V(\fh)=\text{Ind}_{\hat{\fh}_{+}}^{\hat{\fh}}\BC,
\eqno{(1.8.2)}$$ where $\hat{\fh}_{+}=\fh\otimes \BC[t]\oplus\BC\cdot C$, $\hat{\fh}_{+}=\fh\otimes \BC[t]$ perates on $\BC$ by zero, $C$ by 1.
[**1.9. Lattice vertex algebras.**]{} We shall neeed a lattice $L$, that is, a free abelian group with integral bilinear form $(.,.)$ and a 2-cocycle $$\epsilon:\; L\times L\rightarrow \BC^{*}.
\eqno{(1.9.1)}$$ There arise the group algebra $\BC[L]$ with multiplication $e^{\alpha}\cdot e^{\beta}=e^{\alpha+\beta}$, $\alpha,\beta\in L$, and the twisted group algebra, $\BC_{\epsilon}[L]$, equal to $\BC[L]$ as a vector space but with twisted multiplication: $$e^{\alpha}\cdot_{\epsilon}
e^{\beta}=\epsilon(\alpha,\beta)e^{\alpha+\beta}.
\eqno{(1.9.2)}$$ Let $\fh_{L}=\BC\otimes_{\BZ} L$. There arises the Heisenberg vertex algebra $V(\fh_{L})$, see (1.8.2).
As a vector space, the lattice vertex algebra is defined by $$V_{L}=V(\fh_{L})\otimes \BC_{\epsilon}[L],\;
\pr (V(\fh_{L})\otimes e^{\alpha})\equiv (\alpha,\alpha)\text{ mod }2,
\eqno{(1.9.3)}$$ where $\pr$ means the parity, see 1.1.
This vertex algebra is generated by the familiar fields $x(z)=\sum_{i\in\BZ}(x\otimes t^{i})z^{-i-1}$ attached to $x_{(-1)}\otimes 1$ and the celebrated vertex operators $$e^{\alpha}(z)=e^{\alpha}
\exp{(\sum_{j<0}\frac{\alpha_{(j)}}{-j}z^{-j})}
\exp{(\sum_{j>0}\frac{\alpha_{(j)}}{-j}z^{-j})}z^{\alpha_{(0)}},
\eqno{(1.9.4)}$$ attached to $1\otimes e^{\alpha}$.
The action of $x_{(i)}$, $i\neq 0$, ignores $\BC_{\epsilon}(L)$, the action of $x_{(0)}$ is uniquely determined by $$x_{(0)}(1\otimes e^{\alpha})=(\alpha,x)\otimes e^{\alpha}.
\eqno{(1.9.5)}$$ The following commutator and OPE formulas are valid: $$[x(z),e^{\alpha}(w)]=\delta(z-w)(x,\alpha)e^{\alpha}(w),
\eqno{(1.9.6)}$$ $$e^{\alpha}(z)e^{\beta}(w)=(z-w)^{(\alpha,\beta)}:e^{\alpha}(z)e^{\beta}(w):,
\eqno{(1.9.7)}$$ and the reader is advised to consult \[K, (5.4.5b)\] for the meaning of the two-variable field $:e^{\alpha}(z)e^{\beta}(w):$. Note that (1.9.7) allows to compute all operations $(e^{\alpha})_{(n)}e^{\beta}$. We shall need the following particular cases: $$\aligned
(e^{\alpha})_{(-1)}e^{\beta}=&\lim_{z\rightarrow w}
(z-w)^{(\alpha,\beta)}:e^{\alpha}(z)e^{\beta}(w):=\\
&\left\{\aligned
0&\text{ if }(\alpha,\beta)>0\\
\epsilon(\alpha,\beta)e^{\alpha+\beta}&\text{ if }(\alpha,\beta)=0,
\endaligned\right.
\endaligned
\eqno{(1.9.8)}$$ $$[e^{\alpha}(z),e^{\beta}(w)]=0\text{ if } (\alpha,\beta)\geq 0.
\eqno{(1.9.9)}$$
For the future use let us mention that for any sub-semigroup $M\subset L$ there arises the vertex subalgebra $$V_{M,L} \buildrel\text{def}\over = V(\fh_{L})\otimes \BC_{\epsilon}[M]\subset V_{L}.
\eqno{(1.9.10)}$$ naturally graded by $M$.
[**1.10. $N=2$ super-Virasoro algebra.**]{} The celebrated $N=2$ super-Virasoro algebra, to be denoted $N2$ following \[K\], is a supervector space with basis $G_{(n)}$, $Q_{(n)}$, $n\in\BZ$ (all odd), $L_{(n)}$, $J_{(n)}$, $n\in\BZ$, $C$ (all even), and bracket $$\aligned
[L(z),L(w)]&=2\partial_{w}\delta(z-w)L(w)+ \delta(z-w)L(w)'\\
[J(z),J(w)]&=\partial_{w}\delta(z-w)C/3,
\endaligned
\eqno{(1.10.1a)}$$ $$\aligned
[L(z),G(w)]&=2\partial_{w}\delta(z-w)G(w)+
\delta(z-w)G(w)',\\
[J(z),G(w)]&=\delta(z-w)G(w),
\endaligned
\eqno{(1.10.1b)}$$ $$\aligned
[L(z),Q(w)]&= \partial_{w}\delta(z-w)Q(w)+
\delta(z-w)Q(w)',\\
[J(z),Q(w)]&=-\delta(z-w)Q(w),
\endaligned
\eqno{(1.10.1c)}$$ $$[L(z),J(w)]= \partial_{w}^{2}\delta(z-w)\frac{C}{6}+
\partial_{w}\delta(z-w)J(w)+
\delta(z-w)J(w)',
\eqno{(1.10.1d)}$$ $$[Q(z),G(w)]= \partial_{w}^{2}\delta(z-w)\frac{C}{6}
-\partial_{w}\delta(z-w)J(w)+
\delta(z-w)L(w).
\eqno{(1.10.1e)}$$ The vertex algebra structure is carried by the following $N2$-module: $$V(N2)_{c}=\text{Ind}_{N2_{\geq}}^{N2}\BC_{c}.
\eqno{(1.10.2)}$$ where $N2_{\geq}$ is the subalgebra linearly spanned by $G_{(n)},L_{(n)},Q_{(n)},J_{(n)},C$, $n\geq 0$, and on $\BC_{c}$ $G_{(n)},L_{(n)},Q_{(n)},J_{(n)}$ operate by 0, and $C$ as multiplication by $c$.
[**1.10.1.**]{} [*Definition.*]{} An $N2$-structure on a vertex algebra $W$ is a vertex algebra homomorphism $V(N2)_{c}\rightarrow W$. $\qed$
Note that the field $L(z)$ generates the Virasoro algebra; thus an $N2$-structure on a vertex algebra induces a conformal structure, and the grading by eigenvalues of $L_{(1)}$ is conformal, cf. (1.4.1).
[**1.11. Automorphisms.**]{} It is obvious that if $W$ is a vector space with a symmetric non-degenerate bilinear form, then there is an embedding $$\text{O}(W)\hookrightarrow \text{Aut}F_{W},
\eqno{(1.11.1)}$$ where $\text{O}(W)$ is the orthogonal group and $F_{W}$ a vertex algebra defined in (1.6.5); this comes from the standard action of $\text{O}(W)$ on the Clifford Lie algebra $Cl(W)$.
The analogous construction with $\text{O}(W)$ replaced with $\text{Aut}(L)$ and $F_{W}$ with $V_{L}$ does not quite work because of the cocycle (1.9.1). Here is one trivial observation: if we let $\text{Aut}_{\epsilon}(L)$ be the subgroup of $\text{Aut}(L)$ stabilizing $\epsilon$, then there is an (obvious) embedding: $$\text{Aut}_{\epsilon}(L)\hookrightarrow \text{Aut}V_{L}.
\eqno{(1.11.2)}$$
The Lie algebra $N2$ affords an exceptional, mirror symmetry automorphism $$Q(z)\mapsto G(z), G(z)\mapsto Q(z), J(z)\mapsto -J(z),
L(z)\mapsto L(z)+J(z)'.
\eqno{(1.11.3)}$$
[**1.12. Spectral flow.**]{} In all our examples except for $V_{L}$ vertex algebras have come from infinite dimensional Lie algebras. One feature these Lie algebras have in common is that they admit a [*spectral flow*]{}.
Define for any $n\in\BZ$ a linear transformation: $$S_{n}:\;\;
\aligned
Cl\rightarrow Cl,\; \text{s.t. }&b(z)\rightarrow b(z)z^{-n},
c(z)\rightarrow c(z)z^{n},\\
\fa\rightarrow\fa,\; \text{s.t. }&\beta(z)\rightarrow \beta(z)z^{-n},
\gamma(z)\rightarrow \gamma(z)z^{n},\\
N2\rightarrow N2,\; \text{s.t. }&Q(z)\rightarrow Q(z)z^{n},
G(z)\rightarrow G(z)z^{-n},\\
& J(z)\mapsto J(z)-\frac{1}{z}\frac{nC}{3},\\
& L(z)\mapsto L(z)-\frac{1}{z} nJ(z)+\frac{1}{z^{2}}n(n-1)\frac{C}{6},
\endaligned
\eqno{(1.12.1)}$$ where we abused the notation by letting the same letter stand for the maps of different spaces, $Cl$, $\fa$, $N2$, defined in 1.6,7,10 resp. We hope this will not lead to confusion. An untiring reader will check that in each of the cases, $S_{n}$ is an automorphism of the Lie algebra in question.
Maps (1.12.1) generate a $\BZ$-action on each of the algebras known as the spectral flow. In each of the cases, therefore, there arises a family of functors on the category of modules $$S_{n}: \text{Mod}\rightarrow\text{Mod},\;
M\mapsto S_{n}(M),
\eqno{(1.12.2)}$$ action on $S_{n}(M)$ being defined by precomposing that on $M$ with $S_{n}$ of (1.12.1).
The origin of spectral flows (1.12.1) belongs to lattice vertex algebras. In order to see this, let $V_{L}$ be a lattice vertex algebra and $\text{Lie}V_{L}$ the linear span inside $\text{End}V_{L}$ of the coefficients of the fields $v(z)$, $v\in
V_{L}$. It is well known \[K, F-BZ\] that $\text{Lie}V_{L}$ is a Lie subalgebra of $\text{End}V_{L}$.
Furthermore, if $M\subset L$ is a sub-semigroup, then we have, cf. (1.9.10), $$V_{M,L}\subset V_{L},\; \text{Lie}V_{M,L}\subset \text{Lie}V_{L},
\eqno{(1.12.3)}$$ Let $$e^{\alpha}: V_{L}\rightarrow V_{L}
\eqno{(1.12.4)}$$ be multiplication by $e^{\alpha}$. If the restriction of the cocycle $\epsilon(.,.)$ to $M\subset L$ is trivial, that is, $$\epsilon(M,.)=1,
\eqno{(1.12.5)}$$ then the conjugation by map (1.12.4) defines an automorphism $$S_{\alpha}: \text{Lie}V_{M,L}\rightarrow \text{Lie}V_{M,L},\;
X\mapsto (e^{\alpha})^{-1}\circ X\circ e^{\alpha};\; \alpha\in L.
\eqno{(1.12.6)}$$ For example, under this map $$e^{\beta}(z)\mapsto e^{\beta}(z)z^{(\alpha,\beta)},\; \beta\in M
\eqno{(1.12.7)}$$ cf. (1.12.1); the desired power of $z$ owes its appearance to the factor $z^{\alpha_{(0)}}$ in (1.9.4). Likewise, $$x(z)\mapsto x(z)+\frac{(\alpha,x)}{z}.
\eqno{(1.12.8)}$$ All spectral flows (1.12.1) are obtained as follows: embed the corresponding vertex algebra into an appropriate $V_{L}$, thus obtain an embedding of the corresponding Lie algebra $\text{Lie}(\bullet)\rightarrow \text{Lie}V_{L}$, and then restrict “spectral flow in the direction $\alpha$” (1.12.6) to the image. This operation will be of importance for us in 3.10, 5.2.15. Here is another such example.
[**1.13. Boson-fermion correspondence.**]{} Let $\BZ$ be the standard 1-dimensional lattice: this means that if we let $\chi$ be the generator, then $(\chi,\chi)=1$. There arise $V_{\BZ}$, where $\epsilon(.,.)=1$, and the famous vertex algebra isomorphism: $$\aligned
&F\iso V_{\BZ},\; b(z)\mapsto e^{\chi}(z),\\
&c(z)\mapsto e^{-\chi}(z), :b(z)c(z):\mapsto \chi(z)
\endaligned
\eqno{(1.13.1)}$$ The interested reader will check that $S_{N}|_{Cl}$ of (1.12.1) is indeed implemented by conjugation with $e^{-n\chi}$.
Likewise $$\aligned
&F^{\otimes n}\iso V_{\BZ^{n}},\; b_{i}(z)\mapsto e^{\chi_{i}}(z),\\
&c_{i}(z)\mapsto e^{-\chi_{i}}(z), :b_{i}(z)c_{i}(z):
\mapsto \chi_{i}(z),
\endaligned
\eqno{(1.13.2)}$$ where the cocycle on $V_{\BZ^{n}}$ is chosen so as to ensure that $V_{\BZ^{n}}\iso V_{\BZ}^{\otimes n}$.
[**2. The algebra of chiral polyvector fields** ]{}
This section is a reminder on the chiral de Rham complex. Our exposition is close to \[MSV\] but has been influenced by \[GMS\]. We would like to single out 2.3.3, where the title is clarified, and 2.3.5, where a simple cohomology computation is carried out; the results of this computation will play an important role in the proof of Theorem 4.7. Sect. 2.4 is an exposition of a result of \[B\].
[**2.1.**]{} Suppose we have a family of sheaves of vector spaces $\CA_{X}$, one for each smooth algebraic manifold $X$. We shall call $\CA_{X}$ [*natural*]{} if for any étale morphism $\phi: Y\rightarrow X$ there is a sheaf embedding $\CA(\phi): \phi^{-1}\CA_{X}\hookrightarrow\CA_{Y}$ such that given a diagram: $$X\buildrel \phi\over\rightarrow Y\buildrel \psi\over\rightarrow Z$$ the following associativity condition holds: $$\CA(\psi\circ\phi)=\CA(\phi)\circ \phi^{-1}(\CA(\psi)),$$ where $\phi^{-1}$ is understood as the inverse image functor on the category of sheaves of vector spaces.
It should be clear what a natural sheaf morphism $\CA_{X}\rightarrow\CB_{X}$ means.
Here are some obvious examples: $\CO_{X}$, $\Omega^{1}_{X}$, $\CT_{X}$, and sheaves obtained as a result of all sorts of tensor operation performed on these. Note that all these sheaves are sheaves of $\CO_{X}$-modules, and our definition ignores this extra structure. However, one talks about natural sheaves $\CA_{X}$ of different classes of algebras, such as commutative, associative, Lie, vertex, etc., by requiring that $\CA_{\phi}$ preserve this structure.
Constructed in \[MSV\] for any smooth algebraic manifold $X$ there is a sheaf of vertex algebras, $\Omega^{ch}_{X}$. It satisfies the following conditions.
\(i) $\Omega^{ch}_{X}$ is natural as a sheaf of vertex algebras and it carries a bi-grading $\Omega^{ch}_{X}=\oplus_{m,n}\Omega^{ch,m}_{X, n}$ such that each homogeneous component $\Omega^{ch,m}_{X, n}$ is a natural sheaf of vector spaces.
\(ii) There are natural morphisms: $$\Omega^{*}_{X}\hookrightarrow \Omega^{ch}_{X}\hookleftarrow
\Lambda^{*}\CT_{X};
\eqno{(2.1.1)}$$
\(iii) $\Omega^{ch}_{X}$ is not a sheaf of $\CO_{X}$-modules, but it carries a filtration such that there is the following family of natural sheaf isomorphisms $$\text{Gr}\Omega^{ch}_{X}\iso \bigotimes_{n\geq 0} \left(S^{*}_{q^{n+1}}(\CT_{X})\otimes
S^{*}_{q^{n}}(\CT_{X}^{*})\otimes
\Lambda^{*}_{q^{n+1}y}(\CT_{X})\otimes\Lambda^{*}_{q^{n}y^{-1}}
(\CT_{X}^{*})\right).
\eqno{(2.1.2)}$$ where we habitually use the following “ generating functions of families of sheaves”: $$\text{Gr}\Omega^{ch}_{X}=\bigoplus_{m,n}
q^{n}y^{m}\text{Gr}\Omega^{ch,m}_{X,n},
S^{*}_{t}(\CA)=\bigoplus_{n=0}^{\infty}t^{n}S^{n}(\CA),
\Lambda^{*}_{t}(\CA)=\bigoplus_{n=0}^{\infty}t^{n}\Lambda^{n}(\CA).$$ (iv) it follows from (i) that for any $X$ there is a canonical group embedding $$\rho_{X}: \text{Aut}X\rightarrow\text{Aut}\Omega^{ch}_{X}(X).
\eqno{(2.1.3)}$$ Explicit formulas for the latter appeared in \[MSV, (3.1.6)\] as a result of guesswork and were used to define $\Omega^{ch}_{X}$ satisfying (i-iii).
In 2.2 we shall look at some examples that serve as a local model and are needed later; in 2.3 we shall very briefly discuss how these local models are glued together and what effect the gluing has on $N2$-structures and chiral rings.
[**2.2. A local model.**]{}
It is easiest to begin with a local situation in the presence of a coordinate system.
[**2.2.1.**]{} Let $U$ be a smooth affine manifold with a coordinate system $\vec{x}$ by which we mean a collection of functions $x_{1},...,x_{n}\in\CO(U)$, $n=\text{dim }U$, such that the differential forms $dx_{1},...,dx_{n}$ form a basis of the space of 1-forms $\CT^{*}(U)$ over $\CO(U)$. A coordinate system determines a collection of vector fields $\partial_{x_{1}},...,\partial_{x_{n}}$ such that $$\partial_{x_{i}}x_{j}=<\partial_{x_{i}},dx_{j}>=\delta_{ij}.$$ It follows that $[\partial_{x_{i}},\partial_{x_{j}}]=0$ for all $i,j$, and $\partial_{x_{1}},...,\partial_{x_{n}}$ form a basis of the space of vector fields $\CT^{*}(U)$.
Let $\Omega^{ch}(U,\vec{x})$ be the following superpolynomial ring over $\CO(U)$. $$\Omega^{ch}(U,\vec{x})=\CO(U)[x_{i,(-j-1)},\;\partial_{x_{i},(-j)};\;
dx_{i,(-j)}\partial_{dx_{i},(-j)},\;1\leq i\leq n,\, j\geq 1],
\eqno{(2.2.1)}$$ the generators $x_{i,(-j-1)},\partial_{x_{i},(-j)}$ being even, $dx_{i,(-j)}\partial_{dx_{i},(-j)}$ odd. Identifying $x_{i,(-j-1)}$, $\partial_{x_{i},(-j)}$ with different even copies of $dx_{i}$ and $\partial_{x_{i}}$ resp., and $dx_{i,(-j)}$, $\partial_{dx_{i},(-j)}$ with different odd copies thereof, we obtain an identification of superalgebras $$\Omega^{ch}(U,\vec{x})\iso \bigotimes_{n\geq 0}(S^{*}(\CT(U))\otimes
S^{*}(\CT^{*}(U))\otimes\Lambda^{*}(\CT(U))\otimes\Lambda^{*}(\CT^{*}(U))).
\eqno{(2.2.2)}$$ This is a local version of (2.1.2), and the images of embeddings (2.1.1) are generated over $\CO(U)$ by $dx_{i,(-1)}$, $\partial_{dx_{i}, (-1)}$.
The ring $\Omega^{ch}(U,\vec{x})$ carries a canonical vertex algebra structure \[MSV\]. To formulate the result introduce an even derivation $T\in\text{End}\Omega^{ch}(U,\vec{x})$ determined by the conditions $$T(f)=\sum_{i}x_{i,(-2)}\partial_{x_{i}}f,\;
T(a_{(-n)})=na_{(-n-1)},\eqno{(2.2.3)}$$ where $f\in\CO(U)$, $a=x_{i}$, $\partial_{x_{i}}$, or $\partial_{dx_{i}}$, and $n\geq 1$. Note that under identification (2.2.2) the first of these conditions says that $T(\CO(U))\subset\CT^{*}(U)$ and the restriction $T|_{\CO(U)}$ equals the de Rham differential.
[**2.2.2. Lemma.**]{} [*There is a unique vertex algebra structure on $\Omega^{ch}(U,\vec{x})$*]{} $$Y:\;\Omega^{ch}(U,\vec{x})\rightarrow\text{Field}(\Omega^{ch}(U,\vec{x}))$$
*determined by the conditions:*
\(i) $\Omega^{ch}(U,\vec{x})$ is generated by the fields $Y(f,z)$, $f\in\CO(U)$, $Y(\partial_{x_{i},(-1)},z)$, $Y(\partial_{dx_{i},(-1)},z)$, $Y(dx_{j,(-1)},w)$, the list of non-zero brackets amongst them being as follows: $$[Y(\partial_{x_{i},(-1)},z),Y(f,z)]=\delta(z-w)Y(\partial_{x_{i}}f,w),
\eqno{(2.2.4a)}$$ $$[Y(\partial_{dx_{i},(-1)},z),Y(dx_{j,(-1)},w)]=\delta_{ij}\delta(z-w);
\eqno{(2.2.4b)}$$ (ii) $T$-covariance: $$\aligned
& \; [T,a(z)]=a(z)',\\
& a(z)= Y(f,z), Y(\partial_{x_{i},(-1)},z),
Y(\partial_{dx_{i},(-1)},z), Y(dx_{j,(-2)},w); \endaligned
\eqno{(2.2.5)}$$ (iii) vacuum:
$$\; Y(1,z)=\text{Id},\; Y(f,z)g|_{z=0}=fg, Y(a,z)1|_{z=0}=a,\eqno{(2.2.6)}$$ [*where $1,f,g\in\CO(U)$, $a= \partial_{x_{i},(-1)}$, $dx_{i,(-1)}$ or $\partial_{dx_{i}, (-1)}$.*]{}
The uniqueness assertion of this lemma is an immediate consequence of Theorem 4.5 in \[K\]. While proving the existence assertion in general is something of a problem, in many examples, sufficient for our present purposes, this is easy. Before we begin discussing these examples, let us unburden the notation by setting: $$\aligned
f(z)=Y(f,z),\;
&dx_{i}(z)=Y(dx_{i,(-1)},z),\\
\partial_{x_{i}}(z)=Y(\partial_{x_{i},(-1)},z),\;
&\partial_{dx_{i}}(z)=Y(\partial_{dx_{i},(-1)},z).
\endaligned
\eqno{(2.2.7)}$$
[**2.2.3.**]{} [*Example: an affine space.*]{} If $U=\BC$ with the canonical coordinate $x=\vec{x}$, then $\Omega^{ch}(\BC,x)$ is nothing but $FB$ of (1.7.2). Indeed, since in this case $\CO(U)=\BC[x]$, $\Omega^{ch}(\BC,x)$ is generated by the fields $x(z)$, $\partial_{x}(z)$, $dx(z)$, $\partial_{dx}(z)$, which according to (2.2.4a,b) satisfy $$[\partial_{x}(z), x(w)]=\delta(z-w),\;
[\partial_{dx}(z), dx(w)]=\delta(z-w).$$ A quick glance at (1.6.3, 1.7.1) shows that $$b(z)\mapsto \partial_{dx}(z), c(z)\mapsto dx(z),
\gamma(z)\mapsto x(z), \beta(z)\mapsto\partial_{x}(z)$$ identifies $\Omega^{ch}(\BC,x)$ with $FB$.
Likewise, $$\Omega^{ch}(\BC^{N},\vec{x})=FB^{\otimes n}.
\eqno{(2.2.8)}$$ Incidentally, the same formulas define a vertex algebra morphism $$FB^{\otimes N}\hookrightarrow \Omega^{ch}(U,\vec{x}).
\eqno{(2.2.9)}$$ The nature of this morphism is this: a coordinate system $\vec{x}$ determines an étale map $U\rightarrow\BC^{N}$; hence (2.2.9) is a manifestation of the naturality of $\Omega^{ch}_{X}$, see 2.1.
[**2.2.4.**]{} [*Example: localization of an affine space.*]{} Let $f\in\BC[x_{1},...,x_{n}]$, $U_{f}=\BC^{N}-
\{\vec{x}:f(\vec{x})=0\}$, and $\BC[x_{1},...,x_{n}]_{f}$ the corresponding localization. To extend the vertex algebra structure from $\Omega^{ch}(\BC^{N},\vec{x})$ to $$\Omega^{ch}(U_{f},\vec{x})=\BC[x_{1},...,x_{n}]_{f}\otimes_{\BC[x_{1},...,x_{n}]}
\Omega^{ch}(\BC^{N},\vec{x})$$ it suffices to define the field $f^{-1}(z)$. In \[MSV\] an explicit formula for this field was written down using Feigin’s insight. Lemma 2.2.2 is a convenient alternative tool to compute the action of this (and similar) fields. Indeed, in view of the commutation relations (2.2.4a-b) it suffices to know $f^{-1}(z)_{(n)}g$, $g\in\CO(U)$. Due to (2.2.6) we have $$f^{-1}(z)_{(n)}g=\left\{\aligned 0&\text{ if }n\geq 0\\
\frac{g}{f}&\text{ if }n=-1.\endaligned\right.$$ The values $f^{-1}(z)_{(n)}g$, $n\leq -2$, are determined by using (2.2.5). The case where $g=1$ suffices and the repeated application of (2.2.5) gives $$f^{-1}(z)1=e^{zT}f^{-1}.$$ For example, $$f^{-1}(z)_{(-2)}1=T(\frac{1}{f})=
-\sum_{i}x_{i,(-2)}\frac{\partial_{x_{i}}f}{f^{2}}.$$
We shall mostly need localization to the complements of hyperplanes. The corresponding vertex algebras can be realized, thanks to \[B\], inside lattice vertex algebras; this will be reviewed in some detail in sect. 3.
We shall need two morphisms of vertex algebras $$\rho_{1},\rho_{2}:\;V(N2)_{3n}\rightarrow
\Omega^{ch}(U,\vec{x}),\; n=\text{dim}U.$$ The first was used in \[MSV\] and in terms of fields is defined by $$\aligned
Q^{(1)}(z)&=\sum_{i}dx_{i}(z)\partial_{x_{i}}(z),\;
G^{(1)}(z)=\sum_{i}:x_{i}(z)'\partial_{dx_{i}}(z),\\
J^{(1)}(z)&=-\sum_{i}:dx_{i}(z)\partial_{dx_{i}}(z):,
L^{(1)}(z)=\sum_{i}:x_{i}(z)'\partial_{x_{i}}(z):+
:dx_{i}(z)'\partial_{dx_{i}}(z):,\endaligned
\eqno{(2.2.10)}$$ where we let $A^{(1)}=\rho_{1}(A)$, $A$=$Q$, $G$, $J$, or $L$.
The second is obtained by composing the first with automorphism (1.11.3); the result is this: $$\aligned
Q^{(2)}(z)&=\sum_{i}:x_{i}(z)'\partial_{dx_{i}}(z),\;
G^{(2)}(z)=\sum_{i}dx_{i}(z)\partial_{x_{i}}(z),\\
J^{(2)}(z)&=\sum_{i}:dx_{i}(z)\partial_{dx_{i}}(z):,
L^{(2)}(z)=\sum_{i}:x_{i}(z)'\partial_{x_{i}}(z):-
:dx_{i}(z)\partial_{dx_{i}}(z)':.\endaligned \eqno{(2.2.11)}$$ As was noted in 1.10.1, the operators $L^{(i)}_{(1)}$ give two conformal gradings and a simple computation shows that the corresponding chiral rings, 1.4, are as follows: $$\aligned
\text{Ker}L^{(1)}_{(1)} = \CO(U)[dx_{1,(-1)},...,dx_{n,(-1)}],
\text{Ker}L^{(2)}_{(1)} = \CO(U)[\partial_{dx_{1},(-1)},...,\partial_{dx_{n},(-1)}].
\endaligned
\eqno{(2.2.12)}$$
[**2.3.**]{} [*Gluing the local models.*]{}
[**2.3.1.**]{} Localisation procedure explained in 2.2.4 carries over to any $\Omega^{ch}(U,\vec{x})$, see 2.2.1, and defines, in the presence of a coordinate system, a sheaf of vertex algebras $$U\supset V\mapsto \Omega^{ch}_{U,\vec{x}}(V)\buildrel \text{def}\over =
\Omega^{ch}(V,\vec{x})$$ over $U$. By using the action of the group of coordinate changes \[MSV, (3.1.6)\] one obtains canonical identifications $$\Omega^{ch}_{U,\vec{x}}\iso \Omega^{ch}_{U,\vec{y}}$$ for any two coordinate systems $\vec{x}$, $\vec{y}$. This defines a family of sheaves $U\mapsto \Omega^{ch}_{U}$, where $U$ is étale over $\BC^{N}$, natural w.r.t. to étale morphisms.
Finally covering a smooth manifold $X$ by charts $\{U_{\alpha}\}$ étale over $\BC^{N}$ one defines a sheaf $\Omega^{ch}_{X}$ by gluing over intersections according to the diagram $$\Omega^{ch}_{U_{\alpha}}\hookrightarrow
\Omega^{ch}_{U_{\alpha}\cap U_{\beta}}\hookleftarrow
\Omega^{ch}_{U_{\beta}}.$$ Let us recall, briefly but in some more detail, the effect of this procedure on the $N2$-structure.
[**2.3.2.**]{} [*Two $N2$-structures, the chiral de Rham complex and algebra of chiral polyvector fields.*]{} It was computed in \[MSV\] that a coordinate change $\vec{x}\mapsto\vec{y}$, via (2.1.3), induces the following transformation of fields (2.2.10): $$\aligned
Q^{(1)}(z)&\mapsto Q^{(1)}(z)+(d_{DR}(\text{Tr}\log\{(\partial_{x_{i}}y_{j})\}))(z)',\\
G^{(1)}(z)&\mapsto G^{(1)}(z),\\
J^{(1)}(z)&\mapsto J^{(1)}(z)
+(\text{Tr}\log\{(\partial_{x_{i}}y_{j})\})(z)',\\
L^{(1)}(z)&\mapsto L^{(1)}(z),\\
\endaligned
\eqno{(2.3.1)}$$ and of course similar transformation formulas can be written for fields (2.2.11). It follows that $$L^{(i)}(z)_{(1)}, J^{(i)}(z)_{(0)},\; i=1,2,$$ is a well-defined quadruple of operators acting on $\Omega^{ch}_{X}$. Since $[L^{(i)}(z)_{(1)}, J^{(i)}(z)_{(0)}]=0$, there arise two competing bi-gradings by “conformal weight, fermionic charge”: $$\aligned &\Omega^{ch}_{X}=\oplus_{n\geq
0,m\in\BZ}\, ^{(i)}\!\Omega^{ch,m}_{X,n},\; i=1,2,\\
&^{(i)}\!\Omega^{ch,m}_{X,n}=\text{Ker}(L^{(i)}(z)_{(1)}-n\text{Id})\cap
\text{Ker}(J^{(i)}(z)_{(0)}-m\text{Id}).
\endaligned\eqno{(2.3.2)}$$
Formula (2.2.12) shows that the chiral ring, 1.4, now technically a sheaf of chiral rings, associated to the first is the algebra of differential forms: $$\CC^{(1)}=\Omega^{*}_{X}:\; U\mapsto \CO(U)[dx_{1,(-1)},...,dx_{n,(-1)}];
\eqno{(2.3.3a)}$$ the chiral ring associated to the second is the algebra of polyvector fields $$\CC^{(2)}=\Lambda^{*}\CT_{X},\; U\mapsto \CO(U)[\partial_{dx_{1},(-1)},...,\partial_{dx_{n},(-1)}].
\eqno{(2.3.3b)}$$ This is how morphisms (2.1.1) come about.
[**2.3.3.**]{} [*Terminology.*]{} From now on we shall call $\Omega^{ch}_{X}$ equipped with grading (2.3.2) where $i=2$ the algebra of chiral polyvector fields and re-denote it by $\Lambda^{ch}\CT_{X}$. The sheaf $\Omega^{ch}_{X}$ equipped with grading (2.3.2) where $i=1$ will retain the name of the chiral de Rham complex.
Assertions (2.3.3a,b) are one justification of this terminology. Note that (2.1.2) uses bi-grading (2.3.2,i=1); the i=2 analogue is as follows: $$\text{Gr}\Lambda^{ch}\CT_{X}\iso
\bigotimes_{n\geq 0}\left( S^{*}_{q^{n}}(\CT_{X})\otimes
S^{*}_{q^{n+1}}(\CT_{X}^{*})\otimes\Lambda^{*}_{q^{n}y^{-1}}
(\CT_{X})\otimes\Lambda^{*}_{q^{n+1}y}
(\CT_{X}^{*})\right).
\eqno{(2.3.5)}$$ Transformation formulas (2.3.1) imply that $L^{(1)}(z)$ is preserved; hence $\Omega^{ch}_{X}$ always carries a conformal structure, see 1.10.1 for the definition. The situation is different with $\Lambda^{ch}\CT_{X}$: it does not carry a conformal structure compatible with its conformal grading unless $X$ is Calabi-Yau. Indeed, as follows from the last of formulas (2.2.11), $L^{(2)}(z)=L^{(1)}(z)+J^{(1)}(z)'$, and the latter picks the 1st Chern class as a result of transformation (2.3.1).
If, however, $X$ is a projective Calabi-Yau manifold, then it can be derived from (2.3.1), \[MSV\], that the quadruple of fields $Q^{(i)}(z)$, $G^{(i)}(z)$, $J^{(i)}(z)$, $Q^{(i)}(z)$, $i=1,2$, can be made sense of globally, and both $\Omega^{ch}_{X}$, $\Lambda^{ch}\CT_{X}$ acquire an $N2$-structure, see 1.10.1 for the definition. What is especially clear is
[**2.3.4. Lemma.**]{} [*If $\omega$ is a non-vanishing holomorphic form over $X$, and $X$ admits an atlas consisting of charts $\{(U,\vec{x})\}$ such that locally $\omega=dx_{1}\wedge\cdots\wedge dx_{n}$, then formulas (2.2.10,11) define an $N2$-structure on $\Omega^{ch}_{X}$ and $\Lambda^{ch}\CT_{X}$ resp.* ]{}
Indeed, in this case the jacobian $\text{det}(\partial_{x_{i}}y_{j})$ equals 1, and the correction terms in (2.3.1) vanish.
[**2.3.5.**]{} [*An example: $X=\BC^{N}-0$.*]{} As an illsutration, let us compute the cohomology vertex algebra $H^{*}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})$, an example that will prove important later on.
The manifold $\BC^{N}-0$ is quasiaffine and, therefore, it has the standard global coordinate system $x_{i},\partial_{x_{i}}$, $0\leq
i\leq N-1$, inherited from $\BC^{N}$. This places us in the situation of Lemma 2.2.2, and we obtain a morphism of bi-graded sheaves $$\Omega^{ch}_{\BC^{N}-0}\iso
\bigoplus_{n\geq 0}(S^{*}_{q^{n}}(\CT_{\BC^{N}-0})\otimes
S^{*}_{q^{n+1}}(\CT_{\BC^{N}-0}^{*})\otimes\Lambda^{*}_{q^{n}y^{-1}}
(\CT_{\BC^{N}-0})\otimes\Lambda^{*}_{q^{n}y}
(\CT_{\BC^{N}-0}^{*})),
\eqno{(2.3.6)}$$ cf. (2.3.5). For the same reason, the sheaf on the R.H.S. of (2.3.6) is free, hence it suffices to compute $H^{*}(\BC^{N}-0,\CO_{\BC^{N}-0})$. It is a pleasing excersise in Čech cohomology to prove that $$H^{n}(\BC^{N}-0,\CO_{\BC^{N}-0})=
\left\{\aligned
\BC[x_{0},...,x_{N-1}]&\text{ if }n=0\\
\bigotimes_{i=0}^{N-1} \BC[x_{i}^{\pm 1}]/\BC[x_{i}]&\text{ if }n=N-1\\
0&\text{ otherwise}.
\endaligned
\right.
\eqno{(2.3.7)}$$ The first line of (2.3.7) says that all the global sections of $\Lambda^{ch}\CT_{\BC^{N}-0}$ are restrictions from $\BC^{N}$. Therefore, $$H^{0}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})= \Lambda^{ch}\CT(\BC^{N})=
FB^{\otimes N},
\eqno{(2.3.8)}$$ cf. (2.2.8).
Similarly, it follows from the 2nd line of (2.3.7) that $$H^{N-1}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})
= \left(\bigotimes_{i=0}^{N-1} \BC[x_{i}^{\pm 1}]/\BC[x_{i}]\right)\otimes_{\BC[\vec{x}]}
\Lambda^{ch}\CT(\BC^{N}),
\eqno{(2.3.9)}$$ where we use the notation of (2.2.1) with $(U,\vec{x})=(\BC^{N},\vec{x})$.
A moment’s thought shows that, as an $\Lambda^{ch}\CT(\BC^{N})$-module, $H^{N-1}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})$ is obtained from $\Lambda^{ch}\CT(\BC^{N})$ by spectral flow (1.12.2): $$H^{N-1}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})=
S_{1}(\Lambda^{ch}\CT(\BC^{N}))=
S_{1}(FB^{\otimes N}).
\eqno{(2.3.10)}$$ Indeed, by definition 1.7, $\Lambda^{ch}\CT(\BC^{N})=
FB^{\otimes N}$ is generated by a vector $\b1$ annihilated by $x_{i,(j)}$, $\partial_{x_{i},(j)}$, $j\geq 0$, (and we identify $x_{i}=x_{i,(-1)}$); according to (2.3.9), $H^{N-1}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})$ is generated by a vector annihilated by $x_{i,(j-1)}$, $\partial_{x_{i},(j+1)}$, $j\geq 0$; the latter annihilating subalgebra is mapped onto the former by $S_{1}$ of (1.12.1). The odd variables are treated similarly; but notice also that the Clifford algebra has only one irreducible module and so the spectral flow on it is inessential.
Of course, the 3rd line of (2.3.7) implies $$H^{i}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})= 0
\text{ if } i\neq 0, N-1.
\eqno{(2.3.11)}$$
[**2.4. The algebra of chiral polyvector fields over hypersurfaces.**]{}
This is an exposition of a result of \[B\].
[**2.4.1.**]{} Let $$\CL\rightarrow X\eqno{(2.4.1)}$$ be a line bundle, $$\CL^{*}\rightarrow X\eqno{(2.4.2)}$$ its dual, $$t: X\rightarrow\CL\eqno{(2.4.3)}$$ its section with smooth zero locus $Z(t)$. Following \[B\] we shall relate $\Lambda^{ch}\CT_{\CL^{*}}$ and $\Lambda^{ch}\CT_{Z(t)}$ as follows.
Identify $t$ with a fiberwise linear function on $\CL^{*}$. We have the de Rham differential of $t$, $dt\in H^{0}(\CL^{*},\Omega^{1}_{\CL^{*}})$; via (2.1.1), $dt\in H^{0}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})$. It is clear that $$dt_{(0)}: \Lambda^{ch}\CT_{\CL^{*}}\rightarrow
\Lambda^{ch}\CT_{\CL^{*}} \eqno{(2.4.4)}$$ is a derivation with zero square, cf. 1.3.
$\CL^{*}$ carries an action of $\BC^{*}$ defined by fiberwise multiplication. By the naturality, 2.1 (i), this action lifts to an action on $\Lambda^{ch}\CT_{\CL^{*}}$. Hence there arises the grading $$\Lambda^{ch}\CT_{\CL^{*}}=\bigoplus_{n\in\BZ}R^{n}(\Lambda^{ch}\CT_{\CL^{*}}).$$ The operator $dt_{(0)}$ has degree 1 with respect to this grading and, therefore, (2.4.4) is actually a complex of sheaves $R^{*}\Lambda^{ch}\CT_{\CL^{*}}=
(\Lambda^{ch}\CT_{\CL^{*}},dt_{(0)})$ such that $$\cdots\buildrel dt_{(0)}\over\longrightarrow
R^{-1}(\Lambda^{ch}\CT_{\CL^{*}})\buildrel dt_{(0)}\over\longrightarrow
R^{0}(\Lambda^{ch}\CT_{\CL^{*}})\buildrel dt_{(0)}\over\longrightarrow
R^{1}(\Lambda^{ch}\CT_{\CL^{*}})\buildrel dt_{(0)}\over\longrightarrow\cdots
\eqno{(2.4.5)}$$
[**2.4.2. Lemma**]{} (\[B\]) [*The cohomology sheaf $\CH^{n}_{dt_{(0)}}(\Lambda^{ch}\CT_{\CL^{*}})$ of complex (2.4.5) is zero unless $n=0$. The sheaf $\CH^{0}_{dt_{(0)}}(\Lambda^{ch}\CT_{\CL^{*}})$ is supported on $Z(t)$ and naturally isomorphic to $\Lambda^{ch}\CT_{Z(t)}$.*]{}
[**2.4.3.**]{} Observe that $dt\in H^{0}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})$ is of conformal weight 1, as follows e.g. from (2.3.5). Therefore, the differential of complex (2.4.5) preserves conformal weight, and the conformal weight 0 component of (2.4.5) is the following classical complex: $$\cdots\buildrel dt\over\longrightarrow
\Lambda^{i+1}\CT_{\CL^{*}}\buildrel dt\over\longrightarrow
\Lambda^{i}\CT_{\CL^{*}}\buildrel dt\over\longrightarrow
\Lambda^{i-1}\CT_{\CL^{*}}\buildrel dt\over\longrightarrow
\cdots,
\eqno{(2.4.6)}$$ with differential equal to the contraction with the 1-form $d_{DR}t$. Lemma 2.4.2 says, in particular, that this complex computes the algebra of polyvector fields on $Z(t)$, a well-known result perhaps. Note that in the chiral de Rham complex setting, cf. \[B\], this classical construction is somewhat harder to discern because there conformal weight is not preserved by $dt_{(0)}$.
[**2.4.4.**]{} [*The $N2$-structure.*]{} Let $\CL^{*}$ be the canonical line bundle. Then both $\CL^{*}$ and $Z(t)$ are Calabi-Yau – both have a nowhere zero global holomorphic volume form – and both $\Lambda^{ch}\CT_{\CL^{*}}$ and $\Lambda^{ch}\CT_{Z(t)}$ carry an $N2$-structure, see the end of 2.3.3. Let us write down some explicit formulas.
Suppose there is a nowhere zero global holomorphic volume form $\omega$ and $X$ can be covered by charts $s,y_{1},...,y_{N-1}$, $s$ being the coordinate along the fiber, such that $\omega=ds\wedge y_{1}\wedge\cdots\wedge y_{N-1}$. Then, as follows from Lemma 2.3.4, $$\aligned
Q(z)\mapsto s(z)'\partial_{ds}(z)+\sum_{j=1}^{N-1}y_{j}(z)'\partial_{dy_{j}}(z),\;
&G(z)\mapsto ds(z)\partial_{s}(z)+\sum_{j=1}^{N-1}dy_{j}(z)\partial_{y_{j}}(z),\\
J(z)\mapsto -ds(z)\partial_{ds}(z)-\sum_{j=1}^{N-1}dy_{j}(z)\partial_{dy_{j}}(z),\;
&L(z)\mapsto s(z)'\partial_{s}(z)+\sum_{j=1}^{N-1}y_{j}(z)'\partial_{y_{j}}(z)-\\
&-ds(z)\partial_{ds}(z)'-\sum_{j=1}^{N-1}dy_{j}(z)\partial_{dy_{j}}(z)'
\endaligned
\eqno{(2.4.7)}$$ defines an $N2$-structure on $\Lambda^{ch}\CT_{\CL^{*}}$.
\[B, Proposition 5.8\] says that, via Lemma 2.4.2, the $N2$-structure on $\Lambda^{ch}\CT_{Z(t)}$ is determined by $$\aligned
G(z)&\mapsto ds(z)\partial_{s}(z)+\sum_{j=1}^{N-1}dy_{j}(z)\partial_{y_{j}}(z),\\
Q(z)&\mapsto s(z)'\partial_{ds}(z)+\sum_{j=1}^{N-1}y_{j}(z)'\partial_{dy_{j}}(z)-
(s(z)\partial_{ds}(z))'.
\endaligned
\eqno{(2.4.8)}$$
[**3. The lattice vertex algebra realization and applications to toric varieties.**]{}
This section is an exposition of part of Borisov’s free field realization \[B\]. It does not contain any new results except perhaps Lemma 3.8, and in order to construct the spectral sequence appearing in the latter the entire section had to be written up.
[**3.1.**]{} Let $M$ be a rank $N$ free abelian group, $M^{*}=Hom_{\BZ}(M,\BZ)$ its dual. Give $\Lambda=M\oplus M^{*}$ a lattice structure by defining the symmetric bilinear form $$\Lambda\times\Lambda\rightarrow \BZ,
\eqno{(3.1.1)}$$ induced by the natural pairing $M^{*}\times M\rightarrow \BZ$, $(X^{*},X)\mapsto X^{*}(X)$. There arises the lattice vertex algebra $V_{\Lambda}$, 1.9, where we fix the following cocycle, cf.(1.9.1), $$\epsilon(X+X^{*},Y+Y^{*})=(-1)^{X^{*}(Y)},\; X,Y\in M, X^{*},Y^{*}\in M^{*}.
\eqno{(3.1.2)}$$ Next, consider the complexification $\Lambda_{\BC}=\BC\otimes_{\BZ}\Lambda$ onto which form (3.1.2) carries over. There arises the fermionic vertex algebra $F_{\Lambda_{\BC}}$ which we re-denote by $F_{\Lambda}$, see (1.6.5).
Finally, following \[B\] introduce Borisov’s vertex algebra $$\BB_{\Lambda}=V_{\Lambda}\otimes F_{\Lambda}.
\eqno{(3.1.3)}$$
[**Notation.**]{} The notational problem one faces here is that the lattice $\Lambda$ twice manifests itself inside $\BB_{\Lambda}$: first, as an ingredient of $V_{\Lambda}$; second, as that of $ F_{\Lambda}$. We attempt to resolve this issue by letting capital latin letters, $X,Y,Z,..$ ( $X^{*},Y^{*},Z^{*},..$ resp.), denote elements of $M$ ($M^{*}$ resp.) in the context of $V_{\Lambda}$; and let the tilded letters, $\tilde{X},\tilde{Y},\tilde{Z},..$ or $\tilde{X}^{*},\tilde{Y}^{*},\tilde{Z}^{*},..$ denote their respective copies in the context of $ F_{\Lambda}$. Later on this will be related to geometry and then we shall let the lowercase letters denote the respective coordinates. $\qed$
Note that the assignment $M^{*}\mapsto \BB_{\Lambda}$ is functorial. Indeed, if $g\in\text{Hom}(M^{*}_{1},M^{*}_{2})$ is an isomorphism of abelian groups, then $$(g^{-1}, g^{*})\in \text{Hom}(M^{*}_{2},M^{*}_{1})\times
\text{Hom}(M_{2},M_{1})\hookrightarrow\text{Hom}(\Lambda_{2},\Lambda_{1})$$ is an isomorphism of lattices preserving form (3.1.1) and cocycle (3.1.2). According to (1.11.1,2), this isomorphism induces the following isomorphism of vertex algebras $$\hat{g}: \BB_{\Lambda_{2}}\rightarrow \BB_{\Lambda_{1}}
\eqno{(3.1.4a)}$$ $$\aligned
&X^{*}(z)\mapsto g^{-1}X^{*}(z),\tilde{X}^{*}(z)\mapsto g^{-1}\tilde{X}^{*}(z),
e^{X^{*}}(z)\mapsto e^{g^{-1}X^{*}}(z)\\
&X(z)\mapsto g^{*}X(z),\tilde{X}(z)\mapsto
g^{*}\tilde{X}(z),
e^{X}(z)\mapsto e^{ g^{*}X}(z).
\endaligned
\eqno{(3.1.4b)}$$
Therefore, if we introduce the category of lattices $\Lambda$, morphisms being the described isomorphisms, then $$M^{*}\mapsto\BB_{\Lambda},
g\mapsto\hat{g} \eqno{(3.1.5)}$$ is a contravariant functor.
Later we shall have to work with $g$ such that $g(M^{*}_{1})\subset M^{*}_{2}$ but after the extension of scalars to $\BQ$ the induced $g\in\text{Hom}_{\BQ}((M^{*}_{1})_{\BQ},(M^{*}_{2})_{\BQ})$ is an isomorphism. In this case the functorial nature of $M^{*}\mapsto
\BB_{\Lambda}$ is a little more subtle because $g^{-1}$ may have non-integer entries. There are two ways around.
Consider a vertex subalgebra $\BB_{M,\Lambda}\subset
\BB_{\Lambda}$ associated to the sublattice $M\subset \Lambda$, see definition (1.9.10). (This simply means that all the fields $e^{X^{*}}(z)$ are not allowed.) It is clear that $$\BB_{.,.}:\; M^{*}\mapsto\BB_{M,\Lambda},
g\mapsto\hat{g}|_{ \BB_{M,\Lambda}}
\eqno{(3.1.6)}$$ is a contravariant functor because the indicated restriction of (3.1.4a) makes sense for any lattice embedding $g$.
Second, naturally associated to $g$ there is a map $$\hat{g}: \BB_{\Lambda_{2}}\hookrightarrow \BB_{g^{-1}\Lambda_{2}},
\eqno{(3.1.7)}$$ still defined by (3.1.4a), where the lattice $g^{-1}\Lambda_{2}$ is defined to be $g^{*}M_{2}\oplus g^{-1}M^{*}_{2}\subset (\Lambda_{1})_{\BQ}$.
Note that (3.1.6) is a “subfunctor” of (3.1.7).
[**3.2.**]{} Let us introduce the following terminology and notation pertaining to toric variety theory: by a basic cone $\sigma\subset M^{*}$ we shall mean a sub-semigroup spanned over $\BZ_{+}$ by part of a basis (over $\BZ$) of $M^{*}$. Let $<\sigma>$ denote the (uniquely determined) spanning set of $\sigma$ . Given a basic cone $\sigma\subset M^{*}$, let $\check\sigma\subset M$ be its dual cone defined by $\check\sigma=\{X\in M\text{ s.t. } \sigma (X)\geq 0\}$.
A smooth toric variety will always be defined by fixing a lattice $\Lambda$ as in 3.1 and a regular fan $\Sigma$. (Regular means that $\Sigma$ is a collection of basic cones in $M^{*}$.) If we define $$U_{\sigma}=\text{Spec}\BC[\check\sigma],\; \sigma\in\Sigma,
\eqno{(3.2.1)}$$ where $\BC[\check\sigma]$ is the semigroup algebra of $\check\sigma$, then there arises a canonical embedding $$U_{\sigma'}\subset U_{\sigma},\; \sigma'\subset\sigma.
\eqno{(3.2.2)}$$
The toric variety $X_{\Sigma}$ attached to $\Sigma$ is defined by declaring that $$\CU_{\Sigma}=\{U_{\sigma},\sigma\in\Sigma\}
\eqno{(3.2.3)}$$ is its covering and by gluing the charts over intersections $$U_{\alpha}\hookleftarrow U_{\alpha\cap\beta}\hookrightarrow U_{\beta}$$ according to (3.2.2).
Note that the assignment $(\sigma, M^{*})\mapsto U_{\sigma}$ is functorial. Indeed, if we introduce the category whose objects are pairs $(\sigma, M^{*})$ and morphisms $(\sigma_{1}, M^{*}_{1})\rightarrow (\sigma_{2}, M^{*}_{2})$ are abelian group morphisms $g: M^{*}_{1}\rightarrow M^{*}_{2}$ such that $g(\sigma_{1})\subset\sigma_{2}$, then $g^{*}(\check\sigma_{2})\subset\check\sigma_{1}$. Hence $g^*$ induces a ring homomorphism $\BC[\check\sigma_{2}]\rightarrow
\BC[\check\sigma_{1}]$ and thus a morphism $\tilde{g}:
U_{\sigma_{1}}\rightarrow
U_{\sigma_{2}}$. Of course, $$(\sigma, M^{*})\mapsto U_{\sigma},\; g\mapsto\tilde{g}
\eqno{(3.2.4)}$$ is a covariant functor.
This can be globalized: given $(\Sigma_{1}, M_{1}^{*})$and $(\Sigma_{2}, M_{2}^{*})$ with a lattice morphism $g: M_{1}^{*}\rightarrow M_{2}^{*}$ such that for each $\sigma_{1}\in \Sigma_{1}$ there is $\sigma_{2}\in \Sigma_{2}$ containing $g(\sigma_{1})$, there arises a morphism $$\tilde{g}: X_{\Sigma_{1}}\rightarrow X_{\Sigma_{2}}.
\eqno{(3.2.5)}$$
[**3.3.**]{} We would like to define Borisov’s realization \[B\], that is, a vertex algebra embedding $\Lambda^{ch}\CT(U_{\sigma})\hookrightarrow\BB_{M,\Lambda}$ for each basic cone $\sigma\in M^{*}$. To write down an explicit formula for this map, let us choose a basis of $M^{*}$, $X_{0}^{*},..., X_{N-1}^{*}$, such that $\sigma$ is spanned by $X_{0}^{*},..., X_{m-1}^{*}$. This fixes the dual basis $X_{0},..., X_{N-1}$ of $M$.
In order to conform to the notation of sect.2, let $x_{j}\buildrel\text{def}\over = e^{X_{j}}$, $\partial_{x_{i}}$, $0\leq i\leq m-1$, be a coordinate system on $U_{\sigma}$. Thus $$U_{\sigma}=\text{Spec}\BC[x_{0},...,x_{m-1},x_{m}^{\pm 1},...,x_{N-1}^{\pm 1}].$$ Borisov proves that there is a vertex algebra homomorphism $$\CB(\sigma):\; \Lambda^{ch}\CT(U_{\sigma})
\rightarrow \BB_{M,\Lambda}
\eqno{(3.3.1)}$$ determined by the assignment $$\aligned & x_{i}^{\pm 1}(z)\mapsto e^{\pm X_{i}}(z),
x_{j}(z)\mapsto e^{X_{j}}(z),\;
m\leq i\leq N-1,j\leq m-1,\\
& dx_{i}(z)\mapsto :e^{X_{i}}(z)\tilde{X}_{i}(z):.
\endaligned
\eqno{(3.3.2a)}$$ $$\partial_{x_{i}}(z)\mapsto
:(X_{i}^{*}(z)-:\tilde{X}_{i}(z)\tilde{X}_{i}^{*}(z):)e^{-X_{i}}(z):,\;
\partial_{dx_{i}}(z)\mapsto :e^{-X_{i}}(z)\tilde{X}_{i}^{*}(z):,
\eqno{(3.3.2b)}$$ cf. sect. 3.1, Notation. (Note that (3.3.2a-b) are exactly (2.1.3) specialized to the exponential change of variables $x_{i}\rightarrow e^{X_{i}}$; this remark is also borrowed from \[B\].) Formulas (3.3.2a) are manifestly independent of the choice of variables; it is easy then to infer that so are (3.3.2b).
This embedding naturally depends on $\sigma$. To make a precise statement, give
[**3.3.1. Definition.**]{} Fix a number $N$. $\CC$ is a category whose objects are pairs $(\sigma, M^{*})$, $\text{dim}M^{*}=N$, all and morphisms $(\sigma_{1}, M^{*}_{1})\rightarrow (\sigma_{2}, M^{*}_{2})$ are abelian group embeddings $g: M^{*}_{1}\hookrightarrow M^{*}_{2}$ such that $g(\sigma_{1})=\sigma_{2}$. $\qed$
Note that the conditions imposed on morphisms in this definition strengthen those used in (3.2.4). In fact, a short computation shows that the map $\tilde{g}:U_{\sigma_{1}}\rightarrow U_{\sigma_{2}}$ associated with a morphism $g\in\text{Mor}_{\CC}((\sigma_{1}, M_{1}^{*}),
(\sigma_{1}, M_{1}^{*}))$ in (3.2.4) is étale. Hence, by virtue of the naturality of $\Lambda^{ch}\CT_{X}$, see 2.1(i), the composition $$(\sigma, M^{*})\mapsto U_{\sigma}\mapsto\Lambda^{ch}\CT(U_{\sigma})$$ defines a contravariant functor $$\Lambda^{ch}\CT(.):\;(\sigma, M^{*})\mapsto\Lambda^{ch}\CT(U_{\sigma}).
\eqno{(3.3.3)}$$ By forgetting $\sigma$, we regard contravariant functor (3.1.6) as defined on $\CC$. The main property of (3.3.1) is formulated as follows.
[**3.4. Lemma.**]{} [*The assignment $(\sigma,M^{*})\mapsto
\CB(\sigma)$, see (3.3.1), is a functor morphism*]{} $$\CB(\sigma):\; \Lambda^{ch}\CT(.)\rightarrow \BB_{.,.},$$ [*where $\BB_{.,.}$ is the functor defined in (3.1.6).*]{}
[**Notational convention.**]{} Using this fact we shall not distinguish between $\Lambda^{ch}\CT(U_{\sigma})$ and $\CB(\sigma)(\Lambda^{ch}\CT(U_{\sigma}))\subset\BB_{M,\Lambda}$.
[**3.5.**]{} Let $S^{*}\in<\sigma>$ be one of the generators of $\sigma$ and let $\sigma\setminus S^{*}$ denote the cone spanned by $<\sigma>\setminus
S^{*}$. There arises then the restrtiction morphism $$res(\sigma, S^{*}): \Lambda^{ch}\CT(U_{\sigma})\hookrightarrow \Lambda^{ch}\CT(U_{\sigma\setminus S^{*}}),$$ and one would like to extend it to a resolution.
Let $$\CJ^{*}(\sigma,S^{*})=\oplus_{n=0}^{\infty}\CJ^{n}(\sigma,S^{*}),\;
\CJ^{n}(\sigma,S^{*})=\Lambda^{ch}\CT(U_{\sigma\setminus S^{*}})e^{nS^{*}},
\eqno{(3.5.1)}$$ where $\Lambda^{ch}\CT(U_{\sigma\setminus S^{*}})$ is thought of as a vertex subalgebra of $\BB_{M,\Lambda}$, see 3.4, Notational convention, and this makes sense out of $\Lambda^{ch}\CT(U_{\sigma\setminus S^{*}})e^{nS^{*}}$ as a subspace of $\BB_{\Lambda}$.
$\CJ^{*}(\sigma,S^{*})$ is evidently a $\BZ_{+}$-graded vertex subalgebra of $\BB_{\Lambda}$.
Let $$D(\sigma,S^{*})=(e^{S^{*}}(z)\tilde{S}^{*}(z))_{(0)}.
\eqno{(3.5.2)}$$ It is evidently a square zero derivation of $\CJ^{*}(\sigma,S^{*})$, see 1.3. Thus $(\CJ^{*}(\sigma,S^{*}),D(\sigma,S^{*}))$ is a differential graded vertex algebra.
Now look upon $\Lambda^{ch}\CT(U_{\sigma})$ as a differential graded vertex algebra with $\Lambda^{ch}\CT(U_{\sigma})$ placed in degree 0, zero spaces placed everywhere else, and zero differential. Then, by taking the composition $$\Lambda^{ch}\CT(U_{\sigma})\buildrel res(\sigma, S^{*})\over\rightarrow
\Lambda^{ch}\CT(U_{\sigma\setminus S^{*}})=\CJ^{0}(\sigma,S^{*})
\hookrightarrow \CJ^{*}(\sigma,S^{*}),$$ $res(\sigma, S^{*})$ can be interpreted as a morphism of differential graded vertex algebras $$res(\sigma, S^{*}): (\Lambda^{ch}\CT(U_{\sigma}),0)\hookrightarrow (\CJ^{*}(\sigma,S^{*}),D(\sigma,S^{*})).
\eqno{(3.5.3)}$$ This construction is natural. To explain this, let us give the following definition, cf. 3.3.1.
[**3.5.1. Definition.**]{} $\CC_{pnt}$ is a category whose objects are triples $(S^{*},\sigma, M^{*})$ with $S^{*}\in<\sigma>$, $(\sigma, M^{*})\in\text{Ob}(\CC)$, and morphisms $(S_{1}^{*},\sigma_{1}, M^{*}_{1})\rightarrow (S_{2}^{*},\sigma_{2}, M^{*}_{2})$ are abelian group morphisms $g: M^{*}_{1}\rightarrow M^{*}_{2}$ such that $g(\sigma_{1})=\sigma_{2}$ and $g(S^{*}_{1})=S^{*}_{2}$. $\qed$
It is clear that both $$(\Lambda^{ch}\CT(.),0):\; (S^{*},\sigma, M^{*})\mapsto (\Lambda^{ch}\CT(U_{\sigma}),0)
\eqno{(3.5.4)}$$ and $$(\CJ^{*}(.), D(.)):\; (S^{*},\sigma, M^{*})\mapsto (\CJ^{*}(\sigma,S^{*}),D(\sigma,S^{*}))
\eqno{(3.5.5)}$$ are contravariant functors from $\CC_{pnt}$ to the category of differential graded vertex algebras. Indeed, the former is essentially (3.3.3), as to the latter, one has to apply map (3.1.7) restricted to $\CJ^{*}(\sigma,S^{*})$.
[**3.6. Lemma.**]{}
*(i) The assignment $(S^{*},\sigma)\mapsto
res(\sigma, S^{*})$, see (3.5.3), is a functor morphism $$res(.):\; (\Lambda^{ch}\CT(.),0)\mapsto (\CJ^{*}(.),D(.)).$$*
\(ii) For each $(S^{*},\sigma, M^{*})$ map (3.5.3) is a quasiisomorphism.
[**3.7.** ]{} Let us apply Lemma 3.6 to the situation where the fan $\Sigma$ satisfies the following condition: there is $S^{*}\in M^{*}$ such that $S^{*}\in <\sigma>$ for all highest dimension cones $\sigma\in\Sigma$. Let $\Sigma\setminus S^{*}=\{\sigma\setminus S^{*}:\; \sigma\in\Sigma\}$, cf. the beginning of 3.5. This means that the morphism $$X_{\Sigma}\rightarrow X_{\Sigma/\BZ S^{*}}
\eqno{(3.7.1)}$$ induced by the canonical projection $M^{*}\rightarrow M^{*}/\BZ S^{*}$ is a line bundle, and the map $$X_{\Sigma\setminus S^{*}}\hookrightarrow X_{\Sigma}
\eqno{(3.7.2)}$$ induced by the tautological inclusion $\Sigma\setminus S^{*}\subset\Sigma$ is the embedding of the total space of the line bundle without the zero section. We would like to relate the cohomology groups $H^{*}(X_{\Sigma},\Lambda^{ch}\CT_{X_{\Sigma}})$ and $H^{*}(X_{\Sigma\setminus S^{*}},
\Lambda^{ch}\CT_{X_{\Sigma\setminus S^{*}}})$.
The nerve of the covering $\CU_{\Sigma}$ is a simplicial object in the category $\CC_{pnt}$. Applying to it functor (3.5.4) one gets the complex $\Lambda^{ch}\CT(\CU_{\Sigma})$ commonly known as the Čech complex. (A complex, not a bi-complex, because we ignore the trivial differential on $(\Lambda^{ch}\CT(.),0)$.) We denote it by $\check{C}^{*}(\CU_{\Sigma},\Lambda^{ch}\CT_{X_{\Sigma}};d_{\check C})$, where $d_{\check C}$ is the Čech differential.
Likewise applying functor (3.5.5) to the nerve of $\CU_{\Sigma}$, we obtain the bi-complex $\check{C}^{*}(\CU_{\Sigma},\CJ^{*};d_{\check C}, D(S^{*}))$. By definition $$\check{C}^{p}(\CU_{\Sigma},\CJ^{q};d_{\check C}, D(S^{*}))=
\check{C}^{p}(\CU_{\Sigma\setminus S^{*}},\Lambda^{ch}
\CT_{X_{\Sigma\setminus S^{*}}}e^{qS^{*}};d_{\check C}, D(S^{*})).
\eqno{(3.7.3)}$$ According to Lemma 3.6, $$res(\CU_{\Sigma}):
\check{C}^{*}(\CU_{\Sigma},\Lambda^{ch}\CT_{X_{\Sigma}};d_{\check C})
\rightarrow
\check{C}^{*}(\CU_{\Sigma},\CJ^{*};d_{\check C}, D(S^{*}))
\eqno{(3.7.4)}$$ is a quasiisomorphism. More precisely, the bi-complex $\check{C}^{*}(\CU_{\Sigma},\CJ^{*};d_{\check C}, D)$ gives rise to two spectral sequences both converging to its total cohomology. The one where the vertex differential $D(.)$ is used first degenerates in the first term to the Čech complex $\check{C}^{*}(\CU_{\Sigma},\Lambda^{ch}\CT_{X_{\Sigma}};d_{\check C})$ – this follows at once from Lemma 3.6 (ii). Hence both the sequences abut to $H^{*}(X_{\Sigma},\Lambda^{ch}\CT_{X_{\Sigma}})$. By definition, the first and the second terms of the second spectral sequence are as follows: $$\aligned
&(E^{p,q}_{1}, d_{1})=(H^{p}(X_{\Sigma\setminus S^{*}},\Lambda^{ch}
\CT_{X_{\Sigma\setminus S^{*}}}e^{qS^{*}}), D(S^{*})),\\
&D(S^{*})=(e^{S^{*}}(z)\tilde{S}^{*}(z))_{(0)}.
\endaligned
\eqno{(3.7.5)}$$ $$E^{p,q}_{2}=H_{D(S^{*})}^{q}(H^{p}(X_{\Sigma\setminus S^{*}},\oplus_{n=0}^{\infty}\Lambda^{ch}
\CT_{X_{\Sigma\setminus S^{*}}}e^{nS^{*}})).
\eqno{(3.7.6)}$$
Let us summarize our discussion.
[**3.8. Lemma.**]{} [*There is a spectral sequence $\{E^{p,q}_{r},d_{r}\}\Rightarrow
H^{*}(X_{\Sigma},\Lambda^{ch}\CT_{X_{\Sigma}})$ that satisfies (3.7.5,6).*]{}
[**3.9.**]{} The formation of bi-complex (3.7.3), and hence of the corresponding spectral sequences, is functorial in $X_{\Sigma}$. To make this precise, let a triple $S^{*}$, $\Sigma_{2}$, and $M^{*}_{2}$ satisfy the conditions imposed on $S^{*}$, $\Sigma$, and $M^{*}$ in 3.7, and let us give ourselves another pair $\Sigma_{1}$ and $M^{*}_{1}$, $\text{dim}M^{*}_{1}=\text{dim}M^{*}_{2}$, along with a lattice embedding $$g:\; M_{1}^{*}\rightarrow M^{*}_{2}\;\text{ s.t. }
g(\Sigma_{1})=\Sigma_{2}\setminus S^{*}. \eqno{(3.9.1)}$$ According to (3.2.5) this induces a map $$\tilde{g}:\; X_{\Sigma_{1}}\mapsto X_{\Sigma_{2}},
\eqno{(3.9.2)}$$ which is étale, cf. 3.3. Map (3.9.1) gives rise to the lattice $g^{-1}\Lambda^{*}_{2}$ and the embedding of Borisov’s algebras $$\hat{g}:\; \BB_{\Lambda_{2}}\hookrightarrow \BB_{g^{-1}\Lambda_{2}}
\eqno{(3.9.3)}$$ due to (3.1.7). It is rather obvious that maps (3.9.2,3) allow to pull the bi-complex $$\check{C}^{p}(\CU_{\Sigma_{2}\setminus S^{*}},\Lambda^{ch}
\CT_{X_{\Sigma_{2}\setminus S^{*}}}e^{qS^{*}};d_{\check C}, D(S^{*}))
\eqno{(3.9.4)}$$ back onto $X_{\Sigma_{1}}$. Let us write down the relevant formula. An element of bi-complex (3.9.4) is a family of elements $$f_{\sigma}\in\{\Lambda^{ch}\CT_{X_{\Sigma_{2}\setminus
S^{*}}}(U_{\sigma})\}e^{qS^{*}},\; \sigma\in\Sigma_{2}.$$
Define $$\check{C}^{p}(\CU_{\Sigma_{1}},\hat{g}\Lambda^{ch}
\CT_{X_{\Sigma_{2}}}e^{qg^{-1}S^{*}};d_{\check C}, g^{-1}D(S^{*}))
\eqno{(3.9.5)}$$ to be the set of all the families $$f_{\sigma}\in\hat{g}\{\Lambda^{ch}\CT_{X_{\Sigma_{2}}}(U_{g\sigma})\}e^{qg^{-1}S^{*}},\;
\sigma\in\Sigma_{1}.$$ By construction, the map $$\{f_{\sigma},\; \sigma\in\Sigma_{2}\}\mapsto
\{\hat{g}f_{g\sigma},\; \sigma\in\Sigma_{1}\}$$ delivers an isomorphism of (3.9.5) and (3.9.4): $$\aligned
&\check{C}^{p}(\CU_{\Sigma_{2}\setminus S^{*}},\Lambda^{ch}
\CT_{X_{\Sigma_{2}\setminus S^{*}}}e^{qS^{*}};d_{\check C}, D(S^{*}))
\iso\\
&\check{C}^{p}(\CU_{\Sigma_{1}}, \hat{g}\Lambda^{ch}
\CT_{X_{\Sigma_{1}}}e^{qg^{-1}S^{*}};d_{\check C}, g^{-1}D(S^{*})).
\endaligned
\eqno{(3.9.6)}$$
[**3.10.**]{} [*Digression: Borisov’s realization and the spectral flow.*]{} We are now making good on our promise to show how spectral flow (1.12.1) is realized via lattice vertex algebras in the case of the $bc-\beta\gamma$-system. We shall show in 5.2.15 that a simple version of this construction does the same for $N2$.
According to our conventions $\Lambda^{ch}\CT(\BC^{N})\subset\BB_{M,\Lambda}$, and by definition (3.1.2) $M\subset L$ satisfies (1.12.5). Therefore, any $\alpha\in M^{*}$ generates on $\text{Lie}\BB_{M,\Lambda}$ the spectral flow in the direction of $\alpha$, see (1.12.6). A glance at (3.3.2a,b) shows that if $\alpha=\sum_{j}X_{j}^{*}$, then (1.2.6) reads $$S_{\sum_{j}X_{j}^{*}}:
\aligned
&x_{i}(z)\mapsto x_{i}(z)z,\; dx_{i}(z)\mapsto dx_{i}(z)z,\\
&\partial_{x_{i}}(z)\mapsto \partial_{x_{i}}(z)z^{-1},
\partial_{dx_{i}}(z)\mapsto \partial_{dx_{i}}(z)z^{-1},
\endaligned
\eqno{(3.10.1)}$$ cf. 1.12.7-8, and this does coincide with $S_{1}$ of (1.12.1).
It follows that the map $$e^{-\sum_{j}X^{*}_{j}}:\;
e^{\sum_{j}X^{*}_{j}}\Lambda^{ch}\CT(\BC^{N})\rightarrow
\Lambda^{ch}\CT(\BC^{N});
v\mapsto e^{-\sum_{j}X^{*}_{j}}v
\eqno{(3.10.2)}$$ identifies $e^{\sum_{j}X^{*}_{j}}\Lambda^{ch}\CT(\BC^{N})$ with the spectral flow transform of $\Lambda^{ch}\CT(\BC^{N})$: $$e^{\sum_{j}X^{*}_{j}}\Lambda^{ch}\CT(\BC^{N})\iso
S_{1}(\Lambda^{ch}\CT(\BC^{N})),
\eqno{(3.10.3)}$$ see definition of the spectral flow transform (1.12.2).
Therefore, the results of 2.3 can be rewritten as follows: $$H^{n}(\BC^{N}-0,\Lambda^{ch}\CT_{\BC^{N}-0})=\left\{
\aligned
\Lambda^{ch}\CT(\BC^{N})&\text{ if } n=0\\
\Lambda^{ch}\CT(\BC^{N})e^{\sum_{j}X^{*}_{j}}&\text{ if } n=N-1\\
0&\text{ otherwise}.
\endaligned
\right.
\eqno{(3.10.4)}$$
[**4. Chiral polyvector fields over hypersurfaces in projective spaces**]{}
Having put all the preliminaries out of the way we can tackle our main problem – computation of $H^{*}(F,\Lambda^{ch}\CT_{F})$ for a smooth hypersurface $F\subset \BP^{N-1}$.
[**4.1.**]{} Let $\CL\rightarrow\BP^{N-1}$ be the degree $N$ line bundle over $\BP^{N-1}$ and $$\pi:\CL^{*}\rightarrow\BP^{N-1}
\eqno{(4.1.0)}$$ its dual. Let us give the spaces $\BC_{-N}=\BC$ and $\BC^{N}-0$ a $\BC^{*}$-space structure as follows: $$\aligned \BC_{-N}\times\BC^{*}&\rightarrow\BC_{-N},\; u\cdot t=
st^{-N},\\
\BC^{*}\times(\BC^{N}-0)&\rightarrow \BC^{N}-0,\;
t\cdot(x_{0},...,x_{N-1})=(tx_{0},...,tx_{N-1}).
\endaligned
\eqno{(4.1.1)}$$ One has the quotient realization $$\CL^{*}= \BC_{-N}\times_{\BC^{*}}(\BC^{N}-0),\eqno{(4.1.2)}$$ where we impose the relation $$(u;x_{0},...,x_{N-1})\sim (ut^{-N};tx_{0},...,tx_{N-1}),\; t\neq
0. \eqno{(4.1.3)}$$ Let $\BZ_{N}$ act on $\BC^{N}$ as follows $$\aligned &\BZ^{N}\times (\BC^{N}-0)\rightarrow (\BC^{N}-0),\\
&\bar{m}\cdot(x_{0},...,x_{N-1})=(e^{2\pi\sqrt{-1}m/N}x_{0},...,e^{2\pi\sqrt{-1}m/N}x_{N-1}).
\endaligned\eqno{(4.1.4)}$$ Crucial for our purposes is the following isomorphism of smooth algebraic varieties $$\aligned
&p:\;(\BC^{N}-0)/\BZ_{N}\iso\CL^{*}-0,\\
&\text{class of }(x_{0},...,x_{N-1})\mapsto \text{class of
}(1;x_{0},...,x_{N-1}),\endaligned\eqno{(4.1.5)}$$ where $\CL^{*}-0$ denotes $\CL^{*}$ without the zero section. (This is well known and obvious: deleting the zero section means requiring that $u\neq 0$; then one uses the $\BC^{*}$-action to make $u=1$; this leaves only the classes of $(1;x_{0},...,x_{N-1})$ and simultaneously breaks the $\BC^{*}$-action down to the $\BZ_{N}$-action defined in (4.1.4).)
We wish to study Calabi-Yau hypersurfaces in $\BP^{N-1}$. To define any such hypersurface, take $x_{0},...,x_{N-1}$ to be, in accordance with (4.1.1), the homogeneous coordinates on $\BP^{N-1}$. Let $$\fF=\{(x_{0}:\cdots :x_{N-1})\text{ s.t. }f(x_{0},...,x_{N-1})=0\},
\eqno{(4.1.6)}$$ where $f$ is a degree $N$ homogeneous polynomial with a unique singularity at 0. Such an $f$ can be regarded as a section of $\CL$. The corresponding fiberwise linear function $t=t(u;x_{0},...,x_{N-1})$ on $\CL^{*}$ is $$t=uf(x_{0},...,x_{N-1}),\eqno{(4.1.7)}$$ cf. (4.1.2,3). The pull-back of this function onto $(\BC^{N}-0)/\BZ_{N}$ under (4.1.5) is literally $f(x_{0},...,x_{N-1})$: $$p^{*}(t)=f(x_{0},...,x_{N-1})
\eqno{(4.1.8)}$$ as follows from (4.1.5).
[**4.2.**]{} $\CL^{*}$ carries the standard affine covering $\CU=\{U_{j},\;0\leq j\leq N-1\}$ defined by $$U_{j}=\{\text{class of }(u;x_{0},...,x_{N-1})\text{ s.t. } x_{j}\neq 0\}.
\eqno{(4.2.0)}$$ Then $\pi\CU=\{\pi(U_{j}),\;0\leq j\leq N-1\}$ is (also standard) affine covering of $\BP^{N-1}$ and $\pi\CU\cap\fF=\{\pi(U_{j})\cap\fF\}$ is an affine covering of $\fF$.
By definition, the cohomology of the Čech complex $\check{C}^{*}(\pi\CU\cap\fF,\Lambda^{ch}\CT_{\fF})$ equals the cohomology $H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$. A more practical way to compute the latter is provided by Lemma 2.4.2. Indeed, being currently in the situation of this lemma we obtain the bi-complex $\check{C}^{*}(\CU,R^{*}\Lambda^{ch}\CT_{\CL^{*}};
d_{\check{C}},dt_{(0)})$, that is, the Čech complex with coefficients in the complex $(\Lambda^{ch}\CT_{\CL^{*}},dt_{(0)})$ defined in (2.4.5). Associated to this bi-complex there are two standard spectral sequences, $'\!E^{p,q}_{r}$ and $''\!E^{p,q}_{r}$, such that $$\aligned
('\!E^{p,q}_{1},'\!d_{1})&=(H^{p}(\CL^{*},R^{q}( \Lambda^{ch}\CT_{\CL^{*}})),
dt_{(0)}),\\
'\!E^{p,q}_{2}&=
H^{q}_{dt_{(0)}}(H^{p}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})),
\endaligned
\eqno{(4.2.1a)}$$ $$(''\!E^{p,q}_{1},''\!d_{1})=\check{C}^{p}(\CU,\CH^{q}_{dt_{(0)}}
(R^{q}( \Lambda^{ch}\CT_{\CL^{*}})).
\eqno{(4.2.1b)}$$ In the latter $\CH^{q}_{dt_{(0)}}
(R^{*} \Lambda^{ch}\CT_{\CL^{*}})$ denotes the $q$-th cohomology sheaf of complex (2.3.5).
[**4.3. Lemma.**]{} [*Both $'\!E^{p,q}_{r}$ and $''\!E^{p,q}_{r}$ abut to $H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$.*]{}
[**4.4.**]{} [*Proof.*]{} Observe that even though sheaf complex (2.4.5) appears to be infnite in both directions, its differential preserves conformal weight (cf. 2.4.3) and it is easy to see that each fixed conformal weight component of (2.4.5) is finite. (Indeed, the differential $dt_{(0)}$ changes fermionic charge by one and it follows from (2.3.5) that for a fixed conformal weight fermionic charge may acquire only a finite number of values.) This implies, in a standard manner, that both the spectral sequences abut to the cohomology of the total complex $C^{*}(\CU,R^{*}\Lambda^{ch}\CT_{\CL^{*}};d_{\check{C}}+dt_{(0)})$. Lemma 2.4.2 implies that $$''\!E^{p,q}_{1}\iso\left\{\aligned 0&\text{ if }q\neq 0\\
C^{p}(\pi\CU\cap\fF,\Lambda^{ch}\CT_{\fF})&\text{ otherwise }.
\endaligned\right.$$ Hence the second spectral sequence degenerates to the Čech complex over $\fF$ and the lemma follows. $\qed$
[**4.5.**]{} We now wish to compute the 1st term of the 1st spectral sequence recorded in (4.2.1a). Ignoring the double grading we rewrite (4.2.1a) as $$('\!E^{**}_{1},'\!d_{1})=(H^{*}(\CL^{*}, \Lambda^{ch}\CT_{\CL^{*}}),
dt_{(0)}),
\eqno{(4.5.1)}$$ We have $\CL^{*}-0\hookrightarrow\CL^{*}$; this places us in the set-up of 3.7, see e.g. (3.7.1,2), and in order to compute $(H^{*}(\CL^{*}, \Lambda^{ch}\CT_{\CL^{*}}))$ we employ the spectral sequence of Lemma 3.8. – in this particular case we shall be able to compute all its terms. Begin with
[**4.5.1.**]{} [*Toric description of $\CL^{*}$.*]{} Choose $$s=x_{0}^{N}, y_{j}=\frac{x_{j}}{x_{0}},\; 1\leq j\leq N
\eqno{(4.5.2)}$$ to be coordinates of $\CL^{*}$ over the “big cell” $U_{0}$, see (4.2.0). Let $$\aligned
&M_{\CL^{*}}=\BZ S\bigoplus\{\bigoplus_{i=1}^{N-1}\BZ Y_{i}\},\\
&M_{\CL^{*}}^{*}=\BZ S^{*}\bigoplus\{\bigoplus_{i=1}^{N-1}\BZ Y^{*}_{i}\},\\
&\Lambda_{\CL^{*}}=M_{\CL^{*}}\oplus M_{\CL^{*}}^{*},\;
\endaligned
\eqno{(4.5.3)}$$ and the bases $\{S,Y_{1},...,Y_{N-1}\}$ and $\{S^{*},Y^{*}_{1},...,Y^{*}_{N-1}\}$ be dual to each other.
It follows from (4.5.2) and the identifications $s=e^{S}$, $y_{j}=e^{Y_{j}}$ that a fan $\Sigma_{\CL^{*}}$ that defines $\CL^{*}$ can be chosen as follows: the set of its 1-dimensional generators consists of $$S^{*}, NS^{*}-\sum_{i=1}^{N-1}Y^{*}_{i}, Y^{*}_{1}, Y^{*}_{2},..., Y^{*}_{N-1}.
\eqno{(4.5.4)}$$ The set of the highest dimension cones consists of $N$ cones, each generated by $S^*$ and the rest of the vectors in (4.5.4) except one of them. These data determine $\Sigma_{\CL^{*}}$ uniquely.
[**4.5.2.**]{} [*Computation of $(E^{**}_{1}, d_{1})$, $(E^{**}_{2}, d_{2})$,..., $(E^{**}_{\infty}, d_{\infty})$.*]{}
Proceeding along the lines of 3.7 we obtain $$(E(\CL^{*})^{**}_{*}, d_{*})\Rightarrow H^{*}(\CL^{*},
\Lambda^{ch}\CT_{\CL^{*}}).
\eqno{(4.5.4 1/2)}$$ (3.7.5) reads (we skip $\CL^{*}$): $$(E^{p,q}_{1}, d_{1})=(H^{p}(\CL^{*}-0,\Lambda^{ch}\CT_{\CL^{*}-0})e^{qS^{*}},
(e^{qS^{*}}(z)\tilde{S}^{*}(z))_{(0)}).
\eqno{(4.5.5)}$$ Thanks to (4.1.5) and the naturality of $\Lambda^{ch}\CT_{X}$, 2.1 (i), there are canonical ismorphisms $$\aligned
&H^{p}(\CL^{*}-0,\Lambda^{ch}\CT_{\CL^{*}-0})\iso
H^{p}(\BC^{N}- 0/\BZ^{N},\Lambda^{ch}\CT_{\BC^{N}\setminus 0/\BZ^{N}})\\
&\iso
H^{p}(\BC^{N}- 0,\Lambda^{ch}\CT_{\BC^{N}- 0})^{\BZ_{N}}.
\endaligned
\eqno{(4.5.6)}$$ The latter has been already computed, see 2.3.5, formulas (2.3.8,10,11). Therefore, $(E^{p,q}_{1}, d_{1})$ is as follows: $$E^{p,q}_{1}=\left\{
\aligned
\Lambda^{ch}\CT(\BC^{N})^{\BZ_{N}}e^{qS^{*}}&\text{ if }p=0\\
S_{1}(\Lambda^{ch}\CT(\BC^{N})^{\BZ_{N}})e^{qS^{*}}&\text{ if }p=N-1\\
0 &\text{ otherwise}
\endaligned
\right.
\eqno{(4.5.7a)}$$ $$d_{1}=(e^{S^{*}}(z)\tilde{S}^{*}(z))_{(0)}.
\eqno{(4.5.7b)}$$ Note that $(E^{0,*}_{1}, d_{1})$ is canonically a differential graded vertex algebra and $(E^{N-1,*}_{1}, d_{1})$ is canonically a differential graded $(E^{0,*}_{1}, d_{1})$-module. Therefore, as a vertex algebra, $$(E^{*,*}_{1}, d_{1})=(E^{0,*}_{1}, d_{1})\oplus (E^{N-1,*}_{1}, d_{1})
\eqno{(4.5.8)}$$ is the abelian extension of the former by the latter. Therefore, (3.7.6) reads as follows: $$E^{*,*}_{2}=H_{d_{1}}(E^{0,*}_{1}, d_{1})\oplus
H_{d_{1}}(E^{N-1,*}_{1}, d_{1}),
\eqno{(4.5.9)}$$ and again $H_{d_{1}}(E^{0,*}_{1}, d_{1})$ is a vertex algebra, $H_{d_{1}}(E^{N-1,*}_{1}, d_{1})$ its module, and $E^{*,*}_{2}$ is the abelian extension of the former by the latter.
Thanks to (4.5.7a), the dimension consideration shows that $$\aligned
& d_{2}=d_{3}=\cdots d_{N-1}=0,\\
& E^{*,*}_{2}=E^{*,*}_{3}=\cdots =E^{*,*}_{N-3}=E^{*,*}_{N}.
\endaligned
\eqno{(4.5.10)}$$ The same argument shows that the non-zero components of the last non-zero differential are the following maps: $$d_{N}^{(i)}:\; E^{N-1,i}_{2}\rightarrow E^{0,i+N}_{2}.
\eqno{(4.5.11)}$$ As follows from 4.5.1, $\CL^{*}$ is covered by $N$ open affine sets. Therefore, $H^{n}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})=0$ if $n>N-1$. This implies that the maps $ d_{N}^{(i)}$ are isomorphisms if $i>0$ and $ d_{N}^{(0)}$ is an epimorphism. Hence $$E^{*,*}_{\infty}= E^{0,0}_{2}\oplus E^{0,1}_{2}\oplus\cdots
\oplus E^{0,N-2}_{2}\oplus
E^{0,N-1}_{2}\oplus \underbrace{\text{Ker}d_{N}^{(0)}}_{E^{N-1,0}_{\infty}}.
\eqno{(4.5.12)}$$
By the spectral sequence definition, (4.5.12) implies $$\aligned
&H^{i}(\CL^{*},\Lambda^{ch}\CT_{\fF})\iso
H_{d_{1}}(E^{0,i}_{1}, d_{1}),\; 0\leq i\leq N-2,\\
0\rightarrow H_{d_{1}}(E^{0,N-1}_{1}, d_{1})\rightarrow
&H^{N-1}(\CL^{*},\Lambda^{ch}\CT_{\fF})\rightarrow
\text{Ker}d_{N}^{(0)}\rightarrow 0.
\endaligned
\eqno{(4.5.13)}$$
Now let us perform a change of coordinates that will reveal so far invisible structure of result (4.5.7ab, 4.5.13).
[**4.6.**]{} We have used the coordinates attached to $\CL^{*}$ by definition. Now let us employ the map $$\BC^{N}-0\rightarrow \CL^{*}-0,
\eqno{(4.6.1)}$$ that is, the composition of (4.1.5) and the natural projection $\BC^{N}-0\rightarrow (\BC^{N}-0)/{\BZ_{N}}$. We would like to recast the argument of 4.5 in terms inherent in $\BC^{N}-0$.
By invoking (4.6.1) we have placed ourselves in the situation of 3.9. Let us make this explicit.
$\BC^{N}-0$ has the standard coordinate system $x_{i},\partial_{x_{i}}$ and has, therefore, the following toric description: $$M=\oplus_{i=0}^{N-1}\BZ X_{i},\;
M^{*}= \oplus_{i=0}^{N-1}\BZ X_{i}^{*},\; \Lambda=M\oplus M^{*},
\eqno{(4.6.2a)}$$ so that $$X^{*}_{i}(X_{j})=\delta_{ij}.
\eqno{(4.6.2b)}$$ $$\Sigma=\{\sigma_{0},...,\sigma_{N-1}\},\; \sigma_{j}=\{x_{j}\neq 0\}.
\eqno{(4.6.2c)}$$ Map (4.6.1) is induced by the lattice (cf. 3.2) embedding $$g: M^{*}\rightarrow M_{\CL^{*}}^{*}\; (=M_{\CL^{*}-0}^{*})
\eqno{(4.6.3)}$$ dual to the lattice embedding $$\aligned
&g^{*}: M_{\CL^{*}}\rightarrow M,\\
&S\mapsto NX_{0},\; Y_{j}\mapsto X_{j}-X_{0},\; 1\leq j\leq N-1.
\endaligned
\eqno{(4.6.4a)}$$ Indeed, comparing (4.1.5) and (4.5.2) one obtains: under (4.6.1), $s\mapsto x_{0}^{N}$, $y_{j}\mapsto x_{j}/x_{0}$ and it remains to use $x_{i}=e^{X_{i}}$, $s=e^{S}$, $y_{j}=e^{Y_{j}}$ in order to obtain (4.6.4a).
It is immediate to see that $$g(X_{0}^{*})=NS^{*}-\sum_{j=1}^{N-1}Y^{*}_{j},\;
g(X^{*}_{j})=Y^{*}_{j},\; j\geq 1;
\eqno{(4.6.4b)}$$ thus $g(\Sigma)=\Sigma_{\CL^{*}}$. Therefore, we are indeed in the situation of 3.9, and if we write down isomorphism (3.9.6) explicitly, we will have all the assertions of 4.5 recast in terms pertaining to $\BC^{N}-0$.
According to (3.1.7), there arises an isomorphism $$\hat{g}:\BB_{\Lambda_{\CL^{*}}} \rightarrow \BB_{g^{-1}\Lambda_{\CL^{*}}},
\eqno{(4.6.5)}$$ where $$g^{-1}\Lambda_{\CL^{*}}=g^{*}M_{\Lambda_{\CL^{*}}}\oplus
g^{-1}M_{\Lambda_{\CL^{*}}}^{*}\subset\Lambda_{\BQ}.$$ It follows from (4.6.4a) that $$g^{*}M_{\Lambda_{\CL^{*}}}=
\{ \sum_{i=0}^{N-1}m_{i}X_{i}\text{ s.t. }N| \sum_{i=0}^{N-1}\}\subset M.
\eqno{(4.6.6)}$$ Inverting (4.6.4b) we obtain that $g^{-1}M_{\Lambda_{\CL^{*}}}^{*}$ is spanned over $\BZ$ by $$g^{-1}(S^{*})=\frac{1}{N}(X^{*}_{0}+X^{*}_{1}+\cdots +X^{*}_{N-1}),
g^{-1}(Y^{*}_{j})=X^{*}_{j},1\leq j\leq N-1.
\eqno{(4.6.7)}$$
Let us introduce the notation $$X^{*}_{orb}=\frac{1}{N}(X^{*}_{0}+X^{*}_{1}+\cdots +X^{*}_{N-1}).
\eqno{(4.6.9)}$$ The first line of (4.5.7a) and (4.5.7b) rewrites as follows: $$\aligned
&(E^{0,*}_{1},d_{1})=(\oplus_{n=0}^{\infty}
\Lambda^{ch}\CT(\BC^{N})^{\BZ_{N}}e^{nX^{*}_{orb}},D_{orb})\\
&D_{orb}=\frac{1}{N}
\sum_{i=0}^{N-1}(\tilde{X}_{i}(z)e^{X^{*}_{orb}}(z))_{(0)}.
\endaligned
\eqno{(4.6.10)}$$ (For the latter (3.1.4b) and (4.6.7) were used.)
This is a differential (w.r.t. $D_{orb}$) graded vertex algebra and because of its importance and its relation to Landau-Ginzburg models to be discovered later on, we make a digression.
[**4.6.1.**]{} [*Vertex algebra*]{} $\widetilde{\text{LG}}_{orb}$. Introduce the following notation: $$\widetilde{\text{LG}}_{orb}=\oplus_{n=0}^{\infty}\widetilde{\text{LG}}^{(n)}_{orb},
\; \widetilde{\text{LG}}^{(n)}_{orb}=\Lambda^{ch}\CT(\BC^{N})^{\BZ_{N}}e^{nX^{*}_{orb}}
\subset \BB_{g^{-1}\Lambda_{\CL^{*}}}.
\eqno{(4.6.11)}$$ This vertex algebra is filtered by the system of differential vertex ideals $$\widetilde{\text{LG}}_{orb}^{\geq m}=\oplus_{n\geq m}
\widetilde{\text{LG}}_{orb}^{(n)}.
\eqno{(4.6.12)}$$ Hence there arises a projective system of differential vertex algebras $$\widetilde{\text{LG}}_{orb}^{< m}=\widetilde{\text{LG}}_{orb}/
\widetilde{\text{LG}}_{orb}^{\geq m}.
\eqno{(4.6.13)}$$ It is obvious that the natural projection $$\widetilde{\text{LG}}_{orb}\rightarrow
\widetilde{\text{LG}}_{orb}^{< m}
\eqno{(4.6.13)}$$ induces the isomorphisms $$H^{i}_{D_{orb}}(\widetilde{\text{LG}}_{orb})\iso
H^{i}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< m})
\text{ if } i<m-1.
\eqno{(4.6.14)}$$ If $i=m-1$, then (4.6.13) induces the embedding $$H^{m-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb})\hookrightarrow
H^{m-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< m}),
\eqno{(4.6.15)}$$ which is included in the following short exact sequence $$0\rightarrow H^{m-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb})\rightarrow
H^{m-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< m})
\rightarrow B^{m}\rightarrow 0,
\eqno{(4.6.16)}$$ where $B^{m}\subset \widetilde{\text{LG}}_{orb}^{(m)}$ is the group of $m$-couboundaries of $\widetilde{\text{LG}}_{orb}$ w.r.t. $D_{orb}$. If we think of (4.6.15) as a 1-step filtration on $H^{m-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< m})$, then we get from (4.6.16) $$\text{Gr}H^{m-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< m})=
H^{m-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb})\oplus B^{m}.
\eqno{(4.6.17)}$$ $\qed$
Now, to the second line of (4.5.7a). The spectral flow transform appearing there had been earlier interpreted in (3.10.4) inside $\BB_{\Lambda}$, currently in use, by employing exactly $X^{*}_{orb}$ although not its name. According to (3.10.4) we have $$E^{N-1,q}_{1}=\Lambda^{ch}\CT(\BC^{N})^{\BZ^{N}}e^{(q+N)X^{*}_{orb}}.$$ The complete translation of (4.5.7a) is then this: $$(E^{*,*}_{1}, d_{1})=(E^{0,*}_{1}\oplus E^{N-1,*}_{1}, d_{1}) =(\widetilde{\text{LG}}\oplus
\widetilde{\text{LG}}^{\geq N},D_{orb}).
\eqno{(4.6.18)}$$ The rest of 4.5 carries over in a straightforward manner: (4.5.9) is translated as $$(E^{*,*}_{2}, d_{2})=(H^{*}_{D_{orb}}(\widetilde{\text{LG}})\oplus
H^{*}_{D_{orb}}(\widetilde{\text{LG}}^{\geq N}), 0),
\eqno{(4.6.19)}$$ the 1st summand standing for $E^{0,*}_{2}$, the 2nd for $E^{N-1,*}_{2}$; (4.5.13) is translated as $$\aligned
&H^{i}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})\iso H^{i}_{D_{orb}}(\widetilde{\text{LG}}),\; 0\leq i\leq N-2\\
0\rightarrow H^{N-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb})\rightarrow
&H^{N-1}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})\rightarrow B^{N}
\rightarrow 0,
\endaligned
\eqno{(4.6.20)}$$ where we freely use the notation of digression 4.6.1. The first line of (4.6.20) is obvious, and in the second only the identification of $B^{N}$ with $\text{Ker}d^{(0)}_{N}$ needs an explanation. It follows from (4.5.12) that the latter includes in the short exact sequence $$0\rightarrow\text{Ker}d^{(0)}_{N}\rightarrow
E^{N-1,0}_{2}\rightarrow E^{0,N}_{2}\rightarrow 0.
\eqno{(4.6.21)}$$
According to (4.6.19), we have $E^{N-1,0}_{2}=Z^{N}$, $E^{0,N}_{2}=Z^{N}/B^{N}$, where $Z^{N}$, $B^{N}$ are the groups of $N$-cocycles (coboundaries resp.) of the complex $(\tilde{\text{LG}}_{orb}, D_{orb})$. Therefore (4.6.21) can be identified with $$0\rightarrow B^{N}\rightarrow Z^{N}\rightarrow Z^{N}/B^{N}
\rightarrow 0.
\eqno{(4.6.22)}$$ We have proved
[**4.7. Theorem.**]{} [*The cohomology $H^{*}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})$ satisfies:*]{} $$H^{i}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})\iso
H^{i}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< N})\text{ if }i<N-1,
\eqno{(4.7.1)}$$ [*and if $i=N-1$, then there arises the short exact sequence*]{} $$0\rightarrow H^{N-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb})\rightarrow
H^{N-1}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})\rightarrow B^{N}
\rightarrow 0.
\eqno{(4.7.2)}$$ [*In particular,*]{} $$\text{Gr}H^{*}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})\iso
\text{Gr}H^{*}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< N}),
\eqno{(4.7.3)}$$ [*see (4.6.17) with $m=N$ for the definition of*]{} $\text{Gr}H^{*}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< N})$.
[**4.7.1. Remark.**]{} Having put (4.16.14-16) on the table next to (4.6.20) one observes that (4.7.1) has a good chance of being valid for $i=N-1$ as well.
[**4.8.**]{} Recall that our ultimate goal is the cohomology vertex algebra $H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$ and, according to (4.2.1a), Theorem 4.7 only computes the 1st term of the spectral sequence converging to $H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$. It remains to write down explicitly its 2nd term, also see (4.2.1a), and for this we need the differential $dt_{(0)}$ and the grading $R^{q}(.)$ expressed in terms of $\widetilde{\text{LG}}_{orb}$.
Thanks to (4.1.8) the differential is as follows: $$\hat{g}(dt)= df(x_{0},...,x_{N-1}).
\eqno{(4.8.1)}$$ A quick computation using, e.g., (1.9.9) shows that $$[D_{orb}, df(z)_{(0)}]=0.
\eqno{(4.8.2)}$$
The grading $R^{q}(\widetilde{\text{LG}}_{orb})$, $q\in\BZ$, is nicely described as follows: $$R^{q}(\widetilde{\text{LG}}_{orb})=\text{Ker}(X^{*}_{orb,(0)}-qI).
\eqno{(4.8.3)}$$ Indeed, $R^{q}(.)$ was defined in 2.4 as the eigenspace of the $\BC^{*}$-action. In terms of $S,Y_{j}$, the infinitesimal generator of this action is $S^{*}_{(0)}$; for example, $S^{*}_{(0)}s=S^{*}_{(0)}e^{S}=e^{S}=s$, as it should because according to (3.3.2a) $s=e^{S}$ is the coordinate along the fiber. It remains to use (4.6.7).
Incidentally, the $\BZ$-grading built into definition (4.6.11) of $\widetilde{\text{LG}}_{orb}$ is likewise given by the eigenvalues of the operator $\sum_{j}X_{j,(0)}$. The two gradings are compatible because $[X^{*}_{orb}, \sum_{j}X_{j,(0)}]=0$ as follows from (1.8.1).
Thus $\widetilde{\text{LG}}$ is a bi-differential bi-graded vertex algebra, to be denoted in this capacity by $(\widetilde{\text{LG}};D_{orb}, df(z)_{(0)})$, and so is $(\widetilde{\text{LG}}^{<N};D_{orb}, df(z)_{(0)})$. Essentially it remains to summarize Lemma 4.3 and Theorem 4.7.
[**4.9. Theorem.**]{} [*(i) Spectral sequence (4.2.1a) abuts to $H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$ so that*]{}
$$'\!E^{p,q}_{2}=H^{q}_{df(z)_{(0)}}(H^{p}_{D_{orb}}
(\widetilde{\text{LG}}_{orb}^{< N})),\; \text{ if } 0\leq p\leq N-2,
\eqno{(4.9.1)}$$ [*and if $p=N-1$, then $ '\!E^{N-1,q}_{2}$ is included into the long exact sequence stemming from the short exact sequence of complexes, cf. (4.7.2)*]{} $$0\rightarrow (H^{N-1}_{D_{orb}}(\widetilde{\text{LG}}_{orb}), df(z)_{(0)})
\rightarrow ('\!E^{*,N-1-*}_{1}, d_{1})\rightarrow (B^{N}, df(z)_{(0)})
\rightarrow 0.
\eqno{(4.9.2)}$$ [*(ii) In the conformal weight zero component the spectral sequence degenerates in the 2nd term and gives the isomorphism*]{} $$\text{Gr}H^{*}(\fF,\Lambda^{*}\CT_{\fF})\iso
\underbrace{\BC\oplus\cdots\oplus\BC}_{N-1}\oplus
\left(\BC[x_{0},...,x_{N-1}]/<df>\right)^{\BZ_{N}}.
\eqno{(4.9.3)}$$ [**4.9.1. Remark.**]{} For the same reason that was indicated in Remark 4.7.1, isomorphisms (4.9.1) have a good chance of being valid for $p=N-1$ as well. Were this the case, (4.9.2) would be unnecessary. $\qed$
[*Beginning of the proof.*]{} Item (i) is indeed simply (4.2.1a), Lemma 4.3, and (4.7.1-2) of Theorem 4.7 put together. The isomorphism in (ii) is well known classically and the degeneration assertion follows very easily. We prefer, however, to emphasize some additional structure hidden in the spectral sequence and then use it to give, among other things, a self-contained proof of (ii), see 4.13.
[**4.10.**]{} [**Addendum: $N2$-structure.**]{} Condition (4.1.0) ensures that $\CL^*$ is the canonical line bundle and this places us in the situation of 2.4.4: coordinates $s,y_{1},...,y_{N-1}$ of (4.5.2) satisfy the conditions imposed in 2.4.4, formulas (2.4.7) define an $N2$-structure on $\Lambda^{ch}\CT_{\CL^{*}}$ and (2.4.8) define that on $\Lambda^{ch}\CT_{\fF}$. In order to compute this structure in terms of $\widetilde{\text{LG}}_{orb}$ one has to do the following: first, compute the images of (2.4.8) in $\BB_{\Lambda_{\CL^{*}}}$ under Borisov’s map (3.3.2a,b); second, apply map (4.6.5) to the result. This is straightforward and tedious but rewarding, the reward being the coincidence of the result with the free field realization that Witten related with the Landau-Ginzburg model; this coincidence will be verified in Lemma 5.2.14.
In order to record the result it is convenient to use the boson-fermion correspondence, 1.13. Let us introduce the standard lattice $\BZ^{N}$ so that the standard basis $\chi_{0},...,\chi_{N-1}$ is orthonormal. Then, see 1.13, one can make identifications $$\tilde{X}_{i}(z)= e^{\chi_{i}}(z),\;
\tilde{X}_{i}^{*}(z)= e^{-\chi_{i}}(z).
\eqno{(4.10.1)}$$ [**4.10.1. Lemma.**]{} [*The $N2$-structure on $H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$ comes from the following $N2$-structure on*]{} $\widetilde{\text{LG}}_{orb}$: $$\aligned
&G(z)\mapsto \sum_{j=0}^{N-1}
:\left(X_{j}^{*}(z)-\chi_{j}(z)\right)e^{\chi_{j}}(z):,\;
Q(z)\mapsto \sum_{j=0}^{N-1}
:\left(X_{j}(z)+\frac{1}{N}\chi_{j}(z)\right)e^{-\chi_{j}}(z):,\\
&J(z)\mapsto \sum_{j=0}^{N-1}\left(-\frac{1}{N}X^{*}_{j}(z)+X_{j}(z)+
\chi_{j}(z):\right),\\
&L(z)\mapsto \sum_{j=0}^{N-1}
\left( :X_{j}(z)X^{*}_{j}(z):+\frac{1}{2}:\chi_{j}(z)^{2}:
-\frac{1}{2}\chi_{j}(z)'-X_{j}(z)'\right).
\endaligned$$ [**Proof.**]{} $N2$ is generated as a Lie algebra by 2 fields, $G(z)$ and $Q(z)$. Let us do $Q(z)$, the field that acquires the geometrically mysterious factor $1/N$. We have starting with (2.4.8) $$\aligned
&Q(z)\mapsto
s(z)'\partial_{ds}(z)+\sum_{j=1}^{N-1}y_{j}(z)'\partial_{dy_{j}}(z)-
(s(z)\partial_{ds}(z))'=\\
&\sum_{j=1}^{N-1}:y_{j}(z)'\partial_{dy_{j}}(z):-
:s(z)(\partial_{ds}(z))':=\\
&\sum_{j=1}^{N-1}:e^{Y_{j}}(z)'(:e^{-Y_{j}}(z)\tilde{Y}^{*}_{j}(z):)-
:e^{S}(z)(:e^{-S}(z)\tilde{S}^{*}_{j}(z):)':=\\
&\sum_{j=1}^{N-1}:Y_{j}(z)\tilde{Y}^{*}_{j}(z): +:S(z)\tilde{S}^{*}(z):-
\tilde{S}^{*}(z)'=\\
&\sum_{j=1}^{N-1}:(X_{j}(z)-X_{0}(z))\tilde{X}^{*}_{j}(z): +:NX_{0}(z)\frac{1}{N}\sum_{j=0}^{N-1}\tilde{X}^{*}_{j}(z):-
\frac{1}{N}\sum_{j=0}^{N-1}\tilde{X}^{*}_{j}(z)'=\\
&\sum_{j=0}^{N-1}:X_{j}(z)\tilde{X}^{*}_{j}(z):
-\frac{1}{N}\sum_{j=0}^{N-1}\tilde{X}^{*}_{j}(z)'=\\
&\sum_{j=0}^{N-1}
:\left(X_{j}(z)+\frac{1}{N}\chi_{j}(z)\right)e^{-\chi_{j}}(z):.
\endaligned$$ A brief guide to this computation is as follows: the 3rd line is (3.3.2a-b) applied to the 1st line, in the 5th line transformation formulas (4.6.7) are used, and boson-fermion correspondence (4.10.1) is used in the 7th. In addition, the well-known differentiation formula $e^{\alpha}(z)'=:\alpha(z)e^{\alpha}(z):$ has been repeatedly employed. $\qed$
[**4.11. Character and Euler character formulas.**]{} Whenever one has a bi-graded vector space $$V=\bigoplus_{m,n} V^{m}_{n},\; \text{dim}V^{m}_{n}<\infty,$$ one can define its character: $$chV(s,\tau)=\sum_{m,n} e^{2\pi i (ms+n\tau)}\text{dim}V^{m}_{n},
\eqno{(4.11.1)}$$ and if in addition $V$ is a supervector space, one can define its Euler character $$\text{Eu}V(s,\tau)=\sum_{m,n} e^{2\pi i (ms+n\tau)}
s\text{dim}V^{m}_{n},
\eqno{(4.11.2)}$$ where the super-dimension $s\text{dim}V^{m}_{n}$ is defined in the standard manner to be the dimension of the even component of $V^{m}_{n}$ minus the dimension of its odd component. Note that if $V$ carries an odd differential preserving the bi-grading, then $$\text{Eu}H_{d}(V)(s,\tau)=\text{Eu}V(s,\tau).
\eqno{(4.11.3)}$$ As an example, we can consider $\widetilde{\text{LG}}_{orb}^{<N}$ bi-graded by the eigenvalues of $L_{(1)}$, $J_{(0)}$, see Lemma 4.10.1. A straightforward computation (which, however, we postpone until 5.2.16) shows that if we introduce $$E(\tau,s)=\prod_{n=0}^{\infty}\frac{\left(1-e^{2\pi
i\left(\left(n+1\right)\tau+\left(1-1/N\right)s\right)}\right)^{N}\left(1-e^{2\pi
i\left(n\tau+\left(-1+1/N\right)s\right)}\right)^{N}}{\left(1-e^{2\pi
i\left(\left(n+1\right)\tau+s/N\right)}\right)^{N}\left(1-e^{2\pi
i\left(n\tau-s/N\right)}\right)^{N}},$$ then $$\text{Eu}\widetilde{\text{LG}}_{orb}^{<N}(\tau,s)
= \frac{1}{N}\sum_{l=0}^{N-1}\sum_{j=0}^{N-1}
e^{\pi
i\left(N-2\right)\left\{-2js+\left(j^{2}-j\right)\tau+j^{2}\right\}}E(\tau,s-j\tau-l).
\eqno{(4.11.4)}$$ and if we introduce $$\tilde{E}(\tau,s)=\prod_{n=0}^{\infty}\frac{\left(1+e^{2\pi
i\left(\left(n+1\right)\tau+\left(1-1/N\right)s\right)}\right)^{N}\left(1+e^{2\pi
i\left(n\tau+\left(-1+1/N\right)s\right)}\right)^{N}}{\left(1-e^{2\pi
i\left(\left(n+1\right)\tau+s/N\right)}\right)^{N}\left(1-e^{2\pi
i\left(n\tau-s/N\right)}\right)^{N}},$$ then $$ch\widetilde{\text{LG}}_{orb}^{<N}(\tau,s)
= \frac{1}{N}\sum_{l=0}^{N-1}\sum_{j=0}^{N-1}
e^{\pi
i\left(N-2\right)\left\{-2js+\left(j^{2}-j\right)\tau\right\}}
\tilde{E}(\tau,s-j\tau-l).
\eqno{(4.11.5)}$$ The repeated application of (4.11.3) to (4.7.3) shows that result (4.11.4) is valid for $H^{*}(\fF,\Lambda^{ch}\CT_{\fF}) $ as well: $$\text{Eu} H^{*}(\fF,\Lambda^{ch}\CT_{\fF}) =
\frac{1}{N}\sum_{l=0}^{N-1}\sum_{j=0}^{N-1}
e^{\pi
i\left(N-2\right)\left\{-2js+\left(j^{2}-j\right)\tau+j^{2}\right\}}
E(\tau,s-j\tau-l).
\eqno{(4.11.6)}$$ The importance of the latter is that, as was observed in \[BL\], the elliptic genus of $\fF$, $\text{Ell}_{\fF}(\tau,s)$, satisfies $$\text{Ell}_{\fF}(\tau,s)=e^{\pi i (N-2)s}
\text{Eu} H^{*}(\fF,\Lambda^{ch}\CT_{\fF}) .
\eqno{(4.11.7)}$$ We have proved
[**4.12. Corollary.**]{} $$\text{Ell}_{\fF}(\tau,s)=\frac{1}{N}\sum_{l=0}^{N-1}\sum_{j=0}^{N-1}
e^{\pi
i\left(N-2\right)\left\{-js+\left(j^{2}-j\right)\tau+j^{2}\right\}}
E(\tau,s-j\tau-l).
\eqno{(4.11.8)}$$
[**4.13. Chiral rings and $ H^{*}(\fF,\Lambda^{*}\CT_{\fF})$.**]{}
[*4.13.1. End of proof of Theorem 4.9.*]{} Arguments that led to Theorems 4.7, 9 consist of computations of two spectral sequences. Both the sequences are graded by $L_{(1)}$, see e.g. Lemma 4.10.1, and all the differentials preserve this grading. The classical story is about the events unfolding in the conformal weight zero component. Let us re-tell this story.
Therefore, we adopt that point of view according to which our task is to compute the cohomology algebra $ H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ of polyvector fields. Investigated in Theorem 4.9 is spectral sequence (4.2.1a); the origin of its conformal weight zero component is the classical Koszul sheaf complex (2.4.6) and its 1st term is the algebra $ H^{*}(\CL^{*},\Lambda^{*}\CT_{\CL^{*}})$. The computation of this alegbra is accomplished in the classical part of Theorem 4.7 and that is where an important simplification occurs: (4.7.3) restricted to the conformal weight zero component is valid without the passage to the graded objects: $$H^{*}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})_{0}\iso
H^{*}_{D_{orb}}(\widetilde{\text{LG}}_{orb}^{< N})_{0},
\eqno{(4.13.1)}$$ In fact, even taking the $D_{orb}$-cohomology is not necessary: $$H^{*}(\CL^{*},\Lambda^{ch}\CT_{\CL^{*}})_{0}\iso
(\widetilde{\text{LG}}_{orb}^{< N})_{0}.
\eqno{(4.13.2)}$$ Indeed, by definition $$(\widetilde{\text{LG}}_{orb}^{(0)})_{0}=\Lambda^{*}\CT(\BC^{N})^{\BZ_{N}}.
\eqno{(4.13.3)}$$ Further, $$\text{dim}(\widetilde{\text{LG}}_{orb}^{(i)})_{0}=1,\; 1\leq i\leq N-1,
\eqno{(4.13.4)}$$ as easily follows from character formula (4.11.5). The corresponding generator is $$(\widetilde{\text{LG}}_{orb}^{(i)})_{0}= \BC
e^{iX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})},\; 1\leq i\leq N-1,
\eqno{(4.13.5)}$$ where we again use boson-fermion correspondence (4.10.1). Now observe that spaces (4.13.3) and that spanned by elements (4.13.5) are both annihilated by $D_{orb}$, and (4.13.2) follows. We also see that the 1st term of spectral sequence (4.2.1a) is $$\left(('\!E_{1}^{*,*})_{0},d_{1}\right)=\left(
\underbrace {\BC\oplus\cdots\oplus\BC}_{N-1}\oplus
\left(\Lambda^{*}\CT(\BC^{N})\right)^{\BZ_{N}},
df(z)_{(0)}\right).
\eqno{(4.13.6)}$$ Note that in view of (4.13.2), (4.9.1) is valid for $p=N-1$ as well. A quick computation shows that the elements (4.13.5) are annihilated by $df(z)_{(0)}$. As to component (4.13.3), we have:
$(\Lambda^{*}\CT(\BC^{N}),df(z)_{(0)})$ [*is the standard Koszul resolution of the Milnor ring.* ]{}
By virtue of (4.13.6), $$('\!E_{2}^{*,*})_{0}=
\underbrace {\BC\oplus\cdots\oplus\BC}_{N-1}\oplus
\left(\BC[x_{0},...,x_{N-1}]/<df>\right)^{\BZ_{N}}.
\eqno{(4.13.7)}$$ Finally, all the higher differentials vanish simply because the elements listed in (4.13.5,7) are genuine cocycles. This completes the proof of Theorem 4.9. $\qed$
It is amusing to note that we have obtained “vertex” representatives of all the classes of the cohomology $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$. Since, as was explained in 4.8, the eigenvalues of $X^{*}_{orb}+\sum_{j}X_{j,(0)}$ give the cohomological grading, we have: $$(\text{class of}) e^{iX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}\in
H^{i-1}(\fF,\Lambda^{N-i-1}\CT_{\fF}),\; 1\leq i\leq N-1,
\eqno{(4.13.8)}$$ $$(\text{class of})\prod_{j}x_{j}^{m_{j}}=e^{\sum_{j}m_{j}X_{j}}\in
H^{m}(\fF,\Lambda^{m}\CT_{\fF}),\;
m=\frac{1}{N}\sum_{j}m_{j}, 0\leq m_{j}\leq N-2.
\eqno{(4.13.9)}$$
[*4.13.2. Multiplicative structure.*]{} The multiplicative structure of the ring $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ is well known, of course. But let us restore it by the “vertex” methods.
According to (2.3.3b), the chiral ring of $H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$ is isomorphic to $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$. However, what we have at our disposal is the chiral ring $(\widetilde{\text{LG}}_{orb}^{< N})_{0}$ and as a ring it only gives $\text{Gr}H^{*}(\fF,\Lambda^{ch}\CT_{\fF})$, as we have just proved – this is a common problem with spectral sequences.
Nevertheless, having re-examined the way in which spectral sequence (4.2.1a) was defined, one concludes easily that the 0-th component, $(\widetilde{\text{LG}}_{orb}^{(0)})_{0}$, remains unaffected by the passage to the graded object and thus carries the “right” multiplication; therefore, the Milnor ring, $\BC[x]/<df>$, embeds into $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ as a ring, cf. (4.9.3).
Let us now look at elements (4.13.8). The computation $$\aligned
\left(e^{iX^{*}_{orb}-\sum_{j}(X_{i}+\chi_{i})}\right)_{(-1)}
e^{\sum_{j}m_{j}X_{j}}=&\lim_{z\rightarrow w}(z-w)^{i\sum_{j}\frac{m_{j}}{N}}
e^{iX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}=\\
&\left\{\aligned
0&\text{ if }\sum_{j}m_{j}>0\\
e^{iX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}&\text{ if
}\sum_{j}m_{j}=0,
\endaligned\right.
\endaligned
\eqno{(4.13.10)}$$ as follows from (1.9.8), is valid even in $(\widetilde{\text{LG}}_{orb}^{(0)})_{0}$; hence in $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ as well.
Likewise, if $s+t<N$, one obtains $$\aligned
&\left(e^{sX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}\right)_{(-1)}
e^{tX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}=\\
&(-1)^{s}\lim_{z\rightarrow w}(z-w)^{N-s-t}
e^{(s+t)X^{*}_{orb}-2\sum_{j}(X_{j}+\chi_{j})}=0
\endaligned
\eqno{(4.13.11)}$$ inside $(\widetilde{\text{LG}}_{orb}^{(0)})_{0}$, hence inside $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$ as well.
Finally, the same computation shows that $$\left(e^{sX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}\right)_{(-1)}
e^{(N-s)X^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}= (-1)^{s}
e^{NX^{*}_{orb}-2\sum_{j}(X_{j}+\chi_{j})}
\eqno{(4.13.12)}$$ In $(\widetilde{\text{LG}}_{orb}^{< N})_{0}$ the class of this element is zero because it belongs to $\widetilde{\text{LG}}_{orb}^{(N)}$ and this component was cut off in definition (4.6.13). However, an amusing diagram search shows (we skip this computation) that in reality this element is cohomologous to $\pm\prod_{j}x_{j}^{N-2}$, which is a generator of $H^{N-2}(\fF,\Lambda^{N-2}\CT_{\fF})$, see (4.13.9). Therefore, $$\left(\text{class of }e^{sX^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}\right)_{(-1)}
\left(\text{class of }e^{(N-s)X^{*}_{orb}-\sum_{j}(X_{j}+\chi_{j})}
\right)=\pm
\prod_{j}x_{j}^{N-2}
\eqno{(4.13.13)}$$ and gives a non-degenerate pairing $$H^{s-1}(\fF,\Lambda^{N-s-1}\CT_{\fF})\times
H^{N-s-1}(\fF,\Lambda^{s-1}\CT_{\fF})\rightarrow
H^{N-1}(\fF,\Lambda^{N-1}\CT_{\fF}),
\eqno{(4.13.14)}$$ as it should. This completes the description of multiplication on $H^{*}(\fF,\Lambda^{*}\CT_{\fF})$.
[**5. Landau-Ginzburg orbifolds.**]{}
In this section we shall provide the necessary definitions and constructions so as to identify the complex $(\widetilde{\text{LG}}_{orb}^{<N}, df(z)_{(0)})$, which played an important role in Theorem 4.7 and 9, with the chiral algebra of the Landau-Ginzburg orbifold.
[**5.1. Landau-Ginzburg model.**]{} Let $$f\in\BC[x_{0},...,x_{N-1}],\; \text{deg}f=p
\eqno{(5.1.1)}$$ be a homogeneous polynomial such that its partials $\partial_{x_{i}}f$, $0\leq i\leq N-1$, have only one common zero occurring at $\vec{x}=0$. For the time being this $f$ need not be identified with $f$ of (4.1.6), that is, $p$ need not be N, but in the main application this assumption will be made.
By the (chiral algebra of the) Landau-Ginzburg model associated to $\BC^{N}$ and $f$ as in (5.1.1) we understand the differential vertex algebra $$\text{LG}_{f}=(\Lambda^{ch}\CT(\BC^{N}),df(z)_{(0)}).
\eqno{(5.1.2)}$$ Note that the chiral ring of $\text{LG}_{f}$ is the standard Koszul resolution: $$K_{f}^{*}:\; (\text{LG}_{f})_{0}=(\Lambda^{*}\CT(\BC^{N}),df),
\eqno{(5.1.3)}$$ the differential being the contraction with the 1-form $df$, cf. (4.13.6).
The following lemma is an important ingredient in Witten’s approach \[W2\] to the Landau-Ginzburg model; exactly this form of the result appears as formula (3.1.1) in \[KYY\].
[**5.1.1. Lemma.**]{} [*The assignment*]{} $$\rho:
\aligned
&G(z)\mapsto\sum_{i=0}^{N-1}\partial_{x_{i}}(z)dx_{i}(z),\;
Q(z)\mapsto \sum_{i=0}^{N-1}-\frac{1}{p}x_{i}(z)\partial_{dx_{i}}(z)'
-(\frac{1}{p}-1)x_{i}(z)'\partial_{dx_{i}}(z),\\
&J(z)\mapsto \sum_{i=0}^{N-1}-\frac{1}{p}:\partial_{x_{i}}(z)x_{i}(z):+
(\frac{1}{p}-1):\partial_{dx_{i}}(z)dx_{i}(z):,\\
&L(z)\mapsto \sum_{i=0}^{N-1}:\partial_{x_{i}}(z)x_{i}(z)':+
:\partial_{dx_{i}}(z)'dx_{i}(z):
\endaligned$$ [*determines a vertex algebra morphism*]{} $$\rho:\; V(N2)_{N\frac{p-2}{p}}\rightarrow \text{LG}_{f}
\eqno{(5.1.4)}$$ [*such that*]{} $$df(z)_{(0)}\rho\left(V(N2)_{N\frac{p-2}{p}}\right)=0.
\eqno{(5.1.5)}$$
[**5.1.2. Corollary.**]{} [*The vertex algebra $H_{df(z)_{(0)}}(\text{LG}_{f})$ carries an $N2$-structure inherited from that on $\text{LG}_{f}$.*]{}
This follows at once from Lemma 5.1.1 and (1.3.1). Let us look at some basic examples.
[**5.1.3. Theorem**]{} (\[FS\]) [*If $N=1$, $f=x^{p}$, then*]{} $H_{df(z)_{(0)}}(\text{LG}_{f})$ [*is the direct sum of $p-1$ unitary $N2$-modules generated by $1,x,x^{2},...,x^{p-2}$.*]{}
Denote by $U_{i,p}$ the unitary $N2$-module generated according to Theorem 5.1.3 by $x^{i}$, $0\leq i\leq p-2$; here $p$ keeps track of the “central charge”, that is, the value by which the central element $C$ operates on the module; in this case $C\mapsto (p-2)/p$. These modules are pairwise non-isomorphic.
[**5.1.4. Corollary.**]{} [*If $f=\sum_{j}x_{j}^{p}$, then*]{} $$H_{df(z)_{(0)}}(\text{LG}_{f})=\bigoplus_{0\leq j_{0},...,j_{N-1}\leq p-2}
\bigotimes_{t=0}^{N-1}U_{j_{t},p}$$ [*which is a unitary $N2$-module w.r.t. the diagonal action of central charge $N(p-2)/p$.*]{} $\qed$
[**Proof**]{} follows at once from Theorem 5.1.3 because in the case of the “diagonal” $f$ the complex $\text{LG}_{f}$ is the tensor product of the complexes of Theorem 5.1.3 – hence so is its cohomology as follows from the Künneth formula.
[**5.1.5. Remark.**]{} The reader will notice that complex (5.1.2) is nothing but (2.4.5) computed in the purely local situation with the function $t$ replaced with $f$. Furthermore, the cohomology of (2.4.5) gives $\Lambda^{ch}\CT_{Z(t)}$, see Lemma 2.4.2, and $Z(t)$ is exactly the singular locus of $t$: for any affine $U\subset \CL^{*}$ $$Z(t)\cap U=\text{Spec}\left(\CO_{\CL^{*}}(U)/<dt>\right).$$ Thus one is tempted to set $$\Lambda^{ch}\CT_{\text{Spec}M_{f}}(\text{Spec}M_{f})=
H_{df(z)_{(0)}}(\text{LG}_{f}),$$ thereby defining the sheaf $\Lambda^{ch}\CT_{\text{Spec}M_{f}}$, where $M_{f}=\BC[x_{0},...,x_{N-1}]/<df>$.
To put this somewhat differently, we have resolved the singularity of $M_{f}$ by passing to the DGA $K_{f}^{*}$, see (5.1.3), and then chiralized the latter so as to obtain (5.1.2). There seems to be a natural construction \[KV2\] allowing to chiralize in a similar manner other free DGA’s thereby extending algebras of chiral polyvector fields from smooth varieties to spectra of Milnor rings to a wider class of schemes.
[**5.2. Landau-Ginzburg orbifold.**]{}
[**5.2.1.**]{} Familiar in vertex algebra theory is the following pattern: $V$ is a vertex algebra; $g$ is its order $N$ automorphism; $V^{(i)}$ is a “naturally defined” $g^{i}$-twisted $V$-module, $1\leq i\leq N-1$; assuming that the group $\{1,g,...,g^{N-1}\}$ also acts on each $V^{(i)}$, naturally w.r.t to the $V$-action, one forms the vertex algebra of $g$-invariants, $V^{g}$, and its (untwisted) modules $(V^{(i)})^{g}$, $1\leq i\leq N-1$. It sometimes so happens that the space $$V^{g}\oplus (V^{(1)})^{g} \oplus\cdots (V^{(N-1)})^{g}
\eqno{(5.2.1)}$$ carries an “interesting” vertex algebra structure compatible with the described $V^{g}$-module structure. Vertex algebra (5.2.1) is often referred to as an orbifold or a $V$ orbifold.
The most famous example of an orbifold is undoubtedly the Monster vertex algebra $ V^{\text{Mnstr}} $ \[FLM\]. Indeed, $$V^{\text{Mnstr}} = V_{L}^{g}\oplus V_{L,1}^{g},$$ where $L$ is the Leech lattice, $g$ its involution, and $ V_{L,1}$ is an irreducible $g$-twisted $V_{L}$-module (unique by Dong’s theorem \[D2\]).
A number of physics papers, an incomplete list including \[V,VW,W2\] and references therein, suggests that a realization of this pattern in the case where $V=\text{LG}_{f}$, $\text{deg}f=N$, and $g=\exp{(2\pi i \rho J_{0})}$, $\rho J(z)$ being defined in Lemma 5.1.1, may be closely related to “string vacua”. The remainder of the paper is our attempt to understand this idea. We shall, first, construct the candidates for $\text{LG}_{f}^{(i)}$, and in order to do so we shall need a recollection on vertex algebra twisted modules, the notion introduced in \[FFR\]. In our presentation we shall mostly follow \[KR\]. In 5.2.14 we shall conclude that the space $\oplus_{i=0}^{N-1}\left(\text{LG}_{f}^{(i)}\right)^{g}$ with $\text{deg}f=N$ coincides with $\widetilde{\text{LG}}^{<N}_{orb}$ of Theorems 4.7, 4.9 as an $N2$-module. Second, in 5.2.17-5.2.20, we shall make $\oplus_{i=0}^{N-1}\left(\text{LG}_{f}^{(i)}\right)^{g}$ into a vertex algebra such that its chiral ring is isomorphic to the cohomology ring of the polyvector fields, $H_{\fF}(\fF,\Lambda^{*}\CT_{\fF})$, on the corresponding Calabi-Yau hypersurface (4.1.6).
[**5.2.2. Twisting data.**]{}
Let $G$ be an additive subgroup of $\BC$ containing $\BZ$. A vertex algebra $V$ is called $G/\BZ$-graded if $$V=\oplus_{\bar{m}\in G/\BZ}V[\bar{m}],
\eqno{(5.2.2a)}$$ so that $$V[\bar{m}]_{(n)}V[\bar{l}]\subseteq V[\bar{m}+\bar{l}].
\eqno{(5.2.2b)}$$ It is clear that $V[0]\subset V$ is a vertex subalgebra.
To give an example, let $g$ be an order $N$ automorphism of a vertex algebra $V$. Let $G=\frac{1}{N}\BZ$. We have $G/\BZ\iso
\BZ_{N}$.
Then $$V=\oplus_{m=0}^{N-1} V[m]=\{v\in V: gv=e^{2\pi im/N}v\}
\eqno{(5.2.3)}$$ is a $\BZ_{N}$-grading. By definition, in this case $V[0]$ is the vertex subalgebra of $g$-invariants, $V^{g}$.
Let $W$ be a vector space and $\bar{m}\in G/\BZ$, where $G$ is as in 2.1.1. An $\bar{m}$-twisted $\text{End} M$-valued field is a series $$a(z)=\sum_{m\in \bar{m}}a_{(m)}z^{-m-1},$$ where $a_{(m)}\in\text{End} M$ is such that for any $v\in W$, $a_{(m)}v=0$ if $\text{Re } m>>0$. Let $\text{Field}_{G}(W)$ be the linear space of $m$-twisted $\text{End} W$-valued fields for all $m\in G/\BZ$.
[**5.2.3. Definition.**]{} (cf. Definition 1.2 and \[KR, sect.5) A $G$-twisted $V$-module $W$ is a parity preserving linear map $$\rho: V\rightarrow \text{Field}_{G}(W),\;
(\rho a)(z)=\sum_{m}\rho a_{(m)}z^{-m-1}$$ satisfying the following axioms:
\(i) if $a\in V[\bar{m}]$, then $(\rho a)(z)$ is $\bar{m}$-twisted;
\(ii) $\rho (\b1)=\text{id}$;
\(iii) (twisted Borcherds identity) for any $a\in V[\bar{m}]$, $b\in V$ and $F(z,w)=z^{m}(z-w)^{l}$ such that $m\in\bar{m}$, $l\in\BZ$ $$\aligned
&\text{Res}_{z-w}\rho(a(z-w)b,w)i_{w,z-w}F(z,w)\\
&=
\text{Res}_{z}\left((\rho a)(z)(\rho b)(w)i_{z,w}F(z,w)-
(-1)^{\pr(a)\pr(b)}(\rho b)(w)(\rho a)(z)i_{w,z}F(z,w)\right).
\endaligned
\eqno{(5.2.4)}$$
[**5.2.4. Remarks.**]{}
\(i) Note that the $l=0$ case of the twisted Borcherds identity is the following commutator formula $$[\rho a_{(m)},\rho b_{(k)}] =\sum_{j=0}^{\infty}\binom{m}{j}
\rho (a_{(j)}b)_{(m+k-j)} .
\eqno{(5.2.5)}$$ This shows that the coefficients of the fields $(\rho a)(z)$ form a Lie algebra.
\(ii) A $\BZ$-twisted vertex algebra module is called simply a vertex algebra module. In particular, the restriction of a twisted $V$-module $W$ to the vertex subalgebra $V[0]\subset V$ is a $V[0]$-module. If $G$ arises from an order $N$ automorphism $g$ as in (5.2.3), then a $G$-twisted module is called $g$-twisted or twisted by $g$. The restriction of a $g$-twisted $V$-module to the vertex subalgebra $V^{g}$ is a $V^{g}$-module.
\(iii) It should be clear what the phrases “$W$ is an irreducible twisted $V$-module” and “ $W$ is a twisted $V$-module generated by a collection of fields $\{(\rho a_{\alpha})(z)$ from a given vector $m\in M$”.
\(iv) A vertex algebra is canonically a module over itself. Given an arbitrary $G/\BZ$-graded vertex algebra, there is no obvious way to construct a $G$-twisted module, but let us consider some concrete examples.
[**5.2.5.**]{} [*The twisted module $\Lambda^{ch}\CT(\BC^{N})_{\vec{\lambda},\vec{\mu}}$.*]{} Given 2 n-tuples $\vec{\lambda}=(\lambda_{0},...,\lambda_{N-1})\in\BC^{N}$, $\vec{\mu}=(\mu_{0},...,\mu_{N-1})\in\BC^{N}$, let $G$ be the $\BZ$-span of $\lambda_{i}$, $\mu_{i}$, and 1, $0\leq i\leq N-1$. Define the $G/\BZ$ grading on $\Lambda^{ch}\CT(\BC^{N})$ by declaring that $$\aligned
&x_{i}\in \Lambda^{ch}\CT(\BC^{N})[-\bar{\lambda_{i}}],\;
\partial_{x_{i}}\in \Lambda^{ch}\CT(\BC^{N})[\bar{\lambda_{i}}],\\
&dx_{i}\in \Lambda^{ch}\CT(\BC^{N})[-\bar{\mu_{i}}],\;
\partial_{dx_{i}}\in \Lambda^{ch}\CT(\BC^{N})[\bar{-\mu_{i}}],
\endaligned$$ cf. (5.2.2a).
[**5.2.6. Lemma.**]{}
[*(i) There is a unique up to isomorphism structure of a $G$-twisted $\Lambda^{ch}\CT(\BC^{N})$-module* ]{} $$\rho_{\vec{\lambda},\vec{\mu}}:
\Lambda^{ch}\CT(\BC^{N})\rightarrow \text{Field}_{G}W$$ generated by $vac\in W$ such that $$\aligned
&(\rho_{\vec{\lambda},\vec{\mu}} x_{i})_{(-\lambda_{i}+j)}vac=
(\rho_{\vec{\lambda},\vec{\mu}} \partial_{x_{i}})_{(\lambda_{i}+j)}vac=\\
&(\rho_{\vec{\lambda},\vec{\mu}} (dx_{i}))_{(-\mu_{i}+j)}vac=
(\rho_{\vec{\lambda},\vec{\mu}} \partial_{dx_{i}})_{(\mu_{i}+j)}vac=0.
\endaligned
\eqno{(5.2.6)}$$ [*(ii) This module can be constructed by letting $W=\Lambda^{ch}\CT(\BC^{N})$ as a vector space and*]{} $$\aligned
&(\rho_{\vec{\lambda},\vec{\mu}} x_{i})(z)=x_{i}(z)z^{\lambda_{i}},\;
(\rho_{\vec{\lambda},\vec{\mu}} \partial_{x_{i}})(z)=
\partial_{x_{i}}(z)z^{-\lambda_{i}},\\
&(\rho_{\vec{\lambda},\vec{\mu}} dx_{i})(z)=dx_{i}(z)z^{\mu_{i}},\;
(\rho_{\vec{\lambda},\vec{\mu}} \partial_{dx_{i}})(z)=
\partial_{dx_{i}}(z)z^{-\mu_{i}}.
\endaligned
\eqno{(5.2.7)}$$
[**Proof.**]{} The commutation relations (5.2.5) applied to the quadruple of fields $(\rho_{\vec{\lambda},\vec{\mu}} x_{i})(z)$ $(\rho_{\vec{\lambda},\vec{\mu}} \partial_{x_{i}})(z)$, $(\rho_{\vec{\lambda},\vec{\mu}} dx_{i})(z)$, $(\rho_{\vec{\lambda},\vec{\mu}} \partial_{x_{i}})(z)$ imply that their coefficients span the Lie algebra isomorphic to $Cl\oplus\fa$, see 1.6, 1.7. For example, $$[(\rho_{\vec{\lambda},\vec{\mu}} \partial_{x_{i}})_{(\alpha)},
(\rho_{\vec{\lambda},\vec{\mu}} x_{i})_{(\beta)}]=\delta_{\alpha, -\beta-1}.$$ Conditions (5.2.6) become then the vacuum vector conditions for $Cl\oplus\fa$ which of course determine $W$ uniquely even as $Cl\oplus\fa$-module, and uniqueness follows. This fixes recipe (5.2.7), and a standard argument lucidly explained e.g. in \[KR, sect. 5\] allows one to extend (5.2.7) naturally to a twisted module structure $\rho_{\vec{\lambda},\vec{\mu}}:
\Lambda^{ch}\CT(\BC^{N})\rightarrow \text{Field}_{G}W$. $\qed$
[**Notation.**]{} We shall let $\Lambda^{ch}\CT(\BC^{N})_{\vec{\lambda},\vec{\mu}}$ denote the twisted $\Lambda^{ch}\CT(\BC^{N})$-module occurring in Lemma 5.2.6.
[**5.2.7.**]{} [*The twisted module $V_{\Lambda+\eta}$.*]{} While constructing $\Lambda^{ch}\CT(\BC^{N})_{\vec{\lambda},\vec{\mu}}$ requires a little effort, a family of twisted modules over a lattice vertex algebra seems to be built into the definition of the latter. For simplicity we shall only consider the case where the lattice $\Lambda$ is that defined in 3.1. (The following should be regarded as known even though we failed to find the needed results in the literature, but see, e.g., a similar and more involved discussion in \[D1\].)
Fix $\eta\in\BC\otimes_{\BZ}M^{*}$ and consider the abelian group $L_{\eta}=\Lambda+\BC\eta\subset \BC\otimes\Lambda$. Let $\BC[\Lambda_{\eta}]$ be its group algebra. By analogy with the lattice vertex algebra $V_{\Lambda}=V(\fh_{L})\otimes\BC_{\epsilon}[\Lambda]$ introduce $V_{\Lambda_{\eta}}=V(\fh_{L})\otimes\BC[\Lambda_{\eta}]$.
Let $$\rho_{\Lambda}:V_{L}\rightarrow \text{Field}V_{\Lambda}
\eqno{(5.2.8)}$$ be the vertex algebra structure map. Note that $V_{\Lambda_{\eta}}=V(\fh_{L})\otimes\BC[\Lambda_{\eta}]$ is naturally a $V(\fh_{L})$-module: indeed $x(z) 1\otimes
e^{\beta}$, $x\in\fh_{L}$, makes perfect sense if $\beta\in
L_{\eta}$, or indeed if $\beta\in \BC\otimes_{\BZ}L$, see (1.9.5). Similarly, formula (1.9.2) may be extended without any changes to define a $\BC_{\epsilon}[\Lambda]$-action on $V_{\Lambda_{\eta}}=V(\fh_{L})\otimes\BC[\Lambda_{\eta}]$; indeed, cocycle $\epsilon(\alpha,\beta)$ defined in (3.1.2) makes perfect sense for any $\beta\in L+\BC\eta$ because it, so to say, ignores $\eta$. Since any field $\rho_{\Lambda}a(z)$, $a\in V_{L}$, is written in terms of the operators we have just described, cf (1.9.4-5), the very formula for $\rho_{\Lambda}a(z)$ defines it as a “field” operating on $V_{\Lambda_{\eta}}=V(\fh_{L})\otimes\BC[\Lambda_{\eta}]$. For example, $$x(z)e^{\eta}=\left(x(z)+\frac{1}{z}(x,\eta)\b1\right)e^{\eta},\;
x\in\fh_{L}
\eqno{(5.2.9a)}$$ $$e^{\alpha}(z)e^{\eta}=\left( e^{\alpha}(z)z^{(\alpha,\eta)}\b1\right)e^{\eta},
\eqno{(5.2.9b)}$$ where $\b1=e^{0}$ is the vacuum vector of $V_{\Lambda}$. (Note, by the way, that (5.2.9a-b) are analogous to spectral flow formulas (1.12.7-8).) In order to fix the twist, see 5.2.2, let us restrict such fields to the subspace $V_{\Lambda+\eta}\subset V_{\Lambda_{\eta}}$ where only $e^{\alpha+\eta}$, $\alpha\in L$, are allowed. Formulas (5.2.9a-b) allow us to conclude that there is a tautological embedding $$\rho_{\Lambda}V_{\Lambda}\hookrightarrow
\text{Field}_{G_{\eta}}V_{\Lambda+\eta},
\eqno{(5.2.10)}$$ where $G_{\eta}$ is the grading on $V_{\Lambda}$ that assigns degree $-(\alpha,\eta)$ to $e^{\alpha}$, $\alpha\in \Lambda$. Hence there arises the composition $$\rho_{\Lambda+\eta}: V_{\Lambda}\buildrel \rho_{\Lambda}\over\rightarrow
\rho_{\Lambda}V_{\Lambda}\hookrightarrow
\text{Field}_{G_{\eta}}V_{\Lambda+\eta},
\eqno{(5.2.11)}$$ [**5.2.8. Lemma.**]{} [*Map (5.2.11) endows $V_{\Lambda+\eta}$ with a $G_{\eta}$-twisted $V_{\Lambda}$-module structure.*]{}
[**Proof.**]{} Recall that the intuition behind twisted Borcherds identity (5.2.4) – as well as its untwisted version (1.2.2) – is that the 3 expressions appearing in it, $\rho(a(z-w),w)$, $(\rho a)(z)(\rho b)(w)$, and $(-1)^{\pr(a)\pr(b)}(\rho b)(w)(\rho a)(z)$ are Laurent series expansions of the same function in 3 respective domains, see the short discussion after Definition 1.2. This can be made precise: if $W$ is a vector space graded by finite dimensional subspaces, then one can define matrix elements of fields and their products, such as $<v^{*},\rho(a(z-w),w)v>$, $<v^{*},(\rho a)(z)(\rho b)(w)v>$, and $(-1)^{\pr(a)\pr(b)}<v^{*},(\rho b)(w)(\rho a)(z)v>$. If these are indeed Laurent series expansions of the same rational function twisted by $z^{-m}$, $w^{k}$, $m$, $k$ not necessarily integral, with poles on $z=w$, $z=0$, $w=0$, then (5.2.4) holds; cf. \[K, Remark 4.9a\]. In our case a suitable grading is easy to exhibit and a familiar argument along the lines of \[K, sect.5.4\] shows that matrix elements of the products of fields from $\rho_{\Lambda+\eta}(V_{\Lambda})$ are indeed such rational functions. Furthermore, it is easy to see that if we identify $$V_{\Lambda+\eta}\iso V_{\Lambda},\; v\mapsto v e^{-\eta},$$ then these matrix elements, with fixed $v^{*},v\in V_{\Lambda}$, become analytic functions of $\eta$, see (5.2.9a-b). But if $\eta\in M^{*}$, then $V_{\Lambda+\eta}=V_{\Lambda}$ by definition and the matrix elements are indeed equal to each other rational functions. Thanks to analyticity, this must hold for all $\eta$ and the lemma follows.
[**5.2.9.**]{} The constructions of 5.2.5 and 5.2.6 are related to each other. Invoke Borisov’s algebra $\BB_{\Lambda}$, see (3.1.3). Since $\BB_{\Lambda}=V_{\Lambda}\otimes F_{\Lambda}$, the space $$\BB_{\Lambda+\eta}\buildrel \text{def}\over =
V_{\Lambda+\eta}\otimes F_{\Lambda}
\eqno{(5.2.12)}$$ is naturally a twisted $\BB_{\Lambda}$-module. Hence its pull-back w.r.t. Borisov’s embedding $$\Lambda^{ch}\CT(\BC^{N})\rightarrow \BB_{\Lambda}
\eqno{(5.2.13)}$$ is a twisted $\Lambda^{ch}\CT(\BC^{N})$-module.
[**5.2.10. Lemma.**]{} [*If $$\eta=\sum_{j=0}^{N-1}\lambda_{j}X^{*}_{j},
\eqno{(5.2.13)}$$ then the twisted $\Lambda^{ch}\CT(\BC^{N})$-module generated by $\Lambda^{ch}\CT(\BC^{N})$ from the vector $e^{\eta}\otimes\b1\in V_{\Lambda+\eta}$ is isomorphic to $\Lambda^{ch}\CT(\BC^{N})_{\vec{\lambda}, \vec{\lambda}}$, where $\vec{\lambda}=(\lambda_{0},...,\lambda_{N-1})$.*]{}
[**Proof.**]{} According to Lemma 5.2.5 in order to prove the lemma one only has to check that the vector $e^{\eta}$ satifies relations (5.2.6). But this is obvious. For example, (5.2.9b) gives $$\rho_{L+\eta}x_{i}(z)e^{\eta}=
\left(e^{X_{i}}(z)z^{(X_{i},\eta)}\b1\right)e^{\eta}=
\left(e^{X_{i}}(z)z^{\lambda_{i}}\b1\right)e^{\eta},$$ which is exactly the first of conditions (5.2.6). The remaining 3 fields are dealt with in exactly the same manner.
[**5.2.11. Notation.**]{} It is natural to denote the twisted module $\Lambda^{ch}\CT(\BC^{N})_{\vec{\lambda}, \vec{\lambda}}$ realized as in Lemma 5.2.10 by $\Lambda^{ch}\CT(\BC^{N})e^{\sum_{j}\lambda_{j}X_{j}^{*}}$.
[**5.2.12. Landau-Ginzburg orbifold.**]{} We now wish to orbifoldize the differential graded vertex algebra $\text{LG}_{f}$, (5.1.2), with respect to the $\BZ_{p}$-action generated by the operator $g=\exp{(\rho J(z)_{(0)})}$, where $\rho J(z)_{(0)}$ is defined in Lemma 5.1.1. According to the pattern reviewed in 5.2.1, in order to do so one has to exhibit a $g^{i}$-twisted differential $\text{LG}_{f}$-module for each $1\leq i\leq N-1$. (Remark 5.2.4 (ii) explains what “$g^{i}$-twisted” means)
As follows from (5.2.3), the $i$th twisting gradation is determined by the action of $\rho J(z)_{(0)}$ on the generators. Formulas of Lemma 5.1.1 imply $$\aligned
&\rho J(z)_{(0)}x_{i}=-\frac{1}{p}x_{i},\;
\rho J(z)_{(0)}\partial{x_{i}}=\frac{1}{p}\partial_{x_{i}},\\
&\rho J(z)_{(0)}dx_{i}=(1-\frac{1}{p})dx_{i},\;
\rho J(z)_{(0)}\partial{dx_{i}}=(\frac{1}{p}-1)\partial_{x_{i}}.
\endaligned
\eqno{(5.2.14)}$$ Therefore, $\Lambda^{ch}\CT(\BC^{N})_{i\vec{\frac{1}{p}},i\vec{\frac{1}{p}}}$, where $\vec{\frac{1}{p}}=(1/p,...,1/p)$, is a natural candidate for the $\exp{i(\rho J(z)_{(0)})}$-twisted module. So we define, cf. 5.2.11, $$\text{LG}^{(i)}_{f}\buildrel \text{def}\over =
\Lambda^{ch}\CT(\BC^{N})e^{\frac{i}{p}\sum_{j}X_{j}^{*}}.
\eqno{(5.2.15)}$$ As mentioned in 5.2.4 (ii), $\text{LG}^{(i)}_{f}$ is a (untwisted) $\Lambda^{ch}\CT(\BC^{N})^{g}$-module. (Recall that $\Lambda^{ch}\CT(\BC^{N})=\text{LG}_{f}=\text{LG}_{f}^{(0)}$ as vertex algebras.)
Since the Landau-Ginzburg differential $df(z)_{(0)}$ comes from $df\in \Lambda^{ch}\CT(\BC^{N})^{g}$, it operates naturally on $\text{LG}^{(i)}_{f}$ thus making it a differential $\text{LG}_{f}^{g}$-module.
Since (5.1.4) maps $V(N2)_{N\frac{p-2}{p}}$ into $\text{LG}_{f}^{g}$, each $\text{LG}^{(i)}_{f}$ acquires an $N2$-structure. In particular, $g$ operates on $\text{LG}^{(i)}_{f}$ and $\left(\text{LG}^{(i)}_{f}\right)^{g}$ is also a differential $\left(\text{LG}^{(0)}_{f}\right)^{g}$-module.
[**5.2.13. Definition.**]{} Define the Landau-Ginzburg orbifold to be the following differential $\text{LG}_{f}^{g}$-module: $$\text{LG}_{f,\text{orb}}=\left(\bigoplus_{j=0}^{p-1}
\left(\text{LG}^{(j)}_{f}\right)^{g},\; df(z)_{(0)}\right).$$ $\qed$
Note that if $\text{deg}f=p=N$, then $\text{LG}_{f,\text{orb}}$ simply coincides with $(\widetilde{\text{LG}}_{orb}^{<N},df(z)_{(0)})$ appearing in Theorems 4.7, 4.9. This space, however, carries two [*a priori*]{} different $N2$-structures: one computed in Lemma 4.10.1 and having purely geometric origin and another, Landau-Ginzburg structure, recorded in Lemma 5.1.1.
[**5.2.14. Lemma.**]{} [*The two $N2$-structures coincide with each other.*]{}
[**Proof**]{} is, of course, a routine computation consisting in applying Borisov’s formulas (3.3.2a-b) to the 4 fields of Lemma 5.1.1. Let us consider $Q(z)$ and leave the rest as an exercise for the interested reader. We have (and recall that we are using boson-fermion correspondence (4.10.1)): $$\aligned
&Q(z)\mapsto \sum_{i=0}^{N-1}-\frac{1}{N}x_{i}(z)\partial_{dx_{i}}(z)'
-(\frac{1}{N}-1)x_{i}(z)'\partial_{dx_{i}}(z)\mapsto\\
&\sum_{i=0}^{N-1}-\frac{1}{N}:e^{X_{i}}(z)
\left(:\left(-X_{i}(z)-\chi_{i}(z)\right)e^{-X_{i}-\chi_{i}}(z):\right):-\\
&\left(\frac{1}{N}-1\right):\left(:X_{i}(z)e^{X_{i}}(z):\right)
e^{-X_{i}-\chi_{i}}(z):=\\
&\sum_{i=0}^{N-1} \frac{1}{N}:\left(X_{i}\left(z\right)+\chi_{i}\left(z\right)
\right)e^{-\chi_{i}}\left(z\right):+ \left(1-\frac{1}{N}\right):X_{i}\left(z\right)
e^{-\chi_{i}}\left(z\right):=\\
&\sum_{i=0}^{N-1} \frac{1}{N}:\chi_{i}(z)e^{-\chi_{i}}(z):
+:X_{i}(z)e^{-\chi_{i}}(z):,
\endaligned$$ as it should, cf. Lemma 4.10.1. Note that this computation is parallel to that we performed in the proof of Lemma 4.10.1. $\qed$
Lemma 5.2.14 represents the last step in the identification of $\widetilde{\text{LG}}^{<N}_{orb}$ that played an important role in Theorems 4.7, 4.9 with the Landau-Ginzburg orbifold[^3]. The next lemma says that as $N2$-modules the twisted sectors are spectral flow transforms of the untwisted one, see (1.12.1-2) for the definition of the spectral flow.
[**5.2.15. Lemma.**]{} [*The map*]{} $$e^{-j\sum_{i}(\frac{1}{p}X^{*}_{i}-X_{i}-\chi_{i})}:\;
\text{LG}_{f}^{(j)}\rightarrow \text{LG}_{f};\text{ s.t. }
x\mapsto xe^{-j\sum_{i}(\frac{1}{p}X^{*}_{i}-X_{i}-\chi_{i})}.
\eqno{(5.2.16)}$$
*delivers the following isomorphisms:*
(i)
$$\text{LG}^{(j)}_{f}\iso
S_{j}\left(\text{LG}_{f}\right),
H_{df(z)_{(0)}}(\text{LG}^{(j)}_{f})\iso
S_{j}\left(H_{df(z)_{(0)}}(\text{LG}_{f})\right)\;.$$ [*(ii) If $p=N$, then*]{} $$\left(\text{LG}^{(j)}_{f}\right)^{g}\iso
S_{j}\left(\text{LG}_{f}^{g}\right),\;
H_{df(z)_{(0)}}\left(\text{LG}^{(j)}_{f}\right)^{g}\iso
S_{j}\left(H_{df(z)_{(0)}}(\text{LG}_{f}^{g})\right)$$ [*if $p=N$.*]{}
[**Proof**]{} is parallel to the discussion in 3.10, and we will be brief: a glance at the formulas in Lemma 4.10.1 shows that under map (5.2.16) $Q(z)$ gets multiplied by $z^{j}$, $G(z)$ by $z^{-j}$ thereby recovering (1.12.1). This proves the first of ismorphisms (i); the second follows from an equally obvious observation that the differential $df(z)_{(0)}$ is invariant under (5.2.16).
As to (ii), this argument does not quite apply if $p\neq N$ because then $e^{-j\sum_{i}(\frac{1}{p}X^{*}_{i}-X_{i}-\chi_{i}})$ is not $Z_{N}$-invariant and does not give a well-defined map. If, however, $p=N$, then (ii) is proved in the same way as (i).
[**5.2.16. Application: the character formulas.**]{} Lemma 5.2.15 is an intelligent way to obtain (4.11.4-5). Focus on the Euler character formulas. First of all, $\text{LG}_{f}$ being bi-graded by the eigenvalues of $\rho J_{(0)}$ and $\rho L_{(1)}$, definition (4.11.2) is equivalent to $$\text{Eu}(\text{LG}_{f})(s,\tau)=\text{Tr}|_{\text{LG}_{f}}(-1)^{\pr(.)}
e^{2\pi i(sJ_{(0)}+\tau L_{(1)})},
\eqno{(5.2.17)}$$ where $\pr(.)$ is the parity function, see 1.1 – and we are skipping the $\rho$. It is quite standard to deduce that $$\text{Eu}(\text{LG}_{f})(s,\tau)= E(\tau,s),
\eqno{(5.2.18)}$$ where $E(\tau,s)$ is the function appearing right before (4.11.4), after all $\text{LG}_{f}$ is a superpolynomial ring in infinitely many variables. In view of (5.2.16), to obtain the Euler character of a spectral flow transformed module, one has to perform the linear change of variables, determined by (1.12.1), in the Euler character of the original module and then take care of the parity. By virtue of Lemma 5.2.15, in the case of $\text{LG}_{f}^{(j)}$ one has to replace $$\aligned
J_{(0)} &\text{ with } J_{(0)}-(N-2)j,\\
L_{(1)} &\text{ with }L_{(1)}-jJ_{(0)}+j(j-1)(N-2)/2,\\
\pr &\text{ with }\pr +||j\sum_{i}\frac{1}{N}X^{*}_{i}-X_{i}-\chi_{i}||^{2}
\text{ mod } 2.
\endaligned
\eqno{(5.2.19)}$$ The 1st two of these follow from (1.12.1), the last from (5.2.16) and definition of parity (1.9.3). Plugging (5.2.19) in (5.2.17) we obtain $$\aligned
\text{Eu}(\text{LG}_{f}^{(j)})(s,\tau)=&\text{Tr}|_{\text{LG}_{f}}
(-1)^{\pr(.)+j^{2}(N-2)}\times\\
&e^{2\pi i(s(J_{(0)}-(N-2)j)+\tau (L_{(1)}-jJ_{(0)}+j(j-1)(N-2)/2))}.
\endaligned
\eqno{(5.2.20)}$$ Then (5.2.18) can be rewritten as follows $$\text{Eu}(\text{LG}_{f}^{(j)})(s,\tau)=
e^{\pi
i\left(N-2\right)\left\{-2js+\left(j^{2}-j\right)\tau+j^{2}\right\}}
E(\tau,s-j\tau).
\eqno{(5.2.21)}$$ Finally, in order to extract $g$-invariants, one applies the averaging operator $\frac{1}{N}\sum_{l}e^{2\pi i l J_{(0)}}$ which results in shifts $s\mapsto s +l$ and gives (4.11.4).
[**5.2.17. Vertex algebra structure.**]{} Let us focus on the case where $\text{deg}f=p=N$. [*A priori*]{} the orbifold $\text{LG}_{f,\text{orb}}$ carries only uninteresting vertex algebra structure: by definition it can be regarded as an abelian extension of $\text{LG}_{f}^{g}$ by the sum of vertex modules $\oplus_{i}\left(\text{LG}^{(i)}_{f}\right)^{g}$. Cut-off definition (4.6.13), which applies to the present situation, see 5.2.13, fares somewhat better: it endows $\text{LG}_{f,\text{orb}}$ with a filtered vertex algebra structure such that the corresponding graded object is the mentioned above uninteresting vertex algebra. And yet this vertex algebra structure does not seem to be quite right either. Formally, one would expect an orbifold multiplicative structure to be $\BZ_{N}$-graded. Instead, our vertex algebra is graded by the semigroup $\BZ_{+}/(N+\BZ_{+})$. A similar problem has already manifested itself: by virtue of (4.9.3) the chiral ring of $\text{LG}_{f, orb}$ is isomorphic to the cohomology algebra $H_{\fF}(\fF,\Lambda^{*}\CT_{\fF})$ as a vector space but not as a ring, see (4.13.12) and the sentence that follows. Let us propose a way out in the case where $f=\sum_{i}x_{i}^{N}$ and $\text{LG}_{f, orb}$ is replaced with the Landau-Ginzburg cohomology $H_{df(z)_{(0)}}(\text{LG}_{f,
orb})$.
In this case, Corollary 5.1.4 combined with Lemma 5.2.15 provides us with a very explicit description of the cohomology: $$H_{df(z)_{(0)}}(\text{LG}_{f,orb})=\bigoplus_{n=0}^{N-1}
\bigoplus_{j_{0},...,j_{N-1}\leq N-2}
\bigotimes_{t=0}^{N-1}S_{n}\left(U_{j_{t},N}\right)^{g}.
\eqno{(5.2.22)}$$ The point is, the category of unitary highest weight $N2$-modules has a remarkable periodicity property: there are ismorphisms \[FS\] $$T_{m}: S_{mN+n}(U_{j,N})\iso S_{n}(U_{j,N}),\; m\in\BZ,\;0\leq
n\leq N-1.
\eqno{(5.2.23)}$$ Here is what this “Bott periodicity” suggests: since $\text{LG}_{f}^{(n)}$ is naturally defined for any $n\in\BZ$, instead of taking $\oplus_{n\geq
0}H_{df(z)_{(0)}}(\text{LG}_{f}^{(n)})^{g}$ and then quotienting out by the ideal $\oplus_{n\geq
N}H_{df(z)_{(0)}}(\text{LG}_{f}^{(n)})^{g}$, consider $\oplus_{n\in \BZ}H_{df(z)_{(0)}}(\text{LG}_{f}^{(n)})^{g}$ and identify the components $H_{df(z)_{(0)}}(\text{LG}_{f}^{(\alpha)})^{g}$ and $H_{df(z)_{(0)}}(\text{LG}_{f}^{(\beta)})^{g}$ with $\alpha-\beta\in N\BZ$ by using (5.2.23). Note that (5.2.23) is determined up to proportionality and herein lies the difficulty. Nevertheless, the following holds true.
[**5.2.18. Theorem.**]{} [*Collection of isomorphisms (5.2.23) can be normalized so that the linear span $$\CJ=<x-T_{m}^{\otimes
N}x;\;x\in\bigotimes_{t=0}^{N-1}S_{mN+n}\left(U_{j_{t},N}\right)^{g},\;0\leq
j_{t}\leq N-2,0\leq n\leq N-1>$$ is an ideal of the vertex algebra $\oplus_{n\in
\BZ}H_{df(z)_{(0)}}(\text{LG}_{f}^{(n)})^{g}$.*]{}
Proof of this theorem requires a little plunge in the genus zero $N2$ correlation functions and will be skipped. Note that there is an obvious vector space isomorphism $$H_{df(z)_{(0)}}(\text{LG}_{f, orb})\iso \bigoplus_{n\in
\BZ}H_{df(z)_{(0)}}(\text{LG}_{f}^{(n)})^{g}/\CJ,
\eqno{(5.2.24)}$$ and according to Theorem 5.2.18 the vector space $H_{df(z)_{(0)}}(\text{LG}_{f, orb})$ becomes a vertex algebra. The following corollary suggests that this vertex algebra structure is the “right” one.
[**5.2.19. Corollary.**]{} [*The chiral ring of $H_{df(z)_{(0)}}(\text{LG}_{f, orb})$ is isomorphic to the ring $H_{\fF}(\fF,\Lambda^{*}\CT_{\fF})$, where $\fF$ is the Calabi-Yau hypersurface (4.1.6).*]{}
[**5.2.20.**]{} [*Proof.*]{} At the vector space level, Corollary 5.2.19 is nothing but Theorem 4.9 (ii) or, rather, its proof, see sect. 4.13.1, formula (4.13.2), which shows that the differential $D_{orb}$ is zero on the chiral ring. The multiplicative structure of $H_{\fF}(\fF,\Lambda^{*}\CT_{\fF})$ has already been detected inside $\widetilde{\text{LG}}_{orb}$ in sect. 4.13.2. It is clear that the part of that argument which has to do with the untwisted sector (i.e. the Milnor ring) or twisted sectors of “low” degree of twisting, see (4.13.10-4.13.11), carries over to the present situation without any changes. What remains to be done is to check that (4.13.13) holds in $H_{df(z)_{(0)}}(\text{LG}_{f, orb})$.
In order to verify (4.13.13), we have to rely on (4.13.12). For the reader’s convenience, let us copy the latter: $$\aligned
&\left(e^{sX^{*}_{orb}-\sum_{j}(X_{i}+\chi_{i})}\right)_{(-1)}
e^{(N-s)X^{*}_{orb}-\sum_{j}(X_{i}+\chi_{i})}= \\
&(-1)^{s}
e^{NX^{*}_{orb}-2\sum_{j}(X_{i}+\chi_{i})}
\in \text{LG}_{f,orb}^{(N)}.
\endaligned
\eqno{(5.2.25)}$$ We assert that up to proportionality $$T_{1}^{\otimes N} \left(\text{class of }
e^{\sum_{j}(X^{*}_{j}-2(X_{j}+\chi_{j}))}\right)= \text{class of
}e^{(N-2)\sum_{j}X_{j}} \in
H_{df(z)_{(0)}}(\text{LG}_{f,orb}^{(N)}). \eqno{(5.2.26)}$$ If proven, (5.2.26) will recover the desired (4.13.13).
In order to prove (5.2.26), we shall obtain a representation theoretic interpretation of both the elements that determines them up to proportionality.
Since in the case at hand everything is a tensor product of 1-dimensional Landau-Ginzburg models and their cohomology groups, we can drop the subindex $j$, and our task is to show that $$T_{1}(\text{ class of }e^{X^{*} -2(X +\chi )})= \text{class of
}e^{(N-2)X}\in H_{df(z)_{(0)}}(\text{LG}_{f}).
\eqno{(5.2.27)}$$ According to Theorem 5.1.3, $e^{(N-2)X}$ generates in the cohomology the unitary $N2$-module $U_{N-2,N}$. Actually, $e^{(N-2)X}$ is a [*highest weight vector*]{} of $U_{N-2,N}$ meaning that it is annihilated by the Lie algebra span of $$\aligned
&G_{(2)},G_{(3)},...,\\
Q_{(1)}, &Q_{(2)}, Q_{(3)}.
\endaligned
\eqno{(5.2.28)}$$ It follows from Lemma 5.2.15 (i) that $S_{N}(U_{N-2,N})\subset H_{df(z)_{(0)}}(\text{LG}^{(N)}_{f})$ is generated by $e^{X^{*} -2X-N\chi }$. The same lemma implies that the latter is rather a [*twisted highest weight vector*]{} meaning that it is annihilated by the Lie algebra span of $$\aligned
&G_{(N+2)},G_{(N+3)},...,\\
Q_{(-N+1)}, Q_{(-N+2)},..., &Q_{(N+2)}, Q_{(N+3)},...
\endaligned
\eqno{(5.2.29)}$$ Now we would like to find a highest weight vector of $S_{N}(U_{N-2,N})\subset H_{df(z)_{(0)}}(\text{LG}^{(N)}_{f})$. A direct computation shows that $e^{X^{*} -2X-N\chi }$ is annihilated by $G_{(N+1)}$ in addition to subalgebra (5.2.29). Commutation relations (1.10.1) imply then that the vector $$G_{(3)}G_{(4)}\cdots G_{(N-1)}G_{(N)}e^{X^{*} -2X-N\chi }
\eqno{(5.2.30)}$$ is annihilated by the untwisted annihilating subalgebra, (5.2.28), except perhaps the element $G_{(2)}$. It also follows from \[FS, (A.5)\] that the class of (5.2.30) is non-zero in $S_{N}(U_{N-2,N})$. Now we compute $$\aligned
&G_{(N)}e^{X^{*} -2X-N\chi }=
\left(:(X^{*}(z)-\chi(z))e^{\chi}(z):\right)_{(N)}e^{X^{*} -2X-N\chi }=\\
&e^{\chi}(X^{*}_{(0)}-\chi_{(0)})e^{X^{*} -2X-N\chi }=
(N-2)e^{X^{*} -2X-(N-1)\chi }.
\endaligned$$ Likewise, $$G_{(N-1)}G_{(N)}e^{X^{*} -2X-N\chi }=
(N-3)(N-2)e^{X^{*} -2X-(N-2)\chi }.$$ Continuing in the same vein we obtain $$G_{(3)}G_{(4)}\cdots G_{(N-1)}G_{(N)}e^{X^{*} -2X-N\chi } =(N-2)!
e^{X^{*} -2X-2\chi }
\eqno{(5.2.31)}$$ The same computation shows that an application of $G_{(2)}$ annihilates vector (5.2.31). Hence (5.2.31) is a highest weight vector and, since it is determined by this condition up to proportionality, (5.2.27) follows. The $N$th tensor power of the latter gives the desired (5.2.26). $\qed$
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V.G.: Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA; vgorbms.uky.edu
F.M.: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA; fmalikovmathj.usc.edu
[^1]: partially supported by the National Science Foundation
[^2]: this correspondence between $H^{*}(\CL^{*},\Omega^{ch}_{\CL^{*}})$ and $H^{*}(\fF,\Omega^{ch}_{\fF})$ reminds one of the main result in \[S\]
[^3]: the tensor product decomposition of $H_{df(z)_{(0)}}(\widetilde{\text{LG}}^{<N}_{orb})$ arising in the case of a diagonal equation, see Corollary 5.1.4, is perhaps a bridge to Gepner’s models \[G\] also cooked up of the tensor products of $N2$ unitary representations
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $(V,g)$ and $(W,h)$ be compact Riemannian manifolds of dimension at least $3$. We derive a lower bound for the conformal Yamabe constant of the product manifold $( V \times W, g+h)$ in terms of the conformal Yamabe constants of $(V,g)$ and $(W,h)$.'
address:
- |
Fakultät für Mathematik\
Universität Regensburg\
93040 Regensburg\
Germany
- |
Institutionen för Matematik\
Kungliga Tekniska Högskolan\
100 44 Stockholm\
Sweden
- |
Institut Élie Cartan, BP 239\
Université de Nancy 1\
54506 Vandoeuvre-lès-Nancy Cedex\
France
author:
- Bernd Ammann
- Mattias Dahl
- Emmanuel Humbert
title: |
The conformal Yamabe constant of\
product manifolds
---
Introduction
============
The Yamabe functional, constant scalar curvature metrics, and Yamabe metrics
----------------------------------------------------------------------------
For a Riemannian manifold $(M,G)$ we denote the scalar curvature by $\Scal^G$, Laplace operator $\Delta^G$, volume form $dv^G$. In general the dependence on the Riemannian metric is denoted by the metric as a superscript.
For integers $m \geq 3$ we set $a_m {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\frac{4(m-1)}{m-2}$ and $p_m {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\frac{2m}{m-2}$. Let $C^\infty_c(M)$ denote the space of compactly supported smooth functions on $M$. For a Riemannian manifold $(M,G)$ of dimension $m \geq 3$ we define the Yamabe functional by $$\cF^G (u)
{\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\frac{
\int_M \left( a_m |du|_G^2 + \Scal^G u^2 \right) \, dv^G}
{\left( \int_M |u|^{p_m} \, dv^G \right)^{\frac{2}{p_m}}} ,$$ where $u \in C^\infty_c(M)$ does not vanish identically. The [*conformal Yamabe constant*]{} $\mu(M,G)$ of $(M,G)$ is defined by $$\mu(M,G) {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\inf_{u \in C_c^{\infty}(M), u \not\equiv 0} \cF^G(u).$$ The conformal Yamabe constant is usually defined only for compact manifolds, here we allow also non-compact manifolds in the definition. This will turn out to be essential for studying surgery formulas for Yamabe invariants of compact manifolds, see Subsection \[subsec.surg\]. Also notice that the conformal Yamabe constant for non-compact manifolds has been studied for instance in [@kim:00] and [@grosse:p09].
For compact $M$ one easily sees that $\lim_{\ep\to 0} \cF^G (\sqrt{u^2+\ep^2}) = \cF^G (u)$, thus we obtain $$\mu(M,G)
=
\inf_{u\in C^\infty_+(M)} \cF^G (u) > -\infty,$$ where $C^\infty_+(M)$ denotes the space of positive smooth functions. According to the resolution of the Yamabe problem [@trudinger:68; @aubin:76; @schoen:84], see for example [@lee.parker:87] for a good overview article, this infimum is always attained by a positive smooth function if $M$ is a compact manifold.
For a compact manifold $M$ one also defines for any metric $G$ the (normalized) Einstein-Hilbert functional $\cE$ as $$\cE(G)
{\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\frac{\int_M \scal^G\, dv^G}{\vol^G(M)^{\frac{m-2}m}}.$$ These functionals are closely related to each other, namely if $u>0$ and $\witi G = u^{4/(m-2)} G$, then $$\cE(\witi G) = \cF^G (u).$$ From the discussion above it follows that the functional $\cE$ always attains its infimum in each conformal class $[G]$. Such minimizing metrics are called *Yamabe metrics*. Obviously $\witi
G$ is a Yamabe metric if and only if $\la \witi G$ is a Yamabe metric for any $\la>0$. Thus any conformal class on a compact manifold carries a Yamabe metric of volume $1$. Yamabe metrics $\witi G$ are stationary points of $\cE$, restricted to the conformal class, and thus satisfy an Euler-Lagrange equation. This Euler-Lagrange equation says precisely that the scalar curvature of $\witi G$ is constant. One also sees that $\mu(M,G)$ is positive if and only if $[G]$ contains a metric of positive scalar curvature.
We denote the standard flat metric on $\mR^m$ by $\xi^m$. The induced metric on the sphere $S^m \subset \mR^{m+1}$ will be denoted by $\rho^m$. This metric is a Yamabe metric, and the whole orbit of $\rho^m$ under the action of the Möbius group ${\rm Conf}(S^m) =
{\rm PSO}(m+1,1)$ consists of Yamabe metrics. Thus $\mS^m {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}(S^m, \rho^m)$ carries a non-compact space of Yamabe metrics of volume $1$. It follows easily that $\mu(S^m,\rho^m) =
m(m-1) \om_m^{2/m}$ where $\om_m$ is the volume of $(S^m,\rho^m)$.
To determine the conformal Yamabe constant $\mu(M,G)$ it is obviously sufficient to know a Yamabe metric in $[G]$ explicitly. This is often easy to obtain in the following cases:
(i) \[item1\] If $M$ is compact and connected, and if $\mu(M,G)\leq 0$, then it follows from the maximum principle that there is a unique metric $g_0 \in [G]$ of volume $1$ and constant scalar curvature $s_0$. This then implies $\mu(M,G)=s_0$.
(ii) \[item2\] Assume that $(M,G)$ is a connected compact Einstein manifold, and $(M,G)$ is non-isometric to $(S^m,\la \rho^m)$ for any $\la >0$. Then it was proven by Obata [@obata:71.72 Prop. 6.2] that $G$ contains a unique metric $g_0$ of volume $1$ and constant scalar curvature $s_0$. Again $\mu(M,G)=s_0$.
(iii) \[item3\] If $(M,\ti G)$ is a metric of constant scalar curvature $\ti s$ which is close in the $C^{2,\al}$-topology to an Einstein manifold $(M,G)$ as in , then it is proven by [@boehm.wang.ziller:04 Theorem C] that $\ti G$ is also a Yamabe metric, and thus $\mu(M,\ti G) =
\ti s \; \vol(M,\ti G)^{2/m}$. This applies for example to $(M,\ti G) = (S^m,\rho^m)\times (S^m, (1+\ep)\rho^m)$, $\ep$ close to $0$. It also follows from the arguments in [@boehm.wang.ziller:04] that $\ti G$ is then (up to rescaling) the only Yamabe metric in $[\ti G]$. However, it is hard to decide in this situation whether $\ti G$ is (up to rescaling) the only metric of constant scalar curvature in $[\ti G]$. An affirmative answer to this problem was given recently in [@lima.piccione.zedda:p11 Theorem 5] if at least one of the following additional conditions is satisfied: (a) $m\leq 7$; (b) $m\leq 24 $ and $M$ is spin; or (c) $|W|+|\na W|$ is a positive function.
We have seen that in some particular cases, $\mu(M,g)$ can be explicitly calculated. In general the determination of $\mu(M,g)$ is a difficult task, as in most cases it is unclear whether a given constant scalar curvature metric in $[G]$ is a Yamabe metric. The functionals $\cE|_{[G]}$ and $\cF^G |_{C^\infty_+(M)}$ may have non-minimizing stationary points. These stationary points are thus metrics of constant scalar curvature which are not Yamabe metrics. The simplest such example, extensively discussed by Schoen [@schoen:91] for $w=1$, is the metric $G = \rho^v + \la \rho^w$ on $S^v \times S^w$, $v\geq 2$, which has constant scalar curvature $v(v-1) + (1/\la) w(w-1)$, but which is not a Yamabe metric for sufficiently large $\la$. This is due to the fact that $\mu(M,G) \leq \mu(\mS^m)$, which follows from a standard test function argument, whereas $\cE( \rho^v + \la \rho^w )
\to \infty$ as $\la \to\infty$ when $v\geq 2$.
In conclusion, if $(M,G)$ is an explicitly given compact manifold of constant scalar curvature, the calculation of $\mu(M,G)$ is easy if either $(M,G)$ is Einstein or if $\mu(M,G)\leq 0$, but in general it can be a hard problem.
Product manifolds
-----------------
We now consider Riemannian product manifolds, that is for Riemannian manifolds $(V,g)$ and $(W,h)$ of dimensions $v$ and $w$, we equip $M = V \times W$ with the product metric $G = g + h$, or more generally $G = g + \la h$ where $\la > 0$. We ask the following question.
Suppose $V$ and $W$ are compact and equipped with Yamabe metrics $g$ and $h$. Let $\la > 0$. Is then $g + \la h$ also a Yamabe metric?
From the discussion on uniqueness above it follows that the answer is yes,
- if $v,w\geq 3$, $\mu(V,g)\leq 0$ and $\mu(W,h)\leq 0$;
- or if $v,w\geq 3$, $\mu(V,g)> 0$ and $\mu(W,h)< 0$ for $\la>0$ small enough;
- or if $(V,g)$ and $(W,h)$ are both Einstein with $\frac{1}{v} \Scal^g$ close to $\frac{1}{\la w} \Scal^h$.
If the answer to the above question is yes, then one deduces $$\label{naive.product.formula}
\mu(V\times W,g+\la h)
=
\left(\frac{\mu(V,g)}{\vol^g (V)^{2/v}}
+ \frac{\mu(W,h)}{\vol^{\la h} (W)^{2/w}}\right)
\Bigl( \vol^g (V) \vol^{\la h} (W) \Bigr)^{\frac 2{v+w}}.$$ On the other hand if $g$ has positive scalar curvature, then $\cE(g+\la h)\to \infty$ for $\la\to \infty$, thus $g + \la h$ is not a Yamabe metric for large $\la$. This applies, in particular, to the cases $\mu(V,g)>0$, $v\geq 3$, or if $(V,g) = (S^2, \rho^2)$.
An intuitive—but incorrect—argument in the positive case
--------------------------------------------------------
Now we assume $v,w\geq 3$, $\mu(V,g)>0$, and $\mu(W,h)>0$. We already explained why $g+\la h$ is not a Yamabe metric for large (and small) $\la>0$, as a consequence Equation cannot be true for all $\la>0$. Despite of this fact, assume for a moment that were true for all $\la>0$. We then could minimize over $\la$, and we would obtain $$\label{naive.product.concl}
\inf_{\la\in (0,\infty)} \mu(V\times W,g+\la h)
=
(v+w) \left(\frac{\mu(V,g)}v\right)^{\frac v{v+w}}
\left(\frac{\mu(W,h)}w\right)^{\frac w{v+w}}$$
Main result
-----------
Although the *naive derivation* of formula used incorrect assumptions, our main result, Theorem \[main\] will tell us that the *formula itself* is correct up to a factor $$\ep_{v,w} =
\frac{a_{v+w}}{{a_v}^{v/(v+w)}{a_w}^{w/(v+w)}}
< 1.$$ assuming the mild condition .
More precisely, we assume that $V$ and $W$ are compact manifolds of dimension at least $3$, with Yamabe metrics $g$ and $h$ of positive conformal Yamabe constant. In particular, condition is satisfied. Then Theorem \[main\] implies $$\ep_{v,w} \leq
\frac{\inf_{\la\in (0,\infty)} \mu(V\times W,g+\la h)}
{(v+w) \left(\frac{\mu(V,g)}v\right)^{\frac v{v+w}}
\left(\frac{\mu(W,h)}w\right)^{\frac w{v+w}}}
\leq 1 .$$ Note that $\ep_{v,w}\to 1$ for $v,w\to \infty$. See Figure \[fig.ep\] for some values of $\ep_{v,w}$.
$\ep_{v,w}$ w=3 w=4 w=5 w=6 w=7
------------- ---------- ---------- ---------- ---------- ----------
v= 3 0.625 0.7072.. 0.7515.. 0.7817.. 0.8042..
4 0.7072.. 0.7777.. 0.8007.. 0.8367.. 0.8537..
5 0.7515.. 0.8007.. 0.8427.. 0.8631.. 0.8772..
6 0.7817.. 0.8367.. 0.8631.. 0.88 0.8921..
7 0.8042.. 0.8537.. 0.8772.. 0.8921.. 0.9027..
The main theorem also applies to many non-compact manifolds, see Theorem \[main\].
Further comments on related literature
--------------------------------------
Our main motivation to study Yamabe constants of products is the application sketched in Subsection \[subsec.surg\].
Fundamental results on Yamabe constants on products have been found in the interesting article [@akutagawa.florit.petean:07] where it is, among other things, shown that the conformal Yamabe constant of the product $V \times \mR^w$ is a lower bound for $\si(V \times W)$. Here $\si$ denotes the smooth Yamabe invariant defined in Subsection \[subsec.smooth\]. This article also emphasized the importance of the question under which conditions a function $u \in C^\infty(V\times W)$ minimizing $\cF$ is a function of only one of the factors. In [@akutagawa.florit.petean:07] it is proved that if $(V,g)$ is compact and of constant scalar curvature $1$ then the infimum of the Yamabe functional of $V \times \mR^w$ restricted to functions depending only on $\mR^w$ is up to a constant the inverse of an optimal constant in a Gagliardo-Nirenberg type estimate.
In related research Petean [@petean:09a] derived a lower bound for the conformal Yamabe constant of product manifolds $V\times \mR$, where $V$ is compact of positive Ricci curvature. If additionally we require $V$ to be Einstein, any minimizer $u\in C^\infty(V\times \mR)$ of $\cF$ only depends on $\mR$. As a corollary Petean obtained lower bounds for the smooth Yamabe invariant $\si(V \times S^1)$ in this case.
This result of Petean contrasts nicely to Theorem \[main\]. Whereas Petean’s result requires that one of the factor is $1$-dimensional, our Theorem \[main\] requires both factors to be of dimension at least $3$.
In [@petean.ruiz:p10] an explicit lower bound for $\mu(S^2\times \mR^2,\rho^2 + \xi^2)$ is obtained: $\mu(S^2\times \mR^2, \rho^2 + \xi^2)
\geq 0.68 \cdot Y(S^4)$. A similar, but weaker result was obtained in [@petean:01].
Several recent publications study multiplicity phenomena on products $S^v\times W$ equipped with product metric of the standard metric on $S^v$ with a metric of constant scalar curvature $s>0$ on $W$. Explicit lower bounds for the number of metrics of constant scalar curvature $1$ in the conformal class $[g_0]$ are derived, and these bounds grow linearly in $\sqrt{s}$. The case $v=1$ was studied in [@lbb.kaas:a; @lbb.kaas:b], the general case then treated in [@petean:10]. In the recent preprint [@henry.petean:p11] isoparametric hypersurfaces are used in order to obtain new metrics of constant scalar curvature in the conformal class of products of Riemannian manifolds, e.g. the conformal class of $(S^3\times S^3,\rho^3+\lambda \rho^3)$.
Structure of the present article
--------------------------------
In Section \[Section\_product\] we derive the main techniques and the main result of the article. We use mixed $L^{p,q}$-spaces in order to obtain a lower bound of the conformal Yamabe constants in the case that both factors have dimension at least $3$. We start with a proof of an iterated Hölder inequality in Subsection \[iterated\] which is well-adapted for the proof of our product formula in Subsection \[subsec.main\] which is the main result of the article.
In Section \[Section\_applications\] we discuss applications. In Subsection \[subsec.smooth\] we find an estimate for the smooth Yamabe invariant of product manifolds. Subsection \[subsec.surg\] explains our original motivation for the subject, which is to find better estimates for the constants appearing in the surgery formula in [@ammann.dahl.humbert:p08a]. In Subsection \[subsec.stable\] we define a stable Yamabe invariant and show that a similar surgery formula as in the unstable situation holds true.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We want to thank the organizers of the Conference “Contributions in Differential Geometry – a round table in occasion of the 65th birthday of Lionel Bérard Bergery, 2010”, in particular thanks to A. Besse, A. Altomani, T. Krantz and M.-A. Lawn. During that conference we found central ingredients for the present article. We thank P. Piccione for pointing out an error in a preprint version of this article. Further, we would like to thank the anonymous referee for many helpful comments.
B. Ammann was partially supported by DFG Sachbeihilfe AM 144/2-1, M. Dahl was partially supported by the Swedish Research Council, E. Humbert was partially supported by ANR-10-BLAN 0105.
Yamabe constants of product metrics {#Section_product}
===================================
Iterated Hölder inequality for product manifolds {#iterated}
------------------------------------------------
Let $(V,g)$ and $(W,h)$ be Riemannian manifolds of dimensions $v {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\dim V$ and $w {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\dim W$. We set $$(M,G) {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}(V \times W, g + h),$$ so that $m {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\dim M = v + w$. We do not assume that the manifolds are complete. The first result we will need is a kind of iterated Hölder inequality for $(M,G) {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}(V \times W, g + h)$.
\[holder\] For any function $u \in C_c^{\infty}(M)$ we have $$\left( \int_M |u|^{p_m} \, dv^G \right)^{\frac{2}{p_m}}
\leq
\left( \int_V
\left( \int_W |u|^{p_w} \, dv^h \right)^{\frac{2}{p_w} }
\, dv^g \right)^{\frac{w}{m} }
\left( \int_V
\left( \int_W |u|^2 \, dv^h \right)^{\frac{p_v}{2}}
\, dv^g \right)^{\frac{v-2}{m}} .$$
The lemma is actually a special case of the Hölder inequality for mixed $L^{p,q}$-spaces. See [@benedek.panzone:61] for further information on such spaces.
By the Hölder inequality we have $$\int_{W} |u|^{p_m} \, dv^h
\leq
\left( \int_W |u|^{p_w} \, dv^h \right)^{\frac{w-2}{m-2}}
\left( \int_W |u|^{2} \, dv^h \right)^{\frac{v}{m-2}}.$$ We integrate this inequality over $(V,g)$, and use the following Hölder inequality $$\int_V \alpha \beta \, dv^g
\leq
\left(
\int_V |\alpha|^{\frac{m-2}{w}} \, dv^g
\right)^{\frac{w}{m-2}}
\left(
\int_V |\beta|^{\frac{m-2}{v-2}} \, dv^g
\right)^{\frac{v-2}{m-2}}$$ with $$\alpha {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\left( \int_W |u|^{p_w} \, dv^h \right)^{\frac{w-2}{m-2}}
\; \hbox{ and } \;
\beta {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\left( \int_W |u|^{2} \, dv^h \right)^{\frac{v}{m-2}}.$$ This proves Lemma \[holder\]
A Lemma about integration and derivation
----------------------------------------
Second we need a Lemma concerning the interchange of derivation and taking (partial) $L^2$-norm.
\[comparing\_deriv\] Let $u \in C_c^\infty(M)$, $u \not\equiv 0$, and set $$\ga {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\left( \int_W u^2 \, dv^h \right)^{\frac{1}{2}}.$$ Then $$\label{cdr}
\int_V |d\ga|_g^2 \, dv^g
\leq
\int_M |du|_g^2 \, dv^G.$$
Take any vector field $X$ on $M$ tangent to $V$. One has $g$-almost everywhere (except on the boundary of $\ga^{-1}(0)$) $$|X \ga|^2
\leq
\left(
\frac{\int_W u X u \, dv^h}
{ \left(\int_W u^2 \,dv^h \right)^\frac{1}{2} }
\right)^2
\leq
\int_W (X u)^2 \, dv^h ,$$ where we used the Cauchy-Schwartz inequality $$\int_W u X u \, dv^h
\leq
{\left(\int_W (X u)^2 \, dv^h\right)}^{\frac{1}{2}}
{\left(\int_W u^2 \, dv^h\right)}^{\frac{1}{2}}.$$ Integrating over $V$, we deduce that $$\int_V |X \ga|^2 \, dv^g \leq \int_M |X u|^2 \, dv^G.$$ Since this holds for any $X$ tangent to $V$, inequality follows.
Conformal Yamabe constant of product metrics {#subsec.main}
--------------------------------------------
We now state and prove our main theorem. It will turn out that the following modified invariant is convenient when studying products of Riemannian manifolds with non-negative Yamabe constant. If $\mu(M,G) \geq 0$ we set $$\nu(M,G) {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\left( \frac{\mu(M,G)}{m a_m} \right)^m .$$
\[main\] Let $(V,g)$ and $(W,h)$ be Riemannian manifolds of dimensions $v,w \geq 3$. Assume that $\mu(V,g), \mu(W,h) \geq 0$ and that $$\label{assump}
\frac{\Scal^g + \Scal^h}{a_m}
\geq
\frac{\Scal^g}{a_v} + \frac{\Scal^h}{a_w} .$$ Then, $$\mu(M,G) \geq
\frac{m a_m}{(v a_v)^{\frac{v}{m}} (w a_w)^{\frac{w}{m}}}
\mu(V,g)^{\frac{v}{m}} \mu(W,h)^{\frac{w}{m}},$$ or, equivalently, $$\nu(M,G) \geq \nu(V,g) \nu(W,h).$$
Note that we do not assume that the manifolds are complete.
Take any non-negative function $u \in C_c^\infty(M)$ normalized by $$\label{normalization}
\int_M |u|^{p_m} \, dv^G = 1 .$$ We then have $$\frac{1}{a_m} \cF^G(u)
=
\int_M \left( |du|_G^2 +\frac{\Scal^G}{a_m}u^2 \right) \, dv^G.$$ Using $|du|^2_G = |du|^2_g+|du|^2_h$ and $\Scal^G = \Scal^g + \Scal^h$ together with we obtain $$\label{IGu}
\frac{1}{a_m} \cF^G(u)
\geq
\int_M \left( |du|^2_g + \frac{\Scal^g}{a_v} u^2 \right) \, dv^G
+
\int_V
\int_W \left( |du|_h^2 + \frac{\Scal^h}{a_w} u^2 \right) \, dv^h
\, dv^g.$$ We set $\ga {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\left( \int_W u^2 \, dv^h \right)^\frac{1}{2}$. For the first term here, Lemma \[comparing\_deriv\] and the definition of $\mu(V,g)$ imply that $$\label{IGu1}
\begin{split}
\int_M \left( |du|^2_g + \frac{\Scal^g}{a_{v}} u^2 \right) \, dv^G
&\geq
\int_V
\left( |d\ga|_g^2 + \frac{\Scal^g}{a_{v}} \ga^2 \right)
\, dv^g \\
&\geq
\frac{1}{a_v} \mu(V,g)
\left( \int_{V} \ga^{p_v} \, dv^g \right)^{\frac{2}{p_v}} \\
&=
\frac{1}{a_v} \mu(V,g)
\left(
\int_V \left( \int_W |u|^2 \, dv^h \right)^{\frac{p_v}{2}} \, dv^g
\right)^{\frac{v-2}{v} }.
\end{split}$$ For the second term we have $$\label{IGu2}
\begin{split}
\int_V \int_W
\left(|du|_h^2 + \frac{\Scal^h}{a_w} u^2 \right)
\, dv^h \, dv^g
\geq
\frac{1}{a_w} \mu(W,h)
\int_V
\left( \int_W u^{p_w} \, dv^h \right)^{\frac{2}{p_w}}
\, dv^g
\end{split}$$ by the definition of $\mu(W,h)$. Plugging and in we get $$\label{IGu3}
\begin{split}
\cF^G(u)
&\geq
\frac{a_m}{a_v} \mu(V,g)
\left(
\int_V \left( \int_W |u|^2 \, dv^h \right)^{\frac{p_v}{2}} \, dv^g
\right)^{\frac{v-2}{v} } \\
&\qquad +
\frac{a_m}{a_w} \mu(W,h)
\int_V \left( \int_W u^{p_w} \, dv^h \right)^{\frac{2}{p_w}} \, dv^g
\end{split}$$ Set $$r {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}m a_m \nu(V,g)^{\frac{1}{m}} \nu(W,h)^{\frac{1}{m}}$$ For $a,b > 0$ we compute $$\begin{split}
r a^\frac{v-2}{m} b^\frac{w}{m}
&=
r
\left(
\left( \frac{\nu(V,g)^w}{\nu(W,h)^v} \right)^{\frac{1}{m^2}}
a^\frac{v-2}{m}
\right)
\left(
\left( \frac{\nu(W,h)^v}{\nu(V,g)^w} \right)^{\frac{1}{m^2}}
b^\frac{w}{m}
\right) \\
&\leq
r \left[
\frac{v}{m}
\left( \frac{\nu(V,g)^{\frac{w}{v}}}{\nu(W,h)} \right)^{\frac{1}{m}}
a^\frac{v-2}{v}
+
\frac{w}{m}
\left( \frac{\nu(W,h)^{\frac{v}{w}}}{\nu(V,g)} \right)^{\frac{1}{m}}
b
\right] \\
&=
m a_m \nu(V,g)^{\frac{1}{m}} \nu(W,h)^{\frac{1}{m}}
\frac{v}{m}
\left( \frac{\nu(V,g)^{\frac{w}{v}}}{\nu(W,h)} \right)^{\frac{1}{m}}
a^\frac{v-2}{v} \\
&\qquad +
m a_m \nu(V,g)^{\frac{1}{m}} \nu(W,h)^{\frac{1}{m}}
\frac{w}{m}
\left( \frac{\nu(W,h)^{\frac{v}{w}}}{\nu(V,g)} \right)^{\frac{1}{m}}
b \\
&=
a_m v \nu(V,g)^{\frac{1}{v}} a^\frac{v-2}{v}
+ a_m w \nu(W,h)^{\frac{1}{w}} b \\
&=
\frac{a_m}{a_v} \mu(V,g) a^\frac{v-2}{v}
+ \frac{a_m}{a_w} \mu(W,h) b \\
\end{split}$$ where we in the second line used Young’s inequality $$cd \leq \frac{v}{m}c^{\frac{m}{v}} + \frac{w}{m}d^{\frac{m}{w}} ,$$ which is valid for any $c,d \geq 0$. Using the above in with $$a {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\int_V
\left( \int_W |u|^2 \, dv^h \right)^{\frac{p_v}{2}}
\, dv^g
\; \hbox{ and } \;
b {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\int_V
\left( \int_W |u|^{p_w} \, dv^h \right)^{\frac{2}{p_w}}
\, dv^g,$$ we get $$\cF^G(u)
\geq
r
\left( \int_V
\left( \int_W |u|^2 \, dv^h \right)^{\frac{p_v}{2}}
\, dv^g \right)^\frac{v-2}{m}
\left( \int_V
\left( \int_W |u|^{p_w} \, dv^h \right)^{\frac{2}{p_w}}
\, dv^g \right)^\frac{w}{m} .$$ Using Lemma \[holder\] and Relation we deduce $$\cF^G(u)
\geq r
= m a_m \nu(V,g)^{\frac{1}{m}} \nu(W,h)^{\frac{1}{m}}
= \frac{m a_m}{(v a_v)^{\frac{v}{m}} (w a_w)^{\frac{w}{m}}}
\mu(V,g)^{\frac{v}{m}} \mu(W,h)^{\frac{w}{m}}.$$ Since this holds for all $u$, Theorem \[main\] follows.
Applications {#Section_applications}
============
The smooth Yamabe invariant of product manifolds {#subsec.smooth}
------------------------------------------------
Let $M$ be a compact manifold of dimension $m \geq 3$. Then its [*smooth Yamabe invariant*]{} is defined as $$\sigma(M) {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\sup \mu(M,G)$$ where the supremum runs over all Riemannian metrics $G$ on $M$. This invariant of differentiable manifolds has the property that $\si(M) \leq \si(S^m)$ for all $M$ and $\si(M) > 0$ if and only if $M$ admits a metric with positive scalar curvature.
From Theorem \[main\] we obtain the following corollary.
\[cor1\] Let $V,W$ be compact manifolds of dimensions $v,w \geq 3$. Assume $\sigma(V)\geq 0$. Then $$\sigma(V \times W)
\geq
\frac{m a_m}{(v a_v)^{\frac{v}{m}} (w a_w)^{\frac{w}{m}}}
\sigma(V)
^{\frac{v}{m}}
\sigma(S^w)
^{\frac{w}{m}},$$ where $m = v + w$.
We first consider the case $\si(V)>0$. In [@akutagawa.florit.petean:07 Theorem 1.1] it is proven that $$\lim_{t \to\infty} \mu(V \times W, g + t^2 h)
=
\mu(V \times \mR^w, g + \xi^w)$$ if $g$ is a metric on $V$ with positive scalar curvature and $h$ is any metric on $W$. Since $a_v \geq a_m$ we see that holds, so Theorem \[main\] together with $\mu(\mR^w, \xi^w) = \mu(S^w, \rho^w)$ imply the corollary if $\si(V)>0$.
In the case $\si(V)=0$ there is a sequence of metrics $g_i$ on $V$ such that $\vol^{g_i}(V)=1$, $\mu(V,g_i)\leq 0$, and $\mu(V,g_i)\to 0$ as $i \to \infty$. From the solution of the Yamabe problem we can assume that all $g_i$ have constant scalar curvature $\Scal^{g_i}
= \mu(V,g_i)$. Choose $\ep_i>0$ such that $\ep_i \to 0$ and $\ep_i^{-w}\mu(V,g_i)\to 0$ for $i\to \infty$. For a unit volume metric $h$ on $W$ with constant scalar curvature $\Scal^h$, the metric $G_i {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\ep_i^{w} g_i + \ep_i^{-v} h$ has $\vol^{G_i}(V\times W) = 1$ and constant scalar curvature $\ep_i^{-w}\mu(V,g_i)+\ep_i^v \Scal^h \to 0$. It follows that $\mu(V\times W,G_i)\to 0$ and thus $\si(V\times W)\geq 0$.
Surgery formulas {#subsec.surg}
----------------
Assume that $M$ is a compact $m$-dimensional manifold, and that $i:S^k\times \overline{B^{m-k}}\to M$ is an embedding. We define $$N {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}(M\setminus i(S^k\times B^{m-k})\cup_\pa (B^{k+1}\times S^{n-k-1})$$ where $\cup_\pa$ means that we identify $x\in S^k\times
S^{m-k-1}=\pa(B^{k+1}\times S^{m-k-1})$ with $i(x)\in \pa i (S^k\times B^{m-k})$. After a smoothing procedure $N$ is again a compact manifold without boundary, and we say that $N$ is *obtained from $M$ by $m$-dimensional surgery along $i$*.
In [@ammann.dahl.humbert:p08a Corollary 1.4] we found the following result.
\[sigmaanatheo\] Let $N$ be obtained from $M$ via surgery of dimension $k\in\{0,1,\ldots,m-3\}$, then there is a constant $\La_{m,k}>0$ with $$\si(N)\geq \min\{\si(M),\La_{m,k}\}.$$ Furthermore, for $k=0$ this statement is true for $\La_{m,0}=\infty$.
It is helpful to consider how the constant $\Lambda_{m,k}$ was obtained in [@ammann.dahl.humbert:p08a] in the case $k\geq 1$. We showed that Theorem \[sigmaanatheo\] holds for a constant $\La_{m,k}$ satisfying $$\La_{m,k} \geq
\min \left\{ \La^{(1)}_{m,k},\La^{(2)}_{m,k}\right\}.$$ We will not recall the definition $\La^{(1)}_{m,k}$ and $\La^{(2)}_{m,k}$ here in detail, as it is not needed, but we will explain some relevant facts for $\La^{(1)}_{m,k}$ and $\La^{(2)}_{m,k}$.
For $c\in[0,1]$ let $\mH^{k+1}_c$ be the simply connected $(k+1)$-dimensional complete Riemannian manifold of constant sectional curvature $-c^2$, for $c=0$ it is $\mR^{k+1}$ and for $c>0$ it is hyperbolic space rescaled by a factor $c^{-2}$. One defines $$\La_{m,k}^{(0)}{\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\inf_{c \in [0,1]} \mu(\mH^{k+1}_c\times \mS^{m-k-1}).$$
It was shown in [@ammann.dahl.humbert:p08a Corollary 1.4] that $\La^{(1)}_{m,k}\geq\La_{m,k}^{(0)}$ for $k\in \{1,\ldots,m-3\}$. Furthermore $\La^{(2)}_{m,k}\geq \La^{(1)}_{m,k}$ will be shown in our publication [@ammann.dahl.humbert:p11b:lpl2] provided that $k+3\leq m\leq 5$ or $k+4\leq m$. Thus Theorem \[sigmaanatheo\] holds for $\La_{m,k}{\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\La_{m,k}^{(0)}$ if $k+3 \leq m \leq 5$ or $k+4 \leq m$.
Since the manifolds $\mH^{k+1}_c\times \mS^{m-k-1}$ with $c^2 \leq 1$ satisfy Condition and since $\mu(\mH^{k+1}_c) = \mu(\mS^{k+1})$ we obtain the following Corollary from Theorem \[main\].
\[cor2\] If $2 \leq k \leq m-4$, then Theorem \[sigmaanatheo\] holds for $$\La_{m,k}
=
\frac{m a_m}{((k+1) a_{k+1})^{\frac{k+1}{m}}
((m-k-1) a_{m-k-1})^{\frac{m-k-1}{m}}}
\sigma(S^{k+1})^{\frac{k+1}{m}}
\sigma(S^{m-k-1})^{\frac{m-k-1}{m}}.$$
It follows for example: If $M$ is an $m$-dimensional compact manifold, obtained from $S^m$ by performing successive surgeries of dimension $k$, $0\leq k\leq m-4$, $k\neq 1$, then $\si(M)\geq \La_m$, where $\La_6= 54.779$, $\La_7= 74.504$, $\La_8= 92.242$, $\La_9= 109.426$, etc.
Another technique presented in [@ammann.dahl.humbert:p11c:sigmaexp] will allow to control the effect of $1$-dimensional surgeries. In combination with [@petean.ruiz:p10] and [@petean.ruiz:p11] one obtains $\La_{4,1}>38.9$ and $\La_{5,1}>45.1$.
A stable Yamabe invariant {#subsec.stable}
-------------------------
In this section we will define and discuss a “stabilized” Yamabe invariant, obtained by letting the dimension go to infinity for a given compact Riemannian manifold by multiplying with Ricci-flat manifolds of increasing dimension. Very optimistically, such a stabilization could be related to the linear eigenvalue problem obtained by formally letting the dimension tend to infinity in the Yamabe problem. The stable invariant can also be viewed as a quantitative refinement of the property that a given manifold admit stably positive scalar curvature.
For a compact manifold $M$ with $\si(M) \geq 0$ we define $$\Si(M) {\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\left( \frac{\si(M)}{m a_m} \right)^m ,$$ then $$\Si(M) = \sup \nu(M,G)$$ where the supremum runs over all Riemannian metrics $G$ on $M$. The conclusion of Corollary \[cor1\] can be formulated as $$\label{Sigma_product}
\Si (V \times W)
\geq
\Si(V) \Si(S^w).$$
Let $(B,\beta)$ be a compact Ricci-flat manifold of dimension $b$. We could for example choose $B$ to be the $1$-dimensional circle $S^1$, or an $8$-dimensional Bott manifold equipped with a metric with holonomy ${\rm Spin}(7)$. From we then get $$\label{Si_est2}
\frac{\Si(S^{v + bi})}{\Si(S^{bi})}
\geq
\frac{\Si (V \times B^i)}{\Si(S^{bi})}
\geq
\Si(V),$$ where the upper bound comes from $\Si (V \times B^i) \leq
\Si(S^{v + bi})$. We define the stable Yamabe invariant of $V$ as the limit superior of the middle term, $$\overline{\Si} ( V )
{\mathrel{\raise.095ex\hbox{\rm :}\mkern-5.2mu=}}\limsup_{i \to \infty}
\frac{\Si (V \times B^i)}{\Si(S^{bi})}$$
To see that the stable Yamabe invariant is finite we need to study the upper bound in , and the function $v \mapsto \Si(S^v)$. We have $$\si(S^v) = v(v-1) \om_v^{2/v} , \quad
\om_v =
\frac{2 \pi^{\frac{v+1}{2}}}{\Gamma \left( \frac{v+1}{2} \right)},$$ where $\om_v$ is the volume of $\mS^v$, so $$\Si(S^v)
=
4 \pi \left( \frac{\pi(v-2)}{4} \right)^v
\frac{1}{ \Gamma \left( \frac{v+1}{2} \right)^2 } .$$ Stirling’s formula tells us that $$\Gamma(z)
=
\sqrt{\frac{2\pi}{z}} \left( \frac{z}{e} \right)^z
\left( 1 + O \left(\frac{1}{z}\right) \right)$$ and therefore $$\begin{split}
\Si(S^v)
&=
4 \pi \left( \frac{\pi(v-2)}{4} \right)^v
\frac{v+1}{4\pi}
\left( \frac{2e}{v+1} \right)^{v+1}
\left( 1 + O \left(\frac{1}{v}\right) \right) \\
&=
2e \left( \frac{\pi e}{2} \right)^v
\frac{(1 - 2/v)^v}{(1 + 1/v)^v}
\left( 1 + O \left(\frac{1}{v}\right) \right) \\
&=
2e^{-2} \left( \frac{\pi e}{2} \right)^v
\left( 1 + O \left(\frac{1}{v}\right) \right).
\end{split}$$ We see that $$\lim_{i \to \infty} \frac{\Si(S^{v + bi})}{\Si(S^{bi})}
=
\lim_{i \to \infty}
\left( \frac{\pi e}{2} \right)^v
\left( 1 + O \left( \frac{1}{bi} \right) \right)
=
\left( \frac{\pi e}{2} \right)^v,$$ so from we get the following bound on the stable Yamabe invariant $$\left( \frac{\pi e }{2} \right)^v
\geq
\overline{\Si} ( V )
\geq
\Si(V).$$ We conclude that the stable invariant is a non-trivial invariant.
The stable Yamabe invariant is not strictly speaking a stable invariant in the sense that it gives the same value for $V$ and $V \times B^i$. These values are however related by a simple identity, as we will see next. Taking the limit superior as $j \to \infty$ in $$\frac{\Si (V \times B^i \times B^j)}{\Si(S^{bj})}
=
\frac{\Si (V \times B^{i+j})}{\Si(S^{bi+bj})}
\frac{\Si(S^{bi+bj})}{\Si(S^{bj})}$$ we conclude $$\overline{\Si} (V \times B^i)
=
\overline{\Si} (V) \left( \frac{\pi e }{2} \right)^{bi}$$ and further $$\label{prod.prep}
\overline{\Si} (V)
\geq
\Si (V \times B^i)
\left( \frac{\pi e }{2} \right)^{-bi}$$ for all $i \geq 0$.
The next simple proposition tells us that positivity of $\Si(V)$ is equivalent to $V$ having stably metrics of positive scalar curvature.
Let $V$ be a compact manifold. The following three statements are equivalent.
(a) $\overline{\Si} (V) > 0$.
(b) There is $i_0 > 0$ such that $V \times B^{i_0}$ admits a positive scalar curvature metric.
(c) There is a $i_0 > 0$ such that $V \times B^i$ admits a positive scalar curvature metric for all $i \geq i_0$.
The implications $(a) \Rightarrow (b)$ and $(b) \Leftrightarrow (c)$ are easy to show. The implication $(b) \Rightarrow (a)$ is a consequence of .
We also obtain a stable version of Theorem \[sigmaanatheo\] for surgeries of codimension at least $4$. A similar result holds for surgeries of codimension $3$, but with a less explicit constant.
Assume that $N$ is obtained from the compact $m$-dimensional manifold $M$ by surgery of dimension $k$, where $0 \leq k \leq m-4$, then $$\overline{\Si}(N)
\geq
\min\left\{\overline{\Si}(M), \overline{\Si}(S^m),
\left(\frac{\pi e}2\right)^{k+1}\Si (S^{m-k-1})
\right\}.$$
The manifold $N$ after surgery is obtained by a connected sum of $M$ and $S^m$ along embeddings of a $k$-dimensional sphere with trivial normal bundle. Similarly $N \times B^i$ is obtained by a connected sum of $M \times B^i$ and $S^m \times B^i$ by a connected sum along embeddings of $S^k \times B^i$ with trivial normal bundle. Thus [@ammann.dahl.humbert:p08a Theorem 1.3] together with Corollary \[cor2\] tells us that $$\begin{split}
\Si(N \times B^i)
&\geq
\min \left\{ \Si(M \times B^i), \Si(S^m \times B^i),
\left(\frac{\La_{m+bi,k+bi}}{(m+bi)a_{m+bi}}\right)^{m+bi} \right\}\\
&\geq
\min\{\Si(M \times B^i), \Si(S^m \times B^i),
\Si(S^{k+bi+1})\Si(S^{m-k-1})\}
\end{split}$$ and this yields the statement of the theorem.
For the smooth Yamabe invariant the value of the sphere is a universal upper bound. One can ask if the same holds for the stable invariant, is $\overline{\Si}(M) \leq \overline{\Si}(S^m)$ for all $M$?
[10]{}
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